MIMO Channel Rank via the Aperture-Bandwidth ... - IEEE Xplore

GOODMAN et al.: MIMO CHANNEL RANK VIA THE APERTURE-BANDWIDTH PRODUCT 2249
plane wave inbound from the j th transmit antenna. Hence, H1
is the array manifold matrix for a uniform linear array with
receive node spacing d. The fact that the entries of H1 have
equal amplitude and can be expressed as phase shifts is a direct
consequence of the far-field and narrowband assumptions. The
procedure for defining H1 can be repeated for any Hn with
the (n − 1) th scattering group acting in the role of transmit
array.
Each Hn is deterministic, but in the presence of random
scattering, HnΓn−1 is random. The diagonal entries of
Γn−1 are independent complex Gaussian random variables
representing the reflection coefficients of the previous stage’s
scatterers; hence, the signal received at the n th stage is a
linear sum of complex random variables. Furthermore, (5)
implies that the j th diagonal entry of Γn−1 is the amplitude
of a complex sinusoid with spatial frequency kj, which means
that the signal received at the n th stage is a random process
with power spectral density (PSD) defined by the diagonal of
E[|Γn−1| 2 ]. The absolute bandwidth of the random process
is defined by the maximum and minimum kj, which are
determined by the angular spread. The power spectrum of
the random process determines the correlation between signals
arriving at the receiving nodes.
The fact that the signal at each scattering group is a random
process with power spectral density defined by the spread
and distribution of scatterers in the previous stage leads to
a central theme throughout the rest of this paper. Based on
observations concerning the Karhunen-Loeve representation of
random processes over a finite interval, we will see that the
MIMO EDOF at any stage are determined by the product of
spatial frequency spread and receiving aperture. Moreover, the
EDOF of the entire system will be less than or equal to the
EDOF of any single propagation stage.
nT
H2Γ1H1H †
1 Γ†
1 H†
2
The matrix H2 forms linear combinations of the complex
Gaussian elements of Γ1, resulting in H2Γ1 also having
circularly symmetric Gaussian entries. Hence, as in [4], [8],
D
H2Γ1 can be factored as H2Γ1 = R 1/2
2 WΨ1/2 where W
H1
Fig. 2. Single-bounce implementation of the propagation model.
Γ1
is a matrix of i.i.d circularly symmetric Gaussian entries with
unit variance, Ψ1/2 =(E[|Γ1| 2 ]) 1/2 , and the symbol D = means
equally distributed. This leads to
E[I1] =E log2 det InR+
PT
nT
where the fact that R 1/2
2
H2
R 1/2
2 WΨ1/2H1H †
1Ψ1/2W † R 1/2
2
(8)
=(R1/2
2 ) † has been exploited. Using
the property that rank(AB) ≤ min(rank(A), rank(B)), the
EDOF of the system are bounded by (note that rank(R 1/2
2 )=
rank(R2) and rank(R1) =rank(H1))
EDOF ≤ min(rank(R2), rank(WΨ 1/2 ), rank(H1)). (9)
B. Model Properties and Examples
We assume that the receiver has perfect channel state information
but the transmitter does not. Under these conditions,
an appropriate approach is to allocate equal power to each of
the transmitting antennas. This leads to
IN =log2det InR + PT ˜HN+1
nT
˜ H †
N+1 (6)
where IN is the instantaneous mutual information of an Nstage
scattering environment, In is the n-dimensional identity
matrix, and PT is the total power that is divided among all
transmit antennas. Since the matrix ˜ HN+1 contains random
reflection coefficients for N ≥ 1, the mutual information is
random, and we will generally be interested in average mutual
information. Consider the N = 1 case, which is a singlescattering
abstract model [7], [22], [23], [24], [25], [28] shown
in Fig. 2. Here, the average mutual information is
E[I1] =E log2 det InR + PT
Therefore, the instantaneous EDOF of the MIMO system are
limited by the EDOF of any single propagation stage. In
addition, Jensen’s inequality allows (8) to be bounded by
E[I1] ≤ log2 det InR+
PT
R
nT
. (7)
1/2
2 E
WΨ 1/2 H1H †
1Ψ1/2W †
R 1/2
2
=log2det InR + PT
R
nT
1/2
1/2
2 InRR2
=log2det InR + PT
R2 . (10)
nT
Hence, average mutual information of the single-bounce
model is limited by the eigenvalues of R2.
Since the signal must propagate through each stage sequentially,
application of the data processing theorem [33] states
that the mutual information between transmitter and receiver
arrays is upper bounded by the mutual information of any
single propagation stage. In the single-bounce example above,
mutual information is limited by the minimum mutual information
supported by either of the two propagation segments,
whichistosaythat
I1 ≤ I0
(11)
where I0 is the mutual information obtained by replacing
the single-bounce scattering group with receive antennas. For
the general case of N scattering groups and average mutual
information, the data-processing argument requires that
E[IN ] ≤ E[IN−1] ≤···I0. (12)
Thus, we observe that the average mutual information of the
N-group progressive scattering model is limited not only by
the spatial correlation of the final propagation segment, but