JOINT BAYESIAN ESTIMATION OF DOA AND FREQUENCY OF BPSK (BINARY PHASE SHIFT KEYING) SIGNALS IN &-STABLE NOISE

B. Kannan and U! J. Fitzgerald Signal Processing Laboratory, Department of Engineering

University of Cambridge, Cambridge CB2 lPZ , UK e-mail: bk212@eng.cam.ac.uk, phone/fax: (44 01223) 332767/332662

ABSTRACT

The signal processing literature has traditionally been dom- inated by Gaussian noise model assumptions. In this paper, we present a Bayesian approach for DOA (Direction of Ar- rival) and frequency estimation of narrow band signals in additive a-stable noise. Approximating the a-stable noise by Gaussian mixture enables us to use the Bayesian tech- niques to derive a posterior probability density for DOA and frequency parameters from the signal and noise mod- els. This posterior probability is then used in the Metropo- lis-Hasting algorithm to derive samples for the DOA and frequency parameters. The mean square errors (MSEs) of the parameters are compared with ROC-MUSIC (Robust Covariation MUSIC) algorithm. This new algorithm can be used with a significantly lower number of received data to estimate the parameters.

1. INTRODUCTION

Sensor array processing has found important applications in many areas such as radar, sonar, communications and seismic explorations. Determination of DOAs and the fre- quencies of the transmitted signals is one of the main prob- lems in sensor array processing. Various methods have been proposed for estimating DOA and associated parameters for multiple plane-wave signals incident on an array of sensors. These include subspace-based methods [ 1,2] and maximum likelihood techniques [3]. The signal processing literature has traditionally been dom- inated by Gaussian noise model assumptions. However, many classes of noise encountered in the real-world such as underwater acoustic noise, low frequency electromag- netic disturbances, atmospheric noise etc., exhibit outliers that will not fit into a Gaussian noise model. There are sev- eral distributions such as log-normal distribution, Gaussian mixtures, generalised Gaussian models, Cauchy distribution etc., that are used in the literature to model the non-Gaussian noise. Among all these distributions, the a-stable distribu- tion proves to be a more efficient model.

2. PROBLEM FORMULATION In this section, we define the signal, noise models and the array structure. We assume several sources in the far field transmitting narrow band bpsk signals (with a centre fre- quency fa) and the received data at an Uniform Linear Ar- ray (ULA) of sensors are corrupted by additive zero mean symmetric a-stable ( S a S ) [4] noise. In the derivation of the posterior probability, we also assume that we know the number ( M ) of signal sources. Let the ULA have L sensors with an inter-distance d (5 M2, where X is the wave length of the signal). We can represent the received data x(k) in terms of the DOA steering matrix A($), signal vector s(k) and noise vector v(k) as

x(k) = A(q5)s(k) + v(k) for IC = 1,2, ..., N ( 1 )

where x(k) and v(k) are L x l vectors. A(4) is a L x M matrix

A(4) = [a($i) a($2).......a(4~)] (2)

where a(h) is a L x 1 steering vector and can be written as

a($i) = [I cos (h ) .... C O S ( ( L - l)di)lT (3)

with

$i = fOdsin(Oi)/c fori = 1 ,2 , ..., M (4)

where c and Oi are the speed and DOA of the ith signal respectively. s(k) is a M x 1 vector

s (k) = [Sl(k) , s2(k), * . * l SM(k)I* (5 )

with s j ( k ) (j E {1,2, ..Ad}) representing the j t h signal from the far field at time IC. All the transmitted signals are BPSK signals which are modulated by raised cosine pulses. Taking into account all the received data in x(k), s(k) and v(k) for IC = l...,N, we can modify ( 1 ) as

X = A($)S + V (6)

where X, S and V are L x N , M x N and L x N matrices respectively.

0-7803-5682-9/99/$10.00019999 IEEE.

3. MODELLING a-STABLE NOISE BY GAUSSIAN MIXTURES

The characteristic function [4] of an a-stable distribution is represented by:

$(w) = exP{jP - 7 I w Icy [I+ jPsign(wP(w7 (.)I) (7)

where

The parameters a, ,B, y and p are defined as follows: 0 Q (0 < a 5 2) is called the characteristic exponent. A

large value for cy indicates a light tail.

0 /3 (- 1 5 ,B 5 1) is called the symmetry parameter. /3 = 0 corresponds to a symmetric distribution ( S a S ) . ,B < 0 and /3 > 0 correspond to the negatively skewed and positively skewed distribution respectively.

0 p (-00 < p < co) is the location parameter.

0 y (0 < y 5 00) is called the dispersion parameter and is a measure of the deviation of the distribution around the mean.

An a-stable distribution can be approximated by a finite- Gaussian mixture [5] as shown below:

K

where N(n , p, ai) denotes a Gaussian distribution with mean p and standard deviation ai. ri is the scaling fac- tor of the i th component. Given the parameters a, /3, y, p and the number of Gaussian components K , we can use the algorithm in [5] to estimate the ri and oi (for i = l...K). Figure( 1) shows the Gaussian mixture approximation of an S a S for a = 1.1 and y = 0.2. In this paper we assume that we know the parameters of the a-stable noise. Even if we don't know the noise parameters, we can use the algorithm in [6] to estimate them.

4. METROPOLIS-HASTING ALGORITHM

We use the Metropolis-Hasting (M-H) [7] algorithm to get the samples of the marginal posterior probability densities of the DOA vector 8 and frequency vector f from their joint posterior probability density function. The energy @(z) of a posterior distributionp(z I X,M) is defined by

+(z) = -Zlog(p(z I x,M)), z =

Figure 1. Gaussian mixture approximation when 0 = 1.1 and y = 0.2

The Metropolis algorithm can be defined by the following steps:

0 Given the current parameter vector z j at the j t h itera- tion, draw the parameter vector zit: at the ( j + l) th iteration from a symmetric proposal density.

0 Evaluate the energy difference S = @(zj) - @(zi:h)

0 If S 2 0, then assign zj+l = zj+l new'

0 If 6 < 0, then assign zj+' = with probability e6 or assign zj+' = z j with probability (1 - e6).

5. DETERMINATION OF POSTERIOR PROBABILITIES

In this section, we will derive the joint posterior probability densities for frequencies and DOAs for the SaS noise. From Bayes' theorem [8] , we can write the posterior probability density of the parameter vector z (which characterises both the signal and noise models), given the data matrix X and the model order M , as

where z = [zl ... z,], p(z I M ) is the a priori probability density of the parameters, p(X I z, M) is the likelihood function and p(X I M ) is the evidence. Let our parameter vector z be [e, f , b, r, a]. Here b, f and 0 are M x 1 vectors representing the amplitude, frequency and DOA components of M signals respectively. U and r denote the standard deviation vector and the scaling vector of the Gaussian mixture respectively. Then from ( 6 ) and (8), the likelihood function for the received data can be written as

where

Assuming a uniform prior for 8 and f , we can write the negative logarithm of the posterior distribution of 8 and f given all other parameters as

j=1 k = l i=l

6. SIMULATION AND DISCUSSION Various simulations were performed to investigate the per- formance of our method and the results are compared with ROC-MUSIC algorithm. In this simulation, we use two BPSK signals which are modulated by raised cosine pulses. The carrier frequencies of the signals are f l = 7kHz and f 2 = 9 kHz respectively. The corresponding DOAs are 81 = 50" and 82 = 40" respectively. We also assume that the center frequency f o of the signals is 1OkHz. The samples of the frequencies and the DOAs are drawn using the Metropolis- Hasting algorithm. The MSEs of the parameters are cal- culated for various characteristic exponent and dispersion parameters. We use the received data at the first array element to get the initial values for the frequencies in order to have a guaran- teed convergence towards the optimum samples. As we are dealing with narrow band signals, a two-dimensional Gaus- sian probability density with a mean value of [ f o , f o ] is used as the proposal density for frequencies f . Another two di- mensional Gaussian distribution is used as the proposal den- sity for 6. Variances of these proposal distributions are cho- sen according to the steepness of the energy function near the optimum parameter values. For each cy (or y), we performed 100 (=P) Monte Carlo runs taking 1000 samples for each of the parameters. The MSE of the 6 j is calculated as

l P MSE(8,) = - C (6, - iji)' where j = [l, 21 (14)

In ( 14), 6, is the target DOA of the j t h signal and 6ji is the estimated mean value for the DOA of the j t h signal at the ith Monte Carlo run.

i=l P

6.1. Effect of Q on parameter estimation In order to study the effect of Q on our estimation, we vary a while fixing the value of y at 0.2. Using @sols as the en- ergy function, we applied the above MCMC algorithm for 300 received data and 8 array elements. Figure (2) shows the behaviour of the MSE vs cy of the DOA of the first and

Figure 2. Effect of the a on DOA estimation

the second sources. With the same number of received data and ULA, we repeated the simulation for the ROC-MUSIC algorithm. The ROC-MUSIC algorithm failed to resolve the two DOAs in any of the 100 Monte Carlo runs. Figure (3) shows the behaviour of the MSE of the frequency estimates vs Q. As expected, the MSEs are decreasing when a is in- creased.

Figure 3. Effect of a on frequency estimation 2 I

Figure 4. Effect of y on DOA estimation

6.2. Effect of y on parameter estimation We have repeated the simulation for an ScyS noise (a=1.3) with 300 data and 8 array elements using @sas as our en- ergy function. The dispersion factory is varied from 0.2 to 1 to produce different impulsive noise.

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As before, the MSE of estimated DOA and frequency of the first and the second sources vs are plotted in figure (4) and figure (5) respectively. We can see from these figures that the MSEs are decreasing when y is decreased. When we repeated the simulation with ROC-MUSIC [1] algorithm, it failed to resolve both of the DOAs even when y is equal to 0.2.

Figure 5. Effect of y on frequency estimation

7. CONCLUSION In this paper, we presented a Bayesian approach for joint es- timation of DOAs and frequencies of BPSK signals in S a S noise. Although we used BPSK signals in this paper, we can use this algorithm with any of the standard modulated sig- nals. Unlike the conventional subspace-based methods, this new Bayesian approach can be used with a smaller num- ber of training symbols to jointly extract the DOAs and the frequencies of the transmitted signals in additive a-stable noise. We propose to extend this method to detect the num- ber of signals in an unknown impulsive noise environment in addition to parameter estimation.

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C. Molina E. E. Kuruoglu and W. J. Fitzgerald, “Ap- proximation of a-stable probabilty densities using fi- nite mixtures of gaussians,” in Proceedings of EUP- SICO’98, September 1998. George A. Tsihrintzis and Chrysostomos L. Nikias, “Fast estimation of the parameters of alpha-stable im- pulsive interference,” IEEE Transactions on Signal Pro- cessing, vol. 44, no. 6, pp. 1492-1503, June 1996. 0. Stramer and R. L. Tweedie, “Self-targeting candi- dates for Metropolis-Hasting algorithms,” Technical Report, Univeristy of Iowa and Colorado State Univer- sity, Jan. 1998. J. J. K. 0 Ruanaidh and W. J. Fitzgerald, Numer- ical Bayesian methods applied to signal processing, Springer- Verlag, New York, Inc., 1996.

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