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PERFORMANCE ANALYSIS OF SPACE TIME BLOCK CODES IN FLAT FADING MIMO CHANNELS WITH OFFSETS

Manav R Bhatnagar

UniK- University Graduate Center, University of Oslo, Kjeller NO-2027

Email: manav@unik.no and

Vishwanath R University of California at Davis, CA-95616

Email: rama@ucdavis.edu and

Vaibhav Bhatnagar University of Bridgeport, CT-06604

Email: vbhatnag@bridgeport.edu

Abstract- In this paper we consider the effect of imperfect carrier offset compensation

on the performance of Space-Time Block Codes. The symbol error rate (SER) for

Orthogonal Space-Time Block Code (OSTBC) is derived here by taking into account the

carrier offset and the resulting imperfect channel state information (CSI) in Rayleigh

flat fading MIMO wireless channels with offsets.

I. INTRODUCTION

Use of Space-Time codes with multiple transmit antennas has generated a lot of interest for

increasing spectral efficiency as well as improved performance in wireless communications.

Although, the literature on Space-time coding is quite rich now, the Orthogonal designs of

Alamouti [1], Tarokh [2], [3], Naguib and Sheshadri [4] remain popular. The strength of

orthogonal designs is that these lead to simple, optimal receiver structure due to the possibility

of decoupled detection along orthogonal dimensions of space and time.

Presence of a frequency offset between the transmitter and receiver, which could

arrive due to oscillator instabilities, or relative motion between the two, however, has the

potential to destroy this orthogonality and hence the optimality of the corresponding receiver.

Several authors have, therefore, proposed methods based on pilot symbol transmission [11] or

even blind methods [13] to estimate and compensate for the frequency offset under different

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channel conditions. Nevertheless, some residual offset remains, which adversely affects the

code orthogonality and leads to increased symbol error rate (SER).

The purpose of this paper is to analyze the effect of such a residual frequency offset on

performance of MIMO system. More specifically, we obtain a general result for calculating

the SER in the presence of imperfect carrier offset knowledge (COK) and compensation, and

the resulting imperfect CSI (due to imperfect COK and noise). The results of [5]-[10] which

deal with the cases of performance analysis of OSTBC systems in the presence of imperfect

CSI (due to noise alone), follow as special cases of the analysis presented here.

The outline of the paper is as follows: Formulation of the problem is accomplished in

section II. Mean square error (MSE) in the channel estimates, due to the residual offset error

(ROE) is obtained in section III. In section IV, we discuss the decoding of OSTBC data. In

section V, the probability of error analysis in the presence of imperfect offset compensation is

presented and we discuss the analytical and simulation results in section VI. Section VII,

contains some conclusions and in Appendix, we derive the total interference power in the

estimation of OSTBC data.

NOTATIONS

Throughout the paper we have used the following notations: B is used for matrix, b is used for

vector, b∈ b or B, B and b are used for variables, [ ]H is used for hermition of matrix or vector,

[ ]T is used for transpose of matrix or vector, [I] is used for identity matrix, and [ ]* is used for

conjugate matrix or vector.

II. PROBLEM FORMULATION

A. System Modeling

Here for simplicity of analysis we restrict our attention to the simpler case of a MISO system

(i.e. one with m transmit and single receive antennas) shown in Fig.1 in which, the frequency

3

h1

hm

ωo

ωo

Tx

Rx

1

m

offset between each of the transmitters and the receiver is the same, as will happen when the

source of frequency offset is primarily due to oscillator drift or platform motion.

Fig.1 MISO system considered in the problem

Here hk are channel gains between k-th transmit antenna and receive antenna and ωo is the

frequency offset. The received data vector corresponding to one frame of transmitted data in

the presence of carrier offset is given by

= Ω ω +oy ( )Fh e , (1)

where 1 2y y y += Ty [ ...... ]tN N ; Nt and N being the number of time intervals over which,

respectively pilot symbols and unknown symbols are transmitted, F consists of data

formatting matrix [St S] corresponding to one frame; St and S being the pilot symbol matrix

and Space-Time matrix respectively, h=[h1 h2 . . . hm]T denotes the channel gain vector with

statistically independent complex circular Gaussian components of variance σ2, and stationary

over a frame duration, e is the AWGN noise vector 1 2 +e e .....e⎡ ⎤⎣ ⎦T

tN N with a power density of

N0/2 per dimension, and o 0 0 0Ω(ω ) = diagexp(jω ), exp(j2ω ),........., exp(j( + )ω )tN N

denotes the carrier offset matrix.

B. Estimation of Carrier Offset and Imperfect Compensation of the Received Data

Maximum Likelihood (ML) estimation of transmitted data requires the perfect knowledge of

the carrier offset and the channel. The offset can be estimated through the use of pilot symbols

[11], or using blind method [13]. However, a few pilot symbols are almost always necessary,

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for estimation of the channel gains. Considering all this, therefore, we use in our analysis a

generalized frame consisting of an orthogonal pilot symbol matrix (typically proportional to

the identity matrix) and the STBC data matrix. In any case, the estimation of the offset cannot

be perfect due to the limitations over the data rate, and delay and processing complexity.

There could also be additional constraint in the form of time varying nature of the unknown

channel. Thus, a residual offset error will always remain in the received data even after its

compensation based on its estimated value. This can be explained as follows: if ωoˆ denotes

the estimated value of the offset ω0, we have

ω = ω − Δωo oˆ , (2)

where Δω is a residual offset error (ROE) the amount of which depends upon the efficiency of

the estimator. The compensated received data vector will be

c o o(-ω ) = (Δω) + (-ω )ˆ ˆy = Ω y Ω Fh Ω e . (3)

It is reasonable to consider ROE to be normally distributed with zero mean and

variance 2ωσ . We have also assumed that carrier offset and hence ROE remains constant over a

data frame. The problem of interest here is to analytically find the performance of the receiver

in the presence of ROE.

III. MEAN SQUARE ERROR IN THE ESTIMATION OF CHANNEL GAINS IN

THE PRESENCE OF RESIDUAL OFFSET ERROR

Although it is possible to continue with the general case of m transmit antennas, the treatment

and solution becomes cumbersome, especially since the details will also depend on the

specific OSTBC used. On the other hand the principle behind the analysis can be easily

illustrated by considering the special case of two transmit antennas, employing the famous

Alamouti code [1]. Suppose we transmit K orthogonal pilot symbol blocks of 2×2 size and L

Alamouti code blocks over a frame. One such frame is depicted in fig.2.

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3 41 2 2 1 2

* ** * * *4 32 1 2 2 1

trainingdata ( blocks) STBCdata ( blocks)

(1) (1)(1) (1) (1) (1)

(2) (2)(2) (2) (2) (2)

s ss s s sx 0 x 0s ss s s s0 x 0 x

−

−

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥ −− −⎣ ⎦ ⎣ ⎦

L L

L L

K L

Fig.2 Complete frame for two transmit antennas and single receive antenna case

where x is a pilot symbol of unit power and ( )sr k represents the r-th symbol transmitted by

the k-th antenna and x, ( )s ∈r k M-QAM.

The compensated received vector corresponding to K training data blocks (denoted

here by matrix P) can be expressed as

1 2 2 Tc c c c 2 2 o 2 2 2 1= [y y ....y ] = Ω( ) + Ω(-ω )× × ×Δω ˆy Ph eK

K K K K K , (4)

where 2 2Ω( ) ×Δω K K is the ROE matrix and o 2 2Ω(-ω ) ׈ K K is compensating matrix, respectively,

corresponding to K pilot blocks, and 2 1×e K is the noise in pilot data. It may be noted that the

last term in (4) can still be modeled as complex, circular Gaussian and contains independent

components. As the receiver already has the information about P, we can find the ML

estimate of the channel gains as follows [2],[4]:

Hc2

1=| x |

h P yK

. (5)

Substituting the value of cy from (4) into (5), we get

H H2 ×2 o 2 ×2 2 ×12 2

1 1= Ω(Δω) + Ω(-ω )| x | | x |

ˆ ˆh P Ph P eK K K K KK K. (6)

In the low mobility scenario where the carrier offset is mainly because of the oscillator

instabilities, its value is very small and if sufficient training data is transmitted or an efficient

blind estimator is used, the variance 2ωσ of ROE is generally very small ( 2

ωσ <<1), thus we can

comfortably use a first order Taylor series approximation for exponential terms in

2 ×2Ω(Δω) K K as

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exp(j Δω ) = 1+ j ΔωN N . (7)

After a simple manipulation, we can find the estimates of channel gains as

1 1 Ho 2 ×2 2 ×12

2 2

error due tonoiseerror due to residualoffset

h j Δωh 1ˆ ˆ= + + Ω(-ω )h j(1+ )Δωh | x |⎡ ⎤ ⎡ ⎤

= + Δ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

h P e h hK K K

KK K

, (8)

where Δh= 1 Ho 2 ×2 22

2

j Δωh 1+ Ω(-ω )j(1+ )Δωh | x | K K K

KK K

⎡ ⎤⎢ ⎥⎣ ⎦

P eˆ is the total error in estimates. It is

easy to see that there are two distinct interfering terms in (8), due to ROE and AWGN noise.

In the previous work [5]-[10], the interference only due to the AWGN noise is considered.

However, here in (8) we are also taking into account the effect of the interference due to ROE.

The mean square error (MSE) of channel estimate in (8) can be found as follows:

HΔω

1 HΔω o 2 ×2 2 ×12

2

H

1 Ho 2 ×2 2 ×12

2

1MSE = TrE 2

j Δωh1 1Tr E + Ω(-ω )j(1+ )Δωh2 | x |

j Δωh 1+ Ω(-ω ) .j(1+ )Δωh | x |

h, e

h, e

h h

P e

P e

Δ Δ

⎛ ⎞⎡ ⎤= ⎜ ⎟⎢ ⎥

⎣ ⎦⎝ ⎠

⎛ ⎞⎡ ⎤⎜ ⎟⎢ ⎥⎣ ⎦⎝ ⎠

,

, ˆ

ˆ

.K K K

K K K

KK K

KK K

(9)

Assuming, elements of h, Δω and elements of e are statistically independent of each other, the

expectation of cross terms will be zero and the MSE would be simplified as follows:

H1 1

Δω2 2

HH H

o 2 ×2 2 ×1 o 2 ×2 2 ×12 2

2 2 2 2 2ω 0

j Δωh j Δωh1MSE Tr Ej(1 + )Δωh j(1 + )Δωh2

1 1E Ω(-ω ) Ω(-ω )| x | | x |

= (( + (1 + ) )/2)σ σ + (1/ )(N / | x | ).

h,

e P e P e

⎛ ⎞⎡ ⎤ ⎡ ⎤⎜ ⎟= ⎢ ⎥ ⎢ ⎥⎜ ⎟⎣ ⎦ ⎣ ⎦⎝ ⎠

⎛ ⎞ ⎛ ⎞+ ⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠ˆ ˆK K K K K K

K KK K

K K

K K K

(10)

This generalizes the results of mean square channel estimation error in AWGN noise only [5],

[6] to the case where there is also a residual offset present in the data being used for channel

estimation. It is clear that the expression reduces to that in [5], [6], when 2ωσ =0. Fig. 3 depicts

the results in a graphical form for two pilot blocks. It is also satisfying to see that the results

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match closely (except very large ROEs) those based on experimental simulations. The effect

of 2ωσ is seen to be very prominent as it introduces a floor in MSE value, independent of SNR.

Fig.3 Analytical and experimental plots of MSE in the channel estimates for different values of ROE; 2 2 2 2 2 2(2 / 30) , (2 / 50) , (2 /100) , (2 / 250) , (2 /1000) ,0ωσ = π π π π π from uppermost to

down most respectively.

IV. ESTIMATION OF OSTBC DATA

Next, we consider the compensated received data vector corresponding to the OSTBC part of

the frame. Consider the l-th STBC (Alamouti) block, which can be written as [1]

o

o

j(2 -1)ωj(2 -1)Δω(2 -1) 2 * T * Tc c 2 -1 2-j2 ω-j2 Δω

e 0 e 0=[y (y ) ] = + [e e ]0 e 0 e

vvv v

v vvv

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦z Hs

ˆ

ˆ , (11)

where v=K+l, 1 2* *2 1

h h=

h -h⎡ ⎤⎢ ⎥⎣ ⎦

H , and [ ]T2 -1 2s sl l=s . If the channel is known perfectly then the

ML estimation rule for obtaining estimate of s is given as

= arg min - ρs r s , (12)

where H Hρ = and =H H r H z

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In the presence of channel estimation errors, as discussed in section III, the vector r will be

equal to

H H= = ( )r H z H + H zΔ , (13)

where 1 2* *2 1

Δh ΔhΔ =

Δh -Δh⎡ ⎤⎢ ⎥⎣ ⎦

H . Substituting the value of z from (11) into (13), we get

o

o

j(2 -1)ωj(2 -1)ΔωH * T

(2 -1) 2-j2 ω-j2 Δω

I

e 0 e 0= + [e e ]0 e 0 e

⎛ ⎞⎜ ⎟⎡ ⎤⎡ ⎤⎜ ⎟⎢ ⎥⎢ ⎥⎜ ⎟⎣ ⎦ ⎣ ⎦⎜ ⎟⎝ ⎠

ˆ

ˆr H Hsvv

v vvv. (14)

Applying Taylor series approximation for the exponential term in the term (I) in (14), we will

get

o

o

j(2 -1)ωH H H * T

2 -1 2-j2 ω

j(2 -1)Δω 0 e 0= + + [e e ]0 -j2 Δω 0 e

⎡ ⎤⎡ ⎤⎢ ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦

ˆ

ˆr H Hs H Hs Hv

v vv

vv

. (15)

From (13) and (15), r can be expressed as

o

o

j(2 -1)ωH H H * T

Ω (2 -1) 2-j2 ωinterferring term(1) interferring term(2)

interferring term(3)

e 0= ρ + Δ + + [e e ]0 e

⎡ ⎤⎢ ⎥⎣ ⎦

ˆ

ˆr s H Hs H H s Hv

v vv , (16)

where ΩH =j(2 -1)Δω 0

0 -j2 Δω⎡ ⎤⎢ ⎥⎣ ⎦

vv

H . Clearly, estimation of s via minimization of (12) would

be affected by the interfering terms (1)-(3) shown in (16). In the next section, we carry out an

SER analysis by first obtaining expression for the total interference power and its subsequent

effect on the signal to interference ratio (SIR).

V. ERROR PROBABILITY ANALYSIS

In order to obtain an expression for the SIR, and hence for the probability of error, we need to

find the total interference power in (16). To simplify the analysis we restrict our self to those

cases when ωo is typically much smaller than the symbol period and if a sufficiently efficient

estimator like [11], [13] is used for carrier offset estimation, Δω is also very less than the

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symbol period. Under this restriction and assuming channel, noise, training data and S-T data

independent of each other and of zero mean, the correlations between Δh and h, Δh and Δω,

and H and ΩH , which mainly depend upon ωo and Δω, would be so small that these could be

neglected. We make use of this assumption in the following analysis for simplicity, but

without any loss of generality. In this case, the total interference power in (16) is obtained in

Appendix and the average interfering power will be

2 2 2 2 2 2avg s s ω 0Power = 2E σ MSE + 2(2 -1) E σ σ (σ + MSE) + 2(σ + MSE)Nv . (17)

Since, the channel is modeled as complex Gaussian random variable with variance 2σ , hence

2 2

=1

E | h | = σ⎧ ⎫⎨ ⎬⎩ ⎭∑

m

ii

m and the average SIR per channel will be

2 2s s2 2 2 2 2 2

avg s s ω 0

E Eγ = σ = σ2Power 2E σ MSE + (2 -1) E σ σ (σ + MSE) + (σ + MSE)N v

, (18)

where Es is signal power. If there is no carrier offset present, i.e. 2ωσ =0, and channel variance

is unity, i.e. 2σ =1, (18) reduces into the following conventional form [5]-[10]:

s

s 0 0

Eγ =2E MSE + N + N MSE)

. (19)

Hence, (18) is more general form of SIR than (19) and therefore, our analysis presents a

comprehensive view of the behavior of STBC data in the presence of carrier offset. Further,

the expression of exact probability of error for M-QAM data received over J independent flat

fading Rayleigh channels, in the terms of SIR, is suggested in [12] as

(20)

where , , and QAMc

QAM

g γμ =

1+ g γ QAMg = 3/2( -1)M ( )2 2( - )

T = 4 2( - ) +( - )

⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠⎝ ⎠i

il

l l i/ l i 1

l l i

( )

( )( )

( )( )

e

2-11 - μ 1 + μ1 1 μ- 1 +c c c14 1 - - 4 1 - .( - π4=02 2

2-1 -1 2( - )+1Tπ([ - arctanμ ] - sin arctanμ . cos arctanμ )).c c c2 =0 =1 =14 1+g γ 1+g γQAM QAM

P =

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

∑

∑ ∑ ∑

⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠

⎡ ⎤⎣ ⎦

J JJ J l

llM M

lJ J l l il il

ll l i l

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Probability of error in the frame consisting of L blocks of OSTBC data will be

, (21)

where (Pe)i denote the error probability of i-th OSTBC block. As all the interference terms in

(16) consist of Gaussian data and have zero mean and diagonal covariance matrices (see

Appendix), we may assume without loss of generality that all the interference terms are

Gaussian distributed with zero mean and certain diagonal covariance matrices.

VI. ANALYTICAL AND SIMULATION RESULTS

The analytical and simulation results for a frame consisting of two pilot blocks and three

OSTBC block are shown in Fig. 4 - Fig. 6. All the simulations are performed with the 16-

QAM data. The average power transmitted in a time interval is kept unity. The MISO system

of two transmit antennas and a single receive antenna employs Alamouti code. The channel

gains are assumed circular, complex Gaussian with unit variance and stationary over one

frame duration (flat fading). The analytical plots of SIR and probability of error are plotted

under the same conditions as those of experiments.

Fig.4 shows the effect of ROE on the average SIR per channel with -30dB MSE, in

channel estimates. Here, we have plotted (18) with different values of ROE. It is easy to see

that there is not much improvement in SIR with the increase in SNR at large values of ROE,

which is quite intuitive. Hence, our analytical formula of SIR presents a feasible view of the

behavior of OSTBC imperfect knowledge of carrier offset in MIMO channels.

Fig.5 and Fig.6 show the analytical and experimental, probability of error plots with

different values of MSE in channel estimates and with different values of ROE. It is very

much satisfying to see that the analytical results match closely those based on experimental

simulations for small value of residual carrier offsets. However, for the large values of offset

error, the analytical results do not follow the simulation results very tightly because our

assumption of uncorrelatedness between different quantities (section V) gets violated in such

e e1

1P (P )=

= ∑L

iiL

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cases. Nevertheless, our analysis is still able to provide an approximate picture of the behavior

of the S-T data with large residual offset errors.

Fig.4 Plot of avg. SIR per channel vs. SNR for MSE=-30dB (graphs are plotted for 2 2 2 5 20, (2 /1000) , (2 ) /10 , (2 /100)ωσ = π π π from uppermost to down most respectively)

Fig.5 SER vs SNR plots for 16 QAM, with no MSE (graphs are plotted for

2 2 2 2 2 2 2(2 ) /10000, (2 ) / 20000, (2 / 200) , (2 / 300) , (2 / 500) , (2 /1000) ,0ωσ = π π π π π π from uppermost to down most respectively)

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Fig.6 SER vs SNR plots for 16 QAM, with MSE=-40dB (graphs are plotted for

2 2 2 2 2(2 ) / 20000, (2 ) / 80000, (2 / 500) , (2 /1000)ωσ = π π π π from uppermost to down most respectively)

VII. CONCLUSIONS

We have performed a mathematical analysis of the behavior of Orthogonal Space-Time Codes

with imperfect carrier offset compensation in MIMO channels. We have considered the effect

of imperfect carrier offset knowledge over the estimates of the channel gains and resulting

probability of error in the final decoding of OSTBC data. Our analysis also includes the effect

of imperfect channel state information due to AWGN noise, over the decoding of OSTBC

data. Hence, it presents a comprehensive view of the performance of OSTBC with imperfect

knowledge of small carrier offsets (in case of small oscillator drifts or low mobility and an

efficient offset estimator) in flat fading MIMO channels with offsets. The proposed analysis

can also predict the approximate behavior of S-T data with large carrier offsets (in case of

high mobility or highly unstable oscillators and an inefficient offset estimator).

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APPENDIX

DERIVATION OF TOTAL INTERFERENCE POWER IN THE ESTIMATION OF

OSTBC DATA

We will find the expression of the total interference power in (16) here. There are three

interfering term in (16). Initially, we will calculate power of each term separately and finally

we will sum the power of all terms to find the total interference power. Before proceeding to

the power calculation, we can also assume s being a vector of statistically independent

symbols, which are also independent of channel, carrier offset, channel estimation error and

ROE.

A. Power of First Interfering Term

In view of the discussion of section V, we can write

H HE Δ E Δ E E = 0=H Hs H H s , (22)

implying that the first term has zero mean. Further, it can be shown that HsE (E / 2)[ ]ss I= ,

H 2E 2 [ ]= σHH I and HE 2(MSE)[ ]Δ Δ =H H I , where [I] is identity matrix of 2×2. In view

of the uncorrelatedness assumption of ΔH , H and s, and using the results of [14], the

covariance matrix associated with this term can be found as follows:

( )( ) HH H H H H

2s

E Δ Δ E Δ [E E ]Δ

1 0= 2E σ (MSE)

0 1

=

⎡ ⎤⎢ ⎥⎣ ⎦

H Hs H Hs H H( ss )H H

. (23)

B. Power of Second Interfering Term

The mean of the second interfering term, as per the discussion of section V, will be

H HΩ ΩE = E E E = 0H H s H H s , (24)

implying that the second term also has zero mean. Further, it can be shown that

H 2 2 2Ω Ω ωE 2(2 -1) σ [ ]v≅ σH H I and H 2E 2(σ + MSE)[ ]=H H I .In view of the uncorrelated-

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ness assumption of H , ΩH and s, and using the results of [14], the covariance matrix

associated with this term can be found as follows:

( )( ) HH H H H H

Ω Ω Ω Ω

2 2 2 2s ω

E E [E (E ) ]

1 0= 2(2 -1) E σ σ (σ + MSE)

0 1v

=

⎡ ⎤⎢ ⎥⎣ ⎦

H H s H H s H H ss H H

. (25)

C. Power of Third Interfering Term

Assuming e, H and oω being statistically independent of each other, the mean of the third

interfering term will be

o o

o o

j(2 -1)ω j(2 -1)ωH * T H * T

(2 -1) 2 (2 -1) 2-j2 ω -j2 ω

e 0 e 0E [e e ] E E E [e e ] = 00 e 0 e

v v

v v v vv v

⎧ ⎫ ⎧ ⎫⎡ ⎤ ⎡ ⎤⎪ ⎪ ⎪ ⎪=⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥⎪ ⎪ ⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭ ⎩ ⎭

H Hˆ ˆ

ˆ ˆ, (26)

implying that the third term also has zero mean. Further, it can be shown that

(2 -1) *(2 -1) 2 0*

2

eE [e e ] N [ ]

ev

v vv

⎧ ⎫⎡ ⎤⎪ ⎪ =⎨ ⎬⎢ ⎥⎪ ⎪⎣ ⎦⎩ ⎭

I and o o

o o

j(2 -1)ω -j(2 -1)ω

-j2 ω j2 ω

e 0 e 0E [ ]0 e 0 e

v v

v v

⎧ ⎫⎡ ⎤ ⎡ ⎤⎪ ⎪ =⎨ ⎬⎢ ⎥ ⎢ ⎥⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭

Iˆ ˆ

ˆ ˆ . Using the

results of [14], the covariance matrix can be found as follows:

o o

o o

o

o

Hj(2 -1)ω j(2 -1)ωH * T H * T

(2 -1) 2 (2 -1) 2-j2 ω -j2 ω

j(2 -1)ω -(2 -1)H *

(2 -1) 2-j2 ω *2

e 0 e 0E [e e ] [e e ]0 e 0 e

ee 0 eE E E [e e ]e0 e

v v

v v v vv v

vv

v vvv

⎧ ⎫⎛ ⎞⎛ ⎞⎡ ⎤ ⎡ ⎤⎪ ⎪⎜ ⎟⎜ ⎟⎨ ⎬⎢ ⎥ ⎢ ⎥⎜ ⎟⎜ ⎟⎣ ⎦ ⎣ ⎦⎪ ⎪⎝ ⎠⎝ ⎠⎩ ⎭

⎛ ⎞⎧ ⎫⎡ ⎤⎡ ⎤ ⎪ ⎪⎜ ⎟= ⎨ ⎬⎢ ⎥⎢ ⎥⎜ ⎟⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭⎝ ⎠

H H

H

ˆ ˆ

ˆ ˆ

ˆ

ˆ

o

o

j(2 -1)ω

j2 ω

00 e

v

v

⎧ ⎫⎡ ⎤⎧ ⎫⎡ ⎤⎪ ⎪ ⎪ ⎪⎢ ⎥⎨ ⎨ ⎬ ⎬⎢ ⎥⎢ ⎥⎣ ⎦⎪ ⎪ ⎪ ⎪⎩ ⎭⎣ ⎦⎩ ⎭H

ˆ

ˆ

20

1 0= 2(σ + MSE)N

0 1⎡ ⎤⎢ ⎥⎣ ⎦

. (27)

Apparently, all interfering terms are distributed identically with zero mean and their

covariance matrices are proportional to the identity matrix. Further, we note that the power in

the three terms can be simply added, since, these can be shown to be mutually uncorrelated.

Hence, the total interfering power will be

15

2 2 2 2 2 2tot s s ω 0

1 0 1 0 1 0Power = 2E σ MSE + 2(2 -1) E σ σ (σ + MSE) + 2(σ + MSE)N

0 1 0 1 0 1⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

v (28)

ACKNOWLEDGEMENTS

We are extremely thankful to Prof. Are Hjørungnes, UniK, University of Oslo, for his help provided in the

derivation of the expectation in Appendix.

REFERENCES

[1] S. M. Alamouti, “A Simple Transmit Diversity Technique for Wireless Communications”, IEEE Journal on

Selected Areas on Communications, vol. 16, pp. 1451-1458, Oct. 1998.

[2] V. Tarokh, N. Seshadri, and A. R. Calderbak, “Space-Time Codes for High Data Rate Wireless

Communication: Performance Analysis and Code Construction”, IEEE Trans. Inform. Theory, vol. 44, pp.

744-765, Mar. 1998.

[3] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-Time Block Codes from Orthogonal Designs”, IEEE

Trans. Inform. Theory, vol. 45, pp. 1456-1467, May 1999.

[4] A. Naguib, N. Seshadri, and A. R. Calderbank. “Increasing Data Rate over Wireless Channels”, IEEE Signal

Processing Mag., vol. 17, pp. 76-92, May 2000.

[5] L. Tao, L. Jianfeng, H. Jianjun, and Y. Guangxin, “Performance Analysis for Orthogonal Space-Time Block

Codes in the Absence of Perfect CSI”, 14th IEEE Int. Sym. on Personal, Indoor and Mobile Radio Comm., pp

1012-1016, 2003.

[6] K.O. Eunseok, C. Kang, and D. Hong , “Effect of Imperfect Channel Information on M-QAM SER

Performance of Orthogonal Space-Time Block Codes”, IEEE Vehicular Technology Conference, pp.722-726,

2003.

[7] E. G. Larsson, P. Stoica, and J. Li, “Orthogonal Space–Time Block Codes: Maximum Likelihood Detection

for Unknown Channels and Unstructured Interferences”, IEEE Trans. Signal Processing, vol. 51, pp362-372,

Feb. 2003.

[8] D. Mavares and R. P. Torres, “Channel Estimation Error Effects on the Performance of STB Codes in Flat

frequency Rayleigh Channels”, 58th IEEE Vehicular Technology Conference, pp 647 – 651, 2003.

[9] G. Kang and P. Qiu, “An Analytical Method of Channel Estimation Error for Multiple Antennas System”,

IEEE Global Telecommunications Conference, vol.2, pp1114–1118, Dec. 2003.

16

[10] H. Cheon and D. Hong, “Performance Analysis of Space-Time Block Codes in Time-Varying Rayleigh

Fading Channels”, IEEE Int. Conf. on Acoustics, Speech, and Signal Processing, vol3, pp2357 -2360, May

2002.

[11] O. Besson and P. Stoica, “On Parameter Estimation of MIMO Flat-Fading Channels with Frequency

Offsets”, IEEE Trans. Signal Processing, vol. 51, pp. 602-613, Mar. 2003.

[12] M. S. Alouini. and A. J. Goldsmith, “A Unified Approach for Calculating Error Rates of Linearly

Modulated Signals over Generalized Fading Channels”, IEEE Trans. Comm., vol. 47., pp. 1324-1334. Sep.

1999.

[13] M. R Bhatnagar, R. Vishwanath, and M.K. Arti, “On Blind Estimation of Frequency Offsets in Time

Varying MIMO Channels”, IEEE Third Int. Conf. on Wireless and Optical Communications Networks

(WOCN2006), Apr. 2006.

[14] P. H. M. Janssen and P. Stoica, “On the Expectation of the Product of Four Matrix-Valued Gaussian

Random Variables”, IEEE Trans. Aut. Contrl., vol. 33, pp. 867-870, Sep. 1988.

Manav R Bhatnagar was born in Moradabad, India in 1976. He did his B.E. in Electronics in 1997 from North Maharashtra University, Jalgaon, India. He has worked as lecturer in Moradabad Institute of Technology, Moradabad, India from 1998-2003. He has done M.Tech. in Communications Engineering from Indian Institute of Technology Delhi, India in 2005. He is currently doing Phd from UniK, University of Oslo, Norway. His research interests include Signal Processing for MIMO Wireless Communications, Routing in Optical Networks and Image Processing. Vishwanath R was born in Hyderabad, India in 1982.He did his B.E. in Electronics and Communications Engineering from Birla Institute Of Technology, Ranchi, India in 2002. He has done M.Tech. in Communications Engineering from Indian Institute of Technology Delhi, India in 2005. He is currently doing Phd from University of California Davis, CA, USA. His research interests include Routing in Optical Networks, Signal Processing, Wireless Communications and Image Processing. Vaibhav Bhatnagar was born in Moradabad, India in 1982. He did his B.Tech. in Electronics & Communications in 2004 from U.P.T.U., India. He is currently pursuing M.S. in Electrical Engineering from University of Bridgeport, CT, U.S.A.His research interests include Signal Processing in Wireless Communications, Image Processing and Digital Signal Processing.