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BLIND DECODING OF MISO-OSTBC SYSTEMS BASED ON PRINCIPAL COMPONENT
ANALYSIS
Javier V´ ıa, Ignacio Santamar´ ıa, Jes´ us P´ erez and David Ram´ ırez
Dept. of Communications Engineering. University of Cantabria
39005 Santander, Cantabria, Spain
E-mail: {jvia,nacho,jperez,ramirezgd}@gtas.dicom.unican.es
ABSTRACT
In this paper, a new second-order statistics (SOS) based method
for blind decoding of orthogonal space time block coded (OSTBC)
systems with only one receive antenna is proposed. To avoid the in-
herent ambiguities of this problem, the spatial correlation matrix of
the source signals must be non-white and known at the receiver. In
practice, this can be achieved by a number of simple linear preco-
ding techniques at the transmitter side. More specifically, it is shown
in the paper that if the source correlation matrix has different eigen-
values, then the decoding process can be formulated as the problem
of maximizing the sum of a set of weighted variances of the signal
estimates. Exploiting the special structure of OSTBCs, this problem
can be reduced to a principal component analysis (PCA) problem,
which allows us to derive computationally efficient batch and adap-
tiveblinddecodingalgorithms.ThealgorithmworksforanyOSTBC
(including the popular Alamouti code) with a single receive antenna.
Some simulation results are presented to demonstrate the potential
of the proposed procedure.
1. INTRODUCTION
Orthogonal space time block codes (OSTBC) [1, 2, 3] appear as
an important class of space time block codes, which provides full
diversity and low complexity maximum likelihood (ML) decoding.
In recent years, blind decoding of STBCs [4] and OSTBCs has re-
ceived increasing interest, for instance, in [5] the authors propose a
PCA-based blind decoding method which is able to restore the chan-
nel and source signals for the most of the OSTBCs when more than
one receive antenna is available.
In many real applications, the source signals exhibit autocorrela-
tion properties, which can be exploited for developing blind equali-
zationalgorithms[6].Inthispaperwepresentanewcomputationally
efficient method for OSTBC decoding with only one receive anten-
na which exploits the correlation properties of the source signals. In
particular, we prove that a sufficient condition for blind decoding is
that one of the eigenvalues of the correlation matrix has multiplicity
one. In this way, we can propose a blind decoding criterion which
consists on the maximization of a weighted sum of the estimated
signal variances. Furthermore, by exploiting the linear dependence
between the multiple-input single-output (MISO) channel and the
OSTBC equalizers, the criterion can be rewritten as a function of the
estimated channel, which reduces the blind decoding problem to a
single principal component analysis (PCA) [7] problem. Finally, the
direct application of the Oja’s rule [7], provide a fast and efficient
adaptive blind decoding algorithm for OSTBCs.
This work was supported by MEC (Ministerio de Educaci´ on y Ciencia)
under grant TEC2004-06451-C05-02.
2. OVERVIEW OF MISO-OSTBC SYSTEMS
Let us consider a system with Q transmit antennas where the
information symbols to be transmitted are taken M at a time to form
a P × Q data block S[n]. This block is transmitted in P channel
uses, and hence, the average code rate is M/P symbols per channel
use. For linear space time block codes the transmitted data block is
generated as
N
?
where sk[n] and sM+k[n] denote, respectively, the real and imagi-
nary part of the k-th information symbol of the n-th data block, and
the parameter N is N = 2M for general complex constellations
and N = M for real constellations. In particular, for orthogonal
space time block codes (OSTBC) the code matrices fulfil [3], for
k,l = 1,...,N
S[n] =
k=1
sk[n]Ck,
CH
kCl=
?
I
lCk
k = l,
−CH
k = l.
(1)
Considering one receive antenna and a slow flat fading channel,
i.e. a channel with coherence time substantially larger than the data
block length P, the received signal can be written as
x[n] = S[n]h + u[n],
where h is the Q×1 complex channel response and u[n] is a P ×1
noise vector. Defining the “tilde” operator˜A = [?(A)T,?(A)T]T,
and applying it to the above equation yields
˜ x[n] =˜W(h)s[n] + ˜ u[n],
where W(h) = [w1(h)···wN(h)], wk(h) = Ckh and s[n] =
[s1[n],...,sN[n]]T.
It can be derived in a straightforward manner that, under the
conditions (1), the combined code-channel response vectors ˜ wk(h)
satisfy
?
fork,l = 1,...,N,whichimpliesthatassumingzero-mean,tempo-
rally and spatially white Gaussian noise uncorrelated with the data,
the maximum likelihood (ML) estimator of s[n], given h, is [3]
˜ wT
k(h)˜ wl(h) =
?h?2
0
k = l,
k = l,
ˆ sh[n] =
˜
WT(h)˜ x[n]
?h?2
,
(2)
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where the subindex h has been included to denote the dependency
on the true channel. Eq. (2) can be rewritten as
⎡
⎣
whereFk= [Ck
jCk],fork = 1,...,N,˜h = [?(h)T,?(h)T]T,
and the equalizers are given by ˜ wk(h) =˜Fk˜h.
ˆ sh[n] =
⎢
˜ xT[n]˜F1
...
˜ xT[n]˜FN
⎤
⎦
⎥
˜h
?h?2,
(3)
3. BLIND DECODING THROUGH PCA
Assuming zero mean white noise with variance η2, and consi-
dering that the noise is uncorrelated with the data, the correlation
matrix of the data vectors ˜ x[n] is given by
R˜ x= E[˜ x[n]˜ xT[n]] =˜W(h)E[s[n]sT[n]]˜WT(h) +η2
2I,
and taking into account the eigenvalue decomposition of the correla-
tion matrix E[s[n]sT[n]] = QΣ2QT, where Q and Σ are N × N
unitary and diagonal matrices respectively, it is easy to realize that
the eigenvectors associated to the N largest eigenvalues of R˜ x are
given by the columns of the matrix˜
pose the following criterion
?
whichismaximizedforanyestimatedchannelˆhsuchthatthecorres-
ponding code-channel matrix˜
W(ˆh) satisfies [5]
W(h)Q, and then we can pro-
J1(ˆh) = Tr
˜
WT(ˆh)R˜ x˜
W(ˆh)
?
,
(4)
range(˜W(ˆh)) = range(˜W(h)).
(5)
Finally, by combining equations (2), (3) and (4), the function to be
maximized can be rewritten as
J1(ˆh) = ?h?2?ˆh?2E[ˆ sT
ˆh[n]ˆ sˆh[n]] =ˆ˜hT
?
N
?
k=1
˜FT
kR˜ x˜Fk
?
ˆ˜h,
which reduces the problem of estimating the channel to a principal
component analysis (PCA) [7] problem, i.e. the true channel can be
obtained, with a sign and scale indeterminacy, as the channel provi-
ding an estimated output signal ˆ sˆh[n] with maximum variance.
Unfortunately,inthecaseofasinglereceiveantenna,mostofthe
practical OSTBCs provide a non-null subspace of possible estimates
ˆh = ch satisfying (5), which implies that the largest eigenvalue of
the new correlation matrix
N
?
k=1
˜FT
kR˜ x˜Fk,
has a multiplicity larger than one and introduces an additional am-
biguity among the eigenvectors associated to the largest eigenvalue
and their linear combinations. To overcome this problem, in [5] the
authors extend the idea to the case of several receive antennas, whi-
ch reduces drastically the number of OSTBCs provoking this ambi-
guity. Another possibility consists on selecting the estimateˆh whi-
ch maximizes J1(ˆh) and simultaneously optimizes some additional
criterion. For instance, we could use a criterion based on the higher
order statistics (HOS) of the source signal or its finite alphabet pro-
perty in order to eliminate this ambiguity. As an alternative, in the
next section we propose a technique based solely on the exploitation
of the second order statistics (SOS) of the source signal.
4. AVOIDING THE AMBIGUITY WITH CORRELATED
SOURCES
In many real situations, the source signal to be transmitted exhi-
bits a non-white spectrum, which can be use to blindly equalize the
channel [6]. The autocorrelation properties of the source sequences
can be due to a previous precoding step, such as a convolutional co-
de or a partial response system (PRS) [8, 9, 10]. In this section we
propose a modified cost function which exploits the correlation pro-
perties of the source signal to blindly estimate the channel response
with only one receive antenna.
4.1. Uncorrelated source signals with different variances
Let us start by considering that the source signals are uncorre-
lated, which implies that E[s[n]sT[n]] = Σ2is a diagonal matrix
with elements σ2
that σ2
function
J2(ˆh) = Tr
Λ˜
WT(ˆh)R˜ x˜
1,...,σ2
2 ≥ ... ≥ σ2
N. Assuming, without loss of generality,
N, we propose the following modified
?
1 ≥ σ2
W(ˆh)Λ
?
,
(6)
where Λ is a weighting matrix with elements λ1 ≥ λ2 ≥ ... ≥ λN
in its diagonal and zeros elsewhere. Then, combining equations (2)
and (6), the criterion to be maximized now is
J2(ˆh) = ?h?2?ˆh?2E[ˆ sT
ˆh[n]Λ2ˆ sˆh[n]],
(7)
and taking into account that
˜
WT(ˆh)˜W(ˆh)/?ˆh?2=˜WT(h)˜W(h)/?h?2= I,
it is easy to realize that
?E[ˆ sˆh[n]ˆ sT
ˆh[n]]?2
F≤
N
?
k=1
?
σ2
k+
η2
2?h?2
?
,
where the equality is satisfied iff (5) holds. Furthermore, we can find
that
N
?
and the above equality is satisfied iff
E[ˆ sT
ˆh[n]Λ2ˆ sˆh[n]] ≤
k=1
λ2
k
?
σ2
k+
η2
2?h?2
?
,
range(˜Wk(ˆh)) = range(˜Wk(h)),k = 1,...,N,
where˜
vectors ˜ wl(h), for σ2
not necessary) condition to avoid the ambiguity is that the variance
and the corresponding weight of one of the source signals sk[n] be
different from the remaining ones.
Finally, combining equations (3) and (7), the function to be ma-
ximized is
?
k=1
Wk(h) is defined as the matrix containing the equalization
l= σ2
kor λ2
l= λ2
k. In this way, a sufficient (but
J2(ˆh) =ˆ˜hT
N
?
λ2
k˜FT
kR˜ x˜Fk
?
ˆ˜h,
which reduces the estimation problem to a PCA problem where the
true channel is obtained, with the scale and sign indeterminacy, as
the channel providing an estimated weighted output signal with ma-
ximum variance.
Here it is interesting to point out that the selection of the weights
λkoffers a degree of freedom that can be exploited in different ways.
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Initialize the learning rate µ and the estimated channelˆ˜h[0] = 0.
for n = 1,2,... do
for k = 1,2,...,N do
Obtain ˆ yk[n] = λk˜ xT[n]˜Gkˆ˜h[n − 1].
Updateˆ˜h[n] =ˆ˜h[n − 1] + µλk˜GT
Normalizeˆ˜h[n] =ˆ˜h[n]/?ˆ˜h[n]?.
Obtain ˆ sk[n] = ˜ xT[n]˜Fkˆ˜h[n]
end for
end for
Algorithm 1: Summary of the proposed adaptive algorithm.
k˜ x[n]ˆ yk[n].
For instance, the trivial selection Λ = I reduces (6) to (4), which im-
plies that the ambiguities are given by (5). A better alternative con-
sists on selecting the weights such that if σ2
in particular, we can select Λ = Σ following the idea of the mat-
ched filter [8]. As can be seen later, this selection of the parameters
provides a decoding algorithm with good performance.
k> σ2
lthen λ2
k> λ2
l,
4.2. Generalization to correlated source signals
ConsideringthatthecorrelationmatrixE[s[n]sT[n]] = QΣ2QT
is known, the eigenvectors associated to the N largest eigenvalues of
R˜ x, which are given by the columns of the matrix˜
seen as the equalizers ˜ vk(h) of a different OSTBC code with code
matrices
N
?
where qlkis the l-th row and k-th column element of the matrix Q.
Thus, the criterion to be maximized is
?
where V(ˆh) = [v1(ˆh)···vN(ˆh)], vk(ˆh) = Dkˆh, ˜ vk(ˆh) =˜Gkˆ˜h
and Gk = [Dk
jDk]. Following a similar development to that of
the previous subsection, the final criterion to be maximized is
?
k=1
and the ambiguity problem is restricted to the codes and sources sa-
tisfying
W(h)Q, can be
Dk=
l=1
qlkCl,k = 1,...,N,
J3(ˆh) = Tr
Λ˜VT(ˆh)R˜ x˜V(ˆh)Λ
?
,
(8)
J3(ˆh) =ˆ˜hT
N
?
λ2
k˜GT
kR˜ x˜Gk
?
ˆ˜h,
(9)
range(˜Vk(ˆh)) = range(˜Vk(h)),k = 1,...,N,
where˜Vk(ˆh) and˜Vk(h) are defined as the matrices containing the
equalization vectors ˜ vl(ˆh) and ˜ vl(h) respectively, for σ2
λ2
Finally, it is easy to realize that the particular case of Q = I
implies Dk = Ck and˜Gk =˜Fk, and then (6) can be seen as
a particular case of (9). Analogously, in the case of Λ = I (8) is
equivalent to (4).
l= σ2
kor
l= λ2
k.
4.3. Final remarks and implementation
The results of the previous subsection show that the estimation
technique is reduced to the solution of a PCA problem, i.e. the ex-
traction of the main eigenvector of the 2Q × 2Q correlation matrix
N
?
k=1
λ2
k˜GT
kR˜ x˜Gk.
−10−505 1015 2025 30
−25
−20
−15
−10
−5
0
SNR (dB)
MSE (dB)
Coherent ML Receiver
Differential Receiver
Proposed method (T=100)
Proposed method (T=500)
Fig. 1. MSE of a duobinary signal for the Alamouti code.
In practice, the true correlation matrix R˜ xis unavailable and we
must estimate the sample covariance matrix
ˆR˜ x=1
T
T
?
n=1
˜ x[n]˜ xT[n],
where T is the number of available received blocks. An alternative
to this implementation of the algorithm is the direct application of
the Oja’s rule [7]
ˆ˜h[n] =ˆ˜h[n − 1] + µλk˜GT
k˜ x[n]ˆ yk[n],
where µ is the learning rate, and ˆ yk[n] = λk˜ xT[n]˜Gkˆ˜h[n − 1] is
the estimate of the rotated and weighted version of the source sig-
nal sk[n]. Finally, the overall adaptive algorithm is summarized in
Algorithm 1.
5. SIMULATION RESULTS
In this section the performance of the proposed algorithm is eva-
luated through some simulation examples. In all the simulations, the
results of 1000 independent realizations are averaged. The elements
of the flat fading MISO channels are zero-mean, circular, complex
Gaussian random variables with unit variance, the SNR is defined as
10log10(η2
the noise variance.
The source signals are binary i.i.d signals precoded by a filter
with response H(z) = 1+z−1, which is the filter used in duobinary
modulation, then at its output we have a correlated symbol sequence
drawn from the alphabet {−2,0,+2} with probabilities 1/4, 1/2 and
1/4 respectively. Two duobinary signals are the real and imaginary
parts of the complex symbols which are the input of the OSTBC
coder. In this way, the elements of the matrix E[s[n]sT[n]] are 2 in
its main diagonal, 1 in its first diagonals above and below the main
diagonal, and zeros elsewhere. In all the simulations, the weighting
matrix has been selected as Λ = Σ.
In the first example, the Alamouti [1] code has been selected,
and the performance of the proposed batch algorithm is compared
s/η2), where η2
sis the total transmitted energy and η2is
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−10 −50510 1520 25 30
10−4
10−3
10−2
10−1
100
SNR (dB)
BER
Coherent ML Receiver
Differential Receiver
Proposed method (T=25)
Proposed method (T=50)
Proposed method (T=100)
Proposed method (T=500)
Fig. 2. BER of a duobinary signal for the rate=3/4 code.
with the coherent ML receiver and the differential receiver proposed
in [3]. Figure 1 shows the mean squared error (MSE) in the signal
estimate for T = 100 and T = 500 received blocks. As can be seen,
the proposed blind decoder outperforms the differential receiver at
low and moderate SNRs. The noise floor present in the proposed
method can be attributed to the difference between the true correla-
tion matrix and its sample mean (this noise floor rapidly decreases
with the number of available blocks).
In the second example, we have tested the 3/4 OSTBC code for
M = 3 complex symbols, P = 4 time slots and Q = 3 transmit
antennas, which is presented in eq. (7.4.9) of [3]. Figure 2 shows the
final BER after decoding, where we can see that the proposed met-
hod again outperforms the differential receiver in low and moderate
SNRs.
Finally, the proposed adaptive version of the algorithm has been
tested with the 3/4 OSTBC code. The MSE of the channel estimate
for SNR=20dB and three different learning rates (µ = 0,05, µ =
0,1 and µ = 0,2) is shown in Fig. 3. As can be seen, the trade-
off between the speed of convergence and the final residual error is
determined by the learning rate.
6. CONCLUSIONS
In this paper, a new method for blind channel estimation and de-
coding of OSTBC systems with only one receive antenna have been
presented. The proposed method is based solely on second order sta-
tistics (SOS) and the main idea consists on exploiting the correlation
ofthesourcesignals,whichisassumedtobeknown.Wehaveproved
that, if at least one of the eigenvalues of this correlation matrix has
multiplicity one, then the channel and source signals can be extrac-
ted unambiguously, up to a sign and scale factor, by means of a sim-
ple principal component analysis (PCA) procedure. The simulation
results have shown that, for low and moderate SNRs, the proposed
batch and adaptive algorithms are better than the differential recei-
ver and very similar to the coherent receiver. As further lines we can
cite the theoretical study of the optimum selection of the weighting
matrix, as well as the analysis of some linear precoding techniques
which can reduce or avoid the noise floor due to the finite sample
problem.
0100 200 300 400500
−25
−20
−15
−10
−5
0
Number of blocks
MSE (dB)
µ=0.2
µ=0.1
µ=0.05
Fig.3.Performanceoftheproposedadaptivealgorithm.SNR=20dB.
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