PERFORMANCE OF AN ADAPTIVE BLIND
EQUALIZER FOR QAM SIGNALS
Antoinette Beasley and Arlene Cole-Rhodes
Morgan State University
Baltimore, MD 21251
Email: beasley@eng.morgan.edu, acrhodes@ieee.org
ABSTRACT
This paper presents a performance study of a blind
adaptive channel equalization scheme. This scheme is
based on a single cost function, which is created by
combining two well-defined cost functions, the constant
modulus algorithm (CMA) and the alphabet-matched
algorithm (AMA). The combined cost function considers
both the amplitude and the phase of the equalizer output,
which allows for more efficient equalization of QAM
signals. We present results that compare equalization of
a 16-QAM signal using the new cost function with that
obtained using the CMA cost function alone. The
performance measures used for the comparison are the
average mean-squared error (MSE) and the average
symbol error rate (SER).
INTRODUCTION
We consider the problem of recovering an unknown
signal that has been transmitted over an unknown
frequency-selective channel using QAM modulation.
The received symbol has been affected by intersymbol
interference (ISI), which causes harmful distortions and
presents a major difficulty in the recovery process.
Signal detection at the receiver can be improved by
combating this ISI, and we consider the use of an
adaptive blind channel equalizer for this purpose.
The constant modulus algorithm (CMA) is the most
popular adaptive blind equalizer used today because of
its relative simplicity and its good global convergence
properties. It is very effective in equalizing signals that
are of constant modulus. However, for non-constant
modulus signals, CMA equalization can suffer from a
considerable amount of residual ISI, and produce nonminimal convergence. .
Prepared through collaborative participation in the Communications
and Networks Consortium sponsored by the U. S. Army Research
Laboratory under the Collaborative Technology Alliance Program,
Cooperative Agreement DAAD19-01-2-0011. The U.S. Government
is authorized to reproduce and distribute reprints for Government
purposes notwithstanding any copyright notation thereon.
In [1], Barbarossa and Scaglione propose an
alphabet-matched algorithm (AMA), which is able to
provide better equalization of non-constant modulus
signals, and they introduce a scheme that uses CMA to
initialize the alphabet-matched algorithm (AMA). The
AMA cost function tries to force the equalizer output to
belong to the signal constellation of interest, and it is
shown in [1] that this method is able to lower the
residual ISI and improve the convergence rate over that
of CMA alone. However, the AMA equalizer works well
only when the initialization is good. An issue here is
how to automatically switch from the initialization step
using CMA, to the AMA. Using too few iterations of
CMA will lead to ineffective equalization overall, while
using too many iterations increases the runtime without
improving performance.
In this paper we present a new adaptive channel
equalization scheme based on combining the two welldefined cost functions of the constant modulus algorithm
(CMA) and the alphabet-matched algorithm (AMA) into
a single cost function. We show through simulation that
the scheme based on this combined cost function is able
to provide improved results when compared to one using
CMA alone.
SYSTEM MODEL
We consider a communications environment,
which is coherent and synchronous with single-tone
signaling. Carrier timing and waveform recovery have
also been achieved. The received data block is of length
N symbols and the equalizer attempts to remove the
distortions introduced by the channel. Both the channel
and equalizer are constrained to be linear, time-invariant
FIR filters. The time-invariance assumption on the
channel is reasonable because we consider short blocks
of data. Noise in the channel is modeled as zero-mean
additive white Gaussian noise (AWGN). Figure 1
provides a block diagram for the communications
system.
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Figure 1: Channel and Equalizer
The channel is a single-input single-output (SISO)
system with an input/output relationship characterized
by
T −1
x(n) = ∑ h(k ) s (n − k ) + v(n), n = 0,..., N
(1)
k =0
where T is the number of channel zeros and {h(0) ….
h(T-1)} is the channel impulse response, s(n) is the
transmitted symbol sequence, and v(n) is AWGN noise
in a data block of length N. We assume the symbol
sequence in the block of data is independently and
identically distributed (i.i.d.) and is non-Gaussian. The
equalizer output, y(n) at time n is defined as
y ( n) =
L/2
∑ w x(n − l )
l =− L / 2
l
(2)
where {wl : -L/2 ≤ l ≤ L/2} are the equalizer weights.
BLIND EQUALIZATION ALGORITHM
I. Background
The scheme that we propose combines the
conventional CMA with a constellation matched error
(CME) term, similar to the modified CMA (MCMA)
scheme presented by He et. al. in [2]. The authors show
that for QAM signals, this new scheme improves the
convergence rate of the adaptive equalizer and is more
robust than the conventional CMA for low signal-tonoise ratios (SNRs). The cost function presented is a
weighted sum of two individual cost functions, one is
CMA and the other is a constellation matched error
(CME) term. The authors state that the CME term should
satisfy certain desirable properties namely uniformity,
symmetry, and zero/maximum penalties at the
zero/maximum distance from the QAM constellation
points. Uniformity implies that the CME function does
not favor any one data symbol in the alphabet over
another, while symmetry implies that the function is
symmetric around each constellation symbol. It is also
desirable that the CME assigns the zero/maximum
penalty at the zero/maximum distance from a
constellation point. A zero penalty occurs when an
equalized symbol lies directly on a constellation point
and the maximum penalty occurs midway between two
adjacent constellation points. In this work we propose
the use of AMA, presented by Barbarossa and Scaglione
[1] as the CME term.
AMA satisfies both the
uniformity and symmetry properties while satisfying a
minimum/maximum penalty assignment at the
zero/maximum deviation points.
For both the CMA and the AMA, the cost
function J (w) to be minimized is a function of the
T
difference between the equalizer output, y (n) = w x n ,
and the known constellation points. The cost function
for CMA is given by
(
)
2
2
J CMA ( w) = E  y (n) − R2  ,
(3)


4
E c(i )
, and c(i ), i = 1,..., M are the
where R2 =
2
E c(i )
{
{
}
}
known constellation points, M is the number of
constellation points. The notation E{⋅} denotes the
expected value, taken over the entire block of equalizer
output symbols. This cost function attempts to restore
the shape of the constellation by taking into account the
distance between the equalizer output symbols and the
computed constant modulus of the known constellation
symbols. CMA does not take the phase of the
constellation points into consideration. CMA is well
studied and is the most popular algorithm used for blind
adaptive equalization.
The AMA cost function, proposed by Babarossa
and Scaglione [1] is given by
J AMA

( w ) = E 1 −

M
∑e
i =1
− y ( n )−c (i )
2σ
2
2



(4)
The AMA cost function uses the same notation as above
and a parameter, σ, is used to control the width of the
nulls that the function places around each constellation
point. Its value is chosen so that these nulls do not
overlap for adjacent constellation points. The AMA cost
function attempts to restore the shape of the constellation
by considering the distance between the equalizer output
symbol and each of the known constellation symbols. It
assigns an appropriate penalty based on this distance.
When an equalized symbol is sufficiently close to a
constellation point the cost function is minimized. The
opposite is true when an equalized symbol is far from
the constellation points. AMA is known to depend on
good initialization for optimization of the cost function.
In our equalization scheme we combine the two
cost functions described above into a single cost function
similar to that proposed in [2], but modified as given
below:
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J (w) = JCMA(w) + β J AMA(w) .
(5)
This combined cost function attempts to restore the
constellation shape by taking into account both the
amplitude and phase of the equalized symbol. The
parameter β is a fixed scaling factor between the CMA
and AMA terms. Making the substitution of the CMA
and AMA expressions into equation (5), we
obtain
 M − y(n)−c(i)
2
2

2
J (w) = E ( y(n) − R2 ) + β E1 − ∑e 2σ
 i=1
{
}
2


 (6)

The update for the equalizer weights in this
proposed scheme is based on the following stochastic
block gradient descent rule:
w n +1 = w n − µ ∇ w J ( w )
(7)
where µ is the algorithm step-size. The equalizer output
T
is given as y (n) = w x n ,
for equalizer weight
vector, w and a received data vector, x n , which is a
function of time n. The gradient of the cost function is
given in equation (10) as the linear combination of the
gradients of the CMA and AMA cost functions shown in
equations (8) and (9) respectively:
∇ w J CMA ( w) = E 4

(
2
y ( n) − R2
) y ( n)
*
x 

(8)
2
2
M 1 *
− y( n)−c(i ) / 2σ 
∇wJ AMA(w) = E∑ 2 xn ( y(n) −c(i))e

i=1 σ

(9)
Note that the * operator denotes the complex conjugate.
Thus the gradient expression for the combined scheme
is:
∇wJ(w) = ∇wJCMA(w) + β ∇wJAMA(w)
(10)
and it should be noted that the averaged block gradient is
computed at each iteration.
The step-size, µ used in equation (7) is adaptive
and is similar to that proposed in [3]
T
µ =α
xn w
n
∇ w J (w )x n
T
.
(11)
The cost function given in (5) combines the
previously described CMA and AMA equalizers into a
single equalizer. Note that if β is zero, we have the pure
CMA equalizer and if β is very large, (5) is essentially
controlled by the AMA equalizer. For a fixed β, it
should also be noted that when the equalizer is far from
convergence, the AMA cost function remains fairly flat,
and the evolution of the equalizer is controlled by CMA.
Once CMA has converged, the evolution can then be
controlled by the AMA. Thus the combined cost
function eliminates the need to monitor CMA for
convergence, which was an issue discussed in [1].
Further, it provides some additional gains in the region
where CMA has not yet converged, and the AMA
initialization is poor.
II. Experimental Setup
The effectiveness of the combined scheme of
equation (5) is critically dependent on the scaling
parameter, β between the two terms of the cost function.
Thus it is of importance to determine the optimal value
of β. The goal of this section is to find an acceptable
initial range of β values for simulation purposes, and we
note that these initial values may be sub-optimal.
The following experimental setup was used for
testing purposes. Consider a channel with impulse
response ho = [1, 0, 0.5]. This channel is considered to
be a good communications channel because of its fairly
flat magnitude response, and it was previously used for
testing purposes in [3] and [5]. We transmit a data
sequence of length, N=500 symbols using 16-QAM
modulation and use an equalizer of length L=15. Figure
2 provides plots of the average MSE versus SNR (dB)
for different values of the scaling parameter, β. The
MSE is computed as the error between the composite
channel impulse and the impulse, which corresponds to
perfect equalization. For these tests each algorithm was
run for 100 iterations, and then averaged over a total of
500 Monte Carlo runs. The dashed curve presents the
results of using CMA equalization alone with phase
recovery and it is shown for comparison purposes. Here,
CMA does not equalize this QAM data sequence even at
an SNR of 30 dB (see [5]).
Figure 2 shows that the value of the scaling
parameter, β is indeed a determining factor in algorithm
performance. At lower SNR values, specifically 5 to 15
dB, equalization is not achieved but in the cases of
higher SNR where it has been achieved, the β = 30
curve produced the minimal values of the average MSE.
This β value is likely to be sub-optimal since it was
selected to be the best value among a limited range of
tested values. We note that the curves for β = 100 and
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l000 lie above CMA. This is consistent with the
performance of AMA under poor initialization.
Figure 2: Scaling factor curves for Channel ho
It is expected that the optimizing value of β will
change with SNR, so for future simulations a stochastic
approximation (SA) technique will be used offline to
automatically determine the optimal values of β, at
different levels of SNR for a specific channel. An
iterative stochastic gradient method will be used to
adjust the parameter, β to minimize a specified
mathematical loss function. In particular, we require that
value of the scaling parameter β that minimizes the MSE
at a fixed SNR. The SA algorithm allows us to update β
iteratively as follows, while averaging over noisy
measurements of the loss function:
β k +1 = β k − a k gˆ k ( β k ) .
previous section and results are shown for 100 iterations
of each algorithm.
The two different performance measures used for
evaluation are (i) the average MSE taken between the
composite channel (equalizer and channel) impulse and
the impulse corresponding to perfect equalization, and
(ii) the average SER taken between the channel input
and equalizer output sequences. In this case the MSE
and SER are estimated by averaging a total of 1500
Monte Carlo runs.
Figure 3 compares three different schemes via the
average MSE, while Figure 4 provides a comparison
using the average SER. The CMA only curve is
produced using 100 iterations of the CMA equalizer
alone.
The curve corresponding to the original
CMA/AMA equalizer uses CMA to initialize AMA (see
[1]). In this scheme, 100 iterations of CMA were run to
initialize AMA, which is then run for 100 iterations for a
total of 200 iterations. The combined scheme is run for
only 100 iterations using β= 30. We note that the
combined cost function equalizer is comparable in
complexity to that of the original CMA/AMA equalizer.
At lower SNR values, specifically 5 to 15 dB,
equalization performance suffers in all cases. But at
higher SNR values the combined scheme shows
considerable improvement over the CMA only case, and
comparable performance to the original CMA/AMA
case. This is observed in Figure 3 with the average MSE.
Using the average SER, the results of Figure 4 show that
at higher SNR the combination of CMA and AMA
provides a significant improvement over CMA alone and
allows for more efficient equalization.
(12)
β k is the parameter estimate, a k is the algorithm
step-size and gˆ k ( β k ) is the gradient approximation
all at iteration k. It has been determined
experimentally that the optimal value of β for this
channel lies in the range of 10 to 40 for varying
SNRs.
PERFORMANCE EVALUATION RESULTS
In this section we present results of the initial
evaluation of our combined scheme (based on β=30) and
compare it to the original CMA-initialized AMA scheme
[1], and also to CMA alone after phase recovery. The
experimental setup is identical to that described in the
Figure 3:Average MSE vs. SNR dB for Channel ho
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ACKNOWLEDGEMENTS
The authors would like to thank Brian Sadler and
Ananthram Swami of the Army Research Laboratory for
their many helpful suggestions.
REFERENCES
[1] S. Barbarossa, A. Scaglione, “Blind Equalization
using Cost Functions Matched to the Signal
Constellation”, Proc. 31st Asilomar Conf. Sig. Sys.
Comp., Pacific Grove (CA), vol. 1, pp. 550-54, Nov.
1997.
Figure 4: Average SER vs. SNR dB for Channel ho
CONCLUSIONS AND FUTURE WORK
In this work, we have presented initial results of
a new equalization scheme that combines the
conventional CMA cost function with the alphabetmatched algorithm (AMA) in a weighted sum. This
combined cost function performs equalization by taking
into account both the amplitude and phase of the
equalized output symbol. It also takes advantage of the
global convergence properties of CMA, and combines
this with the improved convergence rate of AMA near
its optimum. We have compared the combined scheme
with a CMA only scheme and the original CMA/AMA
equalizer of [1] by measuring the MSE and the SER.
The combined cost function approach has been shown to
provide more efficient equalization of QAM signals than
can be achieved using CMA alone and comparable
performance to the original CMA/AMA equalizer. The
initial results presented here show that this equalizer
provides significant improvements over CMA alone for
QAM signals, while adding minimal complexity. The
value of the scaling factor β used for performance
evaluation was most likely sub-optimal, and may require
further optimization. But we are already able to observe
a significant improvement in performance using this
combined cost function scheme.
In future work we will focus on fully optimizing
the β parameter for improved performance of the
combined single-cost function scheme. We will use a
stochastic approximation algorithm to determine offline
its optimal values for different channels at different SNR
values, and further test and evaluate the equalizer
presented here on different channels to determine its
effectiveness and robustness for QAM constellations.
[2] Lin He, M. G. Amin, C. Reed, Jr., R. C. Malkemes,
“A Hybrid Adaptive Blind Equalization Algorithm for
QAM Signals in Wireless Communications”, IEEE
Trans. on Signal Proc., Vol. 52, No. 7, pp. 2058-2069,
July 2004.
[3] A. Swami, S. Barbarossa, B. M. Sadler, and G.
Spadafora, “Classification of digital constellations under
unknown multipath propagation conditions”, Proc.
SPIE, Digital Wireless Comm. II, Orlando, FL, April
2000.
[4] C.R. Johnson, Jr. et al, “Blind Equalization Using
the Constant Modulus Criterion: A Review”, Proc. IEEE
vol. 86, no. 10, pp. 1927-50, Feb 1995.
[5] A. Beasley, A. Cole-Rhodes, B. Sadler, A. Swami,
“Adaptive Blind Equalization using an AlphabetMatched Algorithm”, Proceedings of the Collaborative
Technology Alliances Conference – Communications &
Networks, pp. 429-433, April 2003.
[6] J. C. Spall. Introduction to Stochastic Search and
Optimization: Estimation, Simulation, and Control.
Hoboken, NJ, Wiley 2003.
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