Cet article traite de l'égalisation aveugle de canaux mobiles qui sont soit sélectifs en fréquence soit multiplicatifs. L'algorithme proposé peut-être utilisé pour compenser la distorsion de canaux á changements rapides où un ensemble restreint de données est disponible pour estimer les coefficients de l'égaliseur

LAMARCA, Meritxell

VAZQUEZ, Gregori

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SEIZIÈME COLLOQUE GRETSI — 15-19 SEPTEMBRE 1997 — GRENOBLE 713
Diversity techniques for blind channel equalization in mobile
communications
Meritxell Lamarca, Gregori Vázquez
Dept. of Signal Theory and Communications, Polytechnic University of Catalonia (UPC)
c/Gran Capitán, 08034-Barcelona (Spain); e-mail:{xell,gregori}@gps.tsc.upc.es; Fax:+34-3-4016447
RÉSUMÉ
Cet article traite de l’égalisation aveugle de canaux mobiles qui
sont soit sélectifs en fréquence soit multiplicatifs. L’algorithme
proposé peut-être utilisé pour compenser la distorsion de canaux
á changements rapides où un ensemble restreint de données est
disponible pour estimer les coefficients de l’égaliseur.
ABSTRACT
This paper deals with blind equalization of mobile channels, which
are either frequency-selective or multiplicative. The proposed algo-
rithm can be used for channel distortion compensation in diversity
systems subject to rapidly varying channels, where a short set of data
is available to estimate the equalizer coefficients.
1 Introduction
Mobile communications operate in a very hostile environment
due to multipath propagation and vehicle displacement. De-
pending on the transmission rate and vehicle speed, either
frequency-selectivity or Doppler spectrum spreading becomes
the major concern ([1]). In this paper, a blind equalization
technique is proposed which can be applied to compensate the
distortion introduced by the multiplicative and the frequency-
selective mobile channels.
The proposed approach relies on the availability of space or
time diversity which enables the use of single-input multiple-
output formulation (SIMO) of the transmission system. It is
based on a criteria which allows for linear equalization of
the received data. In fact, the proposed formulation is more
general than the application suggested here and could also be
applied in other environments. Thus, here it will be shown
to be useful for defining a deterministic criteria for blind
equalization, but it could also be applied to the problem
of channel estimation by means of cyclostationary statistics-
based methods (e.g. [2]).
The suggested algorithm has a low computational load and
exhibits performance similar to that one of other deterministic
criteria proposed in the literature: it obtains relatively good
results for short data sets, it assumes the channel is FIR with
known length (this constraint will be relaxed further on) and
its original derivation does not take into account the additive
noise, although it is of course considered when defining the
method final formulation.
As opposed to methods which have appeared earlier in the
literature, the proposed algorithm is based on the assumption
that the receiver can observe the complete convolution of
the transmitted data and the channel response. In the case
of convolutive channels, the full channel output is available
1This work has been partially supported by the Spanish Research Council (TIC-95-
1022-C05-01, TIC-96-0500-C10-01) and by CIRIT/Generalitat de Catalunya (SGR0096)
if a block transmission scheme is employed and a guard
interval longer than channel response duration is inserted
between consecutive transmitted frames. This is not a major
constraint, given that this guard interval is inserted in most
of burst transmission schemes anyway. In the case of a
multiplicative channel, the benefits of OFDM (Orthogonal
Frequency Division Multiplexing) modulation ([3]) have been
shown in [4]. Since its application will turn the frequency
flat fading into a convolutive distortion, the blind algorithm
proposed here can also be applied in that case.
This paper extends the method proposed in [5], improving
very significantly its robustness in front of the noise. The
relationship of this algorithm with other methods proposed
earlier in the literature is also explored.
2 Problem statement
Figure (1a) shows the discrete-time model for a diversity
receiver: the same information signal T [k] is transmitted
through B diversity branches, it is distorted by different chan-
nel responses Ci [k] and it is degraded by different additive
white Gaussian noise terms W i [k]. Using the z-transforms as-
sociated to these sequences, the received signal can be written
as:
Y i .z/ D T .z/  Ci .z/CW i .z/ i D 1; :::; B (1)
If the noise term is negligible it follows that:
Y i.z/ D T .z/  Ci .z/ i D 1; :::; B
and therefore:
T .z/ D g:c:d:fY i.z/g i D 1; :::; B (2)
where g:c:d: stands for the greatest common divisor. The
algorithm presented here is based on the estimation of the
transmitted data using equation (2). Of course, in order to
apply this equation the complete z-transform Y i .z/ must be
714
available and, therefore, a block transmission scheme with a
guard interval is needed. In the case of OFDM operating, no
efficiency is lost because no guard interval is needed. Indeed,
when working in the transformed domain, the OFDM received
signal could be regarded as the result of a circular convolution
with the DFT (Discrete Fourier Transform) of the channel
distortion ([6]):
Y i [k] D T [k]~ Ci [k] i D 1; :::; B
where ~ stands for circular convolution. Furthermore, over-
sampling the OFDM received signal is equivalent to zero
padding the transformed domain sequences as long as there
was no aliasing introduced when they were sampled at the
symbol rate ([5]). Thus, if the received signal is oversampled,
the circular convolution can be converted into a linear one:
Y i [k] D X Z P [k]~ CiZ P [k] D X [k]  Ci [k]
where Z P stands for ’zero-padding’. This way, the trans-
mission of an OFDM signal through a multiplicative channel
would fit also the model of equation (1).
Figure (1b) shows the linear equalization architecture em-
ployed in this paper. The multiple diversity branches are com-
bined by means of FIR filters Ei [k] to generate an output R[k]:
R .z/ D
BX
iD1
Y i .z/  Ei .z/ D T .z/ 
BX
iD1
Ci .z/  Ei .z/ (3)
Thus, our problem can be stated as that one of designing
the filters Ei [k] in order to retrieve the transmitted data:
R[k] D T [k]. Notice that if B D 2 (dual diversity) and the
optimization criteria is based on forcing R[k] D 0 then the
algorithm in [7] is obtained. In the present case, the perfect
equalization (zero forcing) criteria requires R .z/ D T .z/ and
therefore
BX
iD1
Ci .z/  Ei .z/ D 1 (4)
In the next section a blind algorithm is summarized which
provides the equalizer coefficients Ei [k].
3 Blind algorithm design
The proposed algorithm is based on the following property
(Bezout equation):
Given B polynomials fAi.z/g the equation
BX
iD1
Ai .z/  ai .z/ D 1
has a iff the B polynomials are coprime. Furthermore, the
solution is unique (up to a multiplicative constant) iff
deg

ai .x/
C 1 D deg Ai .x/
B Ä 1 i D 1; :::; B (5)
If the polynomials ai .x/ have a greater degree infinite solu-
tions can be found for this equation. When this property is
applied to equations (3)-(4), it turns out that perfect chan-
nel equalization can be obtained only when channel responses
have no factor in common, a result well known in the liter-
ature ([8]). In case this condition is satisfied, the zero-forcing
equalizer coefficients will be achieved by solving equation (4).
Furthermore, from the previous property also follows that the
zero-forcing equalizer is unique when the equalizer lengths are
selected according to (5) and are non-unique if their filters are
longer, being the difference among the possible solutions their
performance in front of the additive noise ([11]). Thus, design-
ing longer equalizers allowed for performance improvements
in the BER.
Besides, notice that equation (3) says that the equalizer
output R .z/ will always be a multiple of the transmitted data
T .z/, and that
deg fR .z/g D deg fT .z/g C deg
(
BX
iD1
Ci .z/  Ei .z/
)
Therefore, asking for an output of minimum length (R .z/ of
minimum degree) is equivalent to asking for perfect channel
equalization: R[k] D T [k]. This is the design criteria in which
the proposed method is based: design Ei [k] so that R[k] has
minimum length, then R[k] D T [k], being  an unknown
complex constant. The matrix formulation for the method can
be found in [5] and is briefly summarized here in order to
introduce the new method.
As shown in [5], equation (3) can be written using matrix
notation as
Z D Y E
where Y is a generalized Sylvester matrix. Besides, the
perfect equalization case in (4) can be written as
Y D
"
Y
t
Y
o
#
T D Y
t
E
0 D Y
o
E
(6)
where the received data matrix Y has been split in two parts.
The minimum length criteria can be described then as finding
those equalizer coefficients OE such that
0 D Y
o
OE (7)
Thus the method proposed in [5] can be considered as a one
based on the noise subspace of matrix Y
o
. Once the equalizer
has been estimated, the received data can be filtered to yield
an estimation of the transmitted data:
OT D Y
t
OE (8)
4 Proposed algorithm formulation
The previous method has two main drawbacks which are
solved by the new approach proposed here:
– The previous algorithm can only be applied when equal-
izer lengths satisfy (5), for if they were overdimensioned
the algorithm might converge to a non-useful solution
where constant =0 and, therefore, R[k] D 0. Hence,
the advantages of long equalizers in terms of noise can-
not be exploited.
– The previous algorithm does not fully exploit the avail-
able data. Both Y
t
and Y
o
contain information on the
channel and the transmitted data, but the algorithm de-
scribed in [5] designs the equalizer taps based on Y
o
only.
715
According to these considerations, the new algorithm for-
mulation tries to maximize the Signal-to-ISI-plus-noise-ratio
(SINR) at the equalizer output. This SINR can be approxi-
mately estimated (see eq. (6) ) as
\SI N R D
E H Y H
t
Y
t
E
E H Y H
o
Y
o
E
(9)
This is the new cost function to be optimized. Notice that this
new criteria is coherent with the algorithm in [5], given that
it aims to find the solution which maximizes the mean power
of detected symbols under the constraint of minimum length
equalizer output and noise level reduction.
The covariance matrices associated to Y
t
and Y
o
are non-
negative defined and thus the quotient in equation (9) corre-
sponds to a typical Rayleigh quotient form ([10]). Therefore,
it satisfies:
min
h
Y H
t
Y
t
I Y H
o
Y
o
i
6
E H Y H
t
Y
t
E
E H Y H
o
Y
o
E
6 max
h
Y H
t
Y
t
I Y H
o
Y
o
i
That is, the equalizer output SINR is bounded by the minimum
and maximum eigenvalues of the data matrix Y
t
in the norm
of Y
o
. Thus, the equalizer that maximizes (9) can be obtained
as the maximum generalized eigenvector:
Y H
t
Y
t
E D maxY Ho Y o E \SI N R D max (10)
Notice that this new cost function integrates the information
contained in Y
t
and Y
o
. Furthermore, the solution R[k] D 0
would yield a very poor SINR compared to the other solutions
and, therefore, it can be rejected as a solution of the new cost
function. Once the possibility of converging to this solution
has been discarded, the length constraint in (5) can be released
and longer equalizers can be employed. Simulations will show
the performance obtained by increasing the equalizer length.
The equalizer performance can be further improved if
a delay is allowed in the equalized signal. Many sets of
equalizers can be obtained for different delays:
BX
iD1
Ci .z/Ei.z/ D zÄd 0 6 d < deg Ci.z/Ei.z/ (11)
providing different delayed estimates R .z/ D zÄd OT .z/.
Although in average terms some delays will provide better
estimates than others ([9]), the simulations performed showed
that all delays are useful for noise impairment reduction due to
the reduced set of data available. Notice that, if the eigenvalue
is taken as an estimate of the SINR, in order to decide which
of the delays yields the better estimate of OT .z/ only the largest
eigenvalue must be computed for the different values of the
delay d: Unfortunately, this estimate is only reliable in high
SNR scenarios, otherwise the full computation of the equalizer
output must be carried out to find out which delay value is
preferred.
5 Relation with other algorithms
In this section the algorithm in [5] (equations (7)-(8) ) is
compared with the extension of the deterministic method
proposed in [12] to the block transmission case, rather than the
continuous transmission case analyzed in the original paper.
Equation (6) can be written using matrix notation as
Z D Y E D T C E CW E
where the vectors and matrices are associated to the polyno-
mials with the same letters. In order to compare both methods
a singular value decomposition (SVD) must be performed to
the generalized Sylvester matrix Y :
Y D U 6 V H
Then, it can be seen that the proposed method is based on
the signal subspace column vectors of matrix U , whereas the
method proposed in [12] was based on its noise ones.
Furthermore, the algorithm proposed here has a computa-
tional load much lower that one of [12], even if several values
of the delay d in equation (11) are used to reduce variance.
Both methods have in common that they require SVD compu-
tation. However, the method proposed in this paper only one
SVD must be computed and the matrix involved in it has the
same size as the channel length, whereas the algorithm in [12]
requires two SVD of matrices about the same size as the trans-
mitted signal length. Since the frame duration must be chosen
so that
degfT .z/g >> degfCi.z/g
in order to keep efficiency high, the computational load of
the proposed algorithm is much lower than that one of [12].
The dimension of the matrix involved, as well as the fact of
working with the noise subspace singular vectors, has a second
consequence: the algorithm in [12] is more sensitive to noise
than the one proposed here.
The advantage of the method [12] in front of the one pro-
posed here relies in the fact that the estimate provided by the
former one doesn’t need to be the result of a linear equaliza-
tion of the received data, whereas the one proposed here does.
This means that, in principle, better results can be obtained
in ill-conditioned channels where the linear equalization can
have noise enhancement problems (even though in the SIMO
case they are not as bad as in the single channel case ([11]) ).
6 Simulations
Fig.2 and 3 illustrate the performance of the algorithm pro-
posed in this paper. Both plots display the percentage of real-
izations (500 and 1000 were averaged) for which the equalizer
output EbNo was higher than the value indicated in the x-axis.
In all cases the transmitted data consisted of 128 QPSK sym-
bols. Notice that the output EbNo depends changes on each
run due to the algorithm sensitivity to the channel, data and
noise realizations caused by the limited amount of data avail-
able for estimation.
Figure 2 shows the performance of the algorithm in its
application to an OFDM transmission in a frequency-flat
fading channel corresponding to a 25Kb/s transmission at
1GHz with a mobile moving at 100Km/h. In that case two
antennas were used (B D 2) and EbNo=20dB. This figure
716
shows the improvement obtained when the equalizer design
criteria in (7) (I) is replaced by that one of equation (10)
(II). Notice the algorithm performs correctly even though the
multiplicative Rayleigh channel is a very difficult environment
for the blind algorithm, for the multiplicative channel does not
fulfill perfectly the finite length channel hypothesis.
Figure 3 shows the performance obtained when the algo-
rithm is applied to a TDMA block transmission in a frequency
selective channel. In this case, four antennas were simulated
(B D 4) and channel responses:
C1(z)=(1+j)+(-0.1-0.2j)zÄ1+0.4zÄ2+zÄ3+0.5zÄ4
C2(z)=0.1j+zÄ1-0.4jzÄ2+0.2zÄ3-0.5zÄ4
C3(z)=0.1+2zÄ1-4jzÄ2+0.2zÄ3+zÄ4
C4(z)=(1+0.8j)-2jzÄ1-0.4jzÄ2+0.2zÄ3+(1-0.5j)zÄ4
This figure illustrates the improvement obtained by increas-
ing the equalizer length. In this case, EbNo=15dB and the
equalizers of length 2 (I) and 4 (II) were designed using equa-
tion (10).
R [k]
E 1[k]
E 2[k]
E B[k]
···
Y 1[k]
Y 2[k]
Y B[k]
T [k]
C 1[k]
C 2[k]
C B [k]
···
W 1[k]
W 2[k]
W B[k]
Y 1[k]
Y 2[k]
Y B[k]
(a) (b)
Figure 1 — Block diagram of the multichannel system.
(a) Transmission; (b) Equalization
0 2 4 6 8 10 12 14 16 18 20
0
10
20
30
40
50
60
70
80
90
100
EbNo (dB)
%
 ti
m
es
(I) (II)
Figure 2 — Algorithm performance in the transmission of a
OFDM signal in a Rayleigh frequency-flat fading scenario.
References
[1] R.Steele, Mobile Radio Communications, Pentech Press,
1992
0 2 4 6 8 10 12 14 16
0
10
20
30
40
50
60
70
80
90
100
EbNo (dB)
%
 ti
m
es
(II)
(I)
Figure 3 — Algorithm performance in the transmission of a
TDMA signal in a Rayleigh frequency-selective channel
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Diversity techniques for blind channel equalization in mobile communications

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