TEL AVIV UNIVERSITY Gaddi Blumrosen

TEL AVIV UNIVERSITY
The IBY and Alder Fleischman Faculty of Engineering
Combining Orthogonal Space Time Codes with Adaptive
Beamforming
A thesis submitted toward the degree of
Master of Science in Electrical Engineering
by
Gaddi Blumrosen
February 2005

TEL AVIV UNIVERSITY
The IBY and Alder Fleischman Faculty of Engineering
Combining Orthogonal Space Time Codes and adaptive
beamforming
A thesis submitted toward the degree of
Master of Science in Electrical Engineering
by
Gaddi Blumrosen
This research was carried out in the department of Electrical Engineering - Systems
Under the supervision of Dr. Avraham Freedman
February 2005

Abstract
A wireless channel, due to the radio wave propagation in a changing environment,
causes the transmissions to vary in space, time and frequency. A channel, which
causes the transmission to vary at different spatial locations, e.g. different antennas in
antenna array, is called a space-selective fading channel. Multiple transmit and or
receive antennas can be utilized in wireless systems to enhance coverage and capacity
of the wireless channel by exploiting space-selective fading channel either by
diversity or by coherent combining techniques. A special case, which will be
investigated in this work, is a system with multiple transmit antennas and single
receive antenna, an arrangement very common in the downlink of a cellular system,
for example, where the base station can accommodate a number of antennas whereas
the mobile station is limited in that respect. . In this case, diversity can be achieved by
Space Time Codes (STC) techniques while coherent combining is achieved by
Beam-forming (BF) techniques with or without exploitation of Channel State
Information (CSI) available at the transmitter.
In BF, the system is being transformed into a set of parallel scalar channels by Single
Value Decomposition (SVD) and the available transmit power can then be optimally
allocated to each individual scalar channel, called also sub-channel. For this
technique, a certain value of CSI (channel realization) is necessary at the transmitter
and receiver for spatial pre-coding before transmission and spatial post-coding in
reception. Thus, BF is mainly a close- loop method. Adaptive BF techniques are
widely used for tracking the channel fluctuations.
In STC spatial encoding at transmitter and spatial decoding at the receiver are used.
Full CSI knowledge is not assumed but just a general statistical model of the channel,
mostly uncorrelated channel due to rich scattering, is assumed. Thus, STC is an open
loop method.
STC suffer from lack of CSI exploitation and is optimized only in a rich scattering
environment while BF, though it exploits CSI, suffers from CSI quality degradation
and is typically more computation demanding. A lot of research was done recently
trying to combine these two families of techniques, STC and BF, in order to gain the
benefits of both families.
In this work we examine common ways for combining ST processing, mainly by
beamforming techniques and STC. In particular we focus on combining Orthogonal
Space Time Block Codes (OSTBC) and BF through adaptive antenna weight (which
is a linear transformation) in ML approach in different channel scenarios. Later we
derive a simple approximation for the weights which has been shown to be very close
to the ML optimal solution performance in simpler manner (computation gain).
We start this thesis with a describing of recent space-time system models. Then we
explore different Space Time Codes (STC) techniques and BF techniques, mainly
focusing in this work on the case of Multiple transmit antennas and single receive
antenna with imperfect channel estimation at the transmitter and perfect channel
estimation at receiver with transmission of orthogonal STC.
As a first step, the ways of combining these two families of techniques were surveyed.
We have chosen to concentrate on the ML optimal combination OSTBC and BF,
which was described in [25] and denoted in this thesis as the JSO algorithm according
to the initials of the writers, G. Jöngren, M. Skoglund, B. Ottersten.

The JSO algorithm selects an optimal set of the transmitter antenna weights, which
optimizes, in the ML sense, the reception of an OSTBC transmitted code, as a
function of the channel state information and the reliability of the information at the
transmitter. It can be seen as a smart compromise, sometimes simply as a switch,
between OSTBC and BF. As a general ML approach to the problem of weighted
OSTBC transmission, this algorithm is definitely superior to maximization of signal
strength (maximum SNR approaches) where only first SNR moment is used and also
to BF based only on channel correlation (second channel moment) since in slow
fading scenario the nonzero channel mean value, e.g. in Rician channel, can
contribute to performance.
We have studied the JSO algorithm in Rayleigh, Rician and correlated channel fading
and examined, by simulation, its sensitivity to errors in its parameters. We found that
a 10% error is equivalent in Rician fading, according to SNR/BER graphs to 0.3dB
degradation in performance and to 0.2dB in Rayleigh channel. This motivated us for
suggesting a simple approximation function to the exact solution for Rayleigh and
Rice channels which fulfills the same asymptotical properties as the ML optimal
solution.
We have shown that this approximation performs quite close to the optimal JSO
solution for a typical range of channel parameters. With the more manageable and
simpler approximation function, we can easily maintain the behavior of the solution
as a function of CSI parameters and thus the approximation may be used for
implementation and for deriving antenna weights sensitivity to changes in CSI
parameters in a simple manner. The computational benefit of the approximation,
increases with the number of transmit antennas.
Correlated fading causes degradation in performance in both OSTC and BF.
Consequently, JSO for the case of correlated fading, which is a set of non linear
equations, also has degradation in performance. We derived a numeric way for
obtaining the antenna weights for the correlated channel fadings, examined
analytically and by simulation the performance loss of BF, OSTBC and “uncorrelated
fadings JSO” and suggested an approximation to the weights which discard the need
for numerical calculations and should be further studied.
A future work in this field can generalize the approximation for the case of MIMO
channel, add channel coding, simulate the STC techniques derived in this work in
different channel scenarios and standards, study the estimation of the various CSI
parameters as a function of real measurements such as measurement accuracy,
feedback rate and quantization ratio and TDD or FDD.

1. Introduction
A wireless channel, due to the radio wave propagation in a changing environment,
causes the transmissions to vary in space time and frequency. A channel with
variations of the signal in time, e.g. variations of consecutive samples of the signal at
receiver, is called time-selective fading channel. In similar way, a channel which
causes the transmission to vary at different frequency bands, usually due to multi-path
and shadowing affects, is called frequency-selective fading channel and a channel
which causes the transmission to vary at different spatial locations, e.g. different
antennas in antenna array, is called space-selective fading channel [1].
Multiple transmit and or receive antennas can be utilized in wireless systems to
enhance coverage and capacity of the wireless channel by exploiting space-selective
fading channel. This exploitation of space selectivity can be further combined with
exploitation of the variations of transmission in time (time-selectivity) and in
frequency (frequency-selectivity) via the family of Space Time Codes (STC)
techniques or/and the family of Bema-forming (BF) techniques, with or without
exploitation of Channel State Information (CSI).
In BF, the system is being transformed into a set of parallel scalar channels by Single
Value Decomposition (SVD) and the available transmit power is can then be
optimally allocated to each individual channel. For this family, a certain value of CSI
(channel realization) is necessary at the transmitter and receiver for spatial pre-coding
before transmission and spatial post-coding in reception. Thus, BF is mainly a close
loop method. Adaptive BF techniques are widely used for tracking after channel
fluctuations. BF exploit space selectivity by means of coherent gain and thus the
transmit power can be induced to desired directions, usually correspond to directions
of strong valid multi-paths of the desired user. With a smart equalizing and multi-path
combining at the receiver, we exploit the multi-paths and frequency selectivity
(frequency selectivity is induced by multi-paths) and thus enhance performance.
Special cases in that family are Multiple transmit antennas (mostly at base station)
and single receive antenna where a beam is directed to a desired direction (usually
user direction) and Single transmit antenna and Multiple receive antennas, where the
antenna weights “filter” the signal in space, and therefore called also Spatial Filtering.
In STC, spatial encoding at transmitter and spatial decoding at receiver is used. Full
CSI knowledge is not assumed but just a general statistical model of the channel,
mostly uncorrelated channel due to rich scattering, is assumed. Thus, STC is an open
loop method. STC exploits space selectivity by means of diversity order of the
system, called diversity gain, and also by means of coding gain, (related to time
selectivity of the channel).
STC suffer from lack of CSI exploitation and is optimized only in a rich scatterer's
environment while BF, though it exploits CSI, suffers from CSI quality degradation
and is mostly more computation demanding. A lot of research was done recently
trying to combine these two families of techniques, STC and BF, in order to gain the
benefits of each family together. The different methods for combining STC and BF,
differ in optimization criterion -maximization of received SNR, minimizing of
Symbol Error Rate (SER), or maximization the transmission rate, and also in the kind
of CSI available at transmitter and correspondingly, the way CSI is being exploited.

In this work:
We describe different space-time channel and system model representations
taken from recent literature.
We describe the basic of BF and STC techniques, including simulation of the
spatial response of these two techniques.
We give a literature survey of the current methods for combining STC and BF.
In particular we focus on combining OSTBC and BF through adaptive antenna
weight (which is a linear transformation).
For multiple transmit antennas and single receive antenna with OSTBC and
with linear adaptive antenna weights BF we focus on the ML optimal weights
(JSO) for uncorrelated Rayleigh and Rician channel fadings and on Rician
correlated channel fadings (channel with non zero channel mean value and
with diagonal channel covariance matrix).
For Rayleigh and Rician non-correlated channel fading we
o Derive (for Rician fading channel) and explore the JSO solution as a
function of CSI parameters.
o Examine algorithm sensitivity to errors.
o Suggest a simple approximation to the ML solution which fulfills
the same asymptotical properties has the ML optimal solution.
o Analyze graphically and by MMSE the quality of the approximation.
o Obtain in a simple manner the sensitivity of the weights to channel
Parameters
o Compare the error rate performance of the approximation compared to
BF, OSTBC and JSO.
For the Rician correlated channel fading we
o Derive an analytical solution to the ML optimal weights which
acquire only binary search.
o Examine analytically and by simulation the performance loss of BF,
OSTBC and “uncorrelated fadings JSO”
o Suggest a simple approximation to the solution which fulfills the
same asymptotical properties has the ML optimal solution and discard
the need for numerical computations.
We start, in the first chapter, with a description of the space time system model. This
chapter has four sections. The first section deals with antenna array model. In the next
section, we introduce common channel spatial models – from simple single antenna
channel models to multiple-antenna channel. Next we introduce the received and
transmitted Signal model and on the last section, we show common indicators for
evaluating ST system performance.
In the second chapter we give a short overview of ST techniques. The first section
gives a overview of BF techniques, including SVD based BF, and MMSE optimal BF.
In the second section we describe STC, in particular OSTBC (orthogonal STBC).
In the 3 rd chapter we start with a general MMSE approach for combining BF and STC
based on channel feedback quality is developed [25]. Then we give an overview of
existing ways for combining both methods, as a function of CSI, channel statistics,
channel characteristics and SNR, mostly from last years. In particular we examine an
adaptation of STC to channel by linear transformation.
In the 4 th chapter we focus on combining OSTBC with BF. In the first section we
introduce ML solution for the problem. In the second section we give a overview of
existing Space-time Coding schemes with CSI exploitation. And the last section is

dedicated to examine ways of adaptation of STC to the channel by a linear
transformation
In the 5 th chapter we derive ML optimal antenna weights with OSTBC transmission
for different channel scenarios - Rayleigh channel fadings, Rician channel fadings and
correlated channel fadings (channel with non zero mean and non diagonal channel
covariance matrix).
In the 6 th , the last chapter, we conclude the research and define future work which
can be applicable as a proposal for existing standards (e.g., 3 rd and 4 th generation
standards).

Information
Source
2. Space-time system model
2.1 Introduction
Before we describe space-time processing, we have to give an appropriate space-time
system model. The bits of information are and then encoded and distributed toto
different antennas in an antenna array by the special space-time encoder. In reception,
a receive antenna array is used, and the information bits are obtained after ST
decoding and demodulation. Fig 1 below describes the transmission scheme.
We first describe, in the first section, the antenna array model, which has to enable
transmission and reception, of several information sources simultaneously. In the
second section we describe the most complicated and crucial part of space-time
system modeling- space-time channel model. In the third part we give received and
transmitted signal models combined with channel and antenna models. In the last
section we examine ways of evaluation of the space time system performance.
Modulator STC
encoder
h1N
R
h22
h11
1,2
hNTNR
h21
STC
decoder
Figure 1: Spatial Channel model
De-
Modulator
Information
Source

2.2 Antenna array modeling
2.2.1 Introduction
The wireless system antenna is the interface between the transmitter and the receiver
to space. The antenna array is an important tool in controlling the spatial behavior of
transmission/reception signal. Basically it enables several simultaneous
transmissions/receptions from different antennas in the array. The transmission from
each antenna, which can vary in phase and amplitude, is creating the antenna pattern.
In this chapter we will introduce a basic description of antenna array modeling and in
particular the one we use for this work.
2.2.2 Antenna array characteristics
Based on [2], we can characterize an antenna array by four factors.
1. Antenna array geometries.
2. Displacement between elements.
3. Antenna excitation („antenna element gain‟).
4. Antenna excitation phase.
The most common array geometry is the Uniform Linear Array (ULA). In such arrays
the antennas are equally spaced along a straight line. This arrangement is usually the
cheapest and the simplest for implementation and analysis. Other arrangements are
rectangular, circular and triangular. For the sake of simplicity we will focus on ULA
only, when necessary. Hence the term antenna array will refer to ULA only, unless
stated otherwise.
In a ULA, the displacement between elements denoted d, plays an important part in
antenna beam shaping and antenna correlations and influences the performance of
each ST technique. Each antenna element excitation amplitude and phase influences
the beam forming shaping in transmission and of the spatial filtering in reception.
Antenna gain is another important characteristic of the antenna. It is defined as the
ratio of antenna transmission/reception power at the maximum direction and that of
omni antenna. Gr is defined as the power gain of the receive antenna and Gt
is defined
as the power gain of the transmit antenna.
In realistic antenna models, a narrow band assumption (short travel time of signal
compared with symbol period, see [1]), which is being assumed simplifies the
computations complexity.
We will further assume from, far field assumption (the distance between the subantennas
is much smaller than the distance of the source) and thus the field at a given
point in space is a function of only the horizontal angle (see fig. 1).
The electrical field induced by antenna array in direction of is:
h
E
w s(
)
(2.1)
Where:

w is the complex antenna weight vector which describes antenna excitation
gain
s( ) is the steering vector describing the electrical field in the direction of
each single antenna element and has the form of (assuming antenna gain of 1):
2
2
jm1
d sin
j
Nt1
d sin
s(
)
1,...
e ,... e
(2.2)
2.3 Spatial channel modeling
2.3.1 Introduction
Spatial wireless channel modeling is essential for efficient ST processing. Appropriate
ST modeling is essential for determining the best ST processing technique, designing
STC, and getting statistics and measurement for capacity and performance evaluation
as well as for improving the overall performance by better adaptation to channel
conditions. Spatial physical channel models result in a channel response which varies
in space [3, p 43-91]. We will try as much as we can, for simplicity, to eliminate the
use of the spatial dimension as part of the spatial channel modeling, with certain
assumptions. But in most of the cases, it is straightforward to expand the model to
include the spatial dimension.
We start by describing a general spatial physical wireless channel suitable for one
transmit antenna and one receive antenna, later to be called Single-Input-Single-
Output (SISO). The terms Path Loss, fading and multi-path will be explained in the
space, frequency and time dimensions.
Later, we expand the description to a Multiple-Input-Multiple-Output (MIMO)
channel model ([4]-[7]) ,which is the general case for receive diversity Multiple-
Output-Single-Input (SIMO) and transmit diversity Multiple - Input - Single - Output
(MISO), with antenna correlation matrices at the transmit (Tx) and receiving (Rx)
ends.
Next we introduce the term channel feedback quality as was first explored in [8].
We will introduce a new channel parametric model, called virtual channel model, as
was introduced in [9], which is an example for new interesting ST channel model.
In the end of this chapter, we will develop statistical models versus CSI quality for
Gaussian channel assuming flat fading as a function of the correlation between the
real channel and the estimated one and CSI parameters.

2.3.2 Spatial wireless channel models
A signal propagating through the wireless channel arrives to the receiver along a
number of different paths, referred to as multi-paths, caused by scatterers and their
reflection and diffraction. The term scatterers is often used as a general term for all
objects in the environment, between transmitter and receiver. e.g. reflectors are large
scatterers. Different spatial wireless models vary according to scatterers‟ statistics [4],
[3, p 62-92]
Let us overview the physical characteristics of the SISO channel which will later be
generalized to MIMO channel.
Mean Path Loss (Propagation Loss)
Let us define L r as the mean attenuation, which is a function of the receiver
position. In a free space environment the path loss L r is determined by the "inverse
square –law spreading" physical law of electromagnetic wave propagation:
Where:
Pr
is total received power.
P is total transmitted power.
Gr
is the power gain of receive antenna.
G is the power gain of transmit antenna.
- Wavelength.
r - Range of transmission r r
(2.3)
Path Loss is being compensated by receiver Automatic Gain Control (AGC) and
transmitter Power Control. We assume perfect power control and optimal AGC and
thus ignore the effects of Path Loss.
In the case of non line of sights conditions (not free space), the main path loss
expression changes according to different channel models. Some models assume that
the main path loss is inversely proportional to the distance in powers between 3 and 6.
Fading
2
2 Pr
L GtGr Pt4r t
t
Let us denote t as the time index and r as the distance of the transmitter from the
reception array (in ULA the first element), the channel fading, (
tr , ) , can be
expressed by two multiplicative components, s ( , ) and
(2.4)
tr l ( , ) tr
( t, r)
s( t, r) l(
t, r)

Where these components are referred to as:
s ( , ) is large scale fading (called also Long term fading or shadowing)
and caused by reflection and diffraction from constant large scatterers,
sometimes called shadowing effect.
is small scale fading (called also Short term fading) is caused by
variations of scatterers surrounding the transmitter and or the receiver.
tr
l ( , ) tr
Note:
In some notations in literature, large scale fading is referred to as slow fading and
Small scale fading is referred to as fast fading.
With the far field assumption, we can express the fading as function of the angle from
the vertical to the ULA axis (see figure 1) and time
( t, )
s ( t,
)
l
( t,
)
(2.5)
Fig. 2 below (taken from [23]), shows short term fading , long term fading and mean
path loss as a function of source distance in a typical wireless channel.
Figure 2: LOS, slow and fast fading
Fading can be viewed also in time, frequency and space domains.
Time-selective fading
Time-selective fading is caused by variations in time due to scatterers surrounding the
transmitter and or receiver. Coherence time, Tc
, is the time separation for which
time-space correlation function of the channel impulse response at two times is
sufficiently correlated (exceeding pre-determined threshold). The Coherence time is

1
inversely proportional to Doppler spread, i.e. Tc
. If Coherence time is smaller
f d
than the symbol time, the fading is called time selective fading, or fast fading channel.
For the assumption of a large number of scatterers, the wave fronts are with random
amplitudes and angles of arrival and thus the phases are uniformly distributed in
[ 0,
2
) . The fading distribution is denoted by the name Rayleigh fading. The
Rayleigh complex envelope distribution function is given by:
2
y
2
2
y h e
y 0
p(
y)
2
h
y 0
0
(2.6)
Where y is the complex envelope, and is a sample of the stochastic process of the
channel h and h is Rayleigh channel fading standard deviation. The Rayleigh
distribution is:
2
h ~ CN(
0,
h )
(2.7)
Where, CN denotes complex normal distribution.
When some of the scatterers are fixed and there are dominant reflectors, there is a
direct path component (LOS), The complex envelope distribution then becomes
Rician. Rice distribution function is given by:
2 2
( y s
)
2
2
y ys
h e J
0
( ) 2
0
2
y
p y
h
y 0
0
(2.8)
Where y is the complex envelope, s is the mean power of direct path and h is
Rayleigh channel fading standard deviation. Rice distribution can also be expressed
as:
2
h ~ CN(
s,
h )
(2.9)
The spectrum of the fading process is determined by the highest frequency offset,
often related to mobile speed, and denoted as Doppler spread, f d . There are several
models for describing the spectrum. For instance, classical spectrum (as in Jake‟s
model):
1
S y ( f )
f
f
d 1
f d
(2.10)
Frequency-selective fading
Frequency-selective fading is caused by time-shifted replicas of transmitted signal,
multi-paths, due to constant scatterers surrounding the transmitter and on the
propagation path. The interval of delays is called delay spread and is denoted as d
.
s

Jake‟s model describes sum of discrete multi-path model each one has time-selective
fading distribution (e.g., Rayleigh fading distribution).
Coherence bandwidth, f c , is the frequency separation for which frequency-space
correlation function of channel impulse response (see [3, p 52.] for graph of
correlation versus DOA and frequency) at two frequencies is sufficiently correlated
(exceeding predetermined threshold). The Coherence frequency is inversely
1
proportional to Delay spread, i.e., Fc
. When coherence bandwidth is several
d s
times larger than the inverse of Symbol time, the channel is called frequency nonselective
fading channel, or flat fading channel, if not the fading is called, frequency
selective fading.
Space-selective fading
Space-selective fading, the variations of signal in space, is caused by different
receiver departure angles of the multi-paths due to reflection, scattering and
diffraction from scatterers (include obstacles such as buildings) in the propagation
path. In LOS conditions, there is low space selectivity. Space selectivity is determined
by antenna spacing, scatterers‟ distribution (mainly angle spread) and wave length.
Different scatterers‟ distribution models are well summarized in [3, p 63-91]. A
popular assumption (not realistic in correlated fading) is uniform distribution over
[ 0,
2
) or over the Angle Spread.
Coherence distance, c , is the angle or space separation for which channel response
at two antennas is sufficiently correlated (exceeding determined threshold). The
1
Coherence distance is inversely proportional to angle spread, i.e., c
. If
as
coherence distance (proportional to angle spread) is several times larger than the
antenna spacing, the channel is space non-selective fading cahnnel. If not, the fading
is called, space selective fading. Table 1 below (taken from [23]), summarize the
relations between channel parameters.
Spatial Noise
The noise in space time models is sometimes referred to as spatial noise with spatial
statistics which is related to scatterers‟ distribution. For instance, in a multi-user
environment, transmission from different users in different locations, are spatial
sources contributing to noise distribution. Spatial distribution of the scatterers (other
users) which depends on spatial location is referred to as "colored noise". For
simplicity, we will assume from now on, unless mentioned otherwise, that the noise is
independent of the space dimension, and it is a stationary white Gaussian process
(AWGN). We will denote this noise as n
t .

Table 1: Relations between channel parameters
Channel Selectivity Channel spread Measure of Selectivity
Frequency selective Delay spread, ds
Coherence BW,
Time selective
Doppler spread, fd Coherence time, c T
Space selective Angle spread, as Coherence distance, c
2.3.3 SISO channel Modeling
After describing the physical channel characteristics, we can start modeling a general
SISO channel model. The channel response can be expressed according to channel
fading similar to (2.5):
h( t,
)
( t,
)
l ( t,
)
s
( t,
)
(2.11)
With another SISO channel representation, which assumes limited number of discrete
multi-path (number of taps), where each of the paths has time selective fading
corresponding to Doppler spread, the SISO channel can be written as:
h( t,
) s ( t,
)
s,
l ( t,
l
) t
l l
t
l
l
l
(2.12)
Where l, denotes the multi-path index and l are complex, time varying coefficients
representing the channel. e.g. l with Rayleigh distribution.
Note that in case of flat fading, L=1, both SISO representation are the same,
h( t,
) (
t,
)
s ( t,
)
. From now on, we will assume flat fading, unless written
otherwise.
2.3.4 MIMO channel Modeling.
Lets us define a transmitter antenna array with N elements and a receiver antenna
array with r elements. The channel can be expressed as a Nr
N matrix, denoted
as H. Let us look on the antenna array response (steering vector) as part of the
F
T
c
c
c
t
1
d
d
s
1
f
N t
1
a
s
Fc

extended channel response. In SISO modeling where there was only one receive one
transmit antenna, the antenna array response could have been seen as one. With the
assumption that the steering vector is time independent, each element of H, can be
expressed as a function of its steering vector and its fading coefficients e.g. as in [4],
[5], [13]:
hi, j ( t, ) st, i ( ) i, j ( t, ) sr, j ( ) stl,
i ( l ) sl, i, j ( t, l ) srl,
j ( l
) (2.13)
Where 1... N , j 1...
N , l 1...
L and s ( ) s ( ) are transmit and receive
i t
r
steering vectors i'th element, respectively. Each elements, ( t,
)
, is a complex
random variables reflecting the channel statistics. Fig. 1 shows a diagram of a general
MIMO system. In our system model, flat fading, i.e. L=1, is assumed:
h ( t, ) s ( ) ( t, ) s ( )
(2.14)
i, j t, i s, i, j r, j
Correlated MIMO channel
l
t,
i
r,
i
For analyzing the correlated MIMO channel we define the correlation coefficients
between each element as follows:
i j E hi hj
(2.15)
where h is an vector formed by, the operator vec which is the span of H
*
,
N 1
r Nt
*
columns to one column and iE( hihi) 1. Following, the correlation matrix, Rhh
, is
defined as:
H
R [ ] E(
h h)
(2.16)
hh i, j
The additional input parameters to the MIMO model compared to the conventional
SISO models, are the antenna correlation matrices at transmit and receiving ends.
These correlation matrices might be selected to be different for each delay component
in the radio channel. We assume fixed correlation matrices, i.e. time-invariant, but the
model can be extended to support time-varying correlation matrices as in [13].
Another common assumption, as in [4]-[13], based on the fact that correlation is
caused by the nearby surrounding of the antenna array, is that the correlation between
the receive antenna elements are independent of the transmit antennas. Let us define
now, RT as the transmit correlation matrix as of size Nt Nt
, where the index of the
receive array is fixed on one element:
H
RT
E(
H H)
(2.17)
Similarly we define Nr Nr
receive correlation matrix, R , where the index of the
transmit array is fixed on one element
(2.18)
In the general case, of scatterers on the propagation path, we define the elements,
based on general scatterers‟ distribution, same as in [5], as:
(2.19)
R
H
RR E(
HH )
RR
r / 2 di
, j
i2
sin( )
e p(
)
d
i j
RR
i,
j
r / 2
1
i j
hi, j

Where is the distance between the i'th and j'th element in the receive antenna
d i,
j
array, is the angle from the vertical axes of the ULA, r is receive angular spread
and p(
) is scatterers angular distribution. Specific scatterers' angular distribution,
such as uniform, Gaussian and Laplacian are described in [4].
For the extreme case, where the scatterers are uniformly distributed in between 0 and
2, i.e. Rayleigh fading distribution, we obtain the following receive correlation
matrix:
di,
j
R
R i,
j J 0 2
(2.20)
We denote the other extreme case, where there are no scatterers in between 0 and 2,
as Line Of Sight (LOS) condition. From (2.20) the received correlation matrix
becomes, similar to the corresponding array vector response (steering vector) as:
di
, j i2
sin( )
R
e i j
R i,
j
1 i j
(2.21)
Similar expressions as (2.21) and (2.22) can be derived for R .
SVD over the symmetric channel correlation matrices gives:
H H
T T
R E( H H ) V V
H H
R R
R E( HH ) U U
(2.22)
where V and T
are transmit eigenvector and eigenvalue matrices respectively and V
and R
are receive eigenvector and eigenvalue matrices respectively.
As written above, the correlation matrixes, RT and RR
are functions of the angular
distribution inside the angle reception aRs and transmission angular spread aTs
,
respectively. Still, some different spatial scenarios would generate the same channel
response. For instance, it can be shown that a scenario with high correlation, caused
by widely spaced antennas and low angle spread ( as
), is identical to scenario of high
angle spread and closely spaced antennas.
The correlated MIMO channel matrix can now be expressed as (see [5], [7]):
1 1/
2 1/
2
H RR
GRT
tr(
RT
)
(2.23)
Where G is a complex matrix the elements of which are independent and distributed
normally with variance one.
By substituting the appropriate RT and RR
into (2.23), we can obtain the correlated
MIMO correlated Rayleigh channel, , and LOS channel, H .
H Ray
LOS
Under the assumption of Nt Nr
, H can be written, in slow fading channel, as (e.g.,
[13]):
1 1/ 2 1/ 2 H ˆ ˆH
H U R GT V UGV
(2.24)
tr(
)
T
T

1
where Uˆ U , Vˆ V
tr(
)
Later we will see that SVD enables, with appropriate pre and post coding, the transmit
and receive antennas to use independent data streams, called sub-channels.
Rice distribution
A common general MIMO channel distribution is Rice distribution which consists of
weighted summation of a Rayleigh distributed channel, and LOS channel, H ,
based on physical wave superposition:
H aHbH 2 2
Where a, and b are constants normalized to one, a b 1
.
It is common to describe HRician
also as:
H Rician
K
H LOS
K 1
1
H Ray
K 1
a
where K, the Rician constant (Cofactor), equals to, K
b
Channel matrix rank
1/ 2 1/ 2
R T
T
Rician LOS Ray
(2.25)
(2.26)
Let us denote by r, the rank of channel matrix H. If we define, as above, G to be
consisted of independent Gaussian variables, covariance of G is full rank (though the
H
instantaneous value of GG is not necessarily full rank, due to fading effect), and
thus the channel rank is explicitly determined by the minimum rank of RT and RR
.
Consequently, channel rank is determined by the matrix correlation at receive and
transmit ends and thus determines spatial selectivity.
A more realistic approach, in which G is not assumed to be full is analyzed at [5]. In
this approach wide scale reflectors causes keyhole effect for instance, and reduce the
channel rank. Still, different physical scenarios can lead to same channel
characteristics, so the above assumption is justified for analysis purpose.
2.3.5 Channel feedback quality
H Ray
LOS
Channel Side Information (CSI), is based on the channel estimates, the instantaneous
approximated channel value, h and on channel statistics, the estimated channel mean
value, , and the estimated channel covariance matrix, , and some times on
ˆ
R
m
h/ hˆ
hh/ hˆ
higher statistical moments.
A Lot of research was being held recently in the subject of channel feedback quality.
One of the most extensive works, which studied and tested the effects of noisy side
information and quantized side information on the expected SNR and mutual
information, was held by Narula in [8].
2
2

For simplicity, let use the common assumption about the receiver assumed to have
perfect channel knowledge. It is straightforward to extend the following development
to also take channel estimation errors at the receiver into account. In case where there
are constant channel components, i.e. LOS component a in (2.25), we assume the
approximation of the LOS competent, hLOS
, is perfect. This assumption realistic in
slow fading scenarios as was assumed for our system model. CSI can be obtained at
transmitter by:
1. Feed backing the channel estimates which were obtained in the receiver.
2. A direct estimates of the channel at the transmitter according to received
signal at transmitter, usually by dedicated Pilot channel (in Duplex system),
sometimes by blind estimation of CSI parameters.
Measures of the CSI quality is mainly obtained by:
1. Measurement Gaussian noise
Since the side information has the form of random variables [8], representing
noisy estimates of the channel coefficients, we can define the measurement
Gaussian noise as:
n h/ hˆ h
m
(2.27)
CSI quality can be characterized by the measurement noise mean and variance
values – low mean value means good estimation in average.
2. Estimation correlation
Average correlation between actual channel coefficients and their estimates,
defined as:
est
i1
(2.28)
[8] Showed that in a system with CSI, there is an improvement in the expected SNR
over a system with no CSI and it increases with est
.
In [25], the relations between and the conditional channel covariance matrix, R
, are shown to be:
1. If there is perfect side information, 0 and .
R
1
1
2. If there is no side information, R hh/
hˆ
0 and
0
Explicit analysis of est presented in [8] and [25] shows that est
, and thus CSI
quality, suffers from:
cov( h, hˆ
)
i NRNT H
i i
ˆH cov( , )cov( , ˆH
h h h h )
i i i i
1. Short Coherence time (compared with the symbol time or block transmission
time). Caused by fast fading, which implies difficulties in obtaining reliable
measures of the channel, in particular for the mean channel value.
2. Narrow Coherence bandwidth (compared with the symbol bandwidth).
Caused by non-flat fading, implies difficulties in obtaining reliable measures
of the channel, and thus decreases . Combining the multi-paths, related to
the non-flat fading, can increase
level.
est hh/ hˆ
est
hh/
hˆ
est
est
est

3. Quantization error of the feedback
Given N bits of side information, the transmitter can follow a vector-quantized
based approach to determine a locally optimal transmission strategy.
4. Feedback delay and thus outdated measurement (feedback delay) in TDD.
5. Errors due to different estimation in different frequencies (if FDD channel
estimation is used).
2.3.6 Virtual channel model
Analogous to SVD, a new parametric channel model was proposed in [9], called
virtual channel model. The virtual channel modeling, enables great benefits in various
scatterers distributions (realistic MIMO fading channels) by capturing the essence of
physical modeling with reduced computations number and without a need for explicit
channel statistics parameters such as source Direction of Arrival (DOA) (see chapter
4.4).
The virtual representation corresponds to a fixed coordinate transformation with
respect to spatial basis functions defined by fixed virtual angles of arrival/departure,
in a similar manner as DFT (Discrete Fourier Transform).
Figure 3 A schematic illustrating the virtual representation of physical scattering
Parameterized physical model represents H is similar to above MIMO channel:
H l
sR
( R,
l ) sT
( T
, l )
l
(2.29)
Where sR and s T are receive and transmit array steering vectors respectively, Rl
, and
Tl
, are receive and transmit array DOA associated with the l'th path respectively (fig.
3) and R, l d sin( R, l ) / T , l d sin( T , l ) / .
The virtual channel representation is a special case of the parameterized physical
model of (2.23) and can be expressed as:
N
i
N
R T
H
j
[ HV
] i,
j sR
(
R,
i ) sT
( T
, j
) S
R
H
V
S
T
(2.30)

Where the transmit and receive matrixes, S [ s ( ), s ( )... s ( )] and
S [ s ( ), s ( )... s ( )] , of size ( N N ) and ( N N ) , respectively, are full-
T T T,1 T T,2 T T, N
rank matrices defined by the fixed virtual angles and
. These matrices can
T , j R, i
be sampled, for instance, uniformly in the interval [-.5-.5] and thus S R and
become unitary matrices.
ST
H is unitarily equivalent to HV
and captures all channel information. Realistic
propagation environments can be modeled via a superposition of scattering clusters
with limited angular spreads. Thus the virtual channel matrix provides an intuitively
appealing “imaging” representation for such environments: different clusters
correspond to different non-vanishing sub-matrices of H .
2.3.7 Statistical models versus CSI quality.
T
R R R,1 R R,2 R R, N
R R
T T
For further analysis purposes, we will define channel statistical models, as a function
of CSI quality with the assumptions of flat fading, slow fading for the case of transmit
diversity. We will further assume Rice distribution with correlation between transmit
antennas.
Substituting the antenna correlation matrix term in (2.24) to the Rice fading Rayleigh
component in (2.26), we obtain:
h ahLOS
1/
2
bhRay
RT
(2.31)
Subsequently, the distribution of h can be written as:
2 2
h ~ CN(
ahLOS
, b h RT
)
In similar manner, the channel estimate can be expressed as:
(2.32)
hˆ ahˆ
LOS
1/
2
bhˆ
Ray RT
(2.33)
We will further assume that the channel parameters K, a, b and RT
are known and that
hLOS is perfectly estimated, hˆ LOS hLOS
. These assumptions are realistic due to our
system model assumption of slow fading (slow change rate in user location and big
reflectors). The value of hLOS
can practically be obtained by averaging of the
estimated channel and the Rayleigh component variance can be obtained by channel
estimates variance.
(2.32) and (2.33) can now be written as:
1/
2
hˆ ah ˆ
LOS bhRay
RT
ˆ
2 2
h ~ CN(
ahLOS
, b h RT
)
(2.34)
With straightforward calculation, we can now see that the correlation between the
estimated channel h and the real channel, h, is determined only by the Rayleigh
components correlation, i.e. .
For deriving the distribution of the channel conditional first and second moments,
and , we will use the optimum Gaussian estimator:
ˆ
( , ˆ)
( , ˆ
est h h est
hRay
hRay
)
mh / hˆ
Rhh/ hˆ
V
R

m m R R ( hˆ m )
ˆ
h/ h h
hh ˆ
1
hh
hˆ
/ ˆ
hh h hh
hh ˆ
1
hh hhˆ
R R R R R
(2.35)
We have to obtain first, the terms, R R R m , m . R R m , m were
2 2
already obtained above according to (2.32), (2.34) and equal to, R R b R ,
m
. For obtaining R , R , which are identical due to symmetric
h m ah hˆ
LOS
hh ˆ hhˆ,
hh ˆ
similar calculation show that R R .
Rhˆ , h hhˆ
, hh,
hˆ
hˆ
, h hˆ
hh, hh ˆ ˆ, h hˆ
(2.36)
Now we can substitute the obtained terms and obtain the expression of the first two
channel conditional moments:
2 2 2 2 1
m
/ ˆ ah ˆ
LOS b est hRT( b hRT)
( h ahLOS
)
hh
(2.37)
2 2 2 2 2 2 1
2 2
R bRbR( b R ) b R
Note that RT is not always full rank, and thus inverse for RT
can not always be
obtained.
For the case of uncorrelated fadings, i.e. I , we obtain:
(2.38)
(2.39)
To conclude this section, let us look at 3 interesting cases.
The first one, is LOS, where, a=1, b=0. The distribution then becomes, a constant,
h / hˆ
~ CN(
ah
ˆ
LOS ( 1
est ) est
h,
0)
(2.40)
On the other extreme case of Rayleigh fading, b=1, a=0, the distribution then
becomes:
ˆ ˆ 2 2
/ ~ ( est , h ( 1 est ))
(2.41)
With Perfect channel estimation, i.e. , the distribution becomes:
(2.42)
This indicates that the real channel is the estimated one, as expected.
h CN h h
est
1
h / hˆ
h ~ CN(
E(
hˆ
), 0)
CN(
hˆ
, 0)
hh
hˆ
hˆ
(( ˆ ( ˆ H
R E h E h)) ( h E( h)))
considerations, we will use the covariance definition,
and obtain:
hh ˆ
R E(( bhˆ R ) ( bh R )) b E(( R ) hˆ h R ) b R
hh/ hˆ
h/ hˆ
hh/ hˆ
1/ 2 H 1/ 2 2 1/ 2 H H 1/ 2 2 2
Ray T Ray T T Ray Ray T esthT hh ˆ hhˆ
h T est h T h T est h T
m ah (1 ) hˆ
LOS
2 2 2 hest R b(1 ) I
est est
RT N
R NR
and the conditional channel distribution is:
/ hˆ
~ CN(
ah ( 1
) ˆ 2 2 2
h,
b ( 1
) I)
h LOS est est h est
h
T

2.4 Received and transmitted Signal model
Based on previous definitions and assuming slow and flat fading, we can now define
the full system model for MIMO channels. In the system model, a signal is
transmitted from NT transmit antennas to N R receive antennas. The received signal
at the receiver is:
Y HXN (2.43)
Where X is N L space-time code word matrix, expressed as:
T
1 1
1
x
1 x2
x
L
2 2
2
x
1 x2
x
L
X
(2.44)
NT
NT
x
1 x
L
j
Where xi
are the symbols transmitted from the j‟th antenna in the i‟th time slot and L
is the coding block length which unless mentioned else are BPSK modulated, Y is the
N received signal, H is channel matrix in size of with channel fading
R L
NRNT determined by specific channel contribution, N is N R L vector of AWGN with zero
mean and a standard deviation .
The codeword can go through linear transformation, which can be well modeled by a
NT NT
matrix, denoted by WT
.
Thus (2.43) can be written as:
H
Y HWT
X N
(2.45)
It is also common to refer to linear transformation in reception, through N N
matrix, denoted by W , in this case, the received signal becomes W Y .
H
R
W R
The weights, T and , are being determined by techniques, such as Maximum
Ratio Transmission (MRT, see chapter 3.2.2-3.2.3) and Maximum Ratio Combining
(MRC) respectively. These techniques exploit several replicas of the same
information signal over independently fading channels and thus the error probability
is considerably reduced.
W
We will we focus in this work on the case of transmit diversity, i.e., N R 1.
We
assume optimal ML receiver (equal to MMSE receiver in our Gaussian channel) with
imperfect CSI knowledge (see fig. 4) with the following ML criterion:
ˆ H
X argminYHWX X
T
2
F
R
(2.46)
R
R

X
W T 1
W T N
11
1 1
N 1
Figure 4: Received and transmitted channel model scheme
2.5 ST System Performance Evaluation
After describing our system model, we will introduce in this chapter a way of
evaluating system performance by means of error probability at the receiver. We will
later see that most ST techniques are aimed to minimize the receiver error probability.
We have to emphasize that in communication systems it is sometimes used to refer to
system capacity as another measure for system performance, i.e. maximization of the
mutual information between receiver and transmitter enhance system performance as
in e.g., [10], [11]. In certain model assumptions, such as Single User in Gaussian
fading channel BPSK modulation, the two performance measures coincide. These two
measures are also related received SNR maximization.
2.5.1 AWGN error probability bounds
.
.
.
Non Perfect Side Information
1 M
NM
N t N r
In a non-fading scenario, for SISO, the error probability of an un-coded BPSK signal,
denoted also as AWGN SISO bound, is given by:
Perr( b) Q(
2 b)
(2.47)
Eb
Where, b is the received SNR per bit, b , b is the bit energy and is the
N0
noise power spectral density.
In case of multiple antennas, either in reception or transmission, denoted by N, where
receive diversity is being exploited by optimal MRC combining or in case of ideal
E 0 N
.
.
.
N 1
N M
W R 1
W R M
Perfect Side Information
X ˆ

Eb
MRT, the received SNR becomes , b, N N Nband
thus Perr
, N , denoted also
N0
as AWGN MISO or SIMO bound becomes:
P ( ) Q( 2 N
)
err, N b b, N
2.5.2 Error Probability in fading scenarios
(2.48)
2 b
In fading scenarios, such as our system model, the bit energy becomes, b ,
N0
where is the fading coefficient as defined in (2.11).
And thus the error probability density becomes:
Perr( b | ) Q(
2 b
)
(2.49)
For obtaining error distribution, Perr ( b ) , we have to average (2.49) with
distribution in the specific channel. Let us also denote average SNR as b .
For Rayleigh fading, the error probability as was derived in [11], becomes:
b as
1
Perr
1 2
b
1
b
(2.50)
b
Where b , the average SNR, is b N
2
E .
With high average SNR ( 1),
the error probability may be approximated by:
1
Perr
4
b
(2.51)
We see that for Rayleigh fading channel, the error probability decrease inversely with
the SNR. Thus, there is an inverse linear ratio between the SNR and the error
probability. This is in contrary to (2.47), the AWGN channel bound, where the error
probability decreases exponentially, as the SNR increases.
With multiple antennas, again, with the usage of diversity techniques, such as
Maximum Ratio Combining, or Maximum Ratio Transmission, the SNR per bit is
b
2 2
then .
b, N k b k
N0
k1 k1
The error distribution, Perr , N ( b
) , can then be obtained by averaging (2.49) according
to bN
, distribution in the specific channel. For Rayleigh fading, the error probability
as was derived in [11], becomes:
and for high SNR:
N N
b
N
1
k
1N
b N 1k1
b
perr,
N 11 2
1
b k 2 1
k0
b
N
1 2N1 perr,
N
4b N
0
(2.52)
(2.53)

Where , the average SNR per channel, assumed to be identical for all channels,
b
b
2
b
E k
. The diversity gain (coherent Gain) is the slope of the curve of BER
N0
and is a function of average SNR- as the diversity gain increases, the slope becomes
sharper. By smart ST processing with CSI exploitation, the BER is bounded by (2.52)
and in Rician channel, the bound becomes (2.48).
The bounds in (2.48) and in (2.52) are plotted in Fig. 5 for different values of N.
Figure 5: BER performance for BPSK modulation on Rayleigh fading channel (circles) and
SISO and MISO bounds (without fading) with MRT for 1,2,4,8 transmit antennas

3. ST processing techniques Overview
3.1 Introduction
Exploiting the spatial dimension in wireless communication system is enabled by
exploiting antenna arrays and plays essential rule in improving system performance
up to the theoretical capacity limits (see table 2 below). There main ways of
exploiting antenna arrays with spatial processing are:
1. Directional antenna array (in wireless system sectorized antenna array).
Subdivide cellular area into sectors that are covered using directional antennas
looking out from the same base station location. In some cases the sectors are
determined according to channel structure and subscribers distribution.
2. Beamforming (spatial filtering in reception)
Modify the phase and amplitude of transmitted and/or received signals by pre
and post coding respectively, and effectively combine the power of the signals
to produce gain. Spatial filtering (array processing), enables the control of the
outcome array beam pattern to reduce interference.
3. Diversity techniques.
Diversity techniques are based on exploitation of space selectivity for getting
spatial diversity gain. We can integrate several other methods of diversity
exploitation together with space diversity for enhancing the diversity gain:
Space-frequency techniques- use different frequencies for each sub
channel/ antenna.
Space-polarization techniques- use different polarization for each
sub-channel/ antenna.
Space Bandwidth techniques- Change transmitted signal bandwidth
and exploit the strongest multi-paths for obtaining bandwidth diversity
gain.
Space-time (ST) techniques- exploits time selective fading by either
multiplexing transmitted signals in time or by ST coding.
4. Spatial multiplexing.
Spatial multiplexing techniques, transmission of independent data stream from
each antenna element, enables to maximize the transmission rate in MIMO
channel.
We will further focus in this work on Beamforming and Space-time (ST) techniques,
in particular space-time coding (STC) where different code word is transmitted via
each antenna. On the next chapter we will describe ways of combing both techniques.

:
Table 2: The benefits of smart antennas techniques ([12])
Feature Benefit
Signal gain—Inputs from multiple
antennas are combined to optimize
available power required to
establish given level of coverage.
Interference rejection—Antenna
pattern can be generated toward
cochannel interference sources,
improving the signal-to-interference
ratio of the received signals.
Spatial diversity—Composite
information from the array is used
to minimize fading and other
undesirable effects of multi-path
propagation.
power efficiency—combines the
inputs to multiple elements to
optimize available processing gain
in the downlink (toward the user)
3.2 Beamforming
3.2.1 Introduction
Better range/coverage—Focusing the energy sent out
into the cell increases base station range and coverage.
Lower power requirements also enable a greater battery
life and smaller/lighter handset size.
Increased capacity—Precise control of signal nulls quality
and mitigation of interference combine to frequency reuse
reduce distance (or cluster size), improving capacity.
Certain adaptive technologies (such as space division
multiple access) support the reuse of frequencies within the
same cell.
Multi-path rejection—can reduce the effective delay
spread of the channel, allowing higher bit rates to be
supported without the use of an equalizer
reduced expense—Lower amplifier costs, power
consumption, and higher reliability will result.
Beamforming (BF) utilizes transmitted and/or received signals phase and amplitude
induced in each antenna in an antenna array, to form beams in different desired
directions. BF can be seen as moving from antenna space to beam space by Fourier
transform.
There are two main ways for BF. The first one is with fixed beam patterns - the beams
are fixed and predetermined and the transmitter can switch from one to another and is
called Switched-Beam BF. The other way, non-fixed beam patterns, is obtained by
continuous updating of antenna weights according to CSI knowledge and subscriber
spatial distribution. By this way optimal BF can be achieved. When the beamforming
weights are changed adaptively, it is called adaptive BF.
We will start in 3.2.2 with a description of SVD based BF. On 3.2.3, we introduce
MMSE based BF. BF methods performance is briefly discussed in 3.2.4. In 3.2.5, a
physical perspective of BF is shown.

3.2.2 SVD based BF.
BF can be referred to also as an SVD based techniques. The ST system can be
transformed into a set of r parallel scalar channels, where r is the channel rank. The
available transmit power may then be optimally allocated (in maximum capacity
sense) to the individual channels. For this technique, a certain value of channel
realization and statistics is a must in the transmitter and receiver for spatial pre-coding
before transmission and the spatial post-coding in reception. Pre-coding and Postcoding
can be used adaptively to track channel changes. In the case of MISO, the term
Beam Forming (BF) in transmission is often used, while in SIMO, the term Spatial
Filtering in the receiver is often used.
The subspaces spanned by RT and RR
(2.3.4), can be separated to two different
subspaces in the transmission and reception- Signal subspace and noise subspace
respectively [13]. Signal subspace is referred to as signal reflections from large
reflectors (different signals or users), each one has a related eigenvalue and an
eigenvector. Signal subspace can be also obtained by forming a base out of the
steering vectors in different location, and thus refer only to the dominant signal
reflectors (sources). Noise subspace is referring to noise spatial sources each one has
related eigenvalue and eigenvector. Noise subspace can be also obtained by forming a
base from steering vectors in the direction of the spatial noise sources. Thus a good
BF, will try to focus its energy in the direction of the signal corresponding
eigenvectors and will try to avoid transmission in the direction of noise eigenvectors.
If we want to match the weights to the channel statistically, in order to compensate as
much as possible on channel fading, we have to demand in the transmission and
H
H
reception, RT WT
WT
and RR WRWR
, respectively.
By using (2.22), we obtain:
H 1/ 2 1/ 2 H H
RT V TV V T ( VT ) WT
WT
(3.1)
and the transmission weight matrix is:
1/ 2 1/ 2 H H 1/ 2
WT RT ( T V ) VT
(3.2)
similarly, for reception weights, we obtain:
1/
2 1/
2
WR RR
U
R
(3.3)
To examine the effect of the SVD weights on the transmitted signal, and how the
signal is matched to the channel, we have to substitute (3.2) into, (2.45) for example,
in transmit diversity ( RR 1)
correlated Rayleigh fading:
H
H 1/ 2 H 1/ 2 H 1/ 2 H 1/ 2
T T T T T T
Y HW X N G V V X N V V X N X N (3.4)
(3.4) enable the transmitter to adjust to the channel by transmitting the input code in r
( NT
if R T full rank) Gaussian scalar sub-channels at the direction of transmit
v v , , v
correlation eigenvectors,
1, 2 N . The transmitted power can also be
T
optimized (maximize the total capacity) by using algorithms like water pouring for
forming a set of new eigenvalue, , and the transmitted signal, becomes:
(3.5)
ˆ
W X V X v X v X v X
T T N N
H
ˆ1
/ 2
T ˆ
T ˆ ˆ
1 1 2 2

When only one BF (direction) exists, v , 0,
,
0 and the eigenvector is the estimated
channel, the matrix WT
gets the classical meaning of BF (one dimensional BF), and if
the channel estimate is perfect then U X becomes simply X.
3.2.3 Optimal MMSE BF
In this chapter an MMSE approach for BF is introduced. In this approach, a different
weight is induced for each antenna, and forms one vector and thus is sometimes
referred to as one dimensional BF. In order to adopt these weight, (antenna weight
vector), to our system model (antenna weight matrix), the antenna weight matrix has
to be diagonal where its terms are the same as the antenna weights.
In the case of receive diversity (SIMO, similar to [3]), the cost function, obtained for
instance, by sending training sequence (pilot), where H is assumed as being perfectly
estimated, is:
H
2
H H H
R R R
J( W) arg min E W Y X E( W Y X ) ( W Y X )
WR
and in the case of transmit diversity (MISO), the cost function is:
T
(3.6)
(3.7)
From considerations of reciprocallity, reasonable in many channel models, the receive
weights will be the same as the transmit weights. Therefore, with out loss of
generality, we will describe here the optimal weight for receive diversity as were
derived in [3]. Following the method in [5], the gradient:
(3.8)
H
H
Where RY E(
YY ) is the received signal autocorrelation and RYX E(
YX ) is the
cross correlation between received and transmitted signals. The solution of (3.8) is the
Spatial Winner filter:
1
WR
RY
RYX
(3.9)
H
With the power normalization E( XX ) INt
By substituting our system model in (2.45), we obtain:
WR E(( HX N)( HX N) ) E(( HX N) X )
(3.10)
1 1
( Rhh RN)
E( H)
In the case EH ( ) is zero, equal antenna weights are used (from symmetric
considerations).
In the simple case of slow fading, the channel tends to be constant in reservation
period (e.g. LOS) and the location of the reception source is the same as the steering
vector in the source direction. (3.9), then becomes [3]:
(3.11)
When the noise is not directional (like our system model), (3.11) becomes:
1
WR
N
h
(3.12)
1
H
2
H H H
T T T
J( W ) arg min E Y HW X E( Y HW X ) ( Y HW X )
WT
J ( W ) ( E( W Y X ) ( W Y X )) 2 E( YY ) W 2 E( YX )
2R W 2R 0
W
Y R YX
1
RNh R
H 1
N
h R h
R
H H H H H
R R R
H 1
H

The physical interpretation for WR
can be seen as maximal ratio combining
(reception), MRC - optimum weighting in the spatial dimension- canceling the phase
effect of the channel (coherent combining) and weighting the channels according to
their SNR, similar to Wiener filter in frequency or in time. Note that the MMSE
criterion doesn‟t necessarily coincides with received SNR maximization criterion. It
can be explained by MMSE criterion taking into account statistics of higher order
than the SNR. Only in the special case, where CSI is perfectly known at the
transmitter, minimizing the SER is equivalent to maximizing the average SNR.
Similar expression for “Wiener filter” for WT
can be derived in the same way as in
(3.8)-(3.10). BF is also referred to in the literature as Maximum Ratio Transmission
(MRT), due to the coherent combining of the beams, similar to the coherent combing
of multi-path at receiver. In Appendix B, iterative techniques deviated from (3.9), are
shown.
3.2.4 BF coding performance
Straight forward techniques such as MMSE BF or SVD BF, even though each relies
on another optimization criterion, i.e., the first on MMSE and the last on Capacity,
coincides in many scenarios to one solution of antenna weights (e.g., [31]). With
perfect CSI, the performance reaches the bounds in (2.48), (2.52), for the cases of
LOS and Rayleigh fading respectively. In rapid changing channel, where CSI
continuously changes, adaptive BF, is almost a must in order to achieve best
performance with reasonable computations (see Appendix B).
3.2.5 A physical perspective
A physical perspective of BF is usually presented by beam (antenna) patterns
according to:
H H
F( ) f ( ) s ( )
W X
(3.13)
Where f ( ) denotes the radiation pattern of each antenna element, s( ) is the
steering vector of size 1 T , W is weight vector (in reception or transmission) of size
in case of MMSE based BF and of size NT NT
in case of SVD based BF and X
is the transmitted symbol. As described before, the physical meaning of BF is in
steering the beam in the direction of the desired signal source (including multi-paths
in case the channel is frequency selective) and steering nulls in the direction of noise
sources, or undesired signal sources (such as other users). Many scatterers on the
propagation path, results in spreading the radiation in all directions, omni directional
transmissions. Imperfect CSI also results in omni directional transmissions. The
antenna pattern can be derived by plotting the absolute value of the antenna array
radiation as a function of spatial angle, i.e. .Let us assume from
now on, for simplicity that the radiation pattern of each antenna element is equally
distributed in space, i.e. f ( ) 1 .
N
NT 1
H
F( )
F( ) F(
)

MMSE based BF physical perspective
Fig. 6 and fig. 7 below, show antenna patterns which were introduced in simulation in
Line Of Sight (LOS) conditions with one user at predetermined DOA of 20 degrees. It
can be observed, as written in literature, that the antenna pattern directivity increases
linearly with number of antenna elements of the array.
.
150
210
120
240
90
270
Figure 6: Antenna pattern, , for user at DOA of 20 degrees with 2 antennas
Figure 7: Antenna pattern, , for user at DOA of 20 degrees with 8 antennas
1
0.6
0.4
0.2
0.8
180 0
150
210
F(
)
120
240
90
270
1
0.6
0.4
0.2
0.8
60
300
60
300
30
330
30
180 0
F(
)
330

SVD Eigenvector physical perspective
Fig. 8, 9 show the antenna patterns of the eigenvector, i.e. a column of the channel
eigenvectors matrix obtained by SVD, for 2 and 4 antennas respectively, in LOS
conditions with one user at predetermined DOA of 20 degrees. It can be observed that
the beams directions are orthogonal (different beam patterns and directions) and that
one of the eigenvectors (the second one) corresponds to the real channel. In fig. 10
below, all the eigenvectors spatial response is plotted in one graph. The third graph
from left correspond to 20 degrees (-20 degrees). In LOS conditions, only one
eigenvalue is non zero, and the eigenvectors correspond to the zero eigenvalues span
the space orthogonal to the non zero LOS eigenvector.
150
210
120
240
90
270
1
0.8
0.6
0.4
0.2
60
300
30
180 0
150
210
330
Figure 8 : Eigenvector spatial response two antenna elements array.
120
90 1
60
0.5
30
180 0
240
270
300
330
150
210
120
90 1
60
0.5
30
180 0
240
270
300
330
Figure 9: : Eigenvector spatial response for 4 antenna elements array.
150
210
120
240
90
270
1
0.6
0.4
0.2
0.8
60
300
30
180 0
150
210
120
240
90 1
60
0.5
30
180 0
270
300
330
150
210
120
240
330
90 1
60
0.5
30
180 0
270
300
330

3.3 ST coding.
3.3.1 Introduction
150
210
120
240
90
270
Figure 10: All eigenvector spatial response for 4 antenna elements array.
ST diversity techniques, in particular Tx diversity techniques as assumed in our
system model, exploit space selectivity and thus gain spatial diversity. In particular,
we will focus on this section on exploiting Space-time diversity- Exploit space and
time selective fading by multiplexing and or replicating transmitted signal in time and
in space. It can be done, by decreasing the correlation between spatial diversity
branches and thus increase the diversity order of the system.
One way for increasing diversity order is by forcing a different delay to each antenna
element, a technique called also delay diversity. A second way can be achieved by
introducing different phase (random) to each element [18]. A third way, combines
time shifting and altering the shifted symbol phase (conjugate symbol) is called Space
Time Codes (STC). When the coding is coded in blocks, it is called Space Time
Block Codes (STBC). When the blocks are orthogonal it is called Orthogonal Space
Time block Codes (OSTBC). Other space-time coding techniques are layered space–
time architecture spatial multiplexing (e.g. BLAST) which enhance capacity by spacetime
multiplexing, ST trellis codes (STTC) and ST Turbo codes. Common to all the
space–time coding schemes mentioned above is that they do not exploit CSI
knowledge inherently at the transmitter. We will focus mainly in OSBTC and unless
mentioned otherwise by the term STC or STBC we mean OSBTC.
We start this chapter with STBC Design Principles, based on minimization of SER.
Simple orthogonal ST coding was first derived in [20], for two transmit antennas,
later extended to four and eight transmit antennas by [22], with low-complexity
receiver. It was shown that these OSTBC provide the system its maximum diversity
order. We will end this chapter with examination of OSTBC performance as a
function of the number of antennas and with antenna patterns induced by OSTBC.
1
0.8
0.6
0.4
0.2
60
300
30
180 0
330

3.3.2 STBC Design Principles.
As in our system model, (2.45), X is NT L code word matrix and the receiver is
again assumed to be a maximum-likelihood receiver which may decide erroneously in
favor of a signal other than that transmitted. Constellation independent design
criterion can be achieved by proper energy normalization (so the code design applies
equally well to 4-PSK, 8-PSK, and 16-QAM).
Let us examine the probability that codeword matrix X, written in the form of vector,
1 2 NT
1 2 NT
X [ x1
x1
x1
x2x
2 xL
] , was transmitted and ˆ 1 2 NT
1 2 NT
X [ xˆ
ˆ ˆ ˆ ˆ ˆ
1x1
x1
x2x
2 xL
] was
decoded instead.
Let us define the code error matrix B( X , Xˆ
) as:
1 1 1 1
1 1
x xˆ
x xˆ
x
L xˆ
1 1 2 2 L
2 2 2 2
2 2
x x x x x L xL
B X Xˆ
ˆ ˆ
ˆ
1 1 2 2
( , )
(3.14)
NT
NT
NT
NT
x xˆ
xL
xˆ
1 1
L
and the square code error matrix as:
ˆ ˆ H
A(
X,
X ) B(
X,
X ) B ( X,
Xˆ
)
(3.15)
With the assumption of perfect channel knowledge we can express, for perfect
channel knowledge, the conditional Pair wise symbol Error Probability (PEP),
P( X Xˆ | H)
, which as was shown in [19], minimize also the error probability and is
bounded by:
ˆ
2 ˆ 2
P(
X X | H)
exp( d
( X,
X ) / 4
)
(3.16)
Pe 2 ˆ
Where d ( X , X ) is the difference in the received signals in gain and is given by:
2
d ( X , Xˆ
) Y Yˆ
H(
X Xˆ
) ( ( , ˆ H
tr HA X X ) H ) (3.17)
F
(3.17) can be expressed also by its eigenvalues as shown in [19]:
d
2
N
R T
( X , Xˆ
)
N
i1
j1
j
j,
i
2
F
(3.18)
where
) is the orthonormal projection of H column j on the eigenvector
( 1, i NT
, i
space of A( X , X ˆ ) .
by substituting (3.18) into (3.16), with further algebraic manipulation, we derive the
expression of the error probability bound:
EH ( i, j ) j
N R NT
1
2
P
4
e exp
(3.19)
2 2
i1 j111/ 4 1 j/
4
Now according to channel model and its statistical distribution we can derive explicit
error distribution and consequently explicit design criterions, which minimize the
error probability [11]. A detailed design criterions based on channel model, channel
rank, antenna element‟s number, fast or slow fading and Hamming distance between
two codewords, is shown in [19]. For simplicity and as it is the most common
assumption in STBC design, we assume Rayleigh fading model.

The error probability bound in Rayleigh fading model is:
NT 1
Pe
2
j 11 j/ 4
NT
j
j1
2
1/ 4
Where r is the rank of A.
A
j
(3.20)
Since are the coding matrix eigenvalues, the first term in the multiplication in
Nr
NT
(3.20), j , can be interrupted as coding gain (coding factor) and the second
j1
exponent of the second term, r m
A , is the SNR factor of each sub-channel or spatial
diversity gain.
A design criteria for uncorrelated Rayleigh fading, account also for all cases, of slow
fading with low values of rN R was obtained in [19]. Other criteria should be applied
to other cases such as fast fading or large values of rN R
The design criteria for uncorrelated Rayleigh fading consists of two stages:
1. The Rank Criterion
In order to achieve the maximum diversity, the matrix has to maximize the minimum
rank over the set of all two distinct codewords.
2. The Determinant Criterion
Maximize the minimum determinant of the matrix A, along the pairs of distinct
codewords with the minimum rank.
If the channel coefficients are correlated, we face degradation in performance due to
the decrease in system diversity order. With the same expressions and design
criterions are used for the correlated case [19], there is the following penalty in coding
gain and consequently to the over all gain, expressed in Decibels:
L 10/( N N ) log10
det( )
(3.21)
los
h
where E(
H H)
.
Nr Nr
R
T
3.3.3 Orthogonal Space-time codes (OSTBC)
In this section we will examine the properties of a family of STC, OSTBC which are
derived from the performance criterion in (3.20). When the transmitted data is
encoded using a OSTBC, the encoded data is split into NT
streams from each antenna
which tend to be independent as much as and thus designed to achieve the maximum
diversity order. OSTBC enable simple maximum-likelihood decoding algorithm,
which is based only on linear processing at the receiver rather than joint detection.
A simple transmit OSTBC for two antennas was first derived in [20].
The classical mathematical framework of orthogonal designs is applied to construct
space–time block codes [21], [22].
In orthogonal code matrix,
I N i j
H
T
XX j
0 N i j
T
(3.22)
Where X ,
, are the i‟th and j‟th columns respectively.
i X j
rm
A

It is shown that space–time block codes constructed in this way only exist for few
values of NT
. Subsequently, a generalization of orthogonal designs is shown to
provide space–time block codes for both real and complex constellations for any
number of transmit antennas. The OSTBC are designed under the Quasi-Static
assumption, which assumes that the fading is constant during block period and
changes only between blocks.
Alamouti STBC is one of the first and simplest OSTBC. For L=2, NT the coding matrix X is:
2,
N R 1,
x
1
X
*
x2
x 2
x
1
(3.23)
And the received signal can be expressed as:
r1
h1x
1 h2x
2 n1
r2
h1x
2 h2x1
n2
(3.24)
The decoder, with linear processing, produces the two approximated symbols:
xˆ
h r h r
1
2
1 1
2 1
2 2
xˆ
h r h r
1 2
And by substituting (3.23), the symbol obtained for ML decoding is:
xˆ
1
xˆ
2
2 2
h1 h2
x1
h1
n1
h2n
2
2 2
h1 h2
x2
h2
n1
h1n
2
(3.25)
According to code distance properties of OSTBC, we can write A( X , Xˆ
) as a function
of its minimum distance in the following form:
A X Xˆ I
I
( , ) min( ) L L
LL
(3.26)
and for BPSK modulation, where the maximum square root of the difference between
different symbols is 4, the corresponding value of
, is 4 [43].
3.3.4 STC performance
kl
min
Comprehensive comparison between different STC techniques in full rank MIMO
channel with equal number of transmit and receive antennas is summarized in [11]
and indicate that OSTBC is inferior to multiplexing methods such LST, which exploit
channel sub-channels independence and achieve multiplexing gain. Still, in the case
of one receive antenna, with uncorrelated channel fadings due to wide antenna
spacing, many scatterers, the performance of OSTBC is the best out of all other
methods with less encoder and decoder complexity. The graph below in fig. 11, shows
min

the BER improvements with each transmit element for uncorrelated Rayleigh fading
[11]
Figure 11: Bit error probability versus SNR for space–time block codes at 1 bit/s/Hz; one
3.3.5 A physical perspective
receive antenna
Similar to 3.2.5, we will present the corresponding beam (antenna) patterns obtained
according to:
F( ) f ( ) s( )
X
(3.27)
Where f ( ) , s( ) are defined as in 3.2.5 and the STBC matrix X is of size NTL. Simulation results examining the effect of , the coding matrix eigenvalues and
j
NT
, made by Tarokh and were first introduced in [24]. Our simulation is similar to
[24] but with the assumption that the radiation pattern of a single antenna element is
omni directional equally distributed in space, i.e. f ( ) 1
f ( ) 1 .
Same as 3.2.5,
the antenna pattern can be derived by plotting the absolute value of the antenna array
radiation as a function of spatial angle, i.e. F( ) H
F( ) F(
)
.
Fig. 12 and 13 below show antenna patterns which were induced by 2 and 8 antennas
in 2 and 8 continuous symbols, (different columns of the channel STBC matrix), with

OSTBC. It is seen that the antenna patterns are orthogonal, and spread radiation to
space equally with no directivity.
150
210
120
240
90
270
0.5
Figure 12: Polar directional spatial response of OSTC for 2 antennas
Figure 13: Polar directional spatial response of OSTC for 8 antennas
1
2
1.5
60
300
30
180 0
150
210
120
240
90
270
1.5
1
2
0.5
60
30
180 0
300
330
330
T1
T2
T1
T2
T3
T4
T5
T6
T7
T8

4. Combining STC with BF
4.1 Introduction
Information about the channel realization (CSI), if it is available, could be utilized to
maximize the performance. As described in chapter 3, two main techniques, BF and
STC, exist. In BF, a certain level of CSI, channel realization or statistics is a must in
the transmitter and receiver for spatial pre-coding before transmission and spatial
post-coding in reception. This family mainly works on a close loop. BF exploits space
selectivity some times with time and frequency selectivity, optimal only with perfect
CSI and the performance degraded significantly as CSI quality decreases. In STC
family, explicit CSI knowledge does not have to be known to the transmitter but just a
general statistical model of the channel is assumed in STC design, mostly
uncorrelated channel as in rich scattering environment. STC mainly works on open
loop. STC family exploits the space selectivity and time selectivity by means of the
diversity order of the system. The performance has a degradation given by (3.22), for
low rank channels (highly correlated sub-channels from channel conditions or due to
closely spaced antennas constrained by operators' space limitations) in coding gain
and consequently to performance loss to the over all gain. Thus, STC is only optimal
when the statistical channel model, usually uncorrelated fading due to wide antenna
spacing, many scatterers, is realistic.
For integrating the benefits of those two methods, and for achieving maximum
spectral efficiency, some new approaches for combining these two families of
techniques were recently developed and investigated. These different methods for
combining STC and BF, differ in optimization criterion-maximization of received
SNR, minimizing of Symbol Error Rate (SER), or maximization the transmission rate,
and also in the kind of CSI available at transmitter and correspondingly, the way CSI
is being exploited.
The first and simplest optimization criterion is maximum received SNR, e.g. third
generation standards as in [35], and also in [36] and [35]. In [35], the transmit BF
weights are calculated by the mobile in such a way they maximize the received SNR
and then fed back to the base station. [35] uses channel absolute mean value feedback
for maximizing the received SNR. It suffers from lack of channel phase exploitation
and thus loss in lack of directivity of the antenna pattern. [36] uses received SNR
criterion for the case of two transmit antennas and one receive antenna is deployed
with several known multi-paths with delay spread shorter than the symbol duration. It
results in multi-beam BF in the multi-paths direction. In general, the criterion of
received SNR maximization is inferior to other optimization criteria such as
minimizing SER or rate maximization, which include in their optimization criteria
channel statistics of higher order than the SNR. Only in the special case, where CSI is
perfectly known at the transmitter, minimizing the SER is equivalent to maximizing
the average SNR, and the optimal solution is BF only in the directions of the strongest
multi-path [47].
An optimization criterion based on rate maximization was used e.g. in [31], [44]. [31],
showed expressions for the achievable transmission rate as a function of CSI
knowledge at the transmitter. [44] derives spatial linear pre-coding (BF antenna
weights) maximizing information transfer rate for the two extreme cases of feedback.
The first one mean feedback is when channel side information resides in the mean of
the distribution and the covariance modeled as white. This kind of feedback can

epresent for example a Rician channel. For this feedback, the optimum solution is to
use beamforming along the mean direction when the feedback SNR is larger than a
threshold, and to use unitary diversity otherwise. Where the power is distributed
according to a water pouring strategy between the direction of channel mean and the
remaining orthogonal directions receive equal powers. The second is covariance
feedback, in which the channel is assumed to be varying too rapidly to track its mean,
so that the mean is set to zero, and the covariance matrix is nonwhite (non-diagonal).
For covariance feedback, the optimum solution is transmission along the Unitary
eigenvectors the correlation matrix where with appropriate water pouring, the
eigenvectors corresponding to larger eigenvalues receive more power (the power
along some of the eigenvectors may be zero, so that the optimal diversity order may
be not full).
The optimization criterion of minimizing of Symbol Error Rate (SER) or a simpler
but sometimes inferior criterion of Pair-wise symbol Error Probability (PEP) was used
in [25],[47],[31],[48],[10] for deriving optimal transmissions scheme which is mostly
finding optimal antenna weights. [47] has shown that this criterion is equivalent for
single user to maximizing of the transmission rate (it also depend on the
constellation). [25] uses a general ML approach for minimizing SER with Partial CSI
(PCSI) for MIMO in a Gaussian channel. It derives a ML criterion as a function of
channel parameters, from which new ST codes can be derived or with assuming linear
transformation, ML optimal antenna weights. [25] further analyzes asymptotic
behavior of the solution as a function of channel parameters and gives explicit
solution for Rayleigh channel with Orthogonal ST Block Codes (OSTBC) without
correlation between channel coefficients.
The transmission schemes derived from SER minimization dependent on the CSI
feedbacks similarly like in transmission schemes derived from rate maximization
above. [47], [31] derive antenna expressions for the weights for channel mean
feedback (first statistical moment) and channel correlations (second statistical
moment) taking into account different constellations respectively. [47] gives
transmission scheme which is suitable also for any number of transmit antennas. [48]
derive more explicit expressions for the solution in [31] for Alamouti OSTBC with
explicit correlation matrix. [49] study the case of CSI which includes both nonzero
channel mean and non diagonal transmit correlation. Same as [25], [49] work on
minimizing the PEP, where the optimal weights, depending on the channel mean and
correlation, and the SNR, are found is numerically by binary search. Unlike the case
of channel mean or correlation feedback, the solution requires “dynamic waterfilling”
resembling in change in both “water level” and the mode directions in each
iteration.
This new approaches are becoming part of current standardization proposals for
common wireless communication standards. For example, in WCDMA an orthogonal
space–time block code is used with adaptive antenna weights (transmit beamforming).
In section 4.2 we describe the general optimal ML performance criterion as a function
of CSI parameters and CSI quality. In section 4.3, different ST coding which exploit
to some extent information about the channel (channel statistics, CSI) are being
introduced. Section 4.4 introduces approaches, which adopt OSTBC to the channel by
linear transformation.

4.2 ML optimal combination of BF and STC.
4.2.1 Introduction
In section 2.4, we defined the MMSE optimal receiver. In this section, same like in
STBC design, we describe the general ML design criterion which is the ML (MMSE)
solution for optimal antenna coding problem, obtained by minimization of the error
probability at receiver (PEP). It is being shown, that from this general solution,
different techniques can be derived, differ in their optimization criterion (maximum
received SNR, minimum BER) and the way CSI is exploited.
4.2.2 General ML design criterion
With usage of union bound like in [25], we get a bound on PEP, Pe P( X X | h, h)
:
2 2
P ˆ
e exp( d(
X, X ) / 4 )
(4.1)
2
Where d ( X , Xˆ
) is defined similar to (3.15). Although the bound was derived for
PEP, it is shown in [25] that by utilizing this bound we also minimize the over all
error probability. With algebraic manipulation like in [25], we can express (3.15) as:
2
H
d ( X,
Xˆ
) h ( I N A(
X,
Xˆ
)) h
The probability error conditioned on true channel is:
(4.2)
P P( X Xˆ | h, hˆ) p ( h | hˆ) dh
(4.3)
e hh | ˆ
Since we do not assume perfect CSI, it is therefore bounded by:
P exp( d ( X, Xˆ ) / 4 ) p ( h | hˆ ) dh
2 2
e
hh | ˆ
(4.4)
Where with utilizing the channel conditional first and second moments as appear in
2.3.7, we can express the probability of the true channel conditioned on the side
information as:
( h m )
H 1
ˆ
R
ˆ
( h m
ˆ
)
| | |
ˆ e
h h hh h h h
p ˆ ( h | h)
(4.5)
hh |
MN
det( R )
hh| hˆ
Let us introduce the following expression:
ˆ ˆ 2 1
( X , X ) I N A(
X , X ) / 4
R hh|
hˆ
(4.6)
and express V as the bound of integrand in (4.4). By substituting the above equations,
and algebraic manipulation, we obtain the following expression for V:
m
H 1 ˆ 1 1 1
ˆ
R
ˆ
( X , X )
R
ˆ
R
ˆ
m
h | h hh | h
hh | h
hh | h h | hˆ
( , ˆ e
V X X )
2det( R )det( ( , ˆ
ˆ
X X ))
hh | h
(4.7)
By minimization of V with taking the logarithm and emitting parameter-independent
terms, we can derive a performance criterion suitable for the general case:
ˆ
H 1
ˆ 1
1
l( X , X ) m ( , ) log det( ( , ˆ
ˆ R ˆ X X R ˆm
ˆ X X ))
h|
h hh|
h
hh|
h h|
h
(4.8)
ˆ
ˆ

By minimization the performance criterion above over all code-words, we can find
suitable code-words which ensure best performance according to ML. Note that the
problem is not necessarily convex.
While the first term in (4.14) is mainly a function of the channel knowledge obtained
from the actual realization of the channel estimate, the second term, on the other hand,
does not depend on the realization of the channel estimate and therefore strives for a
code design suitable for an open-loop system, which has no side information except
prior knowledge of the distribution of the true channel. This interpretation is further
supported by considering the two special cases- perfect side information,
Rhh/
hˆ
R
1
hh/
hˆ
0,
where the first term is dominant and no side information,
0,
where the second term is dominant and yields to the same criterion
as in 3.3 for designing conventional open-loop space–time codes.
4.3 General Space-time Coding schemes with CSI exploitation
4.3.1 Introduction
We can use the above performance criterion for designing channel estimate dependent
space–time codes. In this section a short survey of several ST techniques which utilize
CSI is being introduced.
We start in 4.3.2 with a Real-time adaptive coding based on optimal solution as was
derived in previous section, (4.8), where according to a certain channel statistics that
is loop-backed to the transmitter, a suitable STC is chosen [26].
In 4.3.3, an open-loop STC, but the design criterion is still based on channel statistics.
In 4.3.4, a Real-time adaptive STC codes with BF, where the transmitted symbols are
linear combination of STC and BF according to a certain bit loading algorithm [27].
In 4.3.5, Dispersion Codes (LDC) [28], are introduced. LDC, though it is an open
loop technique, takes into account through out the design, channel statistics.
In the last section, 4.3.6, partitioning transmit antennas into small groups (suggested
by Tarokh in [29]) is introduced. Again, the particular partition depends on the ST
channel statistics.
4.3.2 Real-time adaptive coding based on optimal solution
One possible approach for designing the corresponding codewords is to minimize the
maximum, taken over all codeword pairs, of (4.8), same like the solution based on the
classic design criterion in [11] but different, since for each new channel estimate that
arrives at the transmitter the optimum codewords depend on the actual channel
estimate.This procedure can be too computationally demanding for the side
information model under consideration, since the optimum codewords depend on the
actual channel estimate and the number of possible channel estimate realizations is
infinite.

[26] suggests that the entire channel estimate dependent space–time code will be precalculated
and then stored in a lookup table, suitable for real-time use. Variations of
this approach are further explored in [26].
4.3.3 OSTBC coding based on true channel statistics
The proposed performance criterion in (4.8), can also be used for designing
conventional space–time codes in various scenarios involving open-loop systems.
Let us assume a case when the side information is statistically independent of the true
channel. The performance criterion reduces to
ˆ H 1
ˆ 1
1
l( X,
X ) m
ˆ
h Rhh
(
X,
X ) Rhhmh
log det( (
X,
X ))
(4.9)
Where, ˆ ˆ 2 1
( X, X ) I N A(
X,
X ) / 4
Rhh
.
There for, an OSTBC can be designed according to (4.14) taking into account CSI
statistics. It can be shown that becomes the OSTBC design criterion is a special case
of (4.14) when the fading is Rayleigh fading.
4.3.4 Real-time adaptive STC coding with BF
A sub optimal solution of (4.14), leads to easy to implementation scheme which was
suggested first in [27]. This scheme combines the advantages of both STC and SVD
by combining BF and STC simultaneously and thus improves the system
performance. In this scheme (Fig 14), it is assumed that the channel is a quasi-static
flat fading channel and that the receiver, can feedback the instantaneous CSI to the
transmitter. The data bits are then being distributed, similarly like TCM, between the
Space-time encoder and the independent AWGN channels created by SVD (BF) using
the bit and power allocation concept. In this concept, same BER is imposed on SVD
bits and STC bits. The inaccuracies of the channel estimation at the transmitter affect
the system performance. If the CSI quality is perfect, all bits will be allocated to the
SVD branch, since with CSI with perfect CSI knowledge, SVD with power allocation
is optimal. On the other case of no CSI knowledge, all bits will be allocated to STC
branch which exploit the independence of the sub-channels.

Figure 14: Real-time adaptive STC coding with BF transmission scheme
4.3.5 Dispersion codes (LDC).
Dispersion codes (LD codes), as was developed in [28], is a space–time transmission
scheme with Quazi Static assumption. LDC work with arbitrary numbers of transmits
and receive antennas and break the data stream into Q sub-streams which are complex
symbols chosen from an arbitrary constellation, e.g r-PSK or r-QAM, and then
dispersed in linear combinations over space and time. The design of LD codes
depends crucially on the choices of the code parameters, and the dispersion matrices.
LDC satisfy the information-theoretic optimality criterion of maximizing the mutual
information between transmitted and received signals, and thus take into account, as
part of mutual information expression, channel parameters statistics. With certain
system and channel assumptions, OSTBC and V-BLAST are special cases of LDC.
LDC has many of the coding and diversity advantages of OSTBC and decoding
simplicity of LSTC.
[11] also shows how the information-theoretic optimization has implications for
performance measures of pair-wise error probability (as was taken in [25]).
4.3.6 Partitioning transmit antennas into small groups (Tarokh)
For large number of transmit antennas and at high bandwidth efficiencies, the receiver
may become too complex whenever the transmit antennas are correlated. Tarokh in
[29] suggested a partitioning of the antennas at transmitter into small groups, and
using individual space–time codes, called the component codes, to transmit
information from each group of antennas. At the receiver, an individual space–time
code is decoded by a novel linear processing technique that suppresses signals

transmitted by other groups of antennas by treating them as interference. A simple
receiver that provides diversity and coding gain over uncoded systems was also
derived. The efficiency of the partitioning of the antennas into small groups is fully
dependant on the channel statistics (e.g. angle spared) and the spatial processing at the
receiver, makes this new transmission scheme a new way for CSI exploitation in STC.
Tarokh solution is a sub-optimal solution for the problem and though it combines
spatial multiplexing techniques (multilayered space–time architecture) and not
OSTBC, with spatial processing at the receiver, it can be also applied to OSTBC
transmission.
4.4 Adaptation of STC to channel by linear transformation
(antenna weights).
4.4.1 Introduction
In this section, we will introduce STC, mainly OSTBC, with linear antenna weighting
which enables utilizing CSI. With these linear weights, the system can be transformed
into a set of parallel scalar channels. The available transmit power may then be
optimally allocated to the individual channels. This approach called, multi-beam
beamforming (two-dimensional, 2-D, beamforming), emerge as a more attractive
choice than 1-D beamforming with uniformly better performance, without rate
reduction, and without complexity increase.
The first linear transformation introduced in 4.4.2, is Pre/Post Coding using Virtual
channel model as was introduced in [31]. An unitary pre-coding is proposed to
provide robustness to unknown channel statistics and non-unitary pre-coding
techniques are proposed to exploit channel structure when channel statistics is known
at the transmitter. In 4.4.3, ML Optimal OSTBC with linear antenna weighting based
on [25] is shown. In 4.4.4, OSTBC with linear antenna weighting, based on only
channel covariance feedback (ML sub-optimal), as was derived by [30], is shown.In
the last section, 4.4.5, OSTBC with linear antenna weighting based on received SNR
maximization is shown. This technique is part of the 3GPP standardization work [34]-
[36]. SNR maximization is inferior to PEP (BER) minimization because average PEP
depend not just on SNR but on higher SNR moments such as SNR variance.
4.4.2 Pre/Post Coding Using Virtual channel
Realistic channels corresponding to scattering clusters exhibit correlated fading, can
significantly compromise the performance of existing ST techniques which are based
on an idealized MIMO channel model representing a rich scattering environment. The
virtual channel model as section 2.3.6, can be used for a Pre and post coding for
enhancing ST techniques [30]. These techniques are similar in a way to SVD

techniques, but differ mainly in the physical interpretation that these techniques have -
the non-vanishing elements of the virtual channel matrix are uncorrelated and capture
the essential degrees of freedom in the channel and provide a simple characterization
of channel statistics. Thus, the virtual channel representation provides a natural
framework for combining beamforming ideas from array processing with space–time
coding techniques. Though in [30], spatial multiplexing techniques were suggested and
analyzed, a similar approach can be applied also to OSTBC. PEP analysis for
correlated channels is greatly simplified in the virtual representation. Using the PEP
analysis, 2 pre-coding schemes are introduced to improve performance in virtual
channels:
Unitary pre-coding - proposed to provide robustness to unknown channel
statistics.
Non-unitary pre-coding - proposed to exploit channel structure when channel
statistics are known at the transmitter.
4.4.3 ML Optimal OSTBC with linear antenna weighting
Optimal linear antenna weight for OSTBC
In this section, we will try to “improve” the codeword X, OSTBC, by linear weighting
like in [25] similar to our system model in (2.45).
From now on, from convenience reasons, we will denote by W the transmit weight
matrix of size NT NT
, written in (2.45) as T , which is shared by all codewords
with the constraint, limiting the average output power:
(4.10)
W
W 1
F
Substituting, (4.15) – (4.16), into (4.8), leads to the performance criterion:
Where is:
H H 1 1 1
kl
h| hˆ hh| hˆ hh| hˆ h| hˆ
l( WW , ) m R R m logdet( )
H
2 1
I N A( WW , kl) / 4
R
hh| hˆ
(4.11)
(4.12)
The dependence on the codeword pair is now only through the scaling factor .
H
Since l( WW , kl ) is a decreasing function of kl the error probability is dominated
by the codeword pairs corresponding to the minimum of kl and min
. Thus, only
one such pair is considered in the optimization procedure. An optimal (optimal in the
sense that it minimizes the criterion function under consideration) could now, at least
H
in principle, be determined by minimizing l( WW , kl
) with respect to W, while
satisfying the power constraint. However, a highly challenging optimization problem,
with a criterion function possessing multiple minima, would need to be solved.
In order to simplify the optimization problem, we do parametric approach same like in
H
2
[25]. By substituting Z WW and min in (4.12), we obtain a new
performance criterion:
(4.13)
4 /
1
H 1 1 1 1
l( Z) m ˆ R ˆ I ˆ ˆ ˆ logdet
h| h hh| h N Z RRmI hh| h hh| h h| h N Z R
hh| hˆ
kl

and in the case of transmit diversity (single receive antenna):
The new convex optimization problem is:
Z arg min l(
Z)
(4.14)
(4.15)
1/
2
The optimal weight, W opt , is further obtained by square root of Z opt , W Z .
Asymptotic properties of the solution
1
H 1 1 1 1
h| hˆ hh| hˆ hh| hˆ hh| hˆ h| hˆ hh| hˆ
l( Z) m R Z R R m logdet Z R
opt
0
( ) 1
tr Z
Z Z
Z
H
Asymptotic properties of this solution, are special cases where a closed-form solution
can be obtained, and enable understanding of the way OSTBC and BF are combined.
4 cases are being examined:
1. No channel knowledge (CSI).
2. Perfect channel knowledge (CSI).
3. Infinite SNR level.
4. Close to zero SNR level.
1
In the first case, where no channel knowledge (CSI) is assumed, i.e., R hh/
hˆ
0,
W becomes a scaled unitary matrix:
W opt I Nt / NT
(4.16)
Thus, the codewords are transmitted without modification. This makes sense
considering the assumptions under which the predetermined space–time code
(OSTBC) was designed. It also makes sense in view of the fact that the transmitter
does not know the channel and therefore has to choose a “neutral” solution.
In the second case, perfect channel knowledge (CSI) is assumed, i.e., 0 . R
As been shown in [25], since in OSTBC the decoding of the constituent data symbols
decouples, the antenna weights should be parallel to the strongest left singular vector
of the channel – beamforming in the direction of left singular vector of the channel
(note that here channel estimates is the true channel) , defined as V . Thus we can
assume that one of the eigen-values of is strictly larger than all the other. The solution
of 4.13 becomes:
Wopt [ VN
, 00]
T
(4.17)
Unlike SVD techniques, where the weights are in the direction of all eigenvectors,
here the weights are in the direction of only the strongest eigenvectors (only one subchannel
is used).
2
In the third case, Infinite SNR level is assumed, i.e. min
Like in the first case, the optimal linear transformation can be chosen to be a scaled
unitary matrix like (4.15). The usefulness of channel knowledge diminishes as the
SNR increases due to optimal combining in the receiver and thus achieving maximum
channel diversity gain.
In the fourth case, low SNR level is assumed, i.e. , the solution
of W also expressed by (4.16).
4 /
2
min
/ 4
0
opt
NT
hh/
hˆ
opt

Case of diagonal conditional covariance matrix
Since the optimization problem in (4.15) is convex, it can be solved with methods
from complex analysis mathematics [25]. Since the complexity of the algorithm is
high, these methods are not easy for implementation and demand great computing
sources. In this paragraph, we consider a simplified fading scenario with substantially
lower algorithm complexity by assuming that the Conditional covariance matrix,
, is diagonal. This assumption is suitable for scenarios where the channel
Rhh/ hˆ
coefficients are not correlated, like in Rice and Rayleigh models. Since we assume
is diagonal matrix, it can be expressed as:
Rhh/ hˆ
R I
hh/
hˆ
Where represents the channel coefficients conditional variance
Let us introduce ˆ as
R
ˆ 1
(4.18)
k 1
(4.19)
(m)
H
Where k denoted the k'th block of size NT NT
on the diagonal of . For
h h h h
the special case of single receive antenna (4.10) becomes:
, can be interpreted as the channel average second moment matrix.
(4.20)
m m | ˆ | ˆ
ˆ 1 H
m
h| hˆ m
h| hˆ
ˆ
Let us further define V and V as the eigenvectors matrix of Z and respectively, and
and as the eigenvalue matrices of Z and :
ˆ ˆ
N
NT
T
ˆ ( ˆ )
ˆ
(4.21)
With the usage of ˆ , , , V and V as defined above and algebraic manipulation,
the following performance criterion to be minimized in (4.14) becomes [25]:
(4.22)
ˆ
1
H H
l( Z)
tr
I N V Vˆ
ˆ
Vˆ
V N
T
R log det
I NT
Subject to the constraint,
The remaining optimization problem is clearly convex and thus the solution may
therefore be obtained by means of the Karush–Kuhn–Tucker (KKT) conditions [29, p.
164] and Newton‟s method. Explicit solution is described in [25] as the following
algorithm:
1. Set l=1.
2. Solve
N
NT
NR
( m)
k
)
diag( k k1
Z VV ˆ Vˆˆ Vˆ
H
H
diag k k1
NT
k1
k1
1,
0
3. Compute:
4. If and repeat from 2.
5. Compute
2
ˆ i
2
l 0, l
0, l l1
W ˆ 1/ 2
V k
2
Nt
N 1
4 ˆ
t l k
10
2
4 1
, i 1,...., N .
i t
opt
1
2
N
T
(4.23)

4.4.4 OSTBC with linear antenna weighting, based on only channel covariance
feedback (ML sub-optimal)
As we seen from previous chapters, the optimal weights are thus obtained by the
conditioned value of the first two moments. If we assume, as in [31], the following
scenario of no direct measurement of the channel is known to the transmitter, but only
the channel covariance matrix, we can assume, in the case of the Rayleigh channel a
zero mean value , m 0 , and no correlation between the channel covariance
hh | ˆ
approximation and its real value, R R .
hh| hˆ
Consequently, the solution can be substantially simplified, especially for the case of
correlated channel coefficients. Note that although [31] does not use conditional
covariance matrix (measure of CSI quality), it has shown that for this scenario of
Gaussian channel with zero conditional channel mean value, same antenna weights
are derived also for non perfect CSI.
Using those relations and multiplying (4.14) by , we obtain:
l( Z) log det I RhhZ (4.24)
Let us further perform Singular Value decomposition (SVD) over the channel
covariance matrix, , where is the unitary eigenvector matrix and
is the corresponding square root eigenvalue matrix.
H
Rhh Uh DhU h
h U
D h
Let us further define, as in [30], the diagonal matrix D as:
2
f
D W U W U
H
T h T h
(4.25)
2
As in [30], maximization of (4.24) force the matrices RhhZ and DD h f to be diagonal.
Now, by substitution of Rhh Uh
DhU h and Z WTWT
into (4.24) and simple
algebraic manipulation, relying on the symmetry of the matrices R and Z and the
unitary of U , we obtain the same minimization expression as in [30]:
h
2
l( D f ) log det I DhD f
hh
H
(4.26)
Explicit expressions, for the k th beam obtained after power loading according to the
power loading algorithm in [30] in the direction corresponding to the k th eigenvector,
which minimizes (4.26), was found to be:
H
Wk D f Uk
(4.27)
The assumptions above, 0 and R
R , make sense in fast fading
mhh | ˆ
hh| hˆ
scenarios. Since both algorithms (in [25] and in [31]) minimize PEP, we expect the
performance of the system of [25], and [31], to be the same in fast fading channel. In
case of slow fading, if we have strong nonzero conditional channel mean value, as in
Rician fading, we expect the solution in (4.14) to enhance the system performance.
R hh
hh
f
H
hh

4.4.5 OSTBC with linear antenna weighting based on received SNR
maximization (3GPP).
Maximum received SNR approach is Sub-optimal approach to the PEP approach
above which may coincide together in certain channel scenarios.
In [32], it was proposed that training sequences will be transmitted periodically from
downlink and uplink for calculation (either in the receiver or in the transmitter) of the
transmission optimal weights by UE received power (SNR) maximization. This
approach was adopted by 3GPP with Alamouti OSTBC in [33]-[34]. The UE
periodically sends quantized estimates of the optimal transmit weights to the BS,
optimized to deliver maximum power to the UE, via a feedback channel.
Fig. (15), depicts this concept.
Figure 15: 3gpp transmission scheme
The weights are obtained by maximization of the received power:
H H
WT arg
max ( WT
H HWT
)
(4.28)
In Gaussian channel with equally spatial distributed noise (white Gaussian noise), the
weights obtained by (4.28), are the same as simple BF weights as was derived in
(3.2.3).
There are two modes in close loop transmit diversity in 3GPP, strongly influenced by
bit rate limitations:
1. Quantize the complex feedback coefficient to 1 bit of magnitude and 3
bits of phase and send them over successive slots. (Same pilot channel
transmitted form each antenna).
2. Feedback only the phase information for the complex coefficients. Set
partitioning is done on the phase constellation, and the transmit
weighting is calculated by filtering over multiple feedback bits.

Though this technique is being optimal only for the Gaussian assumptions, neither the
special structure of OSTBC nor the 2-d BF is not exploited, and thus loss in
performance is maintained.
A more recent approach, similar to the WCDMA standard, but differ in taking into
account the special structure of OSTBC, was suggested to the 3GPP recently for
MIMO 2,2 case [35]. In this approach, the weights have simple closed form of:
1
1
WT
, 1
, WT
, 2
N
2
2
R
N R
h
2i
h1i
j1
j1
1
1
N R
N R
h1i
h2i
j1
j1
(4.29)
This approach suffers from no phase adjustments, and thus lost in gain.

5. ML optimal weights for transmit diversity in various
channel scenarios
5.1 Introduction
In the previous chapter we examined various techniques for combining ST with BF.
In this chapter, we will focus on ML optimal combining of OSTBC and BF, mainly
for the case of transmit diversity. Though with linear weights, the solution for
obtaining optimal weights in (4.12) was shown to be convex, with correlated channel
coefficients and with high number of antennas, the problem complexity is demanding
and iterative numerical methods have to be adopted. Let us assume for simplicity, as
in the "simplified scenario" of [25], that number of antennas at receiver is one. This
assumption is suitable for our system model of transmit diversity (MISO channel).
In this chapter we derive and investigate the JSO (ML optimal) solution in different
fading scenarios. We start with Rayleigh and Rician channels, both in the case of
uncorrelated channel and we end with correlated sub- channel fadings (low rank
channel) with non zero mean. When possible, a more robust, practical algorithm, with
minimum compromise on performance, is further suggested. In each fading scenario,
we first derive a close solution and analyze the eigenvalue of the optimal weight
behavior as a function of channel parameters. Then, according to asymptotic behavior
of the desired solution for optimal weights, "asymptotic solution" based on analytical
functions with robust mechanism that enables simple adaptation to channel
parameters, is being suggested and examined.
5.2 ML optimal antenna weights in Rayleigh fading.
5.2.1 Introduction
As mentioned above, the optimal weights solution in (4.12), is still computationally
demanding. A simpler algorithm can be obtained if we assume independent and
identically distributed (i.i.d.) zero-mean complex Gaussian channel coefficients, .
This assumption reflects a channel with antennas both at the transmitter and the
receiver spaced sufficiently far apart so that the fading is independent and a rich
scattering environment with non-line-of-sight conditions. This scenario corresponds
to diagonal covariance matrix feedback with zero channel mean.
5.2.2 Deriving ML Optimal antenna weights
Like in [25], by substituting the conditional statistical model suitable for Rayleigh
fading developed in (2.37) into the ML optimal antenna weights for the diagonal
covariance case in (4.20), (4.23), and with further algebraic manipulation, we obtain
the following ML optimal antenna weights.
h i,
j

The eigenvectors matrix V can be obtained by SVD over the estimated channel matrix
like in (4.20),
ˆ 1
ˆ ˆ ˆ
(5.1)
The eigenvalues are obtained according to the following procedure:
H
m m V V
h h h h
2
1
1. Let k N T
where, h ( 1
est
), 2
2. Compute
3. Compute . (5.2)
4. If 0 , set 1 2
NT
1 and NT
1 ( NT
1)
5. If 0,
set 0 and 1
6. Compute
The physical interpretation of the following algorithm indicates that when the CSI
quality is above some predetermined threshold, the expression after the comparison in
step four will be executed and all the power is allocated to the direction of the channel
estimate. On the other hand, falling below the threshold means that a part of the total
power is allocated to a beam in the direction of the channel estimate and the
remaining power is divided equally between the NT
1
directions orthogonal to the
channel estimate.
5.2.3 Exploring ML optimal weights solution
Exploring the solution for optimal weights in (5.2), shows that in the case of a MISO
channel, , the highest eigen-value, determines all the other eigen-values as
follows,
2 2 2 2 4 4 2
k (2N 1) ˆ 2 (2 1) ˆ ˆ
T est h k NT est h est h k
i
iN N
T
T
1 N
T
NT
1
1 1
Wopt V 2
2( NT
)
1 2
1/ 2
NT
1
(5.3)
[25] has also shown that increases as the noise level increases ( ), it
decreases as the channel standard deviation increases ( ), it increases with the
channel strength ( h ) and increases with the correlation between the estimated
NT
channel and the true one (
)
ˆ
N
T
NT est
Furthermore, this solution has the following asymptotic properties:
NT
N
T
1
2
h
N
T
2

1.
1
N as
T NT
est0 or 0.
Equal power allocation (STC mode) is
preferable if the channel is not known or the at high SNR ratio.
2. 1 as 1 or . When the channel is known, or in case of low
N
T
est
signal to noise ratio, it is preferable to concentrate the energy in the known
direction.
1
3. N as hˆ 0 or . Again, if the estimated channel magnitude is
T
h
NT
low, or in case of high channel fluctuations, it is preferable to spread the energy
among all beams.
4. 1 as hˆ or 0,
which is the converse of the above statement.
N
T
Fig. 16, 17 below shows the value of as a function of its parameters for 2 transmit
antennas. Fig. 16 shows as a function of and at the point where hˆ 1, 1
. Fig. 17 shows it as a function of hˆ ,and h
. The “beamforming plateau” where
1 can be readily observed for high estand . The function tapers down to 0.5,
N
T
as ˆ
est,
or h go to zero or as h increases.
h
N
T
N
T
Figure 16: Largest eigen-value as a function of the noise level and channel estimation
correlation,
est
hˆ 1, h
1
h

Figure 17: Largest eigenvalue as a function of the channel strength and channel standard
5.2.4 Algorithm sensitivity
deviation,
est0.9, 1
The first question that might arise in implementation of the JSO algorithm is how
sensitive the performance is to an error in the eigenvalues. Fig. 18, presents the Bit
Error Rate vs. the SNR, obtained by simulation, with and without an induced error on
the value of obtained by the JSO algorithm. Fig. 18 shows that 10% error in the
N
T
value of , causes only 0.2dB degradation in performance.
N
T
BER
10 -1
M error = 0%
M error = 10%
M error = 20%
M error = 30%
M error = 40%
10
-10 -5 0 5 10
-2
SNR (dB)
Figure 18: Sensitivity of the BER performance to errors in maximal eigenvalue N
T

5.2.5 An approximation function
The solution presented in (5.2) is not computationally prohibitive, but still it is
cumbersome. For simpler implementation and in order to study the sensitivity of the
solution to errors in the different parameters, we were looking for an analytic
approximation function, which will have the asymptotic properties presented in
section 5.2.2 above, and would not deviate by more than 10% from the optimal
solution. According to the previous section, 5.2.3, that would ensure less than 0.2dB
degradation in performance.
Let us suggest a solution for optimal antenna weights, which satisfies the asymptotic
properties of the general solution in 5.22, based on the analysis in previous paragraph,
from the form:
ˆ ˆ 1
N f ( , , ( ), )
T
est abs h
h
(5.4)
An adaptation of the solution according to specific channel characteristics, Rayleigh
channel in this sub- chapter, either numerically by simulation or analytically, is
further needed.
We suggest the following approximation:
/ | ˆ
CQ min a
bh h|
est
,1
1 N 1
ˆ
T
N
CQ
T NT NT
1 N
ˆ
T
i
iN NT
1
T
(5.5)
The parameters a, b are found in simulation, on typical channel parameters values
according to the following criterion:
2
[ a, b] argmin E
ab ,
NT ˆ N
( a, b)
T
(5.6)
The values received from simulation, a=1.3, b=1.95, with s.d. 0.1. a 1.3, b0.95.
They were found by an exhaustive search over typical range of channel parameters
values. The average estimation error is, 0.0248, its standard deviation is 0.06, and the
maximum estimation error is 0.12. Thus, inaccuracy is less than ~6%, and gain
loss due to estimation error has to be about 0.15 dB.
According to common practice, and for a better realization of the approximation
formula in (5.5), one can normalize the value of the average channel amplitude
Ehˆto be 1. Now, according to the channel model, we can also obtain h as a
constant value, i.e., for Rayleigh fading scenario, we obtain . Following
2
h
1.13
2
the normalization, 1 becomes average signal to noise ratio. If we further define,
h hˆ as the instantaneous channel strength normalized to the standard
ins h
deviation of the channel fluctuations, the term CQ in (5.5) could be expressed as:
b SNR / h
CQ
min( a ins
est
,1)
N
T
(5.7)

Thus eigenvalue approximation depends logarithmically on the ratio between the
received SNR and instantaneous channel strength. However, we chose to perform the
analysis using the values of ˆh , h and explicitly in order to gain a better insight
and to avoid the need of normalization.
Fig. 19, 20 below shows the value of according to (5.5) as a function of its
ˆ
parameters for 2 transmit antennas with same channel parameters as in fig. 17, 18.
Figure 19: Largest eigen-value as a function of the noise level and channel estimation
correlation,
Figure 20: Largest eigen-value as a function of the channel strength and channel standard
deviation,
NT
hˆ 1, h
1
est0.9, 1

5.2.6 Optimal weight sensitivity to Channel Parameters
The analytic approximation function presented above can be used to derive
approximate expressions for the sensitivity of the solution to errors. For a parameter
x (est, ˆh or hthe relative sensitivity was calculated as:
NT x NT x
sen( x,
N
)
T N T N x
x
T
x
(5.8)
Table 3 below summarizes the average and maximal relative change induced by a
10% relative change in each of the parameters.
Table 3: Parameter sensitivity- Relative change in N
induced by 10% change in each
Parameter
est Estimation
Correlation
parameter
Average
change in
Maximal
change in
3.5% 9.1%
Noise level 0.5% 3.3%
Estimated
ˆh
Channel
Strength
0.9% 2.9%
h Channel
std.
0.0% 1.4%
According to this analysis, the estimation correlation, est is definitely the most
sensitive parameter, and it should be known to within 10% in order to limit the
degradation to 0.2dB
5.2.7 New approximation performance
The approximation function presented in section III can easily be used for
implementation. Namely, using (5.5) instead of the procedure outlined in (5.2). In this
section the performance of this implementation is presented, in comparison with the
following:
1. OSTC, wherein only STC is used
2. Conventional BF, where the actual erroneous channel estimates are
used for Beamforming
3. WOSTC, which uses weighted STC, as described in section 4.4.5
according to [35].
4. JSO algorithm described in section 4.4.3 according to [25].
Fig. 21 shows the BER as a function of SNR, for a well-estimated channel (est =
0.9). The BF algorithms provide the best results for low SNR. As shown in [25], for
high SNR OSTC has an advantage, both the JSO and its approximation follow the
best of the two along the whole SNR range. WOSTC performs slightly worse.
NT
T
NT

Fig. 22 shows the results for a lower value of estimation correlation. (0.7). In this case
the conventional Beamforming performs badly. The OSTC schemes have an
noticeable advantage. In this case, due to the lack of high quality CSI, the
conventional BF falls below OSTC in performance. The JSO, WOSTC and the new
algorithm follow the performance of the OSTC.
BER
10 0
10 -1
10 -2
JSO algorithm
New algorithm
OSTC
WOSTC
Conventional BF
10
-10 -5 0 5 10 15
-3
SNR (dB)
Figure 21: BER/SNR graph of new algorithm, JSO algorithm, BF, OSTC, and WOSTC. 2
BER
10 0
10 -1
10 -2
10 -3
transmit antennas, one receive antenna, for .9, 1.
JSO algorithm
New algorithm
OSTC
WOSTC
Conventional BF
est h
-10 -5 0 5 10 15
SNR (dB)
Figure 22: BER/SNR graph of new algorithm, JSO algorithm, BF, OSTC, and WOSTC. 2
transmit antennas, one receive antenna, for 0.7, 1.
est h
For a larger number of transmit antennas, BF does provide an advantage, even in the
low quality CSI case, as demonstrated in Fig. 23. Again in this case the JSO

algorithm follows closely the better of the two. WOSTC fails to obtain the BF
advantage in the low SNR case.
BER
10 0
10 -1
10 -2
10 -3
JSO algorithm
New algorithm
OSTC
WOSTC
Conventional BF
10
-10 -5 0 5 10 15
-4
SNR (dB)
Figure 23: BER/SNR graph of new algorithm, JSO algorithm, BF, OSTC, and WOSTC. 8
transmit antennas, one receive antenna, for 0.7, 1.
In all the cases the suggested approximation performed very close to the optimal JSO
algorithm, and so it is indeed a viable candidate for implementation.
Fig 24 below, shows the BER as a function of the approximation quality represented
by estimation correlation coefficient for Eh { } 1,
h 1 and SNR=7 dB. BF perform
badly for low values of est
(spreading the energy in “wrong” directions), while
OSTBC, as expected does not depend on est
and has a noticeable advantage over
BF. For high values of est
, BF performs better than OSTBC (good information
quality about source direction). Our approximation follows closely the JSO algorithm
which follows the best out of BF and STBC.
BER
10 -1
10 -2
Figure 24: BER as a function of channel estimation correlation coefficient in Rayleigh fading
with SNR=13 dB.
est h
JSO algorithm
New algorithm
OSTC
WOSTC
Conventional BF
10
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-3
est

5.3 ML optimal antenna weights in Rician fading
5.3.1 Introduction
Similar to Rayleigh channel, from the optimal weights solution in (4.15), if we
assume independent and identically distributed (i.i.d.) non-zero-mean complex
Gaussian channel coefficients, , we can obtain close solution to the ML optimal
h i,
j
weight problem. This assumption reflects a channel with antennas both at the
transmitter and the receiver spaced sufficiently far apart so that the fading is
independent. and a rich scattering environment additional to a line-of-sight condition.
This scenario reflects diagonal covariance matrix feedback with non-zero mean
channel feedback
5.3.2 Deriving ML Optimal antenna weights
For deriving the ML optimal solution for Rician fading, we solve the ML optimal
solution for non correlated sub-channel coefficients as in (4.23), in the same
procedure that was used for Rayleigh fading (5.1), (5.2). We first have to find the
expression for , the conditional variance of the channel coefficients (note that also
in Rician fading is a diagonal matrix and can be expressed as R I
)
Rhh/ hˆ
hh/
hˆ
NT
NR
and ˆ , the channel estimation first moment squared matrix expressed as
ˆ 1
H
m ˆm ˆ . The conditional variance , can be obtained by substituting the
h| h h| h
2 1 2 K
relation of Rician fading parameters, b , a into (2.38):
K 1
K 1
1 2 2
h (1 est)
(5.9)
K 1
By substituting m from (2.38) to ˆ 1
H and after rearranging the equation
h hˆ
m
| ˆm h| h h| hˆ
we obtain an expression for ˆ :
ˆ 1 K
2 H ˆH 2 ˆ ˆH
(1 est ) hLOSh LOS 2 est (1 est ) hLOSh esthh
(5.10)
K1 and the other eigenvalues of the single value matrix ˆ are:
ˆ ˆ ˆ 0
1 2 N 1
T
2 (5.11)
2 2
ˆ 1 K K
H ˆ 2
(1 ) 2 (1 ) ˆ
N T
est hLOS est est hLOSh est
h
K 1 K 1
Now we can obtain the ML optimal antenna weights expression for Rician fading, by
substituting the expressions in (5.9)-(5.11) into (4.23) and solving the equation. A
detailed description of the solution is written in APPENDIX A: “ML optimal antenna
weights for Rician channel”.

An alternative, more intuitive way , for deriving the algorithm for Rician fading can
be obtained, if we note that the expression for , obtained after solving (4.23) in the
"simplified scenario” for Rayleigh, has the form of:
2 2
k1 ( 2N
ˆ
ˆ ˆ
T 1)
N
2k1(
2N
1)
k
T
T N
T N T 1
(5.12)
2
2k
Where k1 is equal to NT
. Consequently, since Rician fading is also part of the
"simplified scenario”, by substituting the corresponding expressions for eigenvalues
of ˆ as in (5.11), we obtain the corresponding algorithm for the Rician channel which
is identical to the one which was derived above. Now, finally we can write the
algorithm for ML optimal weights for the Rician fading.
The eigenvectors matrix V can be obtained, as for Rayleigh fading, by SVD over the
estimated channel matrix, as in Rayleigh fading:
ˆ 1
ˆ ˆ ˆ
(5.13)
H
m m V V
h h h h
The eigenvalues are obtained by the following algorithm:
1. Let k1 1 2 2 1
NT
where, h (1 est ), 2
K 1
2. Compute the expression:
ˆ NT 1 K
2 2
H ˆ 2
(1 est ) hLOS 2 est (1 est ) hLOS h est
K1 2
hˆ
3. Compute the expression:
k1 ( 2N
ˆ
T 1)
NT
2k
ˆ
1(
2N
T 1)
NT
2
2k
ˆ 2 2
N
k
T 1
1 1
4. Compute .
5. If 0 , set and
1 2
N
1 T
NT
1 ( NT
1)
6. If 0,
set 0 and 1
7. Compute
Wopt V 1 2
1/ 2
NT
1
5.3.3 Exploring ML optimal weights solution
1
1
(5.14)
Exploring the solution in (5.14), shows similar behavior for the Rayleigh channel, for
all parameters except K, as was excepted since the Rician channel include Rayleigh
component. For the assumption that the constant channel components approximation
is perfect (reasonable assumption in slow fading), the dependence of the eigenvalue
distribution on K, obtained either by simulation or analytically by (5.14), is that as K
increases, will increase, (K proportional to , i.e. K ), till it reaches its
NT
saturation level, 1, 1 as K .
In a slow Rician fading scenario, when the
N
T
LOS component is high enough, it is preferable to concentrate the energy in the
known direction (The LOS direction).
N
NT
T
NT

Fig. 25 below shows the value of as a function of and K for 2 transmit
antennas at the point where h . 1 can be readily observed for high
est
and K
Fig. 26 below shows the value of as a function of , and K=1 for 2 transmit
antennas. Compared with fig 16 (Rayleigh channel), we observe that the eigenvalue
saturation, 1 starts at lower values of and due to the effect of K (K
N
T
proportional to maximum ).
est
N
T
ˆ 1, h 1, 1
N
T
Figure 25: The Largest eigenvalue as a function of K and channel estimation correlation,
Figure 26: The Largest eigenvalue as a function of the noise level and channel estimation
N
T
est
est
NT
NT
0.8
0.6
0.4
1
1
0.8
0.6
1
0.4
1
N
T
0.5
est
0.5
est
0
0
hˆ 1, h1, 1
0
0
K
1
correlation, ˆ h 1, h
1, K 1
5
2
10
3

5.3.4 Algorithm sensitivity
As for Rayleigh channel, we want to know how sensitive the performance is to an
error in the eigenvalues. Fig. 27 presents the Bit Error Rate vs. the SNR, obtained by
simulations, with an induced error on the original value of which was obtained
by the JSO algorithm. The modulation is BPSK, with 2 transmit antennas and
ˆ
est 0.9 , h1,
E h1,
K=5. Simulations with 10% error in the value of N
,
T
same as in section 5.2.3, causes up to 0.3dB degradation in performance. We can see
that as the SNR increases, the sensitivity degradation increases. It can be explained by
higher sensitivity of the system to the eigenvalue in the direction of LOS at high
values of K.
BER
10 -1
10 -2
10 -3
10 -4
M error = 0%
M error = 10%
M error = 20%
M error = 30%
M error = 40%
5.3.5 An approximation function
Similarly to the Rayleigh channel case, we want to obtain an approximation function
for (5.14) that would fulfill the asymptotic properties presented in section 5.33 above,
and would not deviate by more than 10% from the optimal solution. According to the
previous sub-section, that would ensure less than 0.4dB degradation in performance.
As above, we suggest the eigenvalues approximation, based on channel strongest
eigenvalues. The expression for JSO strongest eigenvalue in (5.11) and the Rician
distribution, indicates that the approximation should be in the form of sum of two
components, Rayleigh and Rician components, where the LOS component should be
with the proportion coefficient of K. Further more, the channel variance,
h , is related
only to the Rayleigh component since the Loss component is constant and the channel
N
T
-5 0
SNR (dB)
5 10
Figure 27: Sensitivity of the BER performance to errors in maximal eigenvalue N
T

absolute value is included in the term of K with proper normalization and thus can be
concluded the LOS term. The only dominant factor in the LOS component should be
the Gaussian noise.
The suggested JSO Rician approximation is:
h
b
1 hˆ K
CQ min a c e
d
est ,1
K1K1
ˆ NT
1
NT NT 1
CQ
NT
ˆ i
iNT 1 ˆ N
T
NT 1
(5.15)
The parameters a, b, c, d , were found to be a 1.3, b 0.7, c 1.15, d 0.5 by an
exhaustive search over typical range of channel and system (include number of
transmit antennas) parameters values as follows:
a, b, c, d
2
ˆ
N
T NT
[ a, b] argmin E ( a, b, c, d)
(5.16)
The average estimation error is, 0.05, its standard deviation is 0.1, and the maximum
estimation error is around 0.2. Thus, inaccuracy is about 10%, and gain loss due
to estimation error is less than 0.3 dB in average.
As in Rayleigh channel above, we can present the term CQ in (5.15) as:
SNR
1
1
b K d
min a h
CQ est ins c
e
SNR ,1
K1K1
(5.17)
Where the channel amplitude is normalized to one and the channel estimate is
normalized to the channel standard deviation.
Thus eigenvalue approximation depends logarithmically on the ratio between the
received SNR and instantaneous channel strength. However, we chose to perform the
analysis using the values of ˆh , h and explicitly in order to gain a better insight
and to avoid the need of normalization.
N
T
Figure 28 below shows the value of according to (5.15) as a function of K and est
NT
for 2 transmit antennas for hˆ 1, h1, 1
. Comparing with the related graph for
the JSO in Fig 25, we observe that for medium values of K (K between 2-5) and bad
estimation quality (est below 0.4), our statistical approximation goes to 0.5 (equal
gain) slower, which result in smaller eigen value spread than JSO and focusing the
transmission more in the direction of the estimated channel (for high levels of K,
similar to the LOS component). Figure 29 below shows the value of according to
(5.15) as a function of , for 2 transmit antennas with hˆ 1, 1, 1
. It can
est
be noticed that the saturation level is at lower values of and est due to the
contribution for the saturation of the LOS component, which is perfectly estimated.
h
NT

NT
Figure 28: The Largest eigenvalue a s a function of K and channel estimation correlation,
NT
1
0.9
0.8
0.7
0.6
0.5
1
1
0.8
0.6
0.4
1
0.5
est
0.5
est
Figure 29: Largest eigen-value as a function of the noise level and channel estimation
correlation,
5.36 Optimal weight sensitivity to Channel Parameters
0
0
hˆ 1, h1, 1
0
0
The analytic approximation function presented above can be used to derive
approximate expressions for the sensitivity of the solution to errors. For a parameter
x (est, ˆh
or hthe relative sensitivity was calculated according to (5.8). Table 4
below summarizes the average and maximal relative change induced by a 10%
relative change in each of the parameters. According to this analysis, the estimation
correlation, est is definitely the most sensitive parameter, and it should be known to
within 10% in order to limit the degradation to 0.4dB.
1
K
hˆ 1, h
1
5
2
3
10

Table 4: Parameter sensitivity- Relative change in N
induced by 10% change in each
Parameter
est Estimation
Correlation
parameter
Average
change in
Maximal
change in
1.2259 5.9993
Noise level 0.3349 3.2096
Estimated
ˆh
Channel
Strength
0.2649 1.3387
h Channel
std.
0.8629 1.3387
Channel
Cofactor
0.5157 1.1261
As for the Rican channel, we observe that the sensitivity to the parameters‟ values is
about the same as the Rayleigh channel except the sensitivity to est . which is
considerably lower due to the LOS component which is independent of est.
5.3.7 New approximation performance
NT
The JSO Rician approximation function presented above can easily be used for
implementation, namely, using (5.15) instead of the procedure outlined in (5.14). In
this section the performance of this implementation is presented, in comparison with
the following:
1. OSTBC, wherein only STC is used
2. Conventional BF, where the actual erroneous channel estimates are
used.
3. WOSTC, which uses weighted STC, as described in [35].
4. JSO algorithm described in [25].
Figure 30 below, shows the BER as a function of SNR, for a medium-estimated
channel (est = 0.7) and a Rician channel with a LOS component, K=1. Unlike Fig 22,
where at low SNR BF and OSTC produced the same BER, we see that in Rician
channel BF algorithms provide better results for low SNR due to the LOS component,
which is assumed to be well estimated and thus contributes to the performance. At
high SNR values, OSTBC is still superior in Rician channel as was expected, as was
explained for the Rayleigh channel above, due to the Rayleigh component. We further
note that JSO and its approximation follow the better between OSTC and BF, along
the whole SNR range, which is in this scenario the BF. Still, there is an inaccuracy of
our approximation at high SNR of up to 0.5 dB, still in the limits of our
approximation design.
Figure 31 shows the BER as a function of SNR, for a medium-estimated channel (est
= 0.7) but for high LOS component, K=10. The BF algorithms provide the best
results for all SNR due to the high Rician factor K, which was assumed to be perfectly
T
NT

estimated. Both OSTBC and WOSTC perform worse. For OSTC, it can be explained
by OSTC design assumption of independence between the channel fadings, and thus
OSTC is based on spreading the transmission in all directions and is not affective in a
strong well estimated LOS component. WOSTC was also not designed for LOS
conditions. Looking on the expression for WOSTC weights in (4.29), we notice that
in the extreme case of LOS conditions the obtained weights are equal, unlike a typical
BF which utilizes the antenna phase shifts (forming a steering vector) and thus
correspond to the LOS direction. On the other hand, JSO and its approximation follow
the better between OSTC and BF, along the whole SNR range, which is in this
scenario the BF.
We can observe that JSO algorithm and consequently our approximation as a kind of
switch between OSTC, which enhances diversity order of the system and BF, which
enhances coherent (SNR) gain of the system
Fig 32 below, shows the BER as a function of the approximation quality, measured by
, for Eh { } 1,
est
h 1 and SNR equals to 7 dB. BF perform badly for low values of
est
(spreading the energy in “wrong” directions), while OSTBC, as expected does
not depend on est
and has a noticeable advantage over BF. Again, like in Rayleigh
channel, for high values of est
, BF performs better than OSTBC (good information
quality about source direction). Our approximation emerges with JSO algorithm and
consequently with the best out of BF and OSTC, in the asymptotic values of est
, 0
and 1, and follows quite closely JSO algorithm in the other middle values of est
.
These results can be explained by the limitation of our approximation that on one
hand fulfills the asymptotic properties of JSO, but on the other hand for values of est
in between 0 and 1, is not so accurate (see 5.3.4).
BER
10 0
10 -1
10 -2
10 -3
JSO algorithm
New algorithm
OSTC
WOSTC
Conventional BF
-10 -5 0
SNR (dB)
5 10 15
Figure 30: Rice Channel, BER/SNR graph of new algorithm, JSO algorithm, BF, OSTBC,
and WOSTBC. 2 transmit antennas, one receive antenna, for 0.7, 1,K=1
est h

Figure 31: Rice Channel, BER/SNR graph of new algorithm, JSO algorithm, BF, OSTBC,
and WOSTBC. 2 transmit antennas, one receive antenna, for 0.7, 1,K=10
est h
Figure 32: BER as a function of channel estimation correlation coefficient in Rician fading for
the new algorithm, JSO algorithm, BF, OSTBC, and WOSTBC with SNR=7 dB.

5.4 ML optimal antenna weights with a correlated channel
(Non diagonal channel covariance matrix)
5.4.1 Introduction
The JSO solution for the case of a correlated channel (non diagonal channel
covariance matrix) must solve the non explicit nonlinear equation in (4.14) which
results in numerical calculations. In this section we start by analyzing and simulating
the affect of antenna correlation on performance for BF, OSTBC and JSO for the non
correlated channel. Then we simplify the solution in (4.14) by analytical means and
suggest a non cumbersome numeric solution. We will not obtain as for the
uncorrelated Rayleigh and Rician fading channels (with diagonal channel covariance
matrix) explicit approximation function for the solution, since the true solution is
numerical, but will define the general structure and requirements for an approximation
function of the solution. Note that there is also no explicit solution for the eigenvalues
of the optimal solution as will be shown in 5.4.3 so the approximation function must
be for both the eigenvalues and the eigenvectors of the solution unlike the Rician
approximation which was only for the optimal weights eigenvalues.
5.4.2 Fading (antenna) correlation affect on STC system performance .
The explicit expression for the performance loss of OSTC with fading correlation
loss, which we assume without loss of generality is caused by antenna correlation,
was derived by Tarokh as in (3.21). For the case of transmit diversity, the
performance loss in (3.21) becomes:
Llos 10/(
NT )log10 det([ hh ])
For Rayleigh fading, by simple algebraic manipulation becomes:
H
2
10/( )log10 det(1 )
L N
los T a
(5.18)
(5.19)
where a
is antenna correlation.
For instance for a 0.95 , and a 0 , we obtain performance loss of 5 and 0 dB
respectively.
The performance loss due to correlated channel fadings can be also obtained
numerically by simulations. Fig 33 below shows BER as a function of SNR for BF,
JSO for Rayleigh channel and its approximation („new algorithm‟), WOSTC and Ideal
BF. All plots were obtained for est 0.9, except ideal BF where est
was equal to one
and of with antenna correlation (reflecting channel correlation) of 0.95 (continuous
line) and 0 (dotted line). For instance, for received SNR of 5 dB, the performance
loss of 2.7, 1.8, 1.7, 2.9, 2.9 dB were obtained for BF, OSTC, WOSTC, JSO and new
algorithm respectively. The continuous ideal BF graph indicates on a potential gain
benefit if the correlations are exploited of up to 3 dB.
Fig 34 shows the same scenario as in fig. 33 except the fadings which are Rice
distributed with K=10. The performance loss is still high (the difference between the
continuous and dotted lines), but the ideal BF graph indicates that only a small

potential gain benefit for BF or JSO can be obtained due to the good estimation (a
realistic assumption of our system model) of the dominant LOS component (K= 10).
Figure 33: Affect of antenna correlation in Rayleigh channel. The continuous lines are the BER
for antenna correlation 0.95 while the dotted lines are for the case of no antenna correlation.
Figure 34: Affect of antenna correlation in K=10 Rician channel (scenario as in fig. 21) The
continuous lines are the BER for antenna correlation 0.95 while the dotted lines are for the case
of no antenna correlation.

5.4.3 Deriving ML Optimal antenna weights
The optimization problem for MISO by JSO can be written as:
Z arg min l(
Z)
H 1 1 1 1
where l( Z) m R ZRRmlogdetZR. (5.20)
Optimization methods based on linear programming, as in [37] – [41], can solve the
problem numerically.
Let us, before numeric solution, try to simplify the problem with matrix derivation as
much as possible. We can rewrite lZ ( ) as:
H 1
l( Z) aYa logdet Y
(5.21)
1 1
Where a R m , Y Z R
The constraint of positive semi definite (PSD) of Z, result on PSD of Y, due to the
PSD of R ( R ). Consequently, the constraint of power normalization, , is
tr( Z) 1
transformed to tr( Y) tr( R ) . We will denote the term tr( R ) as b. The
problem in (5.20), can be now written as:
H 1
arg min l( Y ) aYalog det Y
st ..
Z
H
Y Y 0
tr( Y ) b
(5.22)
Since this is convex, and the KKT conditions are valid, (the constraints are analytical
functions), we can use the KKT solution. Lets us denote by L, the Lagrangian with
respect to Y:
H 1
L( Y) a Y a logdet Y tr( Y) b
(5.23)
For solving this KKT problem, we differentiate the Lagrangian with respect to Y and
derive the following set of equations:
H
1 1 1
d / dY L( Y) Y aa Y Y I
0
tr( Y) b
H
Where the solution has to be PSD, Y Y 0.
We used the following matrix derivation calculation, were taken from [50]:
Nt
-1 -1 -1 -T T T -T
d/dX (tr(AX B)) = -(X BAX )T = -(X A B X )
T T T
d/dX (tr(A XB )) = d/dX (tr(BX A)) = AB
T -T T -T
d/dX (ln(det(AXB))) = A (AXB) B = X
Let us write (5.24) as a matrix quadratic equation:
H
d / dY L( Y) Y AY BY C 0
Where
opt
hh| hˆ
0
( ) 1
tr Z
Z Z
Z
H
1
h| hˆ hh| hˆ hh| hˆ hh| hˆ h| hˆ hh| hˆ
hh| hˆ h| hˆ hh| hˆ
1
hh| hˆ
1
hh/ hˆ
H
, , Nt Nt
A I B I C aa
1
hh/ hˆ
(5.24)
(5.25)

2
Analogous to analytical scalar quadratic equation ax bx c 0 , where the solution
can be expressed as given by, x
1
2
1 b 21
c
a
4a 2
b
,
the solution to analytical
a
H
matrix quadratic equation, X AX BX C 0 is:
1
1 1
1 2 H 1
21
2
X A
B A B C A B
4
2
X can also be expressed as:
(5.26)
4 2
1
H 1 H
1
2
1
2 H 1
1
2 2 2
1
X A B A BA A CA A B
(5.27)
If we substitute the matrices into (5.27), we obtain:
1
1
1 1 1 2
H 1 1
1
H 2
Y aa I4 2
Nt aa
I
N
t
4
2
2
(5.28)
where the Lagrangian multiplier can be found by the constraint equation, tr( Y) b :
1
1
H 2
tr I 4 Nt
aa Nt b
2
H
Nt
If we denote the eigenvalues of symmetric matrix , as , the trace
constraint can be expressed as:
iNt
(5.29)
1 4i 2b Nt
i1
(5.30)
We can solve (5.29) numerically, and by substituting back into (5.28), we obtain
the explicit solution for Y.
Nt
If we denote by i the eigenvalues of Y, the constraint of PSD of Y result in
i1
0 , and we have a valid solution. If not, we emit the negative eigenvalues out of
i
the solution of Y, and solve the constraint again for the new Y, b . We follow
this step till we obtain PSD Y.
Finally, we can transfer Y into Z by:
1
hh| hˆ
1
Z Y R
and the optimal antenna weights are obtained by:
W
Z
opt
1
2
aa ii 1
Nt
i2
i
(5.31)
(5.32)

Rician channel with non-diagonal conditional covariance matrix
For the special case of Rician channel model as written in (X), with perfect CSI (
1),
the Rice fading distribution is, h ~ CN( m , R ) and in particular,
est
h/ hˆ hh/ hˆ
2 2
h ~ CN( ahLOS , b h
RT
) .
If we substitute the mean and covariance value into (X), we obtain,
Y
2
I Nt 4 aa I Nt
2
I Nt 4
R ˆm | | ˆm / ˆR hh h h h h h hh/ hˆ
I Nt
(5.33)
Which is similar expression to the one derived in [49] for W with the usage of the
1 1
notations of W F and Y R WR . [49] suggested two fold binary search for the
value of , between lower and upper limits and thus for obtaining a valid KKT
solution for W .
5.4.4 Exploring ML optimal weights solution
5.4.5 Approximation function
A desired approximation function has to be as a function of the antenna correlation or
of another measurement which indicate on channel fadings correlation. It still has to
obey the asymptotic properties of the solution. In general, there are no asymptotic
properties of the antenna weights as a function of antenna correlation, since antenna
correlation affect both BF and STC.
Wopt WR W
where
2 2
1
W R
W
1
H 2
H
H
2 2
H H
opt R R R R
Z W W W W W W W W
The intraction between R W , W .
1
1 1 2
1
1
opt
hh| hˆ hh/ hˆ
Another realistic option, is to find try to find the dependence of on CSI
parameters and to have either explicit solution to the problem, or to have experimental
lookup table of the values of
as a function of CSI measurements.

6. Summary, conclusions and Future Work
6.1 Summary and Conclusions
In [42], a future wireless communication is described, and a forecast that MIMO
channel with combination of ST processing techniques and STC will be dominant in
future wireless communication is given. In this work we examined common ways for
combining beamforming and ST processing techniques with STBC. In particular we
focused on exploring the ML optimal linear antenna weights with OSTBC and on
deriving a simple approximation for the weights.
In this work:
We described different space-time channel models and system
spatial models from recent literature.
We described the basic principles of BF and STC techniques,
including simulation of the spatial response of these two techniques.
We conclude that:
o STC mostly does not exploit CSI and is thus optimized only on
specific channel distribution that was assumed during STC design,
which is mostly rich scatterers‟ environment with efficient space
separation between the antennas.
o BF exploits CSI, but suffers from CSI quality degradation - given the
channel estimation quality, there is an SNR, above which the beamforming
is no longer the optimal [31]. Additionally, usually BF is more
computation demanding.
We gave a literature survey of the current methods for combining
STC and BF. In particular we focused on combining OSTBC and BF
through adaptive antenna weight (which is a linear transformation).
o All methods show performance gain over OSTBC or BF
separately. Combining these two families of techniques gains
the benefits of each family.
o Each way for combining STC and BF:
Was derived from different optimization criterion -
rate maximization, received SNR maximization and
SER minimization.
Was derived with different system model
assumptions- correlated and non-correlated Rician
channel, different modulation, different estimation CSI
quality etc.
Has different computational complexity which depends
also on channel scenario- correlated channel demands
significantly more computations
Has different performance in different channels.
o The ML optimal combination OSTBC and BF was denoted as JSO
algorithm according to the initial names of the writers of [25]. JSO, as
a general ML approach to the problem of weighted OSTBC
transmission, is definitely superior to maximization of signal strength

(maximum SNR approaches) and also to BF based only on channel
mean or correlation feedback.
o The asymptotic properties of the solution indicated that the ML
optimal weights are in between equal gain weight (spreading the
transmitted energy to all directions), like in regular OSTBC, and “one
direction” weight as in BF.
o BF based only on channel correlation is not optimal in slow fading
scenario where a reliable estimation of nonzero channel mean value,
e.g. in Rician channel, can contribute to enhance the performance.
Nevertheless, in fast fading scenarios, where the channel mean value
estimates do not provide reliable information, but the channel
correlations still have meaning, the iterative solution for antenna
weights obtained in [31], for instance, should be used in order to
reduce the computations almost without degradation in performance.
o We have proved that the weights based on channel correlations can be
derived analytically from ML solution.
We focused on this work on JSO solution to the problem for multiple transmit
antennas and single receive antenna with OSTBC and with linear adaptive
antenna weights since it gives a general solution to the problem which
coincides in certain scenarios with other rate maximization, received SNR
maximization or mean and correlation feedback solutions.
For Rayleigh and Rician non-correlated channel fading we
o Derived (for Rician fading channel) and explored the
JSO solution as a function of CSI parameters.
o Examined algorithm sensitivity to errors. Algorithm
eigenvalues‟ 10% error is equivalent in non-correlated
Rician channel fadings to 0.3dB degradation in
performance measured according to SNR/BER graphs
and 0.2dB in non-correlated Rayleigh channel fadings
channel.
o Suggested a simple analytical approximation to the
ML solution in Rician fading, which fulfills the same
asymptotical properties as the ML optimal solution.
This approximation can be seen as a smart compromise,
sometimes simply as a switch, between spreading the
transmitted signal in all of the eigenvectors directions
and between one direction of the channel estimate. It
reflects the tradeoff between enhancing diversity order
of the system and enhancing coherent (SNR, antenna)
gain of the system
o Analyzed graphically and by MMSE the quality of the
approximation.
o Obtained in a simple manner the sensitivity of the
ML optimal weights to channel Parameters
o Compared the error rate performance of the
approximation compared to BF, OSTBC and JSO with
different channel paramters. Our approximation
provides almost the same performance as the JSO with
expected degredation of up to 0.2 dB. The

computational benefits of the approximation increases
as the number of transmit antennas increases.
For the Rician correlated channel fading we
o Examine analytically and by simulation the
performance loss of BF, OSTBC and “uncorrelated
fadings JSO”
o Derived an analytical solution to the ML optimal
weights, in the case of transmit diversity, which may
acquire only binary search.
o Suggest two simple future approximations to the
solution, which fulfills the same asymptotical
properties, has the ML optimal solution and discard the
need for numerical computations with a certain
degradation in performance.
6.1 Future work
Future work, continuing this work can be:
1. Obtain an explicit solution to the approximation suggested
above for MISO correlated channel, simulate it and evaluate
the performance gain in real system.
2. Compare the solutions of [47] for mean feedback based on SER
minimization with JSO.
3. Obtain an approximation for the case of MIMO channel in the
same principles as were derived in our work.
1. Generalize the work for frequency selective channels.
2. To add channel coding (STC rate lower than 1) which will enable exploring
the performance in more realistic SNR values.
3. Simulate the STC techniques derived in this work for different modulations
and standards.
4. Use the virtual channel model in the development of ML solution similar to
the JSO. With the simple linear virtual channel representation, it seems like
the case of correlated channel may have close simple expression.
5. Evaluate the various CSI parameters in our work, est , h, , as a function of
real measurements such as SNR, measurement accuracy, feedback rate
quantization ratio and method of duplexing (TDD, FDD).
K

APPENDIX A: ML optimal antenna weights for Rician
channel.
For obtaining ML optimal antenna weights for non correlated Rician channel, we
follow the solution steps as in (4.23):
1. l=1.
2. Solve:
with respect to .
(a.1)
Let us now multiply equation (a.1) by the factor 2 . After rearrange the terms, we
obtain:
By squaring the two sides of (a.2),
2 2
( (2( N )) (2( N 1) 1)) 4ˆ
and after rearranging and simplifying the equation we get the following:
which can be represented as:
Where:
which can be readily solved:
2
2 4ˆ
Nt
N
t
1 ( Nt
1) 0
22 2 ˆ
(2( N )) (2( N 1) 1) 4
t t N
t t Nt
2
ˆ
t
2 2 2
t N t t t
4( N ) 4 ( N )(2N 1) (2N 1) 1 0
2
a 4( N
)
a bc0 t
2
4 ˆ Nt ( t )(2 t 1)
2
(2 t
2
1) 1
b N N
c N
ˆ ˆ
t t
1,2 2
8( Nt
)
(a.2)
(a.3)
(a.4)
(a.5)
(a.6)
by substituting k N and simplifying (a.6), we obtain the solution for :
t
2
2 2
2
4 N ( Nt )(2Nt 1) 4 N ( Nt )(2Nt 1) 16( Nt ) (2Nt 1) 1
1 T
ˆ ˆ 2 ˆ 2
N k1(2 N 1) 2 1(2 1)
t t N k N
t t N k
t 1
1,2 2
2k1
(a.7)

3.
1 1
i
for i1,...., Nt1and
.
4. If we have to go back to 2.
We can notice that for , we get the same expression for all eigenvalues,
2 Nt
4ˆ
i
1
i Nt
2
l 0, l
0, l l1
i1,...., N 1
t
1 1
i. t
Further more, we note that instead of calculating for i N , the
cumbersome expression,
, we can use the transmit
power constraint and thus
2
N
t
4ˆ
i
1
2
1 ( 1) . N
N t
t
5. After deriving all eigenvalues and eigenvectors, we can simply compute the
1/ 2
weights, W V , in one shot.
opt

APPENDIX B: BF Iterative techniques
BF Weights can be adapted to the channel fluctuation by using CSI. The solutions
were derived from the adaptive filter theory, where the filtering is in the time
frequency and space dimensions. It can be seen as the analytical optimal weights
derived the BF adaptive techniques where each technique emphasizes different factors
in the solution.
The main 4 ways derived from MMSE optimal estimator are:
1. Sample Matrix Inversion (SMI)
2. Least Mean Square (LMS)
3. Eigen-Space Projection Algorithm (EPA)
4. Recursive Least Squares. (RLS)
The LMS (Least Mean Square) is the simplest but its convergence depends on the
Eigenvalue spread and thus in most cases where there are Non Line Of Sight (NLOS)
conditions, suffer from long convergence time. RLS (Recursive Least Squares) in the
other hand requires more computations but has shorter convergence time. EPA
(Eigen-Space Projection Algorithm) is in between LMS and RLS: fewer computations
than RLS but longer convergence time.
SMI
As the MMSE optimal solution is a function of the channel correlation matrix, a
straightforward approach is to evaluate the correlation matrix by averaging:
n NR
1
H 1
H
RR
( n)
y(
n)
y ( n)
( n 1)
RR
( n 1)
y(
n)
y ( n)
N R n 0
n
(b.1)
Then we have to substitute it in one of the analytical expression, e.g. SVD BF. The
problem with this approach is that inverting correlation matrix demands too much
computations for each iteration. Therefore, a simpler algorithm was derived, which
includes inherently the inverse of the matrix, called Sample Matrix Inversion (SMI):
(b.2)
with
1
H 1
1
1
RR
( n 1)
y(
n)
y ( n)
RR
( n 1)
RR
( n)
RR
( n 1)
.
H 1
1
y ( n)
RR
( n 1)
y(
n)
1
( 0)
I,
0
LMS
R R
LMS is the recursive Least squares solution of the optimization problem previously
introduced:
H H H
arg min( W ) J(
W)
arg
min( W ) E(
WR
Y X ) ( WR
Y X )
R
R
(b.3)
The LMS is based on the update step:
w( n)
w(
n 1)
J
( w(
n 1))
(b.4)
Where: is the step size, and J
is the gradient of the MMSE estimation problem.
After calculations, the well known formulation of LMS, can be derived:

H
w(
n)
w(
n 1)
y(
n)
e ( n 1)
H
e(
n 1)
w ( n 1)
y(
n)
x(
n)
(b.5)
Where: y (n)
and x(n)
are the received and transmit signals at discrete time n
respectively.
NLMS is normalized LMS and is derived by normalization of the updated step:
H
y(
n)
e ( n 1)
w(
n)
w(
n 1)
y(
n)
(b.6)
1
In NLMS, the Convergence time is faster: LMS , where max
is the
2
maximum eigenvalue of RT
. The convergence of the NLMS (LMS) algorithm
depends upon the eigenvalues of RT
. So with large eigenvalues spread, the algorithm
converges with slow speed.
R LS
RLS, similarly to LMS, is based on a solution of the optimization problem previously
introduced, but unlike LMS, in RLS algorithm the gradient step size is replaced with a
gain matrix at the n'th iteration, producing the weight update equation:
1
H
w(
n)
w(
n 1)
R ( n)
y(
n)
e ( n 1)
(b.7)
H
R(
n)
0R( n 1)
y(
n)
y ( n)
Where: is the step size, and (n)
is autocorrelation of received signal y, and
is a scalar close to one.
EPA
Eigen-Space Projection algorithm is introduced in [16] as:
w( n)
w(
n 1)
N
PN
( e(
n 1)
y(
n))
S
PS
( e(
n 1)
y(
n))
(b.8)
where P N () and PN
() are operators whose columns span two orthogonal sub-spaces
referred toas noise sub space and signal subspace as were described in [78].
max
R 0

APPENDIX C: Simulation environment
Simulation assumption
In general the simulation is built on the system model assumptions as explained in
chapter 2. The statistics relies on the assumption of ergodic behavior. And at least 100
samples are being obtained for the lowest point in BER/ SNR graphs (sufficient
according to the low of large numbers).
Software description
The software (SW) consists of a system part, an algorithmic part and a debug part.
These three parts are part of each file. Still, for simplicity of description of the
simulation SW, we will divide the SW modules into three main groups:
1. System modules.
2. Algorithmic modules.
3. Debug modules.
System part
The system part of the SW which controls the transmission scheme, is included
mainly in the three System SW files:
beamforming_main.m, Antenna_array_main.m, Antenna_array.m.
Fig 35, shows a simplified flow chart of the main three SW modules.

Yes
End ?
Figure 35: Main System modules.
For minimizing real time delay, the simulation is written throughout vectors and due
to limited memory buffer size, the buffer is restricted to the legth of 1000 samples. A
main loop which runs in beamforming_main divides the samples into the length of
1000 samples. A inner loop in Antenna_array_main, runs according to a specific
paremter determined by the control flags in beamforming_main. The parameter (and
thus the simulation) can be: SNR (producing BER/ SNR graphs) or estimation
correlation coefficient (producing BER / graphs).
Algorithmic part
beamforming_main( )
Global Parameter declarations
SW configuration
System and Channel parameters initialization
No
The algorithmic part of the SW is mainly related to the transmission scheme. The
algorithmic part can be divided to four modules:
1. Transmitter modules:
Tranmitter.m, Alamouti_enc.m, Beamforming_Weight.m, article_eig.m
2. Channel modules:
channel.m , Fading_correlation.m, LOS_correlation.m
3. Receiver modules:
Reciever.m, Alamouti_dec.m
Antenna_array_main( )
Set Parameter value
Plot figure
PARAMETER
Antenna_array ( )
4. New Approximation modules:
CSI_RAY_sensetivity.m (Calculate CSI parameter sensitivity),
CSI_RICE_sensetivity.m (Calculate CSI parameter sensitivity),
est

find_opt_ab_Rice.m (Find MMSE optimal ML estimation parameters for Rician
fading).
find_opt_ab_ray.m (Find MMSE optimal ML estimation parameters for
Rayleigh fading).
Fig 36, below shows how the main SW files compose the transmission scheme.
Debug and analysis part
The debugging and analysis part, which analyze the channel and weights spatial and
temporal response, consist of the following files:
Channel_analyze.m, weight_analyze.m, spatial_SVD_analyze.m,
spatial_analyze_OSTC.m.

ריצקת
ץורע
תרושקת
תיטוחלא
,
בקע
תונוכת
תוטשפתה
ילג
וידר
,
םרוג
תואל
רדושמה
תונתשהל
םוקימב
,
ןמזב
רדתבו
.
ץורע
םרוגש
םייונישל
םייתוהמ
לש
תואה
בחרמב
,
יוניש
דדמנש
אמגודל
ןיב
יתש
תונטנא
תוכומס
,
יורק
ץורע
הנתשמ
בחרמב
,
וא
space selective channel
.
ןתינ
שמתשהל
ךרעמב
תונטנא
הטילקב
רודישבו
ליבשב
לצנל
תא
םייונישה
לש
תואה
בחרמב
םשל
תאלעה
יוסיכה
תלוביקהו
לש
ץורע
תרושקת
יטוחלא
תועצמאב
תוקינכט
תוססבתמש
לע
לוצינ
תוריתי
תואה
לכב
הנטנא
,
וא
לע
ידי
דוחיא
יטנרהוק
לש
תותואה
בחרמב
.
הרקמ
יטרפ
ץופנ
דואמ
תרושקתב
תיטוחלאה
,
ובש
ןודנ
בעב
תדו
רקחמ
וז
,
ונה
הרקמה
לש
ךרעמ
תונטנא
רודישב
,
הנטנאו
תדדוב
הטילקב
.
הרקמב
הז
,
תוריתי
לש
תואה
רדושמה
תגשומ
תועצמאב
תוקינכט
לש
Space Time Codes (STC)
,
דוחיאו
יטנרהוק
לש
תותואה
בחרמב
גשומ
לע
ידי
תוקינכט
לש
Beam-forming (BF)
,
םע
וא
ילב
לוצינ
לש
עדימה
לע
ץורעה
ש
אצמנ
רדשמב
.
ב
-
BF
,
תכרעמה
תרמתומ
ךרעמל
רודיש
לש
רפסמ
םירודיש
םייווק
תועצמאב
קוריפ
לש
Single Value
Decomposition (SVD)
,
קפסהו
רודישה
קלוחמ
ןפואב
יבטימ
ןיב
לכ
דחא
םיצורעהמ
םיווקה
.
רובע
תקינכט
רודיש
וז
,
עדי
םיוסמ
לע
ייצמרופניא
ת
דצה
לש
ץורעה
בייח
תויהל
רדשמב
בו
טלקמ
ליבשב
דוביעה
יבחרמה
םיאתמה
.
הרוצב
וז
,
BF
,
אוה
רקיעב
הטיש
םע
בושמ
(close loop)
.
תוקינכט
דוביע
תולגתסמש
ץורעל
תויתגרדהב
(
tracking
)
תולגוסמו
בוקעל
ירחא
םייוניש
ץורעב
תואצמנ
שומישב
בחר
.
ב
-
STC
,
םישמתשמ
דודיקב
יבחרמ
רדשמב
חונעפו
יבחרמ
רדשמב
.
עדי
לע
מרופניא
ייצ
ת
דצה
לש
ץורעה
אל
ץוחנ
םיקפתסמו
לדומב
יטסיטטס
יללכ
לש
ץורעה
בורל
םע
החנהה
תוכיעדהש
לש
ץורעה
ןיב
לכ
תונטנאה
אל
תויולת
תיטסיטטס
דחא
ינשב
,
ךכ
ןתינש
סחייתהל
ל
-
STC
,
הטישכ
אלל
בושמ
(open
loop)
.
םיעוציבה
לש
תוקינכט
STC
םיעגפנ
האצותכ
יאמ
לוצינ
תייצמרופניא
דצה
ש
ל
ץורעה
.
תמועל
תאז
תומכ
םיבושיחה
םישרדנש
תוקינכטמ
לש
BF
,
ךרדב
ללכ
הבר
,
לכו
האיגש
ךורעשב
לש
ץורעה
הלוכי
םוגפל
םיעוציבב
.
הברה
רקחמ
השענ
הנורחאל
ןויסינב
אוצמל
הקינכט
תדחאמש
תא
תונורתיה
לש
STC
ו
-
BF
.
הדובעב
תירקחמ
וז
,
ונא
םינחוב
רפסמ
םיכרד
ועצוהש
ורשפאיש
דחאל
תא
תונורתיה
לש
STC
ו
-
BF
.
ונא
םיזכרתמ
דחוימב
הרקמב
לש
םידוק
םייבחרמ
יילאנוגותרוא
ם
,
OSTBC
,
ו
-
BF
תועצמאב
םילוקשמ
ייראיניל
ם
לש
תונטנאה
םיגשומש
תועצמאב
ךורעש
תוריבסה
תיברמה
.
ונחתיפ
ךורעיש
טושפ
לש
תולוקשמה
תולתב
םינתשמב
לש
ץורעה
,
עיגמש
םיעוציבל
םימוד
דואמ
ולאל
םיגשומש
תועצמאב
ךורעש
תוריבסה
תיברמה
.
ונלחתה
הזתב
תאז
,
רואיתב
לש
לדומ
ןמז
-
בחרמ
לש
תוכרעמ
תרושקת
תיטוחלא
.
רחאל
ןכמ
ונגצה
אובמ
יטרואית
תוקינכטל
דודיק
ןמז
-
בחרמ
(
STC
)
תוטישלו
לש
בוביר
תומולא
(
BF
.)
ונדקמתה
הדובעב
תאז
רקיעב
הרקמב
יטרפה
ץופנהו
לש
רפסמ
תונטנא
דשמ
תור
םע
הנטנא
תחא
הטילקב
(
transmit

diversity
)
,
םע
עדימ
יקלח
לע
ץורע
רדשמב
(
ךורעש
ץורע
אל
ילאידיא
)
עדימו
אלמ
לע
ץורעה
טלקמב
(
ךורעש
ץורע
אל
ילאידיא
.)
דעצכ
ןושאר
,
ונחתפ
הריקסב
לע
םינפוא
םינוש
בולישל
לש
יתש
תוטישה
.
ונדקמתה
רקיעב
תטישב
ךורעש
תוריבסה
תיברמה
לש
תולוקשמ
הנטנא
ייראיניל
ם
(
BF
)
רודישב
לש
OSTBC
(
הרקמ
יטרפ
לש
STC
)
,
יפכ
ראותש
ב
-
[
55
]
גצומו
הדובעב
תאז
תטישכ
JSO
,
יפל
תויתואה
תויליחתה
לש
תומש
םירבחמה
.
םתירוגלא
ה
-
JSO
,
רחוב
תולוקשמ
הנטנא
םיילמיטפוא
תטישב
ךורעש
תוריבסה
תיברמה
היצקנופכ
לש
םינתשמה
לש
ץורעה
ביטו
ש
ךורע
ץורעה
רדשמב
.
ןתינ
ןיבהל
ןורתפ
הז
כ
"
הרשפ
המכח
"
,
וא
וליפא
גותימ
,
ןיב
OSTBC
ו
-
BF
.
הרקמכ
יללכ
לש
ןורתפ
תועצמאב
ךורעש
תוריבסה
תיברמה
היעבל
,
הטיש
וז
הנה
הפידע
לע
תוטיש
לש
האבה
םומיסקמל
לש
תייגרנא
תואה
רדושמה
טלקמב
,
הבש
ןיא
יוטיב
תוקיטסיטטסל
םירדסמ
םיהובג
לש
תואה
רדושמה
,
וא
הטישמ
תססבתמש
קר
לע
לוצינ
היצלרוקה
ןיב
יביכר
ץורעה
(
de-correlation
)
,
רקיעב
ללגב
תומלעתהה
ךרעהמ
עצוממה
לש
ץורעה
,
תוכיעדבש
תויטיא
םע
ביכר
עצוממ
יתועמשמ
(
LOS
)
,
ונה
ביכרמ
בושח
לוכיש
םורתל
םיעוציבל
.
ונדמל
תא
תטיש
JSO
,
ץורעב
Rayleigh
,
ץורעב
Rice
,
הרקמבו
לש
יביכר
ץורע
םע
היצלרוק
ונחבו
היצלומיסב
,
תא
תושיגר
ןורתפה
תואיגשל
ךורעשב
ירטמרפ
ץורעה
.
ונאצמ
ץורעבש
Rice
,
שי
העיגפ
לש
דע
0.0
םילביצד
םיעוציבב
יפכ
דדמנש
תומוקעמ
BER
-
SNR
האצותכ
מ
-
00
%
האיגש
תולוקשמב
(
םיכרעב
םיימצעה
לש
תולוקשמה
)
ץורעבו
Rayleigh
,
י
ש
העיגפ
לש
דע
0.5
םילביצד
.
תואצות
ולא
ורגתא
ונתוא
עיצהל
ךורעש
טושפ
לש
תולוקשמה
תולתכ
ינתשמב
ץורעה
אלמיש
ירחא
תושירדה
תויטוטפמיסאה
לש
JSO
היהי
חונ
רתוי
שומימל
הווהיו
ילכ
תניחבל
תוגהנתה
ןורתפה
תולתכ
ינתשמב
ץורעה
.
ךורעשה
ונעצהש
,
עיגמ
םיעוציבל
םיבורק
הזל
לש
JSO
רובע
םוחת
ינייפוא
לש
ינתשמ
ץורעה
.
תועצמאב
ךורעיש
בהז
ונחלצה
לבקל
הרוצב
הטושפ
בוריק
לש
תושיגרה
לש
תולוקשמה
םייונישל
ינתשמב
ץורעה
.
ןורתיה
יבושיחה
לש
ךורעיש
הז
לדג
תיתועמשמ
לככ
הלועש
רפסמ
תונטנאה
.
הרקמב
לש
יביכר
ץורע
םייביטלרוק
,
הנשי
העיגפ
םיעוציבב
םג
לש
OSTBC
,
םגו
לש
BF
,
האצותכ
הדירימ
ןוויגב
(
diversity
)
יבחרמה
ןתינש
לוצינל
.
האצותכ
ךכמ
,
םג
ןורתפל
לש
JSO
היהי
הדירי
םיעוציבב
.
ונחב
תא
הדיריה
םיעוצבב
ןפואב
יטילנא
יפלו
תויצלומיס
רובע
BF
,
OSTBC
ןורתפהו
לש
JSO
רובע
הרקמה
רסח
היצלרוקה
.
לוקשמ
תונטנאה
ילמיטפואה
,
תפה
ןור
לש
JSO
,
שרוד
רותפל
הייעב
אל
תיראניל
תכבוסמ
.
ונחלצה
עיגהל
ןורתיפל
יטילנא
הייעבל
רובע
הנטנא
תדדוב
הטילקב
דע
ידכ
שופיח
יראניב
לש
הנתשמ
.
ףסונב
ךכל
ונעצה
תעצה
ךורעש
ןורתפל
הניאש
הכירצמ
בושיח
ירמונ
.
יבלשכ
ךשמה
הדובעל
וז
,
ןתינ
ןוחבל
תא
תואצותה
תוכרעמב
תרושקת
תויתימא
יפל
םיטרדנטס
םילבוקמ
,
הבחרה
לש
הטישה
רובע
ץורע
MIMO
רובעו
תוכיעד
תוריהמ
,
ילואו
ףא
עיצהל
תא
ךורעשה
תודעוול
הניקת
לש
תוכרעמ
תרושקת
תוידיתע
.

ביבא - לת תטיסרבינוא
ןמשיילפ רדלאו יביא ש " ע הסדנהל הטלוקפה
םיילנוגותרוא בחרמ-ןמז
ידוק לש בוליש
תומולא תריצי םע
למשח הסדנהב " הטיסרבינוא ךמסומ"
ראותה תארקל רמג תדובעכ שגוה הז רוביח
ידי -
לע
ןזורמולב ידג
ה
" סשת
טבש

ביבא - לת תטיסרבינוא
ןמשיילפ רדלאו יביא ש " ע הסדנהל הטלוקפה
םיילנוגותרוא בחרמ-ןמז
ידוק לש בוליש
תומולא תריצי םע
למשח תסדנהב
" הטיסרבינוא
ךמסומ"
ידי -
ראותה תארקל רמג תדובעכ שגוה הז רוביח
לע
ןזורמולב ידג
תוכרעמ למשח הסדנהל
הקלחמב התשענ הדובעה
ןמדירפ יבא 'רד
תיחנהב
ה
" סשת
טבש