on the performance of the constant modulus array restricted to the signal subspace
TRANSCRIPT
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7/28/2019 On the Performance of the Constant Modulus Array Restricted to the Signal Subspace
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ON THE PERFORMANCE OF THE CONSTANT MODULUS ARRAY
RESTRICTED TO THE SIGNAL SUBSPACE
J.R. Cerquides and J.A. Fernndez-Rubio
Signal Theory and Communications Department, Polytechnic University of Catalonia
Mdulo D5, Campus Nord UPC, C/Gran Capitn s/n, 08034 Barcelona, SPAIN
Tel.:+34-3-4015938, Fax:+34-3-4016447
E-mail: [email protected]
ABSTRACT
The Constant Modulus Array has a slow rate of
convergence mainly due to both the nonconvex nature
of its cost function and the well known behavior ofstochastic steepest descent algorithms for environments
with a large eigenvalue spread. In this paper we analyze
the solutions of the Constant Modulus Cost Functions,
showing that the weight vectors associated to its
minima lie on the signal subspace. From that
information we develop a modified version of the
Constant Modulus Array, which speeds up the
convergence and reduces the final misadjustment error.
The proposed method is specially useful for arrays
having a large number of sensors and low Signal to
Noise Ratio for the Source of Interest.
1. INTRODUCTION
The Constant Modulus Array (CMA) was introduced
in 1986 by Gooch and Lundell [1], who suggest to
apply the Constant Modulus (CM) criterion originally
designed by Godard [2] to the field of adaptive antenna.
Due to its interesting properties (low computational
load, independence of the array manifold, etc.) it has
become probably the most popular blind beamforming
scheme. However, it is far from being free of
drawbacks. One of its main inconvenient is its slowconvergence, which sometimes makes it inapplicable in
practical environments, specially when the Signal to
Noise Ratio of the incoming signals is poor. This is the
problem we address in our paper. As we will show, the
This work has been partially supported by the National
Research Plan of Spain CICYT, under Grant TIC-95-1022-
C05-01
analysis of the extrema of the cost function will reveal
useful information about the location of the minima and
its relationship with the signal subspace. This fact can
be exploited to speed up the convergence of the
algorithm.
The paper is organized as follows: in section 2 we
review the constant modulus cost functions, introducing
the appropriated vectorial notation; section 3 is devoted
to analyze the nature of the solutions and its
consequences. In section 4 we propose a novel
technique exploiting the results of the performed
analysis and having in mind to avoid some of the
undesirable properties of the original proposal. Section
5 shows the relationship between the proposed
technique and the Generalized SideLobe Canceller
(GSLC). Simulation results of the suggested algorithm
compared with other approximations are shown in
section 6. Finally we end the paper by paying attention
to special scenarios where the proposed algorithm can
be specially useful.
2. THE CONSTANT MODULUS ARRAY
The CMA is one of the simplest blind beamforming
scheme. Although, as happens with Sato and Decision
Directed (DD) algorithms, it can be seen as a particular
case of the more general family of Bussgangalgorithms, it was developed independently by Godard
and later applied to array processing by Gooch and
Lundell. The algorithm tries to minimize a nonconvex
cost function designed to penalize deviations in the
envelope of the array output signal:
[ ]( )J E y n=
2 2
1 (1)
being y[n] the output of the array, obtained as a linear
combination of the received signals,
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[ ] [ ]y n nH= w x (2)where x[n] is the received snapshot and w is the weight
vector, which describes the spatial response of the
array. SuperindexH
hold for the hermitian operator, i.e.:
transpose and conjugate.
The underlying idea under the mathematicalformulation is very simple: the Source Of Interest (SOI)
is assumed to have the constant envelope property1.
This property is lost due to the contribution of noise
and/or interfering processes to the output signal y[n].
The algorithm tries to indirectly remove noise and
interferences by restoring the loss property.
The optimization procedure follows a single, LMS
like, stochastic steepest descent algorithm which yields
a weight vector adaptation equation given by:
[ ] [ ] [ ]( ) [ ] [ ]w w xn n y n y n n+ = 1 12
*
(3)where is the step-size parameter, whose choice is acommitment between convergence speed and
misadjustement noise. Probably the more useful result
about the optimum value of is given by Katia andDuhamel[3], who suggest to select it as:
[ ] [ ] [ ]( ) =
+0 2
1 1
1x n y n y n(4)
where 0 is the normalized step-size parameter, whichmust lie in the open (0,1]. The resulting algorithm is
called the Normalized CMA (NCMA).
3. SOLUTIONS OF THE CONSTANT MODULUS
COST FUNCTION AND ITS NATURE
In a general environment, the snapshot x[n] will be
composed of two terms,
[ ] [ ] [ ]x d nn s n nm mm
M
= +=
1
(5)
where M is the number of incoming signals, dm is the
generalized steering vector (including possible
multipath effects), sm[n] represents the m-th signal and
n[n] models the noise (usually thermal noise) generatedat the sensors. Under this assumption, y[n] can be
rewritten as:
[ ] [ ] [ ] [ ] [ ]y n s n n g s n n nH
m m
H
m
M
m m y
m
M
= + = += =
w d w n1 1
1This property is shared by many manmade
communications signals (i.e.: PSK and FSK modulations,
among others).
(6)
The set of definitions gm = wHdm (m = 1..M) and ny[n] =
wHn[n] is implicit in the above equation. Susbtituting
(6) into (1) and manipulating the expression, we can
finally find:
( )J f g g k k PM M M n= 1 1 1
, , , (7)where km (m = 1..M) represents the kurtosis
2of the m-th
signal, m (m = 1..M) is the standard deviation of the m-th signal, and Pn is the noise power at the array output,
Pn
H
nn= w R w (8)
To determine the behavior of the CM algorithm we
need to find the extrema of J, solving the following
vectorial equation:
=w
0J (9)
However, taking into account expression (7) and the
relationship between the set of coefficients (g1...gM,Pn)
and the weight vector w, it is possible to apply the chainrule to equation (9) obtaining:
= + =
= + =
=
=
w w w
d R w 0
JJ
gg
J
PP
J
g
J
P
m
m
n
n
m
M
m
m
n
nn
m
M
1
1
(10)
Equation (10) has two sets of solutions:
1. IfJ/Pn = 0, then the first term must also be zero,but if we assume that the generalized steering
vectors dm are linearly independent, the only valid
solution is then given by J/gm = 0 for all m. It ispossible to demonstrate that this condition implies
gm = 0 for all m, and consequently, the weight vector
associated to this solution lies completely in the
noise subspace. Also, it is not difficult to
demonstrate that this solution is a minimum if and
only if all the incoming signals show a kurtosis
larger than two, which is not the case for
communications signals. Thus, this solution may be
catalogued as an unwanted one.
2. IfJ/Pn 0, then we can rewrite eq. (8) to read asfollows:
w
R d
R d= =
=
=
J
g
J
P
cmnn m
m
M
n
m nn m
m
M
1
1 1
1
(11)
2The kurtosis of a signal is defined as the quotient between
its fourth order momentum and the squared second order
momentum.
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From observation of eq. (9) we can conclude that
this set of solutions result in a linear combination of
the eigenvectors associated to the signal subspace. A
special case of interest is encountered under the
classical hypothesis Rnn = n2I. In this situation, the
desired solutions of the CM algorithm are linearcombinations of the generalized steering vectors of
the incoming signals.
4. THE PROPOSED ALGORITHM
From the performed analysis it is obvious that we can
directly avoid unwanted solutions if the number of
sources present in the signal scenario is
"approximately" known. We have quoted the word
"approximately" because our proposal does not need to
know exactly the number of incoming sources. It is
enough to provide information about the maximumexpected number of simultaneous sources. We will
denote this number by S (S M).
Under this condition the proposed algorithm is
summarized as follows:
1. Estimate Rxx=E{x[n]xH[n]} from the received data.
2. Solve the generalized eigenvalue problem described
by Rxxvi = iRnnvi for all i=1..N, being N the numberof sensors.
3. Extract the N-S less significant eigenvectors andform the new matrix:
[ ]V v v v= + +S S N1 2, , , (12)where we assume that the eigenvectors vi have been
previously normalized in step 2.
4. Solve the linearly constrained problem:
[ ]( )min E y n H nnw
w R V 0J subject to=
=2
2
1
(13)
where the introduction of a set of N-S restrictions
will improve the convergence rate of the adaptive
process.
The developed algorithm is designed to work overblocks of data rather than on a sample by sample basis,
although it is possible, if required, to follow an adaptive
procedure for obtaining the eigenvectors of the data
correlation matrix [4].
5. REFORMULATION OF THE PROPOSED
TECHNIQUE AND THE GSLC
Equation (13) yields a constrained optimization
problem. A first approach is to employ the Frost
algorithm, preprocessing all snapshots to remove
components lying in the subspace spanned by V.However, this approach does not exploit the subspace
rank reduction to reduce the number of computations. A
general solution to the optimization of the Constant
Modulus Cost Function given some linear restrictions
was given by Griffiths[5]. However, in our special case,
as all restrictions are equal to zero the formulation can
still be simplified. By having in mind the structure of
the GSLC beamformer, shown in figure 1, and taking
into account how it works, it is clear that the simplest
possible choice for w0 is:
w 00 = (14)avoiding the need to perform any computation related
with the upper branch of the beamformer. The blocking
matrix B must be orthogonal to the restrictions. If,
under typical conditions, the noise is assumed to be
uncorrelated between sensors, having equal power in
all of them, the orthogonality condition for B can be
written in terms ofV as:
B V 0H = (15)
Thus, the columns ofB must lie in the signal subspace,
and B can be chosen as:
[ ]B v v v=
1 2
, , ,S
(16)
where vi is the i-th most significant eigenvector ofRxx.
B wa
w0x[n]
z[n]
ya[n]
y0[n] y[n]
#N #S
x1[n]
x2[n]
xN[n]
Figure 1 - Structure of the GSLC beamformer
Steps 3 and 4 of the algorithm proposed in section 4
must be modified according to the new formulation.
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3. Extract the S more significant eigenvectors and formthe matrix B as described in eq. (16).
[ ]B v v v= 1 2, , , S (17)
4. For every new snapshot, x[n],
Project it over the signal subspace to obtain z[n],[ ] [ ]z B xn n
H= (18)Update the weight vector wa[n] following:
[ ] [ ][ ]( )[ ]
[ ]
[ ][ ]w w
zz
0a an n
y n
n
y n
y nn+ =
1
1
2
*
(19)
where the normalized version of the CMA is preferred
for the adaptation process.
6. SIMULATIONS
The signal scenario chosen is shown in table 1. The
selected array is linear, having 30 omnidirectional
sensors. Distance between two of them is half
wavelength.
# of signal Signal type Angle of arrival Input SNR
#1 4-PSK 30 -5 dB
#2 8-PSK 0 -5 dB
#3 Tone, f=0.1 20 10 dB
Table 1 - Signal scenario for the simulations
In figure 2 we can observe the evolution of the output
SINR for both algorithms, proposed and classical
NCMA, when they are optimally initialized. The signal
subspace method is several times faster than its
unconstrained version.
0 200 400 600 800 1000 1200 1400 1600 1800 20003
4
5
6
7
8
9
10Evolution of the output SINR
# iterations
dB
Proposed
NCMA
Optimum
Figure 2 - Evolution of the output SINR for both algorithms:
proposed vs. NCMA
It is difficult to notice, in the representation of the
output SINR, the evolution of the weight vector and the
final misadjustement. The proposed technique also
achieves more precise reception patterns than the usual
NCMA. This fact is shown in figure 3, where both
algorithms are initialized to the optimum weight vector.
The plot shows the error, computed as the distance
between the optimum and the actual weight vectors for
both algorithms.
0 500 1000 1500 2000 25000
0.5
1
1.5
2
2.5x 10
-3
# iterations
Proposed
NCMA
Figure 3 - Distance between the weight vectors and the
optimum
6. CONCLUSIONSThrough the analysis of the solutions of the Constant
Modulus Cost functions we have developed a modified
version of the NCMA algorithm, which exploits the fact
that the optimum weight vector lies on the signal
subspace of the autocorrelation matrix of the snapshots.
The proposed technique speeds up the evolution of the
adaptive beamformer towards the optimum solution,
showing better properties once the algorithm has
converged.
Although the computational load of the proposed
method is higher than in NCMA, there are severalinteresting cases where the suggested algorithm is
specially useful:
when the data set available for beamforming is small for arrays composed by a great number of sensors when convergence speed becomes a critical
parameter in spite of computational load
In a forthcoming paper we will make a full detailed
computational balance of both methods, obtaining
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expressions for the excess of error introduced in each
case.
7. REFERENCES
[1] Godard, D.N., "Self recovering equalization andcarrier tracking in two- dimensional data
communication systems", IEEE Transactions on
Communications, vol. COM-28, pp. 1867-1875,
Nov. 1980.
[2] Gooch, R.P. and Lundell, J.D., "The CM array: an
adaptive beamformer for constant modulus
signals", Proceedings of the International
Conference on Acoustics, Speech and Signal
Processing, Tokyo, pp. 2523-2526, Apr. 1986.
[3] Hilal, K. and Duhamel, P., "A convergence study
of the constant modulus algorithm leading to a
normalized CMA and a block-normalized CMA",
Proceedings of the European Signal Processing
Conference, Bruselas, Belgica, pp. 135-138, Aug.
1992.
[4] Yang, J.F., Kaveh, M., "Adaptive eigensubspace
algorithms for direction or frequency estimation
and tracking", IEEE Transactions on Acoustics,
Speech, and Signal Processing, vol. ASSP-36, pp.
241-251, Feb. 1988.
[5] Rude, M.J. and Griffiths, L.J., "A linearly
constrained adaptive algorithm for constant
modulus signal processing", Proceedings of the
European Signal Processing Conference, vol. I.,
pp. 237-240, Barcelona, Sep. 1990.