applebaum adaptive beamformer with steering vector estimation for ciclostationary signals

7/28/2019 Applebaum Adaptive Beamformer With Steering Vector Estimation for Ciclostationary Signals

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APPLEBAUM ADAPTIVE BEAMFORMER WITH STEERING VECTORESTIMATION FOR CICLOSTATIONARY SIGNALS

Jos Ramn Cerquides Bueno - Juan Fernndez Rubio

Signal Theory and Communications DepartmentPolytechnic University of CataloniaE.T.S.I. Telecommunications

Barcelona - SPAINApdo. 30002

Tel.(34) 3 401 64 58Fax:(34) 3 401 64 47

E-mail: [email protected]

ABSTRACT

The Applebaum beamformer is well known in the field of array signal processing. It can bemathematically shown that, if the correct steering vector is known by the system, the Applebaum loopcan achieve a maximum Signal to Interference-plus-Noise Ratio (SINR). However, little deviationsbetween the values of the ideal and true steering vector result in severe penalizations in the outputSINR. Our proposal is to estimate the real value of the steering vector directly from the receivedsnapshots of data. To this subject, properties of cyclostationarity exhibited in the most commonly usedcommunication signals are extracted and exploited, together with a direct iteration method in order torecover the eigenvector associated with the greatest eigenvalue without needing to compute the cyclic

autocorrelation matrix. The result is an easy to implement, non-computationally complex system,which shows a fast convergence and low sensitivity to estimation errors.

I. Introduction

An adaptive antenna is composed by several sensors located at different spacepositions. One signal is extracted from each one of these sensors. The output of the system isthen constructed as a linear combination of these simple signals, as shown in figure 1.

where xi(n) are the samples of the signalreceived by the ith sensor, and wi is the complexweight assigned to this sensor. Its very usefulto make use of a vectorial notation, where:

w(n)=[w1(n),w2(n),..,wN(n)]T

is the complex weight vector and

x(n)=[x1(n),x2(n),..,xN(n)]T

x

y

z

r

r

r

1

2

N

#1

#2

#N

#1

#2

#N

w

w

w

y

1

2

N

x

x

xN

2

1

*

*

*

Fig. 1

y(n) = w x (n) = w x(n)i=1

N

i* i H


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is the vector of received data. From the antenna point of view, it's equivalent to have anantenna whose reception pattern can be easily changed by adjusting the values of theweights assigned to the sensors.

The problem of beamforming can be stated as finding the optimal values of theseweights in order to optimize (i.e. maximize or minimize) some previously defined costfunction.

Several solutions have been suggested, being able to establish a classification into twolarge groups, attending to the characteristics of the information provided to the system forrejecting undesired interference or noise.

First, we can find the time reference beamformers (TRB), which employ a temporalreference to recover or distinguish between desired and interfering signals. The mainproblem that appears in this kind of beamformers is the need of transmit this reference. Itdoesnt bother the method you choose for the transmission, efficiency of thecommunications system is decreased.

Another possibility is to include in the beamformer information related with thespatial environment, (i.e. Direction Of Arrival (DOA) of the signals). The main advantage isthat no loss of efficiency results. However, as we shall discuss later, these kinds of

beamformers are not free of drawbacks in practical implementations.

Blind beamforming, on the other hand, works by restoring some property derivedfrom the signal structure (i.e. Constant Modulus Algorithm). Usually, it implies a slowerconvergence, and gives place to the appearance of a new problem, known as the "captureproblem" which happens when both signals (desired and interfering) exhibit the same

properties.

Most digital communications signals show a ciclostationary behavior, being able toexploit this fact in order to distinguish between desired signal and interference, thusreducing the capture problem. Our proposal is to take profit of these properties to determinespatial information for the desired signal, and then use this information into an Applebaumloop, the simplest beamformer driven by spatial conditions.

II. Signal model and generalized steering vector

The signals collected by the sensors depend, in general, of its position, its receptionpattern, the angle of incidence and the propagation conditions, as shown in figure two.

In narrowband arrays, and assumingfar field propagation, all the previouslymentioned effects can be recollected in a singlevector with complex elements. This vector iscalled the generalized steering vector and can

be written as:

Fig. 2

v= c a

i=1

P

i i


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being P the number of paths, ci the ith path complex amplitude, and ai the D.O.A. vector

associated to the ith path.

Each one of the impinging signals has its own generalized steering vector. The modelused for the snapshots is then given by:

where:v Generalized steering vector for the desired signals(n) Desired signal

A Matrix of generalized steering vectors for the interfering signalsi(n) Interference's vector

r(n) Additive gaussian noise

III. The Applebaum beamformer

The Applebaum beamformer (fig. 3) isthe simplest structure that makes use of spatialinformation about the desired signal. To leadthe weights to his optimal value, the objectivetaken is to minimize the power of the outputsignal, while the gain of the antenna is fixed forsome generalized steering vector, which may

be known in advance, at the design stage. Themathematical formulation is very popular, andwe will use here only the final equation, whichshows how the weights must be updated:

where v is the generalized steering vector,

and is a parameter that controls theconvergence of the algorithm. To ensure it, is usually taken as (n)=/Pt(n) where

Pt(n)=xH

(n)x(n) is a estimation of the total power received, and is a normalized factor which

must be chosen in the range 0


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IV. Cyclic autocorrelation matrix and steering vector

It's said that a signal x(n) is ciclostationary (in a wide sense) if its mean x(n) and itsautocorrelation are both periodic functions in n with periodM, that is:

These properties hold for many communications signals, especially those used in digitalcommunications. However, an important requirement must be accomplished: the signalmust be oversampled (that is, more than a sample per symbol is needed). Our interest isessentially put in the periodicity shown by the autocorrelation function, becausecommunication signals usually show zero mean.

Thus, given a periodic autocorrelation function Rxx(n,m), it's possible to express it as aFourier series development. That is:

where the coefficients of the development are the so-called cyclic autocorrelation functionswhile the parameter kplays the role of a digital frequency that we will call cycle frequency.The function Rx(m) can be obtained as:

Plots of the magnitude of Rxx(n,m) and Rx(m) for a BPSK baseband signal with ninesamples per period are shown in figures 4 and 5 to help clarify concepts.

It's interesting to note that, when k=0, the cyclic autocorrelation function is equal tothe classical autocorrelation. For other values ofk, the function has valuable information thatdoes not appear in the classical autocorrelation function. Another interesting consideration isthat the cyclic autocorrelation function of a stationary signal will only take value other thanzero when k equals 0. The same happens when the function Rx(m) is computed for aciclostationary signal whose period (say L) is different fromM (exceptions must be done for

some values ofkin the special case ofM being an integer multiple ofL).

Fig. 4. Plot ofRxx(n,m)

x x

xx

* *

xx

(n) = E{x(n)} = E{x(n + lM)}= (n + lM)

R (n,m) = E{x(n)x (n- m)}= E{x(n + lM)x (n+lM-m)}= R (n + lM,m)

xx

k=0

N-1xxk j

2 kn

MR (n, m) =1

NR e(m)

xxk

n=0

N-1

xx-j

2 kn

Mn=0

N-1* -j

2 kn

MR (m) = R e x(n)x (n-m)e(n,m) = E


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Thus, returning to the field of arrays,the cyclic autocorrelation function can be usedas a powerful tool to difference betweendesired signal and interfering ones. Let'scontinue with our previous model for thereceived signals. If we make the next set ofassumptions:

1.- The cyclic autocorrelationfunction of s(n) for some set ofvalues ofM, kand m is not zero(some of these values will bechosen by design, while othersmay be available at the synchronization circuits).

2.- The cyclic autocorrelation function for the interfering signals and the noiseis zero.

These assumptions can be expressed mathematically in the following way:

where Iis the total number of interferences, Nis the number of sensors, Ps is the mean power

of the desired signal, and s(m) is a normalization parameter that measures the quantity ofcyclostationarity of the desired signal. We will call it cyclic autocorrelation coefficient.

The cyclic autocorrelation matrix can be defined as:

Substituting the signal model previously stated in this equation and making use of theestablished assumptions leads to:

showing that the cyclic autocorrelation matrix is a rank-one matrix if k,m and M areconveniently chosen. Another important fact to remark is that the eigenvector associatedwith the only non-zero eigenvalue is exactly the generalized steering vector of the desiredsignal.

V. Estimation of the generalized steering vector

Unfortunately, it's impossible to dispose of the cyclic autocorrelation matrix in thereceiver, because it has been defined as the expected value, that is, taking a average over therealizations. However, if the signals involved are cycloergodic (and this condition holds for

Fig. 5. Plot ofRx(m)

ssk

ssk

i ik

n nk

R (m) = (m)Ps 0

R (m) = 0 q,r =1 I

R (m) = 0 q,r = 1 N

k 0

m,M

q r

q r

fixed

xx

k

n=0

N-1H -j

2 kn

MR (m) = E x(n) x (n-m)e

xxk H

ssk

sR (m) =vv (m)P


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all the communications signals), it's possible to estimate this matrix from the data that isbeing received.

One possibility (although it does not correspond to the maximum likelihoodcriterion) that allows to take into account only the most recent samples is to take an iterativeestimator of the form:

where is called the forgetting parameter.

An easy way of obtaining the value of the generalized steering vector v from theestimated cyclic autocorrelation matrix is given by the direct iteration method. In order toreduce the computational load of the algorithm the next solution is proposed:

1. Accept that [Rx(m)]n can be decomposed in the following form:

where the indexes k and m have been removed on the right hand of theequation only for notational simplicity purposes. Vector vn is assumed to benormalized in some form (i.e. vvn=1 )

2. We do now compute the matrix [Rx(m)]n+1, and project the vector vn over thismatrix. The new vector obtained is called un+1:

3. Normalization ofun+1 yields the values of the new steering vector vn+1 and the

associated eigenvalue n+1:

Under the assumptions taken in the model, the so constructed succession (vn,n)converges to (v,) as n grows.

The final weight adaptation equation can be definitely written as:

VI. Simulations and results

Array geometry: Linear equally spaced array with 6 sensors.

[ ] [ ]n+1xxk

nxxk H -j2 kn

MR (m) = _ R (m) + (1- )x(n) x (n-m)e

[ ]nxxk

n nH

nR (m) v v_

n+1 n n

H -j2 kn

M nu = v + (1- )x(n) x (n-m)

ev

n+1 n+1

n+1n+1

n+1

= u

v =u

_ _

( )H Ht

n*w (n +1) = w (n)+

P (n)v -x(n) y (n)


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Signal scenario: The desired signal is a BPSK signal, which is received inpresence of a interfering unmodulated carrier and additive whitegaussian noise.

Signals parameters:

Desired signal:

BPSK signal with 3 paths.

PATH 1 2 3

SNR 5 dB -5 dB 0 dB

D.O.A. 0 30 20

Number of samples per symbol:M=2

Cyclic parameters chosen: k=m=1

Interfering signal:

Unmodulated carrier.Angle of arrival: -20 dBINR=5 dB

Other parameters of the simulation:

=0.99=0.1v

0=[1,0,0,0,0,0]T Initial estimated steering vector

0=1; Initial estimated eigenvalue

w(0)=[1,0,0,0,0,0]T Initial weight vector

The results of the simulation are shown in figs. 6 and 7. Graph 6 shows the evolutionof the output SINR, while graph 7 corresponds to the reception pattern after 500 samples.

VI. Conclusions

Fig. 6. Evolution of the output SINR Fig. 7. Reception pattern after 500 samples


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A new method of beamforming has been developed. It exploits the fact that, when thesignal subspace has rank one, the only existing signal eigenvector coincides with the desiredsignal generalized steering vector. Cyclostationary properties have been used to constructthe cyclic autocorrelation matrix, allowing us to make disappear information related withthe interfering signals and noise.

The direct iteration method is used to determine the eigenvector of interest, and has

been formulated in a such a way that avoids the need of computing the cyclicautocorrelation matrix, with the consequent reduction of computational load.

The algorithm does not depend on the geometry of the array or the sameness of thereception patterns, and its independent of the phase and magnitude deviations betweendifferent channels.

Finally, in the simulations, the algorithm has proven to have a fast convergencetowards the optimal value of the weights and very low sensitivity to instantaneousestimation errors of the steering vector.

applebaum adaptive beamformer with steering vector estimation for ciclostationary signals

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