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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.

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