10th International Conference on Information Science, Signal Processing and their Applications (ISSPA 2010)

MODEL BASED A DAPTIVE DETECTION A LGORITHM WITH LOW SECONDA RY DATA SUPPORT

Abbas Sheikhi, Ali Zamani, Majid Hatam, and Mahmood Karimi

Electrical & Electronics Eng. Dept., Shiraz Uniersity, Shiraz, Iran.

phone/fax: +(98) 711-2303081, email: azamani@shirazu.ac.ir

ABSTRACT

In this paper the problem of adaptive target detection in

structured Gaussian clutter is considered. The clutter is

modeled as an auto-regressive process with known order

but unknown parameters. To solve this problem, we have

modified a well known adaptive detector (Kelly's GLRT)

in four different forms. In this detector an estimation of

covariance matrix is needed. In order to estimate the co­

variance matrix, we estimate the AR parameters based on

secondary data and use the results in covariance matrix

estimation. Then, we use the estimated matrix in the de­

tector structure. In order to estimate the AR parameters

using more than one set of data, we have extended four

classical AR parameter estimation techniques to use more

data sets. The performance of the proposed detectors have

been evaluated using Monte-Carlo simulations and com­

pared with each other.

1. INTRODUCTION

Adaptive target detection in Gaussian disturbance is a prob­

lem that has been solved in different conditions and dif­

ferent approaches [1, 2, 3]. There are two general classes

of target detection algorithms that are based on statisti­

cal properties of interference returns. In the first category

we consider the case that there is no information avail­

able about interference structure. In the second one, there

is some information available about interference structure

and so we can consider a model for the interference. The

second class is named as Model-based detection.

Due to the advantages of Model-based detection algo­

rithms, we usually prefer them to the other types of de­

tectors. The first benefit of these detectors is that if the

received data matches the assumed model for the prob­

lem, then the performance of these detectors would be

better than the opponent. However, if the data does not

match, the results may be undesirable. The second ad­

vantage of these detectors is that they need less secondary

data to reach the desired performance. It is worth noting

that in some problems, non Model-based detectors can not

be used when the number of secondary data is less than

a minimum. Furthermore, in Model-based methods, the

number of parameters that should be estimated is reduced

and so we can achieve good estimation performance with

less available data [2].

One of the most practical models for interference in

978-1-4244-7167-6/10/$26.00 ©2010 IEEE 177

Model-based detection is Auto-Regressive (AR) model. It

is shown in [4] that in many situations of practical interest,

the interference can be modeled as an AR process of low

order M. AR processes are also important since it is well­

known that any wide-sense stationary Gaussian process

with rational power spectral density can be modeled as

an AR model with proper order. Hence, in this paper the

problem of adaptive target detection using coherent pulse

train in AR clutter is considered.

This paper is organized as follows: The problem for­

mulation is developed in Section 2. This formulation is

used in sections 3 and 4 to derive the detection algorithms.

Section 5 is devoted to discussions on simulation results.

Finally, the paper ends with conclusions.

2. PROBLEM FORMULATION

We consider a coherent pulsed radar system which trans­

mits N pulses and receives their returns. In receiver, the

echo signal is sampled at the output of a matched filter in

the range cell (Primary Data) corresponding to the loca­

tion under test and M neighboring range cells which are

assumed to be target free (Secondary Data). In the pro­

cessing unit, we deal with the problem of detecting the

presence of a target in the range cell under test. The de­

tection problem can be formulated as the following hy­

pothesis test:

{ Ho: HI:

y=n y=a8s+n (1)

where 8 is the Hadamard (elementwise) product of the

two matrices, y = [ YOT YI T YMT r is the

vector of returns of the radar, and

Ym = [ Ym,1 Ym,2 Ym,N J T

is the N-dimensional vector of returns from the m-th range

cell. The Yo denotes the primary data and Ym with m > 0

denotes secondary data. Ym,n is the n-th complex echo

signal from the m-th range cell of the radar and (. ) T de­

notes transpose. In addition,

n = [ noT nIT nMT ] T is the interference vec­

tor of the radar, where nm is the N-dimensional vector of

interferences of the m-th cell. We assume that the inter­

ference vectors (nm 's) are N-tuple vectors containing the

samples of independent and identically distributed sam­

ple functions of an auto-regressive process of known or­

der (MAR) and unknown parameters. In other words we

assume that:

MAR

nm n = '"""" ainm n-i + Wm n , � , ,

i=l

(2)

where wm,n are independent, zero mean, white Gaussian

random variables with variance (}"2. The vector

a = [ ao al aMAH ] contains the coefficients of the AR Process.

The vector of target signals is defined as:

(3)

where 8 = [1 ejf), ej(N-l)f), ( is the N- di­

mensional vector of target signal and 0 is the normalized

target Doppler.

The unknown complex amplitude vector of the target

signal is defined as:

(4)

where 0: is the complex amplitude of the target echo.

In the hypothesis test of (1), we know the structure of

target signal and the statistical properties of the interfer­

ence with some unknown parameters. In this case, the un­

known parameters are the complex target amplitude (0:), target Doppler frequency (0), and the AR parameters of

the interference (a). So, the hypothesis test (1) is a com­

pound hypothesis test. The standard technique for com­

pound hypothesis tests when the probability distribution

function (PDF) of the unknown parameters are not known

and the uniformly most powerful (UMP) Detector does

not exist or can not be found (as in our case), is general­

ized likelihood ratio test (GLRT).

Since Uniformly Most Powerful (UMP) detector can

not be found for this problem, other suboptimal detectors

must be studied. In [4] the Generalized Likelihood Ra­

tio Test (GLRT) is derived and studied. We should note

that no optimality is claimed about GLRT except invari­

ance and asymptotic performance [3]. Therefore, it may

be possible to find different design strategies which could

lead to detectors that achieve higher detection probabili­

ties or a stronger robustness than the GLRT. Based on the

above discussions, we have chosen a heuristic detector de­

sign approach.

3. DETECTORS' STRUCTURES

In this paper we assume that target Doppler frequency

is known and we extend the well known Kelly's GLRT

for this problem. In order to overcome this draw back

(known Doppler frequency), we can use a bank of detec­tors for different Doppler frequencies and decide on maxi­

mum output value of these detectors. So, in this paper, the

target signal (8) is assumed to be known, but the complex

amplitude of the target (0:) is unknown.

Kelly solved this problem for the case of unstructured

covariance matrix using GLRT and derived the following

detector [1]:

178

(5)

where M is Maximum Likelihood estimate of covariance

matrix. As the interference is assumed to be an AR pro­

cess, parameters of this process can be used to find the

covariance matrix. It is known that the elements of the

covariance matrix of an AR process are as follows:

[K1-1] .. = ;2 f (a[i -k]a*[j -k]-",J k=l

a*[N - i + k]a[N -j + k]) (6)

where i = 1,2, ... , N, j = 1,2, ... , N and a rk] 0 for k < 0 and k > MAR [5].

Since we do not know AR parameters initially, we can

replace them with their estimates based on secondary data

using standard AR parameter estimation techniques. But,

these techniques are designed for a single batch of data

[5]. In the next section we will present these standard

techniques and after that, we will extend each technique

to the case where several batches of data are available.

4. EXTENDED AR PARAMETER ESTIMATION

METHODS

There are four well-known methods for AR parameter es­

timation, Yule-Walker method, Covariance method, Mod­

ified Covariance method, and Burg method [5]. In the

following subsections we will extend each of the above

mentioned methods to the multiple data case.

4. 1. Extended Yule-Walker Method

The following set of equations are called the Yule-Walker

equations:

rMAR + RMARaT = 0 (7)

whereRMAH = [ r[o] r[-M�R + 1] l'

r[MAR -1] r[O] T

rMAIl = [ r[l] r[MAR]] and r[k]'s are auto-

correlation coefficients of the data. If we replace each

r[k] with its Maximum Likelihood estimate f[k], then the

equations can be solved to estimate the AR parameters(a)

as follows:

(8)

When more than one batch of data is available, the

estimate of r[k] can be derived in the following way:

1 M N

f[k] = N M L L ydt]y;[t -k] (9)

i=l t=k+l

4.2. Extended Burg Method

The Burg method tries to find the reflection coefficients by minimizing the forward and backward prediction errors.

AR parameters are estimated using the analytical relations

between reflection coefficients and AR parameters. It can

be shown easily that in order to reduce the forward and

backward prediction errors, several batches of data can be

used in the following way:

p ej,p,m[t] = Ym[t] + L ap,i,my[t -i] (10)

i=l

p eb,p,m[t] = Ym[t] + L a;,i,mYm[t - p + i] (11 )

i=l

where p = 1,2, ... ,MAR'

i = 1, ... ,MAR -1 i=MAR

(15)

4.3. Extended Covariance Method (Least Square Method)

The covariance method tries to minimize the forward pre­

diction error. It can be shown easily that in order to reduce

the forward prediction error, several batches of data can be

used as follows:

aLS = _ (y*y)-l (Y*y) (16)

y = [ ydMAR + 1] Yl[N]

YM[MAR + 1] YM[N] ]T (17)

Yl[MAR] yd1]

ydN -1] Yl[N -MAR] y= (18)

YM[MAR] YM[l]

YM[N -1] YM[N -MAR]

179

4.4. Extended Modified Covariance Method

The modified covariance method tries to minimize the sum

of the forward and backward prediction errors. It can be

shown easily that in order to reduce the sum of the for­

ward and backward prediction errors, several batches of

data can be used as follows:

aLS = _ (y*y)-l (Y*y)

y= [Yl[MAR+l] yi[N -MAR]

Yl[N] yi[l] YM[MAR+ 1]

(19)

YM[N] YM[l] YM[N -MAR] r (20)

Yl[MAR] Yl [1]

Yl[N -1] Yl[N -MAR] Yi[2] yi[MAR + 1]

yi[N -MAR + 1] Yi[N] y=

YM[MAR] YM[l]

YM[N -1] YM[N -MAR] YM[2] YM[MAR + 1]

YM[N -MAR + 1] YM[N] (21)

Based on the above discussions, four adaptive detec­

tion algorithms are derived where each is based on one of the extended AR parameter estimation methods. We

name the proposed parametric detectors as PGLRLYule,

PGLRLCov, PGLRTModifiedCov andPGLRT J3urg, fol­

lowing the AR parameter estimation method. In the fol­

lowing section the performance of the proposed detectors

is evaluated.

5. SIMULATION RESULTS AND DISCUSSIONS

The performance of the proposed detectors have been eval­

uated using Monte-Carlo simulations. We will present our

results in power function (probability of detection ver­

sus Signal to Clutter Ratio (SCR)). In all of the simula­

tions, the probability of false alarm is set at Pja = 10-4. The N-dimensional target signal vector in the m-th virtual

radar receiver is assumed to be

s = [ 1 ejf! ej(N-l)f! r, where D, the normalized target Doppler, is a random vari­

able uniformly distributed on [0, 27f]. The complex am­

plitude of the target signal, n, is a zero mean complex

Gaussian random variable.

In our simulations the clutter vectors are samples of

zero-mean, colored, circular, complex, Gaussian station­

ary processes with exponential autocorrelation function

0.9

0.7

0.6

c.." 0.5

OA

0.3

02

0.1

-20

Fig. 1. Pd versus SCR of PGLRLYule detector in AR

clutter.

with one lag correlation coefficient p = 0.95. In other

words the (i, j )-th element of correlation matrix is given

by M = (J20.95Ii-jl.

Fig. 1 shows the performance of PGLRLYule for dif­

ferent values of N and M. The performance shown in this

figure is typical of the performances of all four detectors

that presented in this paper. The simulation results show

that as M (number of secondary data batches) increases

the performance of detector improves. Also they show

that as N (number of pulses) increases the performance

of detector improves. It is worth noting that the former

result can not be always claimed about Kelly's GLRT [1].

Figs. 2 and 3 show the performance of different de­

tectors for N = 8, M = 4 and N = 8, M = 8 respec­

tively. The simulation results show that in these scenar­

ios PGLRLModifiedCov has the best performance, while

other detectors may change their ranking as the amount of available data changes. It can be seen easily that all

the new presented detectors have much better performance

than Kelly's GLRT.

It is worth noting that due to the amount of secondary

data that is available, Kelly's GLRT can not be used in

many of our simulations. As it is noted in [1], Kelly's

GLRT needs at least M > N secondary data and to achieve

satisfactory performance must satisfy M > 2N.

6. CONCLUSIONS

In this paper, the problem of adaptive target detection us­

ing temporally coherent pulse train against auto-regressive

clutter is considered. Based on a well-known detector

(Kelly's GLRT) and extensions to the well-known AR pa­

rameter estimation methods, four adaptive detection algo­

rithms have been proposed that are called PGLRLYule,

PGLRLCov, PGLRLModifiedCov, and PGLRLBurg fol­

lowing the AR parameter estimation method that is used

in each. Simulation results show that the introduced de­

tectors have much better performance than their parent (Kelly's GLRT). The results show that the performances

of the detectors improves as the number of pulses (N) and

number of secondary data batches (M) increase. Simu­

lation results also show the superiority of the proposed

PGLRT ..ModifiedCov against three other introduced de-

180

0.9

0.8

0.7

0.6

Cl." 0.5

OA

0.3

0.2

0.1

-20 25

Fig. 2. Pd versus SCR of PGLRLYule, PGLRLCov,

PGLRT ..ModifiedCov and PGLRT J3urg detectors in AR clutter with N = 8 and M = 4.

SCA(dB)

Fig. 3. Pd versus SCR of PGLRLYule, PGLRLCov,

PGLRT ..ModifiedCov, PGLRT J3urg and Kelly's GLRT

detectors in AR clutter with N = 8 and M = 8.

tectors. In most of the presented cases Kelly's GLRT can

not be used, due to lack of secondary data. Comparison

of presented detector to the other applicable ones in the

current problem will be studied in feature work.

7. REFERENCES

[1] E. J. Kelly, "An adaptive detection algorithm," IEEE

Trans. Aerospace and Electronic Systems, vol. 32,

no. 1, pp. 115-127, Mar. 1986.

[2] G. Alfano, A. De Maio, and A. Farina, "Model-based

adaptive detection of range-spread targets," lEE Proc.

Rad., Son. & Nav., vol. 151, no. 1, pp. 2-10, Feb. 2004.

[3] J. R. Gabriel and S. M. Kay, "On the relationship be­

tween the GLRT and UMPI tests for the detection of

signals with unknown parameters," IEEE Trans. Sig.

Proc., vol. 53, no. 11, pp. 4194-4203, Nov. 2005.

[4] A. Sheikhi, M. M. Nayebi, and M. R. Aref, "Adaptive

detection algorithm for radar signals in autoregressive

interference," lEE Proc. Rad., Son. & Nav., vol. 145,

no. 5, pp. 309-314, Oct. 1998.

[5] S. M. Kay, Modern Spectral Estimation: Theory and

Application. Prentice Hall, 1988.