Blind carrier frequency offset estimation for OFDM/OQAM ... - NTNU

Internal note 11:

**Blind** **carrier** **frequency** **offset** **estimation** **for** **OFDM**/**OQAM**

systems over multipath channel based on correlation function ∗

Gang Lin, Lars Lundheim, Nils Holte

Department of Electronics and Telecommunications

Norwegian University of Science and Technology (**NTNU**)

7491 Trondheim, Norway

E-mail:{lingang; lundheim; holte}@iet.ntnu.no

February 16, 2006

1 Introduction

It is well known that multi**carrier** systems are much more sensitive to **carrier** **frequency** **offset** (CFO)

then single **carrier** systems. The effect caused by CFO **for** classical **OFDM**/QAM systems is analyzed

by Pollet et al. [1]. In [1], the authors indicate that CFO should be less than 2% of the band width of

subchannel to guarantee the signal to interference ratio be higher than 30 dB. An **OFDM**/**OQAM** system

with minimum subchannel space in **frequency** domain is also not robust to CFO [2], even when optimal

pulses are used as pulseshaping filters [3].

Bolcskei presents a blind CFO **estimation** algorithm **for** both **OFDM**/QAM and **OFDM**/**OQAM** systems

based on cyclic correlation functions [4], which is a natural extension of the estimator **for** single

**carrier** QAM transmitting systems [5]. For multipath fading channel, Bolcskei’s method needs the channel

impulse response to be known. Based on the assumption of Rayleigh multipath fading, Park et al.

develop a algorithm which is robust to multipath effect and doesn’t need the channel in**for**mation [6].

However, such an assumption is quite restricted in practice. For example, even a simple stationary

multipath channel doesn’t satisfy such assumption.

Different from **OFDM**/QAM system, **OFDM**/**OQAM** can use time-**frequency** well-localized shaping

pulses [7][8][9]. Thus if we estimate the CFO over each subchannel in the receiver side, the equivalent

channel response can be approximated as flat-fading, then the problem can be greatly simplified. In section

2, we present an **OFDM**/**OQAM** system with **carrier** **frequency** **offset**, then **for**mulate the correlation

functions of the T/2 sampled signal out of receiver filter and prove that subchannel weighting is needed

to recover CFO. Then we present two CFO **estimation** methods **for** null-subchannel inserting systems in

section 3. The asymptotical analysis is drawn in Section 4. At last in Section 5, the simulation results are

shown and confirm that the designed estimators are robust to multipath effects and outper**for**m Bolcskei’s

estimator.

2 System model and correlation functions

The time-discrete model **for** **OFDM**/**OQAM** system with minimum subchannel space is shown in Figure 1.

This model has totally N subchannels which are weighted by factors {w k } N−1

k=0 . The weighting factor w k

should be real-valued to maintain the orthogonality between subchannels.

Each subchannel transmits one QAM symbol a k [n] = a R k [n] + j aI k

[n] per T seconds. The **OQAM**

symbols are **for**med by shifting the imaginary part of QAM symbols by T/2. By summing up all the

∗ This work is supported by BEATS. http://www.iet.ntnu.no/projects/beats

1

Figure 1: The time discrete model of base band **OFDM**/**OQAM** with **frequency** **offset**

2

subchannels, the modulator generates a T/N sampled output sequence

s[l] =

N−1

∑

w m

∑

∞

m=0 n=−∞

(

a

R

m [n] g[l − nN] + j a I m[n] g[l − nN − N/2] ) e j( 2π N l+ π 2 )m (1)

that is also the input to the channel.

We may assume that the input data symbols are statistical independent between different subchannels,

different symbols, and the real and imaginary parts are i.i.d., i.e.

E [ a R m[n 1 ] a R k [n 2 ] ] = E [ a I m[n 1 ] a I k[n 2 ] ] {

σ

2

= a /2, **for** m = k and n 1 = n 2

0, otherwise,

and

E [ a R m[n 1 ] a I k[n 2 ] ] = 0, ∀ m, k, n 1 , n 2 ,

where E[·] stands **for** the statistical expectation and σa 2 is the average power of the input QAM symbols.

Without loss of generality, we may assume that σa 2 = 1. Here we consider time-variant multipath

channel. If the number of subchannels N is large enough, the equivalent channel of the each subchannel

can be approximated as flat-fading with fading factors {µ k [l]} N−1

k=0 . We assume that µ k[l] is a stationary

random process with real-valued correlation function c µk [τ] = E [µ k [l + τ] µ ∗ k [l]] and average power σ2 µ k

=

c µk [0]. The channel model also includes an independent zero-mean white Gaussian additive noise ν[l]

with correlation function c ν [τ] = σν 2 δ[τ]. We assume that data, µ k [l] and ν[l] are mutually independent.

While we don’t assume that the attenuation factors µ k [l] of different subchannels are independent. The

**carrier** **frequency** **offset** f e is normalized with respect to 1/T . Then we can write the received sequence

as

N−1

∑

r[l] ≃ e j 2π N fel

m=0

w m µ m [l]

∞∑

n=−∞

(a R m[n] g[l − nN] + j a I m[n] g[l − nN − N )

2 ] e j( 2π N l+ π 2 )m + ν[l]. (2)

Since the correlation function of received sequence **for** **OFDM**/**OQAM** system over multipath channel

is not shown in Bolcskei’s work [4], we may derive here. The correlation function of the received sequence

r[l] can be written as

c r [l, τ] = E [r[l + τ] r ∗ [l]]

= ej 2π N feτ

2

+ σ 2 ν δ[τ]

= ej 2π N feτ

2

N−1

∑

m=0

w 2 m c µm [τ]

∞∑

n=−∞

( N−1 ∑

m=0

w 2 m c µm [τ] e j 2π N τm ) ∞

∑

(g[l + τ − nN] g[l − nN] + g[l + τ − nN − N/2] g[l − nN − N/2]) e j 2π N τm

n=−∞

(g[l + τ − nN] g[l − nN] + g[l + τ − nN − N/2] g[l − nN − N/2])

+ σ 2 ν δ[τ]. (3)

We can see that c r [l, τ] is periodic to l with a period N/2. If we want to estimate f e based on the

cyclic correlation function c r [l, τ], the phase of ∑ N−1

m=0 w2 m c µm [τ] e j 2π N τm should be known. This leads to

the requirement of channel in**for**mation. Then we can conclude that similar to **OFDM**/QAM systems,

Bolcskei’s method can only be used **for** AWGN channel **for** **OFDM**/**OQAM** systems.

Now we turn to estimate f e based on the sequence out of receiver filter. For the subchannel k in

receiver side, the received sequence is first down-converted by modulator e −j( 2π N l+ π 2 )k , then filtered by

3

the receiver filter f[l] and N/2 times down-sampled to generate a T/2 spaced sequence

b k [s] = r[l] e −j( 2π N l+ π 2 )k ∣

∗ f[l]

=

∞∑

l ′ =−∞

N−1

e j 2π N fe(l−l′ )

∣

l=s N

2

∑

w m µ m [s N 2 ]

m=0

∞ ∑

n=−∞

× e j( 2π N (l−l′ )+ π 2 )(m−k) f[l ′ ] + ν[l] e −j( 2π N l+ π 2 )k ∣

∗ f[l]

N−1

∑

= e j 2π N fel

m=0

w m µ m [s N 2 ]

∞ ∑

∞∑

n=−∞ l ′ =−∞

(

a

R

m [n] g[l − l ′ − nN] + j a I m[n] g[l − l ′ − nN − N 2 ])

∣

l=s N

2

× e j( 2π N (l−l′ )+ π 2 )(m−k) f[l ′ ] e −j 2π N f el ′ + ν[l] e −j( 2π N l+ π 2 )k ∣

∗ f[l]

(

a

R

m [n] g[l − l ′ − nN] + j a I m[n] g[l − l ′ − nN − N 2 ])

∣

l=s N

2

. (4)

Note that although the sequence immediately be**for**e decimator (or immediately out of receiver filter)

contains more in**for**mation than the N/2 down-sampled sequence b k [s], it is not practically feasible since

**for** FFT based **OFDM**/**OQAM** systems [10][11][12][13], such decimator is merged into the FFT modular

and the only signal can be used is b k [s].

Then by defining

and

we can simplify (4) as

N−1

∑

b k [s] = e j 2π N f el

w m µ m [s N 2 ]

m=0

N−1

∑

= e jπf es

w m µ m [s N 2 ]

m=0

N−1

∑

= e jπfes

m=0

w m µ m [s N 2 ]

p (o)

m,k [l] = g[l] ej( 2π N l+ π 2 )(m−k) ∗ f[l] e −j 2π N f el , (5)

ν (o)

k [l] = ν[l] e−j( 2π N l+ π 2 )k ∗ f[l], (6)

∞ ∑

n=−∞

∞ ∑

n=−∞

∞ ∑

n=−∞

∣

(

) ∣∣∣∣

a R m[n] p (o)

m,k [l − nN] + j (−1)(m−k) a I m[n] p (o)

m,k [l − nN − N/2] + ν (o)

k [l] l=s N 2

(

)

a R m[n] p (o)

m,k [sN 2 − nN] + j (−1)(m−k) a I m[n] p (o)

m,k [sN 2 − nN − N/2] + ν (o)

k [sN 2 ]

(

)

a R m[n] p m,k [s − 2n] + j (−1) (m−k) a I m[n] p m,k [s − 2n − 1] + ν k [s], (7)

where p m,k [s] = p (o)

m,k [s N 2 ] and ν k[s] = ν (o)

k

[s N 2

] are the N/2 times down-sampled versions of p(o)

m,k

[l] and

ν (o)

k

[l] respectively.

We can see clearly that p m,k [s] is equivalent shaping pulse from subchannel m to subchannel k (except

channel response). The shaping pulses g[l] and f[l] are designed to guarantee that p m,k [2n] = δ[n] δ[m−k]

**for** f e = 0, thus the sent QAM symbols can be perfectly recovered in the case of ideal noiseless channel.

While **for** f e ≠ 0, such orthogonality is damaged and there exists both inter-symbol interference (ISI)

and inter-channel interference (ICI).

The correlation function is defined as c k [s, τ] = E {b k [s + τ] b ∗ k

[s]}. Then based on the expression of

b k [s] shown in (7), we have

c k [s, τ] = 1 N−1

∑

2 ejπfeτ

+ σ 2 ν p t [τ]

m=0

= 1 N−1

∑

2 ejπfeτ

= 1 2 ejπf eτ

m=0

N−1

∑

m=0

w 2 m c µm [τ N 2 ]

w 2 m c µm [τ N 2 ]

w 2 m c µm [τ N 2 ]

∞ ∑

n=−∞

∞

∑

n=−∞

∞

∑

n=−∞

(

pm,k [s + τ − 2n] p ∗ m,k[s − 2n] + p m,k [s + τ − 2n − 1] p ∗ m,k[s − 2n − 1] )

p m,k [s + τ − n] p ∗ m,k[s − n] + σ 2 ν p t [τ]

p m,k [n + τ] p ∗ m,k[n] + σ 2 ν p t [τ], (8)

4

where p t [s] = g[l] ∗ f[l]| l=s N is N/2 times down-sampled version of the overall response of the cascade of

2

g[l] and f[l].

We can see that c k [s, τ] is not a function of s, i.e. b k [s] is actually wide-sense stationary. Then we

can use c k [τ] to denote the correlation function of b k [s]. By defining

A m,k (τ, f e ) = e jπfeτ

∞ ∑

n=−∞

p m,k [n + τ] p ∗ m,k[n], (9)

we can simplify (8) as

c k [τ] = 1 2

N−1

∑

m=0

w 2 m c µm [τ N 2 ] A m,k(τ, f e ) + σ 2 ν p t [τ]. (10)

By defining P m,k (f) = ∑ ∞

s=−∞ p m,k[s] e −jπfs (note here P m,k (f) is normalized with respect to 1/T

instead of the sampling rate 2/T , thus P m,k (f) should be periodic to f with a period 2 instead of 1), we

∑ ∞

can use the discrete **for**m of Parseval’s relation, i.e.

n=−∞ g ∫

1[n] g2[n] ∗ = 1 1

2 −1 G 1(f) G ∗ 2(f) df, where

G 1 (f) = ∑ ∞

n=−∞ g 1[n] e −jπfn and G 2 (f) = ∑ ∞

n=−∞ g 2[n] e −jπfn , to rewrite (9) as

A m,k (τ, f e ) = e jπf eτ

∞∑

p m,k [n + τ] p ∗ m,k[n] = 1 2

n=−∞

∫ 1

−1

|P m,k (f)| 2 e jπ(f+f e)τ df. (11)

Now we consider the case of that the transmitter f[l] and receiver g[l] are identical real-valued symmetric

pulses and band-limited to [−1/T, 1/T ] (as usually assumed in **OFDM**/**OQAM** systems), **for** example,

square root raised cosine pulse with a roll-off factor less or equal to one. Then it is proved in appendix A

that ∑ N−1

m=0 A m,k(τ, f e ) is real-valued and only a function of τ. Then going back to (10), in the case

of non-weighting system and ideal transmitting channel, i.e. w m = 1, µ m [l] ≡ 1, we have that is a

real-valued constant **for** τ given. Then both the phase and amplitude of c k [τ] contain no in**for**mation of

f e .

3 Estimation algorithm

In the previous section, we have proved that in the case of w k = 1 and µ k [l] ≡ 1, the correlation functions

c k [τ] contain no in**for**mation of f e . Thus subchannel weighting, i.e. distributing individual subchannel

different power, is needed to recover CFO, which is similar to Bolcskei’s method [4]. Here we consider the

case of null-subchannel inserting, i.e. setting w k = 0 **for** some subchannels, while the other factors chosen

as 1. If the null-subchannels are sparsely distributed and the kth subchannel is one null subchannel, it can

be easily verified that if 0 ≤ f e < 1, then c k−1 [τ], c k [τ], c k+1 [s, τ] and c k+2 [s, τ] contain the in**for**mation

of f e . This implies that we can estimate f e from subchannels k − 1, k and k + 1. While the useful signal

power of subchannels k − 1, k + 1 and k + 2 is much lower than k. Thus we use only the kth subchannel

to estimate CFO.

Here we consider a special case that only subchannel k is set as null-subchannel. We also assume

5

|f e | < 1. Then by using (30), we have

where

c k [τ] = 1 2

= 1 4

= 1 4

= 1 4

N−1

∑

m=0

m≠k

∫ 1

−1

∫ 1

−1

∫ 1

−1

c µm [τ N 2 ] A m,k(τ, f e ) + σ 2 ν p t [τ]

⎛

⎞

N−1

⎜ ∑

⎝ c µm [τ N 2 ] |P m,k(f)| 2

m=0

m≠k

⎛

⎞

N−1

⎜ ∑

⎝ c µm [τ N 2 ] |P m,k(f − f e )| 2

⎛

⎜

⎝

m=0

m≠k

k+2

∑

m=k−2

m≠k

≃ 1 4 c µ k

[τ N 2 ] ∫ 1

= 1 4 c µ k

[τ N 2 ] (2

c µm [τ N 2 ] |P m,k(f − f e )| 2 ⎞

( k+2 ∑

−1

m=k−2

∫ 1

−1

⎟

⎠ e jπ(f+fe)τ df + σ 2 ν p t [τ]

⎟

⎠ e jπfτ df + σ 2 ν p t [τ]

⎟

⎠ ejπfτ df + σ 2 ν p t [τ]

)

|P m,k (f − f e )| 2 ∣

− ∣P (d)

k,k (f − f e) ∣ 2 e jπfτ df + σν 2 p t [τ]

G 2 (f) cos (πfτ) df −

= 1 4 c µ k

[τ N 2 ] (

4 p t [τ] − e jπf eτ/2

= − c µ k

[τ N 2 ]

4

= − c µ k

[τ N 2 ]

4

∫ 1

−1

∫ 1

−1

)

G 2 (f − f e ) G 2 (f) e jπfτ df + σν 2 p t [τ]

G 2 (f − f e /2) G 2 (f + f e /2) e jπfτ df

∫ 1

e jπfeτ/2 G 2 (f − f e /2) G 2 (f + f e /2) cos (πfτ) df +

−1

e jπf eτ/2 M g (f e , τ) +

M g (f e , τ) =

∫ 1

−1

)

+ σ 2 ν p t [τ]

(

c µk [τ N )

2 ] + σ2 ν p t [τ]

(

c µk [τ N )

2 ] + σ2 ν p t [τ], (12)

G 2 (f − f e /2) G 2 (f + f e /2) cos (πfτ) df. (13)

It is obvious that M g (f e , τ) is real-valued. In Figure 2, we show the curves of Mg(f e , τ) **for** the cases

that G(f) is the square root raised cosine pulses with a roll-off factor equal to 0.2 and 1.0 respectively.

We can see that the amplitude of M g (f e , τ) decreases quickly with increasing τ.

3

3

2

2

1

1

0

0

−1

2

−1

2

1

0

fe

−1

−2

−10

−5

0

τ

5

10

1

0

fe

−1

−2

−10

−5

0

τ

5

10

(a) α = 0.2

(b) α = 1.0

Figure 2: The curves **for** M g (f e , τ)

Then we plot the curves when τ = 0, 1, 2 in Figure 3.

6

We can see that M g (f e , 2) is positive **for**

f e ∈ [−1, 1]. We also note that even though M g (f e , 0) and M g (f e , 1) decrease quickly with increasing

|f e |, their ratio keeps about the same, especially **for** the roll-off factor α = 1.0.

4

3.5

Mg(fe,0)

Mg(fe,1)

Mg(fe,2)

Mg(fe,0)/Mg(fe,1)

4

3.5

Mg(fe,0)

Mg(fe,1)

Mg(fe,2)

Mg(fe,0)/Mg(fe,1)

3

3

2.5

2.5

Mg(fe,τ)

2

Mg(fe,τ)

2

1.5

1.5

1

1

0.5

0.5

0

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

fe

0

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

fe

(a) α = 0.2

(b) α = 1.0

Figure 3: The curves **for** M g (f e , τ) **for** τ = 0, 1, 2

In practice, c k [τ] should be estimated based on data records with length M. If the undefined samples

are set to zero, we can estimate the correlation function by ĉ k [τ] = 1 ∑ M−1

M s=0 b k[s + τ] b ∗ k

[s]. It is obvious

that such **estimation** is asymptotically unbiased. Below we present two **estimation** methods:

Estimator 1:

Since p t [l] is Nyquist pulse, p t [τ] = 0 **for** non-zero even τ. We also assume that c µk [τ N 2

] is real-valued.

For slow fading, c µk [τ N 2 ] ≃ c µ k

[0] = σµ 2 k

, thus it is positive **for** small value of τ. Then we can estimate f e

based on the phase of c k [2] as

ˆf e = 1 π angle {−ĉ k[2]} , (14)

where angle {·} stands **for** the operation of taking phase in radian.

It is obvious that the acquisition range should be f e ∈ (−1, 1). For fast fading channel, c µk [τ N 2 ] may

become negative or have a very small absolute value, then this estimator fails.

Estimator 2:

Based on Figure 3 and the relationship of c k [τ] to M g (f e , τ) shown in (12), we can see that the power

of c k [0] and c k [1] is much higher than that of c k [2]. Thus better per**for**mance is expected if we estimate

f e based on c k [0] and c k [1]. First based on (12), we have

c k [0] = − σ2 µ k

4 M g (f e , 0) + ( σµ 2 k

+ σν

2 )

c k [1] = − c µ k

[ N 2 ]

4

e jπf e/2 M g (f e , 1) +

(

c µk [ N )

2 ] + σ2 ν p t [1]. (15)

We can see that both c k [0] and c k [1] suffer from the interference of noise. For slow fading channel,

c µk [ N 2 ] ≃ σ2 µ k

(**for** stationary channel they are actually equal), then we have

(

p t [1] c k [0] − c k [1] ≃ σ2 µ k

4 M g (f e , 1) e jπfe/2 − M )

g (f e , 0)

M g (f e , 1) p t[1] . (16)

Now the effect of noise has been successfully eliminated. Since M g (f e , τ) and p t [τ] can be deferred

from G(f) directly, CFO can be estimated as

(

ˆf e = arg φ(f e ) = ˆφ

)

, (17)

{

}

where φ(f e ) def

= angle e jπfe/2 − M g(f e ,0)

M p g(f e,1) t[1] and ˆφ = angle {p t [1] ĉ k [0] − ĉ k [1]}.

7

Now the **estimation** problem turns to solve a nonlinear equation φ(f e ) = ˆφ. The property of φ(f e )

should be studied. In Figure 3, the curves of φ(f e ) and φ ′ (f e ) are shown **for** g[l] is the square root raised

cosine pulse with a roll-off factor α = 1.0, where φ ′ (f e ) is the derivative of φ(f e ) and will be used **for**

the calculation of theoretical mse later. We can see that φ(f e ) increases monotonously with increasing

f e in the section of −π < φ(f e ) < π. Thus ˆf e is uniquely determined **for** −π < ˆφ < π. Here just **for** the

purpose of simulation, matlab function fzero() is used to do this work. In practice, a look-up table may

be used.

While this doesn’t mean that the acquisition range is f e ∈ (−2, 2). In fact from Figure 3, we can

see that both M g (f e , 0) and M g (f e , 1) are close to zero **for** |f e | > 1.5, thus is not possible to get reliable

**estimation**. In the following simulation, we set |f e | < 1.

1

0.8

6

phase (π)

0.6

0.4

0.2

0

−0.2

−0.4

5

4

3

2

−0.6

−0.8

−1

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

fe (1/T)

1

0

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2

fe (1/T)

(a) φ(f e )

(b) φ ′ (f e )

Figure 4: The curve of φ(f e ) and φ ′ (f e )

4 Asymptotical analysis

In this section, we will derive the theoretical per**for**mance of both estimator 1 and 2. Recalling that

ĉ k [τ] = 1 ∑ M−1

M s=0 b k[s + τ] b ∗ k [s], it is obvious that lim M→∞ E [ĉ k [τ]] = c k [τ]. Since b k [s + τ] b ∗ k

[s] satisfies

so called mixing conditions, the **estimation** variance |ĉ k [τ] − c k [τ]| 2 → 0 as M → ∞. Thus the amplitude

of estimate error ∆c k [τ] def

= ĉ k [τ] − c k [τ] should be small **for** large M.

Estimator 1:

First we analysis estimator 1. The estimated CFO can be continued as

ˆf e = 1 π angle {−ĉ k[2]} = 1 π Im {ln(−ĉ k[2])}

= 1 π Im {ln(−c k[2] − ∆c k [2])}

(a)

≃ 1 {

π Im ln(−c k [2]) + ∆c }

k[2]

c k [2]

= f e + ∆c k[2] c ∗ k [2] − ∆c∗ k [2] c k[2]

2 j π |c k [2]| 2 , (18)

where (a) follows from the first order Taylor approximation.

Thus based on the fact that lim M→∞ E [∆c k [τ]] = E [ĉ k [τ]] − c k [τ] = 0, we have immediately conclude

that lim M→∞ E [ ˆfe − f e

]

= 0, i.e. the **estimation** of fe is asymptotically unbiased. This will be verified

by simulation results in the next section.

8

Now we will try to calculate the mse of ˆf e . Based on (18), we have

E [ | ˆf e − f e | 2] 1

=

4 π 2 |c k [2]| 4 E[ |∆c k [2] c ∗ k[2] − ∆c ∗ k[2] c k [2]| 2]

1

(

=

2 π 2 |c k [2]| 4 |c k [2]| 2 E [ |∆c k [2]| 2] − Re

{(c ∗ k[2]) 2 E [ (∆c k [2]) 2]})

1

(

=

2 π 2 |c k [2]| 2 E [ |∆c k [2]| 2] {

− Re e −j2πf e

E [ (∆c k [2]) 2]}) . (19)

Estimator 2:

Now we analysis estimator 2. First we define β = p t [1] c k [0] − c k [1] and ∆β = p t [1] ∆c k [0] − ∆c k [1],

and ∆f e = ˆf e − f e . Based on (17), we have

φ(f e + ∆f e ) = Im {ln(β + ∆β)} .

Then by using the first order Taylor approximation in both sides and the fact that φ(f e ) ≃ Im {ln(β)}

(they are actually equal **for** stationary channel), we have

{ } ∆β

φ ′ (f e ) ∆f e = Im = ∆β β∗ − ∆β ∗ β

β 2 j |β| 2 ,

where φ ′ (f e ) stands **for** the derivative of φ(f e ).

Thus the mse of estimator 2 is

E [ |∆f e | 2] 1

[

=

4 |β| 4 |φ ′ (f e )| 2 E |∆β β ∗ − ∆β ∗ β| 2] ,

(

1

=

2 |β| 2 |φ ′ (f e )| 2 E [ |∆β| 2] { β

∗

− Re

β E[ (∆β) 2]}) ,

1

(

=

2 |β| 2 |φ ′ (f e )| 2 E [ |∆β| 2] {

− Re e −j2φ(fe) E [ (∆β) 2]}) . (20)

We may rewrite ∆β in a **for**m of matrix as

∆β = [ p t [1] −1 ] [ ]

∆c k [0]

.

∆c k [1]

Then the expectation in (20) can be written as

E [ |∆β| 2] = [ p t [1] −1 ] [ E [ |∆c k [0]| 2] E [ ∆c k [0] ∆c ∗ k [1]]

] [ ]

E [ ∆c k [1] ∆c ∗ k [0]] E [ pt

|∆c k [1]| 2] [1]

−1

E [ (∆β) 2] = [ p t [1] −1 ] [ E [ (∆c k [0]) 2] E [ ∆c k [0] ∆c k [1] ] ] [ ]

E [ ∆c k [1] ∆c k [0] ] E [ pt

(∆c k [1]) 2] [1]

. (21)

−1

From (19) and (20), we can see that the only work left is to calculate E [ ∆c k [τ 1 ] ∆c ∗ k [τ 2] ] and

E [ ∆c k [τ 1 ] ∆c k [τ 2 ] ] . It is shown in Appendix B that lim M→∞ M E [ ∆c k [τ 1 ] ∆c ∗ k [τ 2] ] = S ek (0, τ 1 , τ 2 ) and

lim M→∞ M E [ ∆c k [τ 1 ] ∆c k [τ 2 ] ] = ˜S ek (0, τ 1 , τ 2 ), where S ek (0, τ 1 , τ 2 ) and ˜S ek (0, τ 1 , τ 2 ) are given by (48)

and (50) **for** stationary channel, and (51) and (52) **for** time-variant channel respectively. By noting that

both S ek (0, τ 1 , τ 2 ) and ˜S ek (0, τ 1 , τ 2 ) are composed by three terms: the term independent of σν, 2 the term

inversely proportional to σν 2 and the term inversely proportional to σν, 4 we can write the asymptotical

mse **for** estimator 1 and 2 as

mse 1 (f e , SNR) = 1 (

A 1 (f e ) + B 1(f e )

M

SNR + C )

1(f e )

SNR 2

mse 2 (f e , SNR) = 1 (

A 2 (f e ) + B 2(f e )

M

SNR + C )

2(f e )

SNR 2 (22)

9

espectively, where SNR def

= σ 2 a/σ 2 ν, and the parameters A 1 (f e ), B 1 (f e ), C 1 (f e ) and A 2 (f e ), B 2 (f e ), C 2 (f e )

are independent of SNR.

First we note that the mse of both estimators is inversely proportional to the data record length M.

We also note that both estimators suffer from floor effects which are determined by A 1 (f e ) and A 2 (f e ),

except **for** estimator 1 at f e = 0. Such floor effects are caused by self noise. Although we would expect

that both estimators are independent of SNR, the reality is not that simple way. It is not difficult to

write the expressions **for** A 1 (f e ), B 1 (f e ), C 1 (f e ) and A 2 (f e ), B 2 (f e ), C 2 (f e ) explicitly. Here we would

rather plot the numerical results. The numerical results **for** the case of roll-off factor α = 1 are shown in

Figure 5.

10 1 fe (1/T)

10 0

A 1

A 2

B 1

10 −1

B 2

C 1

C 2

10 −2

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

Figure 5: The curves of A 1 (f e ), B 1 (f e ), C 1 (f e ) and A 2 (f e ), B 2 (f e ), C 2 (f e )

5 Simulation results

First we present some preliminary simulation conditions which are used through out all simulations:

(1) 16**OQAM** modulation with uni**for**mly distributed input symbols and σ 2 a = 1;

(2) g[l] and p[l] are square root raised cosine pulses with a roll off factor α = 1.0;

(3) The number of subchannels is set as N = 16;

(4) Each result is obtained by averaging over 1000 Monte Carlo trials.

Simulation 1: per**for**mance of estimators versus SNR over AWGN channel

In this simulation, we set M = 256 (corresponding to 128 **OFDM** symbols). The simulation results

are shown in Figure 6. We can see that estimator 2 is better than estimator 1 except in the section of

small f e . Although the estimator per**for**mance should be independent to SNR theoretically, we observe

that mse is still affected by noise level due to the inefficient length of data records. We also note that

there exists no floor effect **for** estimator 2 when f e = 0.

Simulation 2: per**for**mance versus f e over AWGN channel

Now we simulate the per**for**mance of estimators versus f e . We still set M = 256. The simulation

results are shown in Figure 7. We can see that estimator 1 is sensitive to f e **for** high SNR, especially **for**

f e close to zero. While estimator 2 is largely insensitive to f e , especially in the section of −0.2 < f e < 0.2.

We also note that both estimators suffer threshold effect **for** large value of f e , especially **for** estimator 1.

Such threshold effect will disappear **for** large enough M.

Simulation 3: per**for**mance versus M over AWGN channel

In this simulation, we try to find the per**for**mance versus the length of data records. The simulation

results are shown in Figure 8. We can see that **for** both estimator 1 and 2, larger M means lower bias

and mse. In figure 8(h), we note that the threshold effect of estimator 2 disappear with increasing M **for**

SNR = 40 dB.

Simulation 4: per**for**mance versus SNR over stationary multipath fading channel

10

4

3

Estimator 1: simulated

Estimator 2: simulated

10 −2

Estimator 1: theoretical

Estimator 1: simulated

Estimator 2: theoretical

Estimator 2: simulated

2

10 −3

1

10 −4

bias

0

−1

5 x 10−3 SNR (dB)

mse

10 −1 SNR (dB)

10 −5

−2

10 −6

−3

10 −7

−4

−5

0 10 20 30 40 50 60

10 −8

0 10 20 30 40 50 60

(a) Bias: f e = 0.0

(b) Mse: f e = 0.0

0.01

0.005

Estimator 1: simulated

Estimator 2: simulated

Estimator 1: theoretical

Estimator 1: simulated

Estimator 2: theoretical

Estimator 2: simulated

10 −1 SNR (dB)

0

10 −2

bias

−0.005

mse

−0.01

−0.015

10 −3

−0.02

0 10 20 30 40 50 60

SNR (dB)

0 10 20 30 40 50 60

(c) Bias: f e = 0.2

(d) Mse: f e = 0.2

0.01

0.005

0

Estimator 1: theoretical

Estimator 1: simulated

Estimator 2: theoretical

Estimator 2: simulated

−0.005

−0.01

bias

−0.015

mse

10 −1 SNR (dB)

10 −2

−0.02

−0.025

Estimator 1: simulated

Estimator 2: simulated

−0.03

−0.035

−0.04

0 10 20 30 40 50 60

SNR (dB)

10 −3

0 10 20 30 40 50 60

(e) Bias: f e = 0.4

(f) Mse: f e = 0.4

0.1

0

−0.1

bias

−0.2

−0.3

mse

10 0 SNR (dB)

10 −1

Estimator 1: theoretical

Estimator 1: simulated

Estimator 2: theoretical

Estimator 2: simulated

−0.4

Estimator 1: simulated

Estimator 2: simulated

−0.5

0 10 20 30 40 50 60

SNR (dB)

10 −2

0 10 20 30 40 50 60

(g) Bias: f e = 0.8

(h) Mse: f e = 0.8

11

Figure 6: Bias and mse versus SNR **for** different f e over AWGN channel

1

0.8

0.6

0.4

Estimator 1: estimated

Estimator 2: estimated

Estimator 1: theoretical

Estimator 1: estimated

Estimator 2: theoretical

Estimator 2: estimated

0.2

bias

0

mse

10 −1

−0.2

−0.4

−0.6

−0.8

10 −2

−1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

fe (1/T)

10 0 fe (1/T)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

(a) Bias: SNR = 00 dB

(b) Mse: SNR = 00 dB

1

0.8

0.6

0.4

Estimator 1: estimated

Estimator 2: estimated

10 −1

Estimator 1: theoretical

Estimator 1: estimated

Estimator 2: theoretical

Estimator 2: estimated

0.2

bias

0

mse

10 −2

−0.2

−0.4

−0.6

10 0 fe (1/T)

10 −3

−0.8

−1

10 −4

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

fe (1/T)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

(c) Bias: SNR = 20 dB

(d) Mse: SNR = 20 dB

1

0.8

0.6

Estimator 1: estimated

Estimator 2: estimated

10 −1

0.4

0.2

10 −2

bias

0

mse

10 0 fe (1/T)

10 −3

−0.2

−0.4

−0.6

−0.8

−1

10 −4

10 −5

10 −6

Estimator 1: theoretical

Estimator 1: estimated

Estimator 2: theoretical

Estimator 2: estimated

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

fe (1/T)

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

(e) Bias: SNR = 40 dB

(f) Mse: SNR = 40 dB

1

0.8

Estimator 1: estimated

Estimator 2: estimated

0.6

0.4

10 −2

0.2

bias

0

−0.2

−0.4

−0.6

mse

10 0 fe (1/T)

10 −4

10 −6

Estimator 1: theoretical

Estimator 1: estimated

Estimator 2: theoretical

Estimator 2: estimated

−0.8

−1

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

fe (1/T)

10 −8

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

(g) Bias: SNR = 60 dB

(h) Mse: SNR = 60 dB

12

Figure 7: Bias and mse versus f e **for** different SNR over AWGN channel

mse

0.01

0.008

0.006

0.004

0.002

0

Estimator 1: SNR = 00dB

Estimator 2: SNR = 00dB

Estimator 1: SNR = 20dB

Estimator 2: SNR = 20dB

Estimator 1: SNR = 40dB

Estimator 2: SNR = 40dB

Estimator 1: SNR = 60dB

Estimator 2: SNR = 60dB

mse

10 −2

10 −3

10 −4

−0.002

−0.004

−0.006

−0.008

10 −5

10 −6

10 −7

Estimator 1: theoretical (00dB)

Estimator 1: simulated (00dB)

Estimator 2: theoretical (00dB)

Estimator 2: simulated (00dB)

Estimator 1: theoretical (40dB)

Estimator 1: simulated (40dB)

Estimator 2: theoretical (40dB)

Estimator 2: simulated (40dB)

−0.01

500 1000 1500 2000 2500 3000 3500 4000

M

10 −1 M

500 1000 1500 2000 2500 3000 3500 4000

(a) Bias: f e = 0.0

(b) Mse: f e = 0.0

0.02

0.015

0.01

0.005

Estimator 1: SNR = 00dB

Estimator 2: SNR = 00dB

Estimator 1: SNR = 20dB

Estimator 2: SNR = 20dB

Estimator 1: SNR = 40dB

Estimator 2: SNR = 40dB

Estimator 1: SNR = 60dB

Estimator 2: SNR = 60dB

10 −2

Estimator 1: theoretical (00dB)

Estimator 1: simulated (00dB)

Estimator 2: theoretical (00dB)

Estimator 2: simulated (00dB)

Estimator 1: theoretical (40dB)

Estimator 1: simulated (40dB)

Estimator 2: theoretical (40dB)

Estimator 2: simulated (40dB)

mse

0

mse

10 −3

−0.005

−0.01

−0.015

10 −1 M

10 −4

−0.02

500 1000 1500 2000 2500 3000 3500 4000

M

500 1000 1500 2000 2500 3000 3500 4000

(c) Bias: f e = 0.2

(d) Mse: f e = 0.2

0.02

0.01

0

10 −2

Estimator 1: theoretical (00dB)

Estimator 1: simulated (00dB)

Estimator 2: theoretical (00dB)

Estimator 2: simulated (00dB)

Estimator 1: theoretical (40dB)

Estimator 1: simulated (40dB)

Estimator 2: theoretical (40dB)

Estimator 2: simulated (40dB)

mse

−0.01

−0.02

−0.03

Estimator 1: SNR = 00dB

Estimator 2: SNR = 00dB

Estimator 1: SNR = 20dB

Estimator 2: SNR = 20dB

Estimator 1: SNR = 40dB

Estimator 2: SNR = 40dB

Estimator 1: SNR = 60dB

Estimator 2: SNR = 60dB

mse

10 −3

−0.04

−0.05

500 1000 1500 2000 2500 3000 3500 4000

M

10 −4

10 −1 M

500 1000 1500 2000 2500 3000 3500 4000

(e) Bias: f e = 0.4

(f) Mse: f e = 0.4

mse

0

−0.1

−0.2

−0.3

−0.4

Estimator 1: SNR = 00dB

Estimator 2: SNR = 00dB

Estimator 1: SNR = 20dB

Estimator 2: SNR = 20dB

Estimator 1: SNR = 40dB

Estimator 2: SNR = 40dB

Estimator 1: SNR = 60dB

Estimator 2: SNR = 60dB

mse

10 0

10 −1

10 −2

M

Estimator 1: theoretical (00dB)

Estimator 1: simulated (00dB)

Estimator 2: theoretical (00dB)

Estimator 2: simulated (00dB)

Estimator 1: theoretical (40dB)

Estimator 1: simulated (40dB)

Estimator 2: theoretical (40dB)

Estimator 2: simulated (40dB)

−0.5

500 1000 1500 2000 2500 3000 3500 4000

M

10 −3

500 1000 1500 2000 2500 3000 3500 4000

(g) Bias: f e = 0.8

(h) Mse: f e = 0.8

13

Figure 8: Bias and mse versus M **for** different f e over AWGN channel

In this simulation, we set M = 256. First we simulate the estimator per**for**mance over stationary

multipath channel. Here we consider a three paths channel, which has an impulse response

h[l] =

2∑

λ d δ[l − d], (23)

d=0

where λ d is the complex-valued amplitude.

Just **for** the purpose of simulation, we set [ λ 0 λ 1 λ 2

]

=

[

0.8729 0.4364 −0.2182

]

and subchannel

4 as the null-subchannel. The **frequency** response of channel is shown in Figure 9.

1.2

1

0.8

|H(f)|

0.6

0.4

0.2

0

0 2 4 6 8 10 12 14 16

f (1/T)

Figure 9: The **frequency** response of transmitting channel

To calculate the theoretical mse, the **frequency** response of subchannel k is approximated by µ k =

∑ 2

d=0 λ d e −j2πkd/16 .

Here we also compare estimator 1 and 2 to Bolcskei estimator [4]. The **for**mula (15) in [4] is used

**for** **estimation** of **carrier** **frequency** **offset** θ e and timing **offset** n e . Note that **for** **OFDM**/**OQAM** system,

M = N, which was clearly indicated in (3) of [4]. Replacing M by N, we copy (15) in [4] as

C r [k, τ] = 1 N ej2πθ eτ e −j 2π N kn e

Γ N [τ] A (g,g) [

τ, k N

) [σ

2

c,R + (−1) k σ 2 c,I]

+ cρ [τ] δ[k], (24)

where

Γ N [τ] =

[

A (g,g) τ, k )

=

N

N−1

∑

k=0

∞∑

l=−∞

|w k | 2 e j 2π N kτ

g[l] g[l − τ] e −j 2π N kl , (25)

and σc,R 2 and σ2 c,I are respectively the average power of the real and imaginary parts of sent QAM symbols.

The **frequency** **offset** θ e in [4] is normalized with respect to N/T , while our expression f e is with respect

to 1/T , thus f e = N θ e .

Now we assume σc,R 2 = σ2 c,I and the transmitter pulse g(t) is band-limited to [−1/T, 1/T ]. By using

Parseval’s relation, we can rewrite A [ ) (g,g) τ, k N shown in (25) as

[

A (g,g) τ, k )

=

N

=

∞∑

l=−∞

∫ 0.5

−0.5

g[l] g[l − τ] e −j 2π N kl

G(f) G(f + k N ) e−j2πfτ df, (26)

where G(f) = ∑ ∞

l=∞ g[l] e−j2πfl .

Since g[l] is the discrete **for**m of g(t) with a sampling interval T/N, G(f) should be band-limited

to [−1/N, 1/N]. This leads to A (g,g) [ τ, k N

)

is nonzero only if k = 0, ±1. While **for** k = ±1, σ

2

c,R +

14

(−1) k σc,I 2 = 0 because of the assumption of σ2 c,R = σ2 c,I . Thus we conclude that C r[k, τ] can be used

**for** **estimation** only **for** k = 0. In this case, only **frequency** **offset** can be recovered and the accuracy will

be affected by the term c ρ [τ] caused by noise. The weighting factor w k is still important to keep Γ N [τ]

nonzero **for** τ ∈ [0, N − 1]. For the case of subchannel 4 is the only null-subchannel, one can verify that

Γ N [τ] ≠ 0 **for** all τ ∈ [0, N − 1]. At last, the estimator can be expressed as

ˆθ e = 1

2π

∑L τ

τ=1

1

{Ĉr

τ angle [0, τ]

Γ N [τ]

}

. (27)

Note that since A (g,g) [τ, 0) is real-valued and positive, we don’t need to consider it’s effect.

In simulation we set L τ = 15, then the acquisition range of Bolcskei estimator is θ e ∈ [− 1

30 , 1 30

], i.e.

f e ∈ [−0.53, 0.53]. Thus we simulate Bolcskei estimator only **for** f e = 0.0, 0.2 and 0.4. The simulation

results is shown in Figure 10.

As expected, both estimator 1 and 2 are robust to multipath effects. We also observe that the mse **for**

multipath channel is even smaller that AWGN channel. This can be explained by that since subchannel

4 is set as the null-subchannel, while as shown in Figure 9, the amplitude of H(f) is slightly higher than

1, which means that the equivalent SNR is higher than AWGN case.

We find that both estimator 1 and 2 per**for**m better than Bolcskei estimator, except that Bolcskei estimator

is slightly better than estimator 1 **for** low SNR section over AWGN channel. The more important

difference is that Bolcskei estimator is not robust to unknown multipath channel, while both estimators 1

and 2 are robust to multipath effects. This has been verified by both theoretical prediction and simulation.

Simulation 5: per**for**mance versus SNR over Rayleigh multipath fading channel

Then we simulate the per**for**mance of estimator 1 and 2 over Rayleigh multipath fading channel. We

still set M = 256. The amplitude λ d in (23) will be relative to time instant l. In our simulation, λ d [l]

are identical distributed **for** all paths and modelled as a lowpass autoregressive precess with fifth order

multiple poles at ρ, which can be expressed as

λ d [l] = 5 ρ λ d [l − 1] − 10 ρ 2 λ d [l − 2] + 10 ρ 3 λ d [l − 3] − 5 ρ 4 λ d [l − 4] + ρ 5 λ d [l − 5] + n s [l], (28)

where n s [l] is the complex-valued zero-mean Gaussian white noise with variance σn.

2

The **frequency** response of such a filter is H(f) = 1/(1 − ρ e −j2πf ) 5 and the 3 dB bandwidth B µ T is

related to ρ by the equation ( 5√ 2 − 1) ρ 2 − 2 [ 5√ 2 − cos(2πB µ T )] ρ + 5√ 2 − 1 = 0. Since σλ 2 d

= c λd [0] =

∫

σn

2 0.5

−0.5 |H(f)|2 df, we set σn 2 1

=

3 ∫ 0.5

d=0 σ2 λ d

= 1.

Then we approximate the equivalent channel response of subchannel k as flat-fading with fading

−0.5 |H(f)|2 df to let ∑ 2

factor µ k [l] = ∑ 2

d=0 λ d[l] e −j 2π N kd . Since λ d [l] are i.i.d., we have c µk [τ] = ∑ 2

d=0 c λ d

[τ] = 3 c λ0 [τ], which

is independent to k. Here we simulate three cases: B µ T = 0.001 (very slow fading), B µ T = 0.01 (slow

fading). The curves of (estimated) c µk [τ] are shown in Figure 11 **for** different fading speed. We can see

that **for** B µ T = 0.001 and 0.01, c µk [16] is positive and c µk [8] ≃ c µk [0] = σµ 2 k

= 1, thus both estimator 1

and 2 will work fine. While **for** B µ T = 0.05 (fast fading), such conditions will not be satisfied, thus both

estimator 1 and 2 will not work.

The simulation results are shown in Figure 12. We can see that **for** B µ T = 0.001 and 0.01, there

exists only slight degradation compared with AWGN channel. Thus we can conclude that they are both

robust to slow Rayleigh fading. Also as discussed in Appendix B, **for** both estimator 1 and 2 the floor

mse over slow Rayleigh fading channel with σµ 2 k

= 1 will be approximately 2 times higher than that over

AWGN channel. We can see that such theoretical prediction are verified by simulation.

A

Proof of ∑ N−1

m=0 A m,k(τ, f e ) is real-valued and independent of f e

Proof. We first define

T m,k (f) = |P m,k (f)| 2 (29)

that is the integral kernel of (57).

From the definition of p m,k [l] shown in (5) and the relationship of p m,k [s] = p (o)

the spectrum of p m,k [s] as

m,k [s N 2

], we can write

P m,k (f) = G(f − (m − k)) G(f + f e ) e j π 2 (m−k) , f ∈ [−1, 1], (30)

15

0.2

10 −1 SNR (dB)

bias

0.15

0.1

0.05

0

Estimator 1: simulated (AWGN)

Estimator 1: simulated (SMP)

Estimator 2: simulated (AWGN)

Estimator 2: simulated (SMP)

Bolcskei estimator: simulated (AWGN)

Bolcskei estimator: simulated (SMP)

−0.05

0 10 20 30 40 50 60

SNR (dB)

mse

10 −2

10 −3

10 −4

10 −5

10 −6

10 −7

10 −8

Estimator 1: theoretical (AWGN)

Estimator 1: simulated (AWGN)

Estimator 1: theoretical (SMP)

Estimator 1: simulated (SMP)

Estimator 2: theoretical (AWGN)

Estimator 2: simulated (AWGN)

Estimator 2: theoretical (SMP)

Estimator 2: simulated (SMP)

Bolcskei estimator: simulated (AWGN)

Bolcskei estimator: simulated (SMP)

0 10 20 30 40 50 60

(a) Bias: f e = 0.0

(b) Mse: f e = 0.0

10 −1 SNR (dB)

bias

0.04

0.02

0

−0.02

mse

10 −2

Estimator 1: theoretical (AWGN)

Estimator 1: simulated (AWGN)

Estimator 1: theoretical (SMP)

Estimator 1: simulated (SMP)

Estimator 2: theoretical (AWGN)

Estimator 2: simulated (AWGN)

Estimator 2: theoretical (SMP)

Estimator 2: simulated (SMP)

Bolcskei estimator: simulated (AWGN)

Bolcskei estimator: simulated (SMP)

−0.04

−0.06

−0.08

Estimator 1: simulated (AWGN)

Estimator 1: simulated (SMP)

Estimator 2: simulated (AWGN)

Estimator 2: simulated (SMP)

Bolcskei estimator: simulated (AWGN)

Bolcskei estimator: simulated (SMP)

10 −3

−0.1

0 10 20 30 40 50 60

SNR (dB)

0 10 20 30 40 50 60

(c) Bias: f e = 0.2

(d) Mse: f e = 0.2

0.05

0

−0.05

10 −1 SNR (dB)

Estimator 1: theoretical (AWGN)

Estimator 1: simulated (AWGN)

Estimator 1: theoretical (SMP)

Estimator 1: simulated (SMP)

Estimator 2: theoretical (AWGN)

Estimator 2: simulated (AWGN)

Estimator 2: theoretical (SMP)

Estimator 2: simulated (SMP)

Bolcskei estimator: simulated (AWGN)

Bolcskei estimator: simulated (SMP)

bias

mse

10 −2

−0.1

−0.15

Estimator 1: simulated (AWGN)

Estimator 1: simulated (SMP)

Estimator 2: simulated (AWGN)

Estimator 2: simulated (SMP)

Bolcskei estimator: simulated (AWGN)

Bolcskei estimator: simulated (SMP)

−0.2

0 10 20 30 40 50 60

SNR (dB)

10 −3

0 10 20 30 40 50 60

(e) Bias: f e = 0.4

(f) Mse: f e = 0.4

0.1

0

−0.1

bias

−0.2

−0.3

−0.4

Estimator 1: simulated (AWGN)

Estimator 1: simulated (SMP)

Estimator 2: simulated (AWGN)

Estimator 2: simulated (SMP)

mse

10 0 SNR (dB)

10 −1

Estimator 1: theoretical (AWGN)

Estimator 1: simulated (AWGN)

Estimator 1: theoretical (SMP)

Estimator 1: simulated (SMP)

Estimator 2: theoretical (AWGN)

Estimator 2: simulated (AWGN)

Estimator 2: theoretical (SMP)

Estimator 2: simulated (SMP)

−0.5

0 10 20 30 40 50 60

SNR (dB)

10 −2

0 10 20 30 40 50 60

(g) Bias: f e = 0.8

16

(h) Mse: f e = 0.8

Figure 10: Bias and mse versus SNR over stationary multipath fading channel **for** different f e

1

0.8

B µ

T = 0.001

B µ

T = 0.01

B µ

T = 0.05

0.6

c µk [τ]

0.4

0.2

0

−50 −40 −30 −20 −10 0 10 20 30 40 50

τ

Figure 11: The curves of c µk [τ] **for** different fading factor B µ T

where G(f) is the prototype filter and band limited to [−1, 1] (normalized with respect to 1/T ).

Then substituting (30) into (57), we have

N−1

∑

m=0

A m,k (τ, f e ) = 1 2

∫ 1

−1

(a)

= 1 ∫ 1

2

−1

∫ 1

( N−1

)

∑

|P m,k (f)| 2 e jπ(f+fe)τ df

m=0

( N−1

)

∑

|P m,k (f − f e )| 2 e jπfτ df

m=0

( N−1

)

∑

G 2 (f − (m − k) − f e ) G 2 (f) e jπfτ df

m=0

= 1 2 −1

∫ 1

(b)

= G 2 (f) e jπfτ df =

−1

∫ 1

−1

G 2 (f) cos (πfτ) df, (31)

( ∑N−1

where (a) follows from the fact that

m=0 |P m,k(f)| 2) e jπ(f+fe)τ is periodic to f with a period 2, and

(b) follows from the fact that ∑ N−1

m=0 G2 (f − (m − k)) ≡ 2 since G 2 (f) is Nyquist pulse.

Since G ( f) is real-valued, we can conclude immediately that ∑ N−1

m=0 A m,k(τ, f e ) is real-valued and only

a function of τ.

B Derivation of explicit expressions **for** lim M→∞ M E [ ∆c k [τ 1 ] ∆c ∗ k [τ 2] ]

and lim M→∞ M E [ ∆c k [τ 1 ] ∆c k [τ 2 ] ]

First by defining e k [s, τ] = b k [s + τ] b ∗ k [s] − c k[τ], which is actually the **estimation** error of correlation

function by one sample, we have ĉ k [τ] = 1 ∑ M−1

M s=0 e k[s, τ] + c k [τ]. It will be verified soon that both the

(cross) correlation function R ek [λ, τ 1 , τ 2 ] = E [ e k [s + λ, τ 1 ] e ∗ k [s, τ 2] ] and the (cross) conjugate correlation

function ˜R ek [λ, τ 1 , τ 2 ] = E [ e k [s + λ, τ 1 ] e k [s, τ 2 ] ] are not functions of time instant s. Then based on the

17

ias

0.01

0.008

0.006

0.004

0.002

0

Estimator 1: simulated (AWGN)

Estimator 2: simulated (AWGN)

Estimator 1: simulated (B µ

T=0.001)

Estimator 2: simulated (B µ

T=0.001)

Estimator 1: simulated (B µ

T=0.01)

Estimator 2: simulated (B µ

T=0.01)

mse

10 −1 SNR (dB)

10 −2

Estimator 1: theoretical (AWGN)

Estimator 1: simulated (AWGN)

Estimator 1: theoretical (Rayleigh)

Estimator 1: simulated (B µ

T=0.001)

Estimator 1: simulated (B µ

T=0.01)

Estimator 2: theoretical (AWGN)

Estimator 2: simulated (AWGN)

Estimator 2: theoretical (Rayleigh)

Estimator 2: simulated (B µ

T=0.001)

Estimator 2: simulated (B µ

T=0.01)

−0.002

−0.004

10 −3

−0.006

−0.008

−0.01

0 10 20 30 40 50 60

SNR (dB)

10 −4

0 10 20 30 40 50 60

(a) Bias: f e = 0.0

(b) Mse: f e = 0.0

bias

0.03

0.02

0.01

0

Estimator 1: simulated (AWGN)

Estimator 2: simulated (AWGN)

Estimator 1: simulated (B µ

T=0.001)

Estimator 2: simulated (B µ

T=0.001)

Estimator 1: simulated (B µ

T=0.01)

Estimator 2: simulated (B µ

T=0.01)

mse

10 −1 SNR (dB)

10 −2

Estimator 1: theoretical (AWGN)

Estimator 1: simulated (AWGN)

Estimator 1: theoretical (Rayleigh)

Estimator 1: simulated (B µ

T=0.001)

Estimator 1: simulated (B µ

T=0.01)

Estimator 2: theoretical (AWGN)

Estimator 2: simulated (AWGN)

Estimator 2: theoretical (Rayleigh)

Estimator 2: simulated (B µ

T=0.001)

Estimator 2: simulated (B µ

T=0.01)

−0.01

−0.02

10 −3

−0.03

0 10 20 30 40 50 60

SNR (dB)

0 10 20 30 40 50 60

(c) Bias: f e = 0.2

(d) Mse: f e = 0.2

bias

0.1

0.08

0.06

0.04

0.02

0

−0.02

Estimator 1: simulated (AWGN)

Estimator 2: simulated (AWGN)

Estimator 1: simulated (B µ

T=0.001)

Estimator 2: simulated (B µ

T=0.001)

Estimator 1: simulated (B µ

T=0.01)

Estimator 2: simulated (B µ

T=0.01)

mse

10 −1 SNR (dB)

10 −2

Estimator 1: theoretical (AWGN)

Estimator 1: simulated (AWGN)

Estimator 1: theoretical (Rayleigh)

Estimator 1: simulated (B µ

T=0.001)

Estimator 1: simulated (B µ

T=0.01)

Estimator 2: theoretical (AWGN)

Estimator 2: simulated (AWGN)

Estimator 2: theoretical (Rayleigh)

Estimator 2: simulated (B µ

T=0.001)

Estimator 2: simulated (B µ

T=0.01)

−0.04

−0.06

−0.08

−0.1

0 10 20 30 40 50 60

SNR (dB)

10 −3

0 10 20 30 40 50 60

(e) Bias: f e = 0.4

(f) Mse: f e = 0.4

bias

0.1

0

−0.1

−0.2

Estimator 1: simulated (AWGN)

Estimator 2: simulated (AWGN)

Estimator 1: simulated (B µ

T=0.001)

Estimator 2: simulated (B µ

T=0.001)

Estimator 1: simulated (B µ

T=0.01)

Estimator 2: simulated (B µ

T=0.01)

mse

10 0 SNR (dB)

10 −1

Estimator 1: theoretical (AWGN)

Estimator 1: simulated (AWGN)

Estimator 1: theoretical (Rayleigh)

Estimator 1: simulated (B µ

T=0.001)

Estimator 1: simulated (B µ

T=0.01)

Estimator 2: theoretical (AWGN)

Estimator 2: simulated (AWGN)

Estimator 2: theoretical (Rayleigh)

Estimator 2: simulated (B µ

T=0.001)

Estimator 2: simulated (B µ

T=0.01)

−0.3

−0.4

−0.5

0 10 20 30 40 50 60

SNR (dB)

10 −2

0 10 20 30 40 50 60

(g) Bias: f e = 0.8

18

(h) Mse: f e = 0.8

Figure 12: Bias and mse versus SNR over Rayleigh multipath fading channel **for** different f e

definition of ∆c k [τ] = ĉ k [τ] − c k [τ] and the mixing conditions of e k [s, τ], we have

lim M E[ ∆c k [τ 1 ] ∆c ∗ k[τ 2 ] ] =

M→∞

lim

M→∞

= lim

M→∞

1

M

1

M

M−1

∑

M−1

∑

s 1 =0 s 2 =0

M−1

∑

M−1

∑

s 1 =0 s 2 =0

M−1

1 ∑ ∑s 1

= lim

M→∞ M

s 1=0

(

∑ ∞

= R ek [λ, τ 1 , τ 2 ]

=

lim M E[ ∆c k [τ 1 ] ∆c k [τ 2 ] ] =

M→∞

λ=−∞

∞∑

λ=−∞

lim

M→∞

= lim

M→∞

E [ e k [s 1 , τ 1 ] e ∗ k[s 2 , τ 2 ] ]

R ek [s 1 − s 2 , τ 1 , τ 2 ]

λ=s 1 −M+1

) (

R ek [λ, τ 1 , τ 2 ]

lim

M→∞

R ek [λ, τ 1 , τ 2 ] = S ek (0, τ 1 , τ 2 )

1

M

1

M

M−1

∑

M−1

∑

s 1=0 s 2=0

M−1

∑

M−1

∑

s 1=0 s 2=0

M−1

1 ∑ ∑s 1

= lim

M→∞ M

s 1 =0

(

∑ ∞

= ˜R ek [λ, τ 1 , τ 2 ]

=

λ=−∞

∞∑

λ=−∞

M−1

1 ∑

M

s 1 =0

1

E [ e k [s 1 , τ 1 ] e k [s 2 , τ 2 ] ]

λ=s 1 −M+1

˜R ek [s 1 − s 2 , τ 1 , τ 2 ]

) (

˜R ek [λ, τ 1 , τ 2 ]

lim

M→∞

M−1

1 ∑

M

s 1=0

˜R ek [λ, τ 1 , τ 2 ] = ˜S ek (0, τ 1 , τ 2 ), (32)

1

)

)

where

S ek (f, τ 1 , τ 2 ) def

=

˜S ek (f, τ 1 , τ 2 ) def

=

∞∑

λ=−∞

∞∑

λ=−∞

R ek [λ, τ 1 , τ 2 ] e −j2πfλ

˜R ek [λ, τ 1 , τ 2 ] e −j2πfλ . (33)

Now the work left is to calculate R ek [λ, τ 1 , τ 2 ] and ˜R ek (λ, τ 1 , τ 2 ), then to derive S ek [f, τ 1 , τ 2 ] and

˜S ek (f, τ 1 , τ 2 ).

B.1 Derivation of R ek [λ, τ 1 , τ 2 ] and ˜R ek (λ, τ 1 , τ 2 )

Now we need to derive the explicit expressions **for** R ek [λ, τ 1 , τ 2 ] and ˜R ek (λ, τ 1 , τ 2 ). Recalling that e k [s, τ] =

b k [s + τ] b ∗ k [s] − c k[τ], we have

R ek [λ, τ 1 , τ 2 ] = E [ e k [s + λ, τ 1 ] e ∗ k[s, τ 2 ] ]

= E [ (b k [s + λ + τ 1 ] b ∗ k[s + λ] − c k [τ 1 ]) (b ∗ k[s + τ 2 ] b k [s] − c ∗ k[τ 2 ]) ]

= E [ b k [s + λ + τ 1 ] b ∗ k[s + λ] b ∗ k[s + τ 2 ] b k [s] ] − c k [τ 1 ] c ∗ k[τ 2 ]

˜R ek [λ, τ 1 , τ 2 ] = E [ e k [s + λ, τ 1 ] e k [s, τ 2 ] ]

= E [ (b k [s + λ + τ 1 ] b ∗ k[s + λ] − c k [τ 1 ]) (b k [s + τ 2 ] b ∗ k[s] − c k [τ 2 ]) ]

= E [ b k [s + λ + τ 1 ] b ∗ k[s + λ] b k [s + τ 2 ] b ∗ k[s] ] − c k [τ 1 ] c k [τ 2 ]. (34)

19

It will be verified soon that the fourth order statistics E [ b k [s + τ 3 ] b k [s + τ 2 ] b ∗ k [s + τ 1] b ∗ k [s]] is not a

function of s, thus the fourth order statistics in the expressions of R ek [λ, τ 1 , τ 2 ] and ˜R ek [λ, τ 1 , τ 2 ] can be

deferred from E [ b k [s + τ 3 ] b k [s + τ 2 ] b ∗ k [s + τ 1] b ∗ k [s]] directly. By using (7), we have

E [ b k [s + τ 3 ] b k [s + τ 2 ] b ∗ k[s + τ 1 ] b ∗ k[s] ]

[(

= E

×

×

×

×

×

×

e jπf e(s+τ 3 )

∞∑

n=−∞

(

N−1

∑

m=0

w m µ m [(s + τ 3 ) N 2 ]

(

)

a R m[n] p m,k [s + τ 3 − 2n] + j (−1) (m−k) a I m[n] p m,k [s + τ 3 − 2n − 1]

e jπf e(s+τ 2 )

∞∑

n=−∞

(

N−1

∑

m=0

w m µ m [(s + τ 2 ) N 2 ]

(

)

a R m[n] p m,k [s + τ 2 − 2n] + j (−1) (m−k) a I m[n] p m,k [s + τ 2 − 2n − 1]

e −jπf e(s+τ 1 )

∞∑

n=−∞

(

N−1

∑

m=0

w m µ ∗ m[(s + τ 1 ) N 2 ]

(

)

a R m[n] p ∗ m,k[s + τ 1 − 2n] − j (−1) (m−k) a I m[n] p ∗ m,k[s + τ 1 − 2n − 1]

e −jπfes N−1

= e jπf e(τ 3 +τ 2 −τ 1 )

×

∞∑

∑

w m µ ∗ m[s N 2 ]

m=0

n 1 ,n 2 ,n 3 ,n 4 =−∞

N−1

∑

m 1,m 2,m 3,m 4=0

E

∞ ∑

n=−∞

+ ν k [s + τ 3 ]

+ ν k [s + τ 2 ]

+ ν ∗ k[s + τ 1 ]

)]

(

)

a R m[n] p ∗ m,k[s − 2n] − j (−1) (m−k) a I m[n] p ∗ m,k[s − 2n − 1] + νk[s]

∗

w m1 w m2 w m3 w m4 E

[µ m1 [(s + τ 3 ) N 2 ] µ m 2

[(s + τ 2 ) N 2 ] µ∗ m 3

[(s + τ 1 ) N 2 ] µ∗ m 4

[s N ]

2 ]

[(

)

a R m 1

[n 1 ] p m1 ,k[s + τ 3 − 2n 1 ] + j (−1) (m1−k) a I m 1

[n 1 ] p m1 ,k[s + τ 3 − 2n 1 − 1]

(

)

× a R m 2

[n 2 ] p m2,k[s + τ 2 − 2n 2 ] + j (−1) (m2−k) a I m 2

[n 2 ] p m2,k[s + τ 2 − 2n 2 − 1]

(

)

× a R m 3

[n 3 ] p ∗ m 3,k[s + τ 1 − 2n 3 ] − j (−1) (m3−k) a I m 3

[n 3 ] p ∗ m 3,k[s + τ 1 − 2n 3 − 1]

(

)]

× a R m 4

[n 4 ] p ∗ m 4,k[s − 2n 4 ] − j (−1) (m4−k) a I m 4

[n 4 ] p ∗ m 4,k[s − 2n 4 − 1]

+ σ2 N−1

∑

ν

2 ejπf eτ 2

p t [τ 3 − τ 1 ] wm 2 N

∞

c µm [τ 2

2 ] ∑

p m,k [n + τ 2 ] p ∗ m,k[n]

m=0

n=−∞

+ σ2 N−1

∑

ν

2 ejπfe(τ2−τ1) p t [τ 3 ] wm 2 c µm [(τ 2 − τ 1 ) N 2 ]

m=0

+ σ2 N−1

∑

ν

2 ejπf eτ 3

p t [τ 2 − τ 1 ] wm 2 N

∞

c µm [τ 3

2 ] ∑

m=0

n=−∞

+ σ2 N−1

∑

ν

2 ejπf e(τ 3 −τ 1 ) p t [τ 2 ] wm 2 c µm [(τ 3 − τ 1 ) N 2 ]

m=0

∞ ∑

n=−∞

p m,k [n + τ 2 − τ 1 ] p ∗ m,k[n]

p m,k [n + τ 3 ] p ∗ m,k[n]

∞

∑

n=−∞

p m,k [n + τ 3 − τ 1 ] p ∗ m,k[n]

+ E [ν k [s + τ 3 ] ν k [s + τ 2 ] ν ∗ k[s + τ 1 ] ν ∗ k[s]] . (35)

)

)

)

20

Then we can continue the summation in the first term in the right-hand side of (35) as

=

N−1

∑

m 1,m 2,m 3,m 4=0

×

∞∑

n 1 ,n 2 ,n 3 ,n 4 =−∞

w m1 w m2 w m3 w m4 E

[µ m1 [(s + τ 3 ) N 2 ] µ m 2

[(s + τ 2 ) N 2 ] µ∗ m 3

[(s + τ 1 ) N 2 ] µ∗ m 4

[s N ]

2 ]

[(

)

E a R m 1

[n 1 ] p m1 ,k[s + τ 3 − 2n 1 ] + j (−1) (m1−k) a I m 1

[n 1 ] p m1 ,k[s + τ 3 − 2n 1 − 1]

(

)

× a R m 2

[n 2 ] p m2,k[s + τ 2 − 2n 2 ] + j (−1) (m2−k) a I m 2

[n 2 ] p m2,k[s + τ 2 − 2n 2 − 1]

(

)

× a R m 3

[n 3 ] p ∗ m 3,k[s + τ 1 − 2n 3 ] − j (−1) (m3−k) a I m 3

[n 3 ] p ∗ m 3,k[s + τ 1 − 2n 3 − 1]

(

)]

× a R m 4

[n 4 ] p ∗ m 4,k[s − 2n 4 ] − j (−1) (m4−k) a I m 4

[n 4 ] p ∗ m 4,k[s − 2n 4 − 1]

( [ (a

R

E k [n] ) ] 4

− 3 N−1 ∑

wm 4) 4 E

[µ m [(s + τ 3 ) N 2 ] µ m[(s + τ 2 ) N 2 ] µ∗ m[(s + τ 1 ) N 2 ] µ∗ m[s N ]

2 ] m=0

×

∞∑

n=−∞

N−1

∑

+ 1 4

[

∑ ∞

×

×

p m,k [n + τ 3 ] p m,k [n + τ 2 ] p ∗ m,k[n + τ 1 ] p ∗ m,k[n]

N−1

∑

m 1=0 m 2=0

n=−∞

[ ∞

∑

n=−∞

N−1

∑

+ 1 4

(

∑ ∞

×

m 1 =0 m 2 =0

n=−∞

+ 1 4

(

∑ ∞

×

N−1

∑

wm 2 1

wm 2 2

E

[µ m1 [(s + τ 3 ) N 2 ] µ m 1

[(s + τ 2 ) N 2 ] µ∗ m 2

[(s + τ 1 ) N 2 ] µ∗ m 2

[s N ]

2 ]

(p m1,k[2n + τ 3 ] p m1,k[2n + τ 2 ] − p m1,k[2n + τ 3 + 1] p m1,k[2n + τ 2 + 1])

(

p

∗

m2 ,k[2n + τ 1 ] p ∗ m 2 ,k[2n] − p ∗ m 2 ,k[2n + τ 1 + 1] p ∗ m 2 ,k[2n + 1] )]

N−1

∑

m 1 =0 m 2 =0

n=−∞

wm 2 1

wm 2 2

E

[µ m1 [(s + τ 3 ) N 2 ] µ m 2

[(s + τ 2 ) N 2 ] µ∗ m 2

[(s + τ 1 ) N 2 ] µ∗ m 1

[s N ]

2 ]

p m1 ,k[n + τ 3 ] p ∗ m 1 ,k[n]

N−1

∑

) ( ∞

∑

n=−∞

p m2 ,k[n + τ 2 − τ 1 ] p ∗ m 2 ,k[n]

wm 2 1

wm 2 2

E

[µ m1 [(s + τ 3 ) N 2 ] µ m 2

[(s + τ 2 ) N 2 ] µ∗ m 1

[(s + τ 1 ) N 2 ] µ∗ m 2

[s N ]

2 ]

p m1 ,k[n + τ 3 − τ 1 ] p ∗ m 1 ,k[n]

) ( ∞

∑

n=−∞

p m2 ,k[n + τ 2 ] p ∗ m 2 ,k[n]

We can also rewrite the last term in the right-hand side of (35) as

)

)

]

. (36)

E [ν k [s + τ 3 ] ν k [s + τ 2 ] ν ∗ k[s + τ 1 ] ν ∗ k[s]] = = σ 4 ν (p t [τ 1 − τ 3 ] p t [τ 2 ] + p t [τ 3 ]p t [τ 1 − τ 2 ]) . (37)

21

Then by substituting (36) and (37) into (35), we have

E [ b k [s + τ 3 ] b k [s + τ 2 ] b ∗ k[s + τ 1 ] b ∗ k[s] ]

= e jπfe(τ3+τ2−τ1) { (

E

×

∞∑

n=−∞

N−1

∑

+ 1 4

[

∑ ∞

×

×

[ (a

R

k [n] ) ] 4

− 3 N−1 ∑

wm 4) 4 E

[µ m [(s + τ 3 ) N 2 ] µ m[(s + τ 2 ) N 2 ] µ∗ m[(s + τ 1 ) N 2 ] µ∗ m[s N ]

2 ] m=0

p m,k [n + τ 3 ] p m,k [n + τ 2 ] p ∗ m,k[n + τ 1 ] p ∗ m,k[n]

N−1

∑

m 1 =0 m 2 =0

n=−∞

[ ∞

∑

n=−∞

N−1

∑

+ 1 4

(

∑ ∞

×

m 1=0 m 2=0

n=−∞

n=−∞

wm 2 1

wm 2 2

E

[µ m1 [(s + τ 3 ) N 2 ] µ m 1

[(s + τ 2 ) N 2 ] µ∗ m 2

[(s + τ 1 ) N 2 ] µ∗ m 2

[s N ]

2 ]

(p m1 ,k[2n + τ 3 ] p m1 ,k[2n + τ 2 ] − p m1 ,k[2n + τ 3 + 1] p m1 ,k[2n + τ 2 + 1])

(

p

∗

m2,k[2n + τ 1 ] p ∗ m 2,k[2n] − p ∗ m 2,k[2n + τ 1 + 1] p ∗ m 2,k[2n + 1] )]

N−1

∑

wm 2 1

wm 2 2

E

[µ m1 [(s + τ 3 ) N 2 ] µ m 2

[(s + τ 2 ) N 2 ] µ∗ m 2

[(s + τ 1 ) N 2 ] µ∗ m 1

[s N ]

2 ]

p m1,k[n + τ 3 ] p ∗ m 1,k[n]

) ( ∞

∑

n=−∞

p m2,k[n + τ 2 − τ 1 ] p ∗ m 2,k[n]

+ 1 N−1

∑

N−1

∑

wm 2 4

1

wm 2 2

E

[µ m1 [(s + τ 3 ) N 2 ] µ m 2

[(s + τ 2 ) N 2 ] µ∗ m 1

[(s + τ 1 ) N 2 ] µ∗ m 2

[s N ]

2 ] m 1=0 m 2=0

( ∞

) (

∑

∞

)}

∑

× p m1,k[n + τ 3 − τ 1 ] p ∗ m 1,k[n] p m2,k[n + τ 2 ] p ∗ m 2,k[n]

n=−∞

+ σ2 N−1

∑

ν

2 ejπfeτ2 p t [τ 3 − τ 1 ] wm 2 N

∞

c µm [τ 2

2 ] ∑

m=0

n=−∞

+ σ2 N−1

∑

ν

2 ejπf e(τ 2 −τ 1 ) p t [τ 3 ] wm 2 c µm [(τ 2 − τ 1 ) N 2 ]

m=0

+ σ2 N−1

∑

∞

ν

2 ejπf eτ 3

p t [τ 2 − τ 1 ] wm 2 N

c µm [τ 3

2 ] ∑

m=0

n=−∞

+ σ2 N−1

∑

ν

2 ejπfe(τ3−τ1) p t [τ 2 ] wm 2 c µm [(τ 3 − τ 1 ) N 2 ]

m=0

p m,k [n + τ 2 ] p ∗ m,k[n]

∞ ∑

n=−∞

)

p m,k [n + τ 2 − τ 1 ] p ∗ m,k[n]

p m,k [n + τ 3 ] p ∗ m,k[n]

∞ ∑

n=−∞

p m,k [n + τ 3 − τ 1 ] p ∗ m,k[n]

+ σ 4 ν (p t [τ 1 − τ 3 ] p t [τ 2 ] + p t [τ 3 ]p t [τ 1 − τ 2 ]) . (38)

]

Then we can see that the only term related to time instant s in the expression of E [ b k [s + τ 3 ] b k [s +

τ 2 ] b ∗ k [s+τ 1] b ∗ k [s]] is E [ µ m1 [(s + τ 3 ) N 2 ] µ m 1

[(s + τ 2 ) N 2 ] µ∗ m 2

[(s + τ 1 ) N 2 ] µ∗ m 2

[s N 2 ]] . We will treat stationary

and time-variant channel separately.

22

B.1.1

Stationary channel

For stationary channel, based on (38), we immediately have

E [ b k [s + τ 3 ] b k [s + τ 2 ] b ∗ k[s + τ 1 ] b ∗ k[s] ]

{ ( [ (a

= e jπfe(τ3+τ2−τ1) R

E k [n] ) ] 4

− 3 N−1 ∑ ∑

|w m µ m |

4) ∞ 4 p m,k [n + τ 3 ] p m,k [n + τ 2 ] p ∗ m,k[n + τ 1 ] p ∗ m,k[n]

m=0

n=−∞

[

+ 1 N−1 ∑ ∞

]

∑

m µ m )

4

m=0(w 2 (p m,k [2n + τ 3 ] p m,k [2n + τ 2 ] − p m,k [2n + τ 3 + 1] p m,k [2n + τ 2 + 1])

×

n=−∞

[ N−1 ∑

m=0(w m µ ∗ m) 2 ∞ ∑

n=−∞

(

p

∗

m,k [2n + τ 1 ] p ∗ m,k[2n] − p ∗ m,k[2n + τ 1 + 1] p ∗ m,k[2n + 1] )]

(

+ 1 N−1

) ( ∑ ∑

∞ N−1

)

∑ ∑

∞

|w m µ m | 2 p m,k [n + τ 3 ] p ∗

4

m,k[n] |w m µ m | 2 p m,k [n + τ 2 − τ 1 ] p ∗ m,k[n]

m=0

n=−∞

m=0

n=−∞

(

+ 1 N−1 ∑ ∞

) (

∑

N−1 ∑ ∞

)}

∑

|w m µ m | 2 p m,k [n + τ 3 − τ 1 ] p ∗

4

m,k[n] |w m µ m | 2 p m,k [n + τ 2 ] p ∗ m,k[n]

m=0

n=−∞

m=0

n=−∞

+ σ2 N−1

∑

∑

∞ ν

2 ejπfeτ2 p t [τ 3 − τ 1 ] |w m µ m | 2 p m,k [n + τ 2 ] p ∗ m,k[n]

m=0

n=−∞

+ σ2 N−1

∑

∑

∞ ν

2 ejπf e(τ 2 −τ 1 ) p t [τ 3 ] |w m µ m | 2

m=0

n=−∞

+ σ2 N−1

∑

∑

∞ ν

2 ejπfeτ3 p t [τ 2 − τ 1 ] |w m µ m | 2

m=0

n=−∞

+ σ2 N−1

∑

∑

∞ ν

2 ejπf e(τ 3 −τ 1 ) p t [τ 2 ] |w m µ m | 2

m=0

n=−∞

p m,k [n + τ 2 − τ 1 ] p ∗ m,k[n]

p m,k [n + τ 3 ] p ∗ m,k[n]

p m,k [n + τ 3 − τ 1 ] p ∗ m,k[n]

+ σ 4 ν (p t [τ 1 − τ 3 ] p t [τ 2 ] + p t [τ 3 ]p t [τ 1 − τ 2 ]) . (39)

Then substituting (8) and (39) into (34), we have

R ek [λ, τ 1 , τ 2 ]

{ ( [

= e jπf e(τ 1 −τ 2 ) (a

R

E k [n] ) ] 4

− 3 N−1 ∑ ∑

|w m µ m |

4) ∞ 4 p m,k [n + λ + τ 1 ] p ∗ m,k[n + λ] p ∗ m,k[n + τ 2 ] p m,k [n]

m=0

n=−∞

[

+ 1 N−1 ∑ ∞

]

∑

m µ m )

4

m=0(w 2 (p m,k [2n + τ 1 ] p m,k [2n − λ] − p m,k [2n + τ 1 + 1] p m,k [2n − λ + 1])

×

n=−∞

[ N−1 ∑

m=0(w m µ ∗ m) 2 ∞ ∑

n=−∞

(

p

∗

m,k [2n + τ 2 − λ] p ∗ m,k[2n] − p ∗ m,k[2n + τ 2 − λ + 1] p ∗ m,k[2n + 1] )]

(

+ 1 N−1

) ( ∑ ∑

∞ N−1 ∑ ∑

∞

|w m µ m | 2 p m,k [n + λ + τ 1 − τ 2 ] p ∗

4

m,k[n] |w m µ m | 2

m=0

n=−∞

m=0

+ σ2 N−1

∑

∑

∞ ν

2 e−jπf eλ p t [λ + τ 1 − τ 2 ] |w m µ m | 2 p m,k [n − λ] p ∗ m,k[n]

m=0

+ σ2 N−1

∑

∑ ∞

ν

2 ejπfe(λ+τ1−τ2) p t [λ] |w m µ m | 2

m=0

n=−∞

n=−∞

p m,k [n + λ + τ 1 − τ 2 ] p ∗ m,k[n]

n=−∞

p m,k [n − λ] p ∗ m,k[n]

+ σ 4 ν p t [λ + τ 1 − τ 2 ] p t [λ], (40)

23

)}

and

˜R ek [λ, τ 1 , τ 2 ]

{ ( [

= e jπf e(τ 1 +τ 2 ) (a

R

E k [n] ) ] 4

− 3 N−1 ∑ ∑

|w m µ m |

4) ∞ 4 p m,k [n + λ + τ 1 ] p m,k [n + τ 2 ] p ∗ m,k[n + λ] p ∗ m,k[n]

m=0

n=−∞

[

+ 1 N−1 ∑ ∞

]

∑

m µ m )

4

m=0(w 2 (p m,k [2n + λ + τ 1 ] p m,k [2n + τ 2 ] − p m,k [2n + λ + τ 1 + 1] p m,k [2n + τ 2 + 1])

×

n=−∞

[ N−1 ∑

m=0(w m µ ∗ m) 2 ∞ ∑

n=−∞

(

p

∗

m,k [2n + λ] p ∗ m,k[2n] − p ∗ m,k[2n + λ + 1] p ∗ m,k[2n + 1] )]

(

+ 1 N−1

) ( ∑ ∑

∞ N−1 ∑ ∑

∞

|w m µ m | 2 p m,k [n + λ + τ 1 ] p ∗

4

m,k[n] |w m µ m | 2

m=0

n=−∞

m=0

+ σ2 N−1

∑

∑

∞ ν

2 ejπf e(τ 2 −λ) p t [λ + τ 1 ] |w m µ m | 2 p m,k [n + τ 2 − λ] p ∗ m,k[n]

m=0

n=−∞

+ σ2 N−1

∑

∑

∞ ν

2 ejπfe(λ+τ1) p t [τ 2 − λ] |w m µ m | 2

m=0

n=−∞

p m,k [n + λ + τ 1 ] p ∗ m,k[n]

n=−∞

p m,k [n + τ 2 − λ] p ∗ m,k[n]

+ σ 4 ν p t [λ + τ 1 ] p t [λ − τ 2 ]. (41)

)}

B.1.2

We can see that both R ek [λ, τ 1 , τ 2 ] and ˜R ek [λ, τ 1 , τ 2 ] are not a function of s.

Time-variant channel

For time-variant channel, we may also assume that µ k [l] is a zero-mean complex-valued Gaussian process,

and the real and imaginary parts are i.i.d., i.e. E [Re {µ k [l + τ]} Re {µ k [l]}] = E [Im {µ k [l + τ]} Im {µ k [l]}] =

c µk [τ]/2 and E [Re {µ k [l + τ]} Im {µ k [l]}] = 0, ∀ l, τ. Since the shaping filter g[l] is band-limited to [−1, 1],

p m,k [s] is non-zero only if −2 ≤ m − k ≤ 1 **for** 0 ≤ f e < 1. Thus we only need to consider the terms that

satisfying |m − k| ≤ 2 in (38). Then by using the approximation of µ m1 [l] ≃ µ k [l] and µ m2 [l] ≃ µ k [l], we

have

E

[µ m1 [(s + τ 3 ) N 2 ] µ m 1

[(s + τ 2 ) N 2 ] µ∗ m 2

[(s + τ 1 ) N 2 ] µ∗ m 2

[s N ]

2 ]

≃ E

[µ k [(s + τ 3 ) N 2 ] µ k[(s + τ 2 ) N 2 ] µ∗ k[(s + τ 1 ) N 2 ] µ∗ k[s N ]

2 ]

= c µk [(τ 3 − τ 1 ) N 2 ] c N

µ k

[τ 2

2 ] + c N

µ k

[τ 3

2 ] c µ k

[(τ 2 − τ 1 ) N 2 ]

≃ 2 σµ 4 k

,

where the last approximation flows from the fact that c µk [l] ≃ σµ 2 k

**for** small value of l over slow fading

channel.

Actually the approximation of c µk [l] ≃ σµ 2 k

is reasonable since the bandwidth of µ[s N 2

] is much

narrower than p m,k [s] **for** slow fading channel.

24

Then substituting this result into (38), we have

E [ b k [s + τ 3 ] b k [s + τ 2 ] b ∗ k[s + τ 1 ] b ∗ k[s] ]

≃ 2 σ 4 µ k

e jπf e(τ 3 +τ 2 −τ 1 )

{ (

E

+ 1 4

[ N−1 ∑

m=0

[ N−1 ∑

× wm

2

m=0

(

+ 1 N−1 ∑

4

m=0

(

+ 1 N−1 ∑

4

m=0

+ σ2 ν σ 2 µ k

2

+ σ2 ν σ 2 µ k

2

+ σ2 ν σ 2 µ k

2

+ σ2 ν σ 2 µ k

2

∞∑

wm

2 n=−∞

∞∑

n=−∞

∞∑

wm

2 n=−∞

wm

2 n=−∞

[ (a

R

k [n] ) ] 4

− 3 N−1 ∑

wm

4) 4 m=0

∞∑

n=−∞

p m,k [n + τ 3 ] p m,k [n + τ 2 ] p ∗ m,k[n + τ 1 ] p ∗ m,k[n]

(p m,k [2n + τ 3 ] p m,k [2n + τ 2 ] − p m,k [2n + τ 3 + 1] p m,k [2n + τ 2 + 1])

(

p

∗

m,k [2n + τ 1 ] p ∗ m,k[2n] − p ∗ m,k[2n + τ 1 + 1] p ∗ m,k[2n + 1] )]

∞∑

p m,k [n + τ 3 ] p ∗ m,k[n]

N−1

∑

e jπf eτ 2

p t [τ 3 − τ 1 ]

N−1

∑

e jπfe(τ2−τ1) p t [τ 3 ]

N−1

∑

e jπfeτ3 p t [τ 2 − τ 1 ]

N−1

∑

e jπf e(τ 3 −τ 1 ) p t [τ 2 ]

) ( N−1

p m,k [n + τ 3 − τ 1 ] p ∗ m,k[n]

∞∑

wm

2

m=0 n=−∞

∞∑

wm

2

m=0 n=−∞

∞∑

wm

2

m=0 n=−∞

∞∑

wm

2

m=0 n=−∞

∞∑

∑

wm

2

m=0 n=−∞

) ( N−1 ∑ ∞∑

wm

2

m=0 n=−∞

p m,k [n + τ 2 ] p ∗ m,k[n]

p m,k [n + τ 2 − τ 1 ] p ∗ m,k[n]

p m,k [n + τ 3 ] p ∗ m,k[n]

p m,k [n + τ 3 − τ 1 ] p ∗ m,k[n]

p m,k [n + τ 2 − τ 1 ] p ∗ m,k[n]

p m,k [n + τ 2 ] p ∗ m,k[n]

+ σ 4 ν (p t [τ 1 − τ 3 ] p t [τ 2 ] + p t [τ 3 ]p t [τ 1 − τ 2 ]) . (42)

)

]

)}

Then substituting (8) and (42) into (34), we have

R ek [λ, τ 1 , τ 2 ]

= 2 σ 4 µ k

e jπf e(τ 1 −τ 2 )

{ (

E

+ 1 4

[ N−1 ∑

m=0

[ N−1 ∑

× wm

2

m=0

(

+ 1 N−1 ∑

4

m=0

+ σ4 µ k

∞∑

wm

2 n=−∞

∞∑

n=−∞

[ (a

R

k [n] ) ] 4

− 3 N−1 ∑

wm

4) 4 m=0

∞∑

n=−∞

p m,k [n + τ 1 + λ] p ∗ m,k[n + λ] p ∗ m,k[n + τ 2 ] p m,k [n]

(p m,k [2n + τ 1 ] p m,k [2n − λ] − p m,k [2n + τ 1 + 1] p m,k [2n − λ + 1])

(

p

∗

m,k [2n + τ 2 − λ] p ∗ m,k[2n] − p ∗ m,k[2n + τ 2 − λ + 1] p ∗ m,k[2n + 1] )]

∞∑

wm

2 n=−∞

∑ ∞∑

wm

2

m=0 n=−∞

4 ejπfe(τ1−τ2) ( N−1

+ σ2 ν σ 2 µ k

2

+ σ2 ν σ 2 µ k

2

p m,k [n + λ + τ 1 − τ 2 ] p ∗ m,k[n]

N−1

∑

e −jπfeλ p t [λ + τ 1 − τ 2 ]

N−1

∑

e jπf e(λ+τ 1 −τ 2 ) p t [λ]

) ( N−1

p m,k [n + τ 1 ] p ∗ m,k[n]

wm

2

m=0 n=−∞

∞∑

wm

2

m=0 n=−∞

∞∑

∞∑

∑

wm

2

m=0 n=−∞

) ( N−1 ∑ ∞∑

wm

2

m=0 n=−∞

p m,k [n − λ] p ∗ m,k[n]

p m,k [n + λ + τ 1 − τ 2 ] p ∗ m,k[n]

]

p m,k [n − λ] p ∗ m,k[n]

)}

p m,k [n − τ 2 ] p ∗ m,k[n]

+ σ 4 ν p t [λ + τ 1 − τ 2 ] p t [λ], (43)

25

)

and

˜R ek [λ, τ 1 , τ 2 ] = E [ b k [s + λ + τ 1 ] b ∗ k[s + λ] b k [s + τ 2 ] b ∗ k[s] ] − c k [τ 1 ] c k [τ 2 ]

{ ( [

= 2 σµ 4 k

e jπf e(τ 1 +τ 2 ) (a

R

E k [n] ) ] 4

−

4) 3 N−1 ∑ ∞∑

wm

4 p m,k [n + λ + τ 1 ] p m,k [n + τ 2 ] p ∗ m,k[n + λ] p ∗ m,k[n]

m=0 n=−∞

[

+ 1 N−1

]

∑ ∞∑

wm

2 (p m,k [2n + λ + τ 1 ] p m,k [2n + τ 2 ] − p m,k [2n + λ + τ 1 + 1] p m,k [2n + τ 2 + 1])

4

m=0 n=−∞

[ N−1 ∑ ∞∑

× wm

2 (

p

∗

m,k [2n + λ] p ∗ m,k[2n] − p ∗ m,k[2n + λ + 1] p ∗ m,k[2n + 1] )]

m=0 n=−∞

(

+ 1 N−1

) ( ∑ ∞∑

N−1

)}

∑ ∞∑

wm

2 p m,k [n + λ + τ 1 ] p ∗

4

m,k[n] wm

2 p m,k [n + τ 2 − λ] p ∗ m,k[n]

m=0 n=−∞

m=0 n=−∞

( N−1

) (

+ σ4 µ ∑ ∞∑

N−1

)

∞∑

k

4 ejπf e(τ 1 +τ 2 )

wm

2 p m,k [n + τ 1 ] p ∗ m,k[n] wm

2 p m,k [n + τ 2 ] p ∗ m,k[n]

m=0 n=−∞

n=−∞

+ σ2 ν σ 2 µ k

2

+ σ2 ν σ 2 µ k

2

N−1

∑

e jπf e(τ 2 −λ) p t [λ + τ 1 ]

N−1

∑

e jπfe(λ+τ1) p t [τ 2 − λ]

∞∑

wm

2

m=0 n=−∞

∞∑

wm

2

m=0 n=−∞

∑

m=0

p m,k [n + τ 2 − λ] p ∗ m,k[n]

p m,k [n + λ + τ 1 ] p ∗ m,k[n]

+ σ 4 νp t [λ + τ 1 ]p t [λ − τ 2 ]. (44)

We can see that similar to the case **for** stationary channel, both R ek [λ, τ 1 , τ 2 ] and ˜R ek [λ, τ 1 , τ 2 ] are

independent of s. We also note that there exists a term independent of λ in both R ek [λ, τ 1 , τ 2 ] and

˜R ek [λ, τ 1 , τ 2 ], which means a sharp peek in the spectrum. While such sharp peak will be cancelled out.

For estimator 1, only the term **for** τ 1 = τ 2 = 2 is used and we have

( N−1

) ( ∑ ∞∑

N−1

)

∑ ∞∑

e jπf e(τ 1 −τ 2 )

wm

2 p m,k [n + τ 1 ] p ∗ m,k[n] wm

2 p m,k [n − τ 2 ] p ∗ m,k[n]

m=0 n=−∞

m=0 n=−∞

{

( N−1

) (

∞∑

N−1

)}

∞∑

− Re e jπfe(τ1+τ2−2) p m,k [n + τ 1 ] p ∗ m,k[n]

p m,k [n + τ 2 ] p ∗ m,k[n]

N−1

∑

=

∣

1

=

∣2

1

=

∣2

(a)

=

(b)

=

∞∑

wm

2

m=0 n=−∞

1

∣2

1

∣2

1

=

∣2

= 0,

∫ 1

N−1

∑

−1 m=0

∫ 1

N−1

∑

−1 m=0

∫ 1

N−1

∑

−1 m=0

∫ 1

N−1

∑

−1 m=0

∫ 1 N−1

∑

−1 m=0

∑

wm

2

m=0 n=−∞

p m,k [n + 2] p ∗ m,k[n]

∣

wm 2 |P m,k (f)| 2 e j2πf df

∣

wm 2 |P m,k (f)| 2 e j2πf df

∣

wm 2 |P m,k (f)| 2 e j2πf df

∣

wm 2 |P m,k (f)| 2 e j2πf df

∣

wm 2 |P m,k (f)| 2 e j2πf df

∣

2

2

2

2

2

2

⎧

⎨

− Re

⎩ ej2πf e

⎧(

⎨

1

− Re

⎩ 2

⎧(

⎨

1

− Re

⎩ 2

⎧(

⎨

1

− Re

⎩ 2

1

−

∣2

1

−

∣2

∫ 1

∫ 1

( N−1 ∑

m=0

N−1

∑

−1 m=0

∫ 1

N−1

∑

−1 m=0

∫ 1

N−1

∑

−1 m=0

∫ 1 N−1

∑

−1 m=0

N−1

∑

−1 m=0

26

wm

2 n=−∞

∑

wm

2

m=0 n=−∞

∞∑

) 2

⎫

⎬

p m,k [n + 2] p ∗ m,k[n]

⎭

) 2

⎫

⎬

wm 2 |P m,k (f)| 2 e j2π(f+fe/2) df

⎭

) 2

⎫

⎬

wm 2 |P m,k (f − f e /2)| 2 e j2πf df

⎭

) 2

⎫

⎬

wmG 2 2 (f − (m − k) − f e /2) G 2 (f + f e /2) e j2πf df

⎭

wmG 2 2 (f − (m − k) − f e /2) G 2 (f + f e /2) e j2πf df

∣

2

wmG 2 2 (f − (m − k)) G 2 (f + f e ) e j2πf df

∣

2

where (a) follows from the expression of P m,k (f) in (30), and (b) follows from the fact that the definite

integral ∫ 1

−1 G2 (f − (m − k) − f e /2) G 2 (f + f e /2) e j2πf df is real-valued ∀ m ∈ Z, |f e | < 1.

Then we can see that the terms independent of λ are cancelled out and we need not consider them.

Through similar procedure, it can be shown that the terms independent of λ don’t affect the asymptotical

mse of estimator 2 also.

B.2 Derivation of S ek [f, τ 1 , τ 2 ] and ˜S ek (f, τ 1 , τ 2 )

Now we are ready to calculate S ek (f, τ 1 , τ 2 ). We also treat stationary and time-variant channel separately.

B.2.1

Stationary channel

Now we will calculate S ek [f, τ 1 , τ 2 ] and ˜S ek (f, τ 1 , τ 2 ) based on the expressions of R ek [λ, τ 1 , τ 2 ] and ˜R ek (λ, τ 1 , τ 2 )

over stationary channel (shown in (40) and (41) respectively).

We define

∞∑

B m,k (τ, f e ) = (p m,k [2n + τ] p m,k [2n] − p m,k [2n + τ + 1] p m,k [2n + 1]) . (45)

and

n=−∞

Then it is proven in Appendix that B m,k (τ, f e ) can be expressed in **frequency** domain as

B m,k (τ, f e ) = 1 2

∫ 1

−1

P m,k (f) P m,k (1 − f) e jπfτ df. (46)

This **for**mula will be used to calculate both S ek [f, τ 1 , τ 2 ] and ˜S ek (f, τ 1 , τ 2 ).

Derivation of S ek [f, τ 1 , τ 2 ]:

First we calculate S ek [f, τ 1 , τ 2 ]. By using Parseval’s relation, we have

= 1 8

= 1 8

∞∑

n=−∞

∫ 1

−1

∫ 1

−1

p m,k [n + λ + τ 1 ] p ∗ m,k[n + λ] p ∗ m,k[n + τ 2 ] p m,k [n]

(e jπfλ ∫ 1

(∫ 1

−1

∞∑

n=−∞

∫ 1

) (∫ 1

) ∗

P m,k (f 1 ) Pm,k(f ∗ 1 − f) e jπf 1τ 1

df 1 P m,k (f 2 ) Pm,k(f ∗ 2 − f) e jπf 2τ 2

df 2 df

−1

−1

)

P m,k (f 1 ) Pm,k(f ∗ 1 − f) Pm,k(f ∗ 2 ) P m,k (f 2 − f) e jπ(f 1τ 1 −f 2 τ 2 ) df 1 df 2 e jπfλ df,

−1

∞∑

n=−∞

p m,k [n − λ] p ∗ m,k[n] = 1 2

p m,k [n + λ + τ 1 − τ 2 ] p ∗ m,k[n] = 1 2

∫ 1

−1

∫ 1

Also by noting that G 2 (f) = ∑ ∞

τ=−∞ p t[τ] e −jπfτ , we have

p t [λ] = 1 2

p t [λ + τ 1 − τ 2 ] = 1 2

∫ 1

−1

∫ 1

−1

−1

G 2 (f) e jπfλ df

|P m,k (f)| 2 e −jπfλ df

(

|P m,k (f)| 2 e jπf(τ 1−τ 2 ) ) e jπfλ df.

(

G 2 (f) e jπf(τ1−τ2)) e jπfλ df.

Based on the definition of B m,k (τ, f e ) shown in (45), we have

∞∑

n=−∞

∞∑

(p m,k [2n + τ 1 ] p m,k [2n − λ] − p m,k [2n + τ 1 + 1] p m,k [2n − λ + 1]) = (−1) λ B m,k (τ 1 + λ, f e )

(

p

∗

m,k [2n + τ 2 − λ] p ∗ m,k[2n] − p ∗ m,k[2n + τ 2 − λ + 1] p ∗ m,k[2n + 1] ) = B ∗ m,k(τ 2 − λ, f e ).

n=−∞

27

Then at last we have

S ek (f, τ 1 , τ 2 )

= ejπfe(τ1−τ2)

4

(∫ 1

×

−1

∫ 1

−1

( [ (a

R

E k [n] ) ] 4

− 3 N−1 ∑

|w m µ m |

4) 4

m=0

)

P m,k (f 1 ) Pm,k(f ∗ 1 − f) Pm,k(f ∗ 2 ) P m,k (f 2 − f) e jπ(f 1τ 1 −f 2 τ 2 ) df 1 df 2

+ 1 8 ejπf e(τ 1 −τ 2 ) e jπ(f+1)τ 1

×

∫ 1

−1

( N−1

)

∑

(w m µ m ) 2 P m,k (f + 1 − f 1 ) P m,k (f 1 − f)

m=0

( N−1

∗

∑

(w m µ m ) 2 P m,k (f 1 ) P m,k (1 − f 1 ))

e −jπf1(τ1+τ2) df 1

m=0

+ 1 8 ejπfe(τ1−τ2) ∫ 1

+ σ2 ν

4

∫ 1

−1

−1

+ σ2 ν

4 ejπf(τ1−τ2) ∫ 1

+ σ4 ν

2

∫ 1

−1

( N−1 ∑

m=0

) ( N−1

)

∑

|w m µ m | 2 |P m,k (f 1 )| 2 |w m µ m | 2 |P m,k (f 1 − f)| 2 e jπf1(τ1−τ2) df 1

m=0

( N−1

) ∑

|w m µ m | 2 |P m,k (f 1 − f − f e )| 2 G 2 (f 1 ) e jπf1(τ1−τ2) df 1

m=0

( N−1

) ∑

|w m µ m | 2 |P m,k (f − f 1 − f e )| 2 G 2 (f 1 ) e −jπf1(τ1−τ2) df 1

−1 m=0

G 2 (f − f 1 ) G 2 (f 1 ) e jπf1(τ1−τ2) df 1 . (47)

Then we immediately have

S ek (0, τ 1 , τ 2 )

= ejπf e(τ 1 −τ 2 )

4

( [ (a

R

E k [n] ) ] 4

−

4) 3 N−1 ∑

(∫ 1

) (∫ 1

)

|w m µ m | 4 |P m,k (f)| 2 e jπfτ 1

df |P m,k (f)| 2 e −jπfτ 2

df

m=0

−1

−1

+ 1 ∫ 1

8 (−1)τ 1

e jπf e(τ 1 −τ 2 )

+ 1 ∫ 1

8 ejπf e(τ 1 −τ 2 )

+ σ2 ν

2

+ σ4 ν

2

∫ 1

−1

∫ 1

−1

−1

−1

N−1

∑

(w

∣ m µ m ) 2 P m,k (f) P m,k (1 − f)

∣

m=0

( N−1

) 2

∑

|w m µ m | 2 |P m,k (f)| 2 e jπf(τ 1−τ 2 ) df

m=0

( N−1

) ∑

|w m µ m | 2 |P m,k (f − f e )| 2 G 2 (f) e jπf(τ 1−τ 2 ) df

m=0

2

e −jπf(τ 1+τ 2 ) df

G 4 (f) e jπf(τ 1−τ 2 ) df. (48)

Now we get the explicit expression of S ek (0, τ 1 , τ 2 ).

Derivation of ˜S ek [f, τ 1 , τ 2 ]:

First we have

= 1 8

= 1 8

∞∑

n=−∞

∫ 1

−1

∫ 1

−1

p m,k [n + λ + τ 1 ] p ∗ m,k[n + λ] p m,k [n + τ 2 ] p ∗ m,k[n]

(e jπfλ ∫ 1

(∫ 1

−1

∫ 1

) (∫ 1

) ∗

P m,k (f 1 ) Pm,k(f ∗ 1 − f) e jπf 1τ 1

df 1 P m,k (f − f 2 ) Pm,k(−f ∗ 2 ) e jπf 2τ 2

df 2 df

−1

−1

)

P m,k (f 1 ) Pm,k(f ∗ 1 − f) P m,k (−f 2 ) Pm,k(f ∗ − f 2 ) e jπ(f1τ1−f2τ2) df 1 df 2 e jπfλ df,

−1

28

and

∞∑

p m,k [n + λ + τ 1 ] p ∗ m,k[n] = 1 2

n=−∞

∞∑

n=−∞

p m,k [n + τ 2 − λ] p ∗ m,k[n] = 1 2

∫ 1

−1

∫ 1

−1

)

(|P m,k (f)| 2 e jπfτ1 e jπfλ df

)

(|P m,k (f)| 2 e jπfτ2 e −jπfλ df.

Also by noting that G 2 (f) = ∑ ∞

τ=−∞ p t[τ] e −jπfτ , we have

p t [λ + τ 1 ] = 1 2

p t [λ − τ 2 ] = 1 2

∫ 1

−1

∫ 1

−1

(

G 2 (f) e jπfτ1) e jπfλ df

(

G 2 (f) e −jπfτ 2 ) e jπfλ df.

Based on the definition of B m,k (τ, f e ) shown in (45), we have

∞∑

(p m,k [2n + λ + τ 1 ] p m,k [2n + τ 2 ] − p m,k [2n + λ + τ 1 + 1] p m,k [2n + τ 2 + 1]) = (−1) τ 2

B m,k (λ + τ 1 − τ 2 , f e )

n=−∞

∞∑ (

p

∗

m,k [2n + λ] p ∗ m,k[2n] − p ∗ m,k[2n + λ + 1] p ∗ m,k[2n + 1] ) = Bm,k(λ, ∗ f e ).

n=−∞

Then at last we have

˜S ek (f, τ 1 , τ 2 )

= ejπfe(τ1+τ2)

4

(∫ 1

×

−1

∫ 1

−1

+ 1 8 (−1)τ 2

e jπf e(τ 1 +τ 2 )

×

( [ (a

R

E k [n] ) ] 4

− 3 N−1 ∑

|w m µ m |

4) 4

m=0

)

P m,k (f 1 ) Pm,k(f ∗ 1 − f) P m,k (−f 2 ) Pm,k(f ∗ − f 2 ) e jπ(f 1τ 1 −f 2 τ 2 ) df 1 df 2

∫ 1

−1

( N−1

)

∑

(w m µ m ) 2 P m,k (f 1 ) P m,k (1 − f 1 )

m=0

( N−1

∗

∑

(w m µ m ) 2 P m,k (f 1 − f) P m,k (f + 1 − f 1 ))

e jπf 1(τ 1 −τ 2 ) df 1

m=0

+ 1 8 ejπfe(τ1+τ2) e −jπfτ2 ∫ 1

+ σ2 ν

4 e−jπfτ2 ∫ 1

+ σ2 ν

4 ejπfτ1 ∫ 1

−1

+ σ4 ν

2 ejπfτ1 ∫ 1

−1

−1

−1

( N−1 ∑

m=0

) ( N−1

)

∑

|w m µ m | 2 |P m,k (f 1 )| 2 |w m µ m | 2 |P m,k (f 1 − f)| 2 e jπf1(τ1+τ2) df 1

m=0

( N−1

) ∑

|w m µ m | 2 |P m,k (f 1 − f − f e )| 2 G 2 (f 1 ) e jπf1(τ1+τ2) df 1

m=0

( N−1

) ∑

|w m µ m | 2 |P m,k (f − f 1 − f e )| 2 G 2 (f 1 ) e −jπf1(τ1+τ2) df 1

m=0

G 2 (f − f 1 ) G 2 (f 1 ) e −jπf1(τ1+τ2) df 1 . (49)

29

Then we immediately have

˜S ek (0, τ 1 , τ 2 )

= ejπf e(τ 1 +τ 2 )

4

( [ (a

R

E k [n] ) ] 4

− 3 N−1 ∑

(∫ 1

) (∫ 1

)

|w m µ m |

4) 4 |P m,k (f)| 2 e jπfτ 1

df |P m,k (f)| 2 e jπfτ 2

df

m=0

−1

−1

N−1

2

∑

(w

−1 ∣ m µ m ) 2 P m,k (f) P m,k (1 − f)

e jπf(τ1−τ2) df

∣

1

m=0

( N−1

) 2

∑

|w m µ m | 2 |P m,k (f)| 2 e jπf(τ1+τ2) df

−1 m=0

+ 1 8 (−1)τ2 e jπfe(τ1+τ2) ∫ 1

+ 1 8 ejπfe(τ1+τ2) ∫ 1

+ σ2 ν

2

+ σ4 ν

2

B.2.2

∫ 1

−1

∫ 1

−1

( N−1

) ∑

|w m µ m | 2 |P m,k (f − f e )| 2 G 2 (f) e jπf(τ1+τ2) df

m=0

G 4 (f) e jπf(τ1+τ2) df. (50)

Now we get the explicit expression of ˜S ek (0, τ 1 , τ 2 ) over stationary channel.

Time-variant channel

Through similar procedure of the stationary case and without consider the terms independent of λ in

R ek [λ, τ 1 , τ 2 ] and ˜R ek [λ, τ 1 , τ 2 ], we have

S ek (0, τ 1 , τ 2 )

= σ4 µ k

e jπfe(τ1−τ2)

2

and

∫

+ σ4 1

µ k

4 (−1)τ 1

e jπf e(τ 1 −τ 2 )

∫

+ σ4 1

µ k

4 ejπf e(τ 1 −τ 2 )

+ σ2 ν σµ 2 k

2

+ σ4 ν

2

∫ 1

−1

∫ 1

−1

˜S ek (0, τ 1 , τ 2 )

= σ4 µ k

e jπf e(τ 1 +τ 2 )

2

( [ (a

R

E k [n] ) ] 4

− 3 N−1 ∑

(∫ 1

) (∫ 1

)

wm

4) 4 |P m,k (f)| 2 e jπfτ 1

df |P m,k (f)| 2 e −jπfτ 2

df

m=0 −1

−1

−1

−1

N−1

∑

wm 2 P

∣

m,k (f) P m,k (1 − f)

∣

m=0

( N−1

) 2

∑

wm 2 |P m,k (f)| 2 e jπf(τ 1−τ 2 ) df

m=0

( N−1

) ∑

wm 2 |P m,k (f − f e )| 2 G 2 (f) e jπf(τ 1−τ 2 ) df

m=0

2

e −jπf(τ 1+τ 2 ) df

G 4 (f) e jπf(τ 1−τ 2 ) df, (51)

+ σ4 µ k

4 (−1)τ2 e jπfe(τ1+τ2) ∫ 1

+ σ4 µ k

4 ejπfe(τ1+τ2) ∫ 1

+ σ2 ν σµ 2 k

2

+ σ4 ν

2

∫ 1

−1

∫ 1

−1

( [ (a

R

E k [n] ) ] 4

− 3 N−1 ∑

(∫ 1

) (∫ 1

)

wm

4) 4 |P m,k (f)| 2 e jπfτ1 df |P m,k (f)| 2 e jπfτ2 df

m=0 −1

−1

−1

−1

N−1

∑

wm 2 P

∣

m,k (f) P m,k (1 − f)

∣

m=0

( N−1

) 2

∑

wm 2 |P m,k (f)| 2 e jπf(τ1+τ2) df

m=0

( N−1

) ∑

wm 2 |P m,k (f − f e )| 2 G 2 (f) e jπf(τ 1+τ 2 ) df

m=0

2

e jπf(τ1−τ2) df 1

G 4 (f) e jπf(τ1+τ2) df. (52)

30

We can see that these expressions are quite similar to the stationary case. It is interesting to compare

two special case with the same attenuation: AWGN channel(i.e. µ k = 1) and Rayleigh fading channel

with σµ 2 k

= 1. From (48), (41), (51) and (52), we can see that both cases have the same per**for**mance

versus SNR while the floor mse of Rayleigh fading channel is 3 dB higher than AWGN channel.

C Derivation of **frequency** domain expression **for** B m,k (τ, f e )

Proof. First by noting that P m,k (f) = ∑ ∞

s=−∞ p m,k[s] e −jπsf , we can use the decimator **for**mula (see

**for**mular 4.1.13 in [14]) to get

P 1 m,k (f) def

=

∞∑

n=−∞

p m,k [2n] e −j2πnf = 1 2

(

Pm,k (f) + P m,k (f − 1) ) . (53)

It is more difficult to get the **frequency** response of the decimated filter p m,k [2n + 1]. We can view

p m,k [2n+1] as the 2 times decimated version of p m,k [s+1], while ∑ ∞

s=−∞ p m,k[s+1] e −jπsf = e jπf P m,k (f),

then we have

P 2 m,k (f) def

=

∞∑

n=−∞

p m,k [2n + 1] e −j2πnf = 1 2 ejπf ( P m,k (f) − P m,k (f − 1) ) . (54)

We can use the discrete **for**m of Parseval’s relation, i.e. ∑ ∞

n=−∞ g 1[n] g 2 [n] = ∫ 0.5

−0.5 G 1(f) G 2 (−f) df,

where G 1 (f) = ∑ ∞

n=−∞ g 1[n] e −j2πfn and G 2 (f) = ∑ ∞

n=−∞ g 2[n] e −j2πfn . For the case of τ = 2 q, i.e. τ

is even, by using (53) and (54), we can rewrite (9) as

B m,k (τ, f e ) =

=

= 1 2

∞∑

n=−∞

∫ 0.5

−0.5

∫ 0.5

−0.5

Similarly, **for** τ = 2q + 1, we have

B m,k (τ, f e ) =

=

= 1 2

∞∑

n=−∞

∫ 0.5

−0.5

∫ 0.5

−0.5

By combining (55) and (56), we have

B m,k (τ, f e ) = 1 2

= 1 2

= 1 2

∫ 0.5

−0.5

∫ 0.5

(p m,k [2n] p m,k [2n + 2q] − p m,k [2n + 1] p m,k [2n + 2q + 1])

(P 1 m,k (f) P 1 m,k (−f) − P 2 m,k (f) P 2 m,k (−f)) e −j2πfq df

(P m,k (f) P m,k (1 − f) + P m,k (−f) P m,k (f − 1)) e −jπfτ df. (55)

(p m,k [2n] p m,k [2n + 2q + 1] − p m,k [2n + 1] p m,k [2n + 2q + 2])

(

P 1 m,k (f) P 2 m,k (−f) e −j2πfq − P 2 m,k (f) P 1 m,k (−f) e −j2πf(q+1)) df

(−P m,k (f) P m,k (1 − f) + P m,k (−f) P m,k (f − 1)) e −jπfτ df. (56)

(

P m,k (f) P m,k (1 − f) e jπfτ + P m,k (−f) P m,k (f − 1) e jπ(f+1)τ ) df

−0.5

P m,k (f) P m,k (1 − f) e jπfτ df + 1 2

∫ 1

−1

∫ 1.5

0.5

P m,k (f) P m,k (1 − f) e jπfτ df

P m,k (f) P m,k (1 − f) e jπfτ df. (57)

31

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32