1
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: [email protected] and
Vishwanath R University of California at Davis, CA-95616
Email: [email protected] and
Vaibhav Bhatnagar University of Bridgeport, CT-06604
Email: [email protected]
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.
5
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.
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