1

Adaptive Radar Signal Detection in Autoregressive Interference using Kalman-based Filters

Mehdi Dorostgan 1, Mohammad Reza Taban

2,1*

1 Department of Electrical Engineering, Yazd University, P.O.Box: 89195-741, Yazd, Iran

2 Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan 84156-83111,

Iran *Corresponding author: Email: mrtaban@cc.iut.ac.ir , Tel: 0098 9131519702

Mehdi Dorostgan: Email: mdorostgan@gmail.com, Tel: 0098 9132513243

Abstract: This paper deals with the adaptive detection of radar target signal with unknown amplitude embedded in Gaussian interference which has been modelled as an AR process. Considering such model for the interference decreases the number of parameters that must be estimated and therefore less or even no secondary data is needed to obtain a detector with desired performance. Herein the detection is based on only the primary data. The authors resorting to the modern Kalman filtering technique develop the conventional GLRT-based detection in the presence of AR interference and propose two new detectors; AREKF based on extended Kalman filter and ARUKF based on unscented Kalman filter. The performance assessment conducted by Monte Carlo simulation compares the proposed detectors with the existing detectors based on generalised likelihood ratio test and Kalman filter. The results show that the ARUKF detector significantly has better detection performance than that of other detectors for the low number of primary data and high signal to noise ratio (SNR).

Keywords: Radar, Adaptive detection, Autoregressive interference, primary data, Kalman filter.

1. Introduction

In this paper, we consider coherent pulsed radar that transmits a train of pulses to detect targets around the radar. In the

receiver, after pre-processing and sampling of the received signal, a data vector is formed containing N successive returned

samples from the range cell under test (CUT). One of the most important challenges that we face with the problem of radar

target detection, is the lack of accurate information of interference (clutter plus noise). Even if we suppose that the interference

has Gaussian model, usually there is no enough information about the covariance matrix of interference and so the optimum

Neyman-Pearson detector is not implementable.

A conventional method to face with the mentioned problem is to use the generalised likelihood ratio test (GLRT)-based

detectors such as Kelly [1], adaptive matched filter (AMF) [2] and two dimensional detector [3], in which the covariance

matrix of interference is estimated by maximum likelihood (ML) approach. In these and similar types of detectors, the

covariance matrix is generally estimated from a number of independent and identically distributed sample vectors which are

achieved by receiving data from the range cells adjacent to the CUT. For appropriate estimation, these samples, often called

training or secondary data, must contain only the same interferences as the CUT. It is well known that the accuracy of the ML

estimation relies on the amount of data used for the estimation. Herein we need K>2N secondary data vectors to obtain a good

estimation [4]. Unfortunately in some applications such as airborne radar because of clutter heterogeneity or outliers, providing

a sufficient number of secondary data vectors to achieve a detector with satisfactory performance may be impractical [5].

2

There are some solutions addressed in open literature that need to have less or even no secondary data for target

detection such as resort to covariance matrix estimation with Bayesian approach [6]. Another approach is to apply structural

information (in addition to the Hermitian property) about the covariance matrix such as circulant structure [7], Toeplitz

property [8] and especially autoregressive (AR) model [9, 10]. Assuming AR model for interference is equivalent to consider a

model with specific analytical structure for the interference covariance matrix and this idea is addressed herein. Fortunately,

for interference modelled as an AR process, the model order is rather low in the practical applications of radar and active sonar

and is in the range from 2 to 5 in the former applications and has been chosen up to 8 in the latter applications [11].

Considering such model for the interference decreases the number of parameters that must be estimated and therefore less or

even no secondary data is needed to obtain a detector with desired performance.

In recent years, using the clutter with the AR model has received substantial attentions in the literature. In [9] Kay has

presented a GLRT-based adaptive detector using an AR model for the interference and assuming the target signal to be

completely known. In [12], a GLR detector of radar target with unknown amplitude embedded in interference with AR model

is introduced in which only the primary data from a pulsed radar with single antenna is used. This detector is developed in [13]

using the secondary data and in [14] using a multichannel radar and in [15] using a multi-input multi-output (MIMO) radar. In

[16] the problem of detecting range-distributed targets in the presence of structured interference modelled as an AR Gaussian

process is investigated with no secondary data and therein both the clutter homogeneity and heterogeneity in the cells under

test are considered. Assuming the interference with the AR model in [17], first the model coefficients are estimated by some

methods such as Yule-Walker. Then the inverse covariance matrix of the interference is obtained by those coefficients and AR-

model-based adaptive detection is performed by GLRT detector. In [18] the GLRT-based detection of a moving target

embedded in non-homogeneous environments with distributed MIMO radars is considered and therein interference is modelled

as an AR process. Moving target detection (MTD) is extended in [19] to distributed MIMO radar on moving platforms where

the effects of platform motion is considered and the order of AR process is estimated. In [20] AR spectral estimation and its

application on weak target detection are investigated based on the fractal properties of sea clutter in the power spectrum

domain considered as an AR process. The multichannel AR model for the interference is used in space-time adaptive

processing (STAP) for phased array radar applications. Using this model in [21, 22], the parametric detectors containing AMF,

Rao and GLRT are proposed.

The aforementioned detectors show that modelling the interference as an AR process is conventional and acceptable in

adaptive radar target detection. In this category of detectors, unknown parameters are usually estimated by ML approach.

Although the ML estimation has useful properties such as asymptotically efficiency and consistency, its accuracy depends on

the length of the data which is operationally finite and inadequate. Therefore, it may be possible to achieve better results using

optimal filtering techniques. Linear relationship between the received signal and interference as well as recursive form of the

interference modelled as an AR process motivate us to form a Gauss-Markov model and estimate the AR model coefficients by

3

Kalman filter (KF). Although the KF is widely used in some fields such as tracking [23, and references therein] and so on [24,

25], it has received little attention in the context of radar target detection. We will show that using KF in the estimation

procedure significantly improves the detection performance especially when there is no secondary data and primary data is

limited. Another advantage of KF used in the on-line radar detection is that because of sequential processing of the estimator,

if the estimation operation stops during its execution (for any reason such as expiring the processing time), the result of

processing is not corrupted and an estimation of intended parameters is available.

In this paper resorting to the modern Kalman filtering technique, we develop the conventional GLRT-based detection in

the presence of AR interference. We consider a coherent pulsed radar with single antenna that uses N successive returned

samples from the CUT for adaptive detection of a likely slow fluctuating target embedded in interference modelled as an AR

complex Gaussian process. Herein the detection is performed with no secondary data and the unknown parameters contain the

complex amplitude of the target signal, as well as variance of the white noise and coefficients of the AR model. In detector

proposed in [26], the AR model coefficients are estimated using the KF. The problem of this method explained further in the

next section is that the observation equation in Gauss-Markov model naturally is nonlinear and is approximated to a linear

equation by substituting some state variables from the previous stage. In this paper to deal with this problem we use the

extended KF (EKF) and unscented KF (UKF) and will show that these methods outperform the methods presented in [26] and

[12]. Also, the detector based on the UKF significantly has better detection performance than that of other detectors for the low

number of primary data and high signal to noise ratio (SNR).

The rest of paper is organised as follows. The next section is dedicated to model the signal and interference and to

briefly review the conventional GLRT-based detection in the presence of AR interference. In Section III, we address the

estimation of the AR model coefficients using the KF and proposed detectors based on the EKF and UKF. In Section IV, the

performance of the proposed detectors is investigated by Monte Carlo simulation and compared with other detectors. Finally,

some concluding remarks are given in Section V.

2. Problem statement and GLRT-based detection

Consider a coherent pulsed radar with single antenna that transmits a train of pulses. The receiver forms a primary data

vector containing N successive returned samples from the CUT in one coherent processing interval (CPI) for adaptive target

detection in that CUT. Hence, we deal with a binary hypothesis test problem as follows

0

1

H    :     

H    :       

y n

y n s (1)

where 1

1 2[              ]T N

Ny y y y is the data vector received from the CUT that T denotes transpose operator. This vector

under H0 only contains the interference vector 1

1 2[              ]T N

Nn n n n and under H1 includes target vector in addition to

4

the interference. The target is modelled as 1N s , where is a complex amplitude assumed deterministic and unknown

and θ 2θ ( 1)θ 1[1                 ]j j j N T Ne e e s is a known vector corresponding to the known Doppler shift of target. On the radars,

the known Doppler shift assumption is common [1, 4, 27]. This assumption is due to the use of filter banks in the classical

radars and the extraction of the signal spectrum by FFT in the digital radars based on new signal processing techniques. The

interference vector assumed to be an AR complex Gaussian process with known model order  M M N , given by

1

 M

n j n j n

j

n a n w

(2)

This interference is represented by 2 ~  ( , , )ARN Mn a where the coefficient vector 1 2[     ]T

Ma a a a and variance 2

belonging to the complex white Gaussian noise nw , both are deterministic and unknown. The probability density function

(PDF) of the received vector under two hypotheses H0 and H1 and the ratio of likelihood functions can be written as follows

[12]

2

    |  , , ,i if H y

y a

2

2 21 1

1 1   exp{ } ,   0,1   

N Mi i

n j n jN Nn M j

y a y i

 

(3)

2

1 

2

0 

   |  , , ,( )

      |  , , , 0   

f HL

f H

y

y

y ay

y a (4)

Where 0

1

                 H  ( 0)

      H  ( 1)

ni

n

n n

y under iy

y s under i

,

0

1

0       H  ( 0)

       H  ( 1)i

under i

under i

, and   ns is the nth element of vector s . Note that the

implementation of optimum Neyman-Pearson detector requires the perfect knowledge of the interference covariance matrix as

well as the target amplitude [28]. Herein due to the unknown parameters 2{ , , } a in the above ratio, the realisation of the

optimum Neyman-Pearson detector is not possible. However, with proper estimating of these parameters under any hypothesis,

we can obtain a suboptimum detector with the below test statistic

1 

0  

2

1  11 1  

 2

0  0 0

ˆ ˆˆ   |  , , ,   

ˆ ˆ      |  , , , 0   

H

subopt H

f HL

f H

y

y

y ay

y a (5)

by using the ML estimates of under H1 and also 2 under H1 and H0 , the test statistic (5) yields [12]

1 

0  

2 0

 2

1

ˆ    

ˆ   

H

H

(6)

22

0 0

1ˆˆ  

N u Ya

(7)

22

1 1

1ˆˆ  

N ' '

u Y a

(8)

5

where

1[           ]T

M Ny y u (9)

1

1

 

   

M

N N M

y y

y y

Y

(10)

'u Hu , '

Y HY (11)

where H is the projection matrix of the null space of ψ , which is defined as

    

H

H

ψψH I

ψ ψ (12)

θ 2θ ( 1)θ[1                 ]j j j N M Te e e ψ (13)

herein the superscript H denotes the conjugate transpose operator and I is the identity matrix. Substituting the ML estimations

0a and 1a in (7) and (8), the noise variance under H0 and H1 namely 2

0 and 2

1 are obtained. Then by substituting 2

0 and

2

1 into (6), the GLRT detector based on the ML estimation of unknown parameters is derived and called ARGLR [12].

In this paper, using Kalman-based filter for estimating the coefficients 0a and

1a , the test (6) is realised and a new

detector is proposed. Of course, in [26] the coefficients of AR model are also estimated by the conventional KF approximately,

and the achieved detector has been called ARKD. Since the data model is non-linear, in this paper we use the EKF and UKF

for estimating the AR model and will show that the obtained detectors have better performance.

3. Detection based on Kalman estimates of interference parameters

In this section, we use the EKF and UKF to estimate the unknown coefficients of the AR model used in (7) and (8) and

substitute these estimates in (6) which results in two new detectors. Modelling the interference as an AR process provides a

group of equations that are appropriate to realise Kalman family filters with high ability and accuracy for estimating the

unknown parameters of AR model. To achieve KF, the first stage is to define state variables and obtain suitable Gauss-Markov

equations. As mentioned 1 2[     ]T

Ny y y y denotes the data vector received from the CUT and contains the interference

1 2[     ]T

Nn n n n . Under both hypotheses we can write    ,   1,  , n n nn y s n N (note that under H0 , 0 ) and substitute

it into (2) that results in the following recursive relation

1 1 1 2 2 2n n n n n M n M n M n ny a y s a y s a y s s w

(14)

Recall in this equation nw is the complex white Gaussian noise of the AR model with variance 2 , and ns are the

complex amplitudes and the nth element of the target vector respectively. For indices 0n , ny and ns are considered equal to

zero. Herein and 1a to Ma are unknown parameters, so the state vector at time n is considered as

6

1 2[       ] [      ]T T T

n n n Mn n n na a a x a . As seen in (14), ny cannot be displayed as a linear combination of state variables which is

the main idea of using the EKF and UKF. In fact, the measurement equation has a non-linear form as follows

( ) n n n ny h w x (15)

On the other hand to obtain the other equation of Gauss-Markov model, we assume that during a CPI the AR model

coefficients belonging to the interference of the CUT are almost time invariant. Also assuming the slow target fluctuation, the

target amplitude is fixed. So the state vector is assumed to be fixed within the CPI and the system dynamic equation can be

written as

1  n n n x x v (16)

where nv is a complex white Gaussian noise with covariance matrix

nQ as follows

2

'2

0

0

n M

n

n

q

q

IQ (17)

where 2

nq and ' 2

nq are variances of the deviations of AR model coefficients and target amplitude respectively. It is clear that

considering nv makes small changes of the AR model coefficients and target signal amplitude during the implementation of

the KF steps to compensate. Herein, the Gauss-Markov equations can be expressed as follows

1    

 

n n n

n n n ny h w

x x v

x 1,  , n N (18)

In the rest of this section, considering the nonlinear form of Gauss-Markov equations we estimate the state vector nx

based on the EKF and UKF algorithms only using the primary data.

3.1. Detector based on EKF

Suppose that | 1ˆ

n nx is the prediction of the state vector nx based on the received data until time n-1 , namely

1 2 1[     ]T

ny y y . In the EKF, non-linear function n nh x in (18) is changed to a linear equation in terms of the state variables

using Taylor expansion around the point | 1ˆ

n n nx x as follows

| 1

| 1 | 1

ˆ  

ˆ ˆ  ( )

n n n

n

n n n n n n n n

n

hh h

x x

x x x xx

(19)

in which assuming that the variation | 1ˆ( )n n nx x becomes small, the second and higher order terms in the expansion are

supposed negligible and ignored. By defining

7

| 1ˆ  

 

n n n

n

n

n

h

x x

Hx

(20)

and by definition n nh x in (15), we have

1 | 1 1

2 | 1 2

| 1

1 | 1 1 2 | 1 2

| 1

         

                       

ˆ

ˆ

ˆ

ˆ

   

ˆ

ˆ 

T

n n n n

n n n n

nn M n n n M

n n n n n n n

Mn n n M

y s

y s

y s

s s

s

a a

a

s

H (21)

Thus, the equations of the EKF in the measurement-update algorithm denoted by|

ˆn nx is as follows

| | 1 | 1ˆ ˆ ˆ(  ( ) ) n n n n n n n n ny h x x L x

1

2

| 1 | 1             H

n n n n n n n n n

L P H H P H

| | 1       H

n n n n n n P I L H P

(22)

where 2

n is the variance of the white noise nw in AR model (2) and (18) and can be time-varying in the proposed method.

According to the linear form of the system equation in (18), the equations of the EKF in the time-update algorithm or

prediction is as follows

| 1 1| 1ˆ ˆ

n n n n x x

| 1 1| 1n n n n n P P Q (23)

In the above equations, P denotes the covariance matrix of state estimate in each stage. Since in a practical sense, the variance

2

n and covariance matrix nQ are not available, during the implementation of the KF algorithm, we can adaptively estimate

these values with the moment estimation. Considering (15), 2

n can be written as

2  H

n n n n n n nE y h y h x x (24)

where E denotes expectation value operator. The moment estimation of 2

n is obtained as follows [9]

2

| 1 | 1

1

1ˆ ˆˆ

n H

n k k k k k k k k

k n K

E y h y hK

x x (25)

From the simulation investigation, we suggest 3K which should be temporarily assumed 1K n for 2n , to prevent

k from being negative. Similarly, considering (16), nQ can be expressed as follows

1 1    H

n n n n nE Q x x x x (26)

so the moment estimation of nQ is obtained as follows

8

1| 1 | 1| 1 |

1

1ˆ ˆ ˆ ˆ ˆ  n

H

n k k k k k k k k

k n K

EK

Q x x x x (27)

where K is supposed as before.

After running the EKF, the AR model coefficients (of the interference) are estimated and by placing them in (7) and (8), the

noise variance of the AR model is estimated under both hypotheses as

22

0 |

1ˆˆ    N N

N u Y a (28)

22

1 |

1ˆˆ   N N

N ' '

u Y a (29)

Finally by placing 2

0 and 2

1 in (6), the first proposed detector is obtained. We call it AREKF.

3.2. Detector based on UKF

Unlike the EKF, the UKF has no need to equation linearization, and the required means for the implementation of the

KF algorithm is obtained by simulation. Assuming that we have l state variables, to run each step of the filter we use

2 1M l weighted samples (called sigma points) where their weights are given as follows

0   

wl

1          ,  1, 2,   , 2

 2   iw i l

l

(30)

Also in each step of algorithm, the sigma points are defined as

0, 1| 1 1| 1ˆ

n n n n χ x

, 1| 1 1| 1 1| 1ˆ         , 1, 2,   ,  i n n n n n n

i

l i l

χ x P

, 1| 1 1| 1 1| 1ˆ        i n n n n n n

i

l

χ x P ,  1,   , 2i l l

(31)

where    i

A denotes the ith column of matrix A and the square root of the covariance matrix 1| 1n n P is obtained by Cholesky

factorization. is a scaling parameter and an appropriate choice of it can reduce estimation error. In [29] this parameter is

proposed as follows

23 l (32)

where is a constant that determines the spread of the samples around the estimates of state vector at each step and is

usually set to a small positive value (e.g., 410 1 ).

According to the system dynamic presented in (18), the weighted samples do not change and the equations of the UKF

in the time-update algorithm is as follows

9

, | 1 , 1| 1        ,  0,1  ,  , 2i n n i n n i l χ  χ

2

| 1 , | 1

0

ˆl

n n i i n n

i

w

x  χ

2

| 1 , | 1 | 1 , | 1 | 1

0

ˆ ˆ( )( )l

H

n n i i n n n n i n n n n n

i

w

P  χ x  χ x Q

(33)

As the same way the equations of the UKF in the measurement-update algorithm is as follows y

| | 1 | 1ˆ ˆ ˆ  (  )n n n n n n n ny y x x K

2

| 1   , | 1

0

ˆl

n n i i n n

i

y w

  , | 1 , | 1i n n n i n nh χ

1

,  | 1 ,  | 1 n n n n n

 x

K

2

,  | 1 , | 1 | 1 , | 1 | 1

0

ˆ ˆ( )( )l

H

n n i i n n n n i n n n n

i

w y

 x χ x

2

,  | 1 , | 1 | 1 , | 1 | 1

0

ˆ ˆ( )( )l

H

n n i i n n n n i n n n n

i

w y y

1

| | 1 ,  | 1 ,  | 1 ,  | 1  H

n n n n n n n n n n

 x x

P P  

(34)

Here, like the EKF, during the implementation of the UKF algorithm, we can adaptively estimate the variance 2  n and

covariance matrix nQ with the moment estimation.

Like before, after running the UKF, the AR model coefficients (of the interference) are estimated and by placing them

in (24) and (25), the noise variance of the AR model is estimated under both hypotheses. Finally substituting the obtained 2

0

and 2

1 by the UKF in (6), the second proposed detector is achieved. We call it ARUKF.

The state space model in (18) is a nonlinear model with time variant parameters and this complexity makes that the

convergence and stability verification of the algorithm are not analytically easy to do. Therefore, the stability of algorithm has

been studied using simulation and with statistical surveys in different conditions. The result of investigations shows the high

stability of the proposed methods.

In the next section, using the simulation we investigate the performance of the proposed detectors and compare it with

the performance of the detectors ARGLR and ARKD and show that the ARUKF detector outperforms its counterparts

especially in low sample support.

4. Performance evaluation

In this section resorting to Monte Carlo simulation, the performance of the proposed detectors is evaluated and is

compared with the performance of ARKD and ARGLR. In modelling of ground clutter by an AR process, usually the model

order is assumed less than 4. In our simulations, we consider the interference as an AR Gaussian process of order 2 with

10

coefficients vector [ 0.25 0.25    0.3]Ti a in [12]. Assuming an arbitrary variance for the interference, the AR model of the

interference and thus its experimental covariance matrix NR are specified.

NR is obtained by averaging a sufficient number of

simulated interference vectors. The signal to interference ratio is defined as follows [12]

2 1| |   H

NSNR s R s (35)

Hence, assuming that the target vector to be known, for each SNR the target amplitude is achieved in simulations. Also the

target phase is determined as a random value with uniform distribution over the range [ , ) . For evaluating and comparing

the detection performance, the receiver operation characteristics (ROC) curves of the detectors are used. In order to evaluate

more precisely, the curves of detection probability versus signal-to-noise ratio are plotted as well.

Before comparing the detectors with each other, first we study the performance of each of the two proposed detectors

by changing the number of primary data N. For this purpose, the curves of detection probability versus the probability of false

alarm and also versus the signal to noise ratio are plotted for different values of N.

Fig. 1 shows the probability of detection (Pd) versus the probability of false alarm (Pfa) of AREKF detector for different

values of N, 1 rad and 10 SNR dB . As it can be seen, with the increase in the number of primary data, due to more

accurate estimation of the interference parameters, the performance of detector is improved. Fig. 2 shows the AREKF

detection probability versus SNR for different values of N, 1 rad and 210faP . This figure shows that at first by

increasing the SNR, the probability of detection is improved, while if this ratio exceeds a certain limit, owing to the decline in

relative power of the AR interference, the estimation error of the AR model parameters increases and the probability of

detection is dropped. However, with increasing number of primary data this event occurs later or in other words, for larger

amounts of SNR it happens.

Next, in figs. 3 and 4, the performance of ARUKF detector by changing the number of primary data is considered. In fig.

3 the curve of Pd versus Pfa for different values of N, 1 rad and 10 SNR dB is plotted. As expected, owing to the

dependence of the estimation accuracy on the number of primary data, the more increase in the number of primary data, the

more improvement is achieved in the detector performance. In fig. 4 the curve of Pd versus SNR for different values of N,

1 rad and 210faP is plotted. Fortunately here, even for the low number of primary data, by increasing SNR up to

25 dB , the detection probability will not be dropped. Indeed, even for the small number of primary data, the estimation error of

the AR model parameters in the UKF is not increased by raising SNR. This is significant advantage of ARUKF detector

compared with the previous detectors based on KF such as AREKF and ARKD.

In the following, using the ROC curves as well as the curves of detection probability versus SNR, the performance of

the proposed detectors AREKF and ARUKF is compared with the performance of detectors ARGLR and ARKD. Figs. 5, 6

11

and 7 show the ROC curves of these detectors for different number of primary data 10N , 30N and 50N respectively,

1 rad and 10 SNR dB . It is observed that for low number of primary data the KF-based detectors outperform the

ARGLR detector and especially the ARUKF always has considerably better performance than that of other detectors. We see

that with increasing the number of primary data, the ARGLR performance will gradually become better than the performance

of ARKD and AREKF. This is due to the fact that in ARGLR the AR model parameters entirely are estimated based on the

ML method and hence with increasing the number of data, these estimates will tend to the efficient estimates [30].

In Figs. 8, 9 and 10, the curves of detection probability versus SNR of mentioned detectors for different number of primary

data 10N , 30N and 50N , 1 rad and 210faP are plotted. Here again we see that the performance of ARUKF

detector significantly is better than that of others. However with the increase in the number of primary data, the parameters

estimation error of the AR model is reduced and therefore the performance of the other detectors especially ARGLR,

approaches to the ARUKF. Interestingly here, like considered and discussed in Figs. 2 and 4, the detection probability of

ARUKF will not fall by increasing SNR even for small numbers of primary data. This is another important advantage of the

ARUKF detector compared with two other KF-based detectors.

We also investigate the convergence of the KF-based detectors. For this purpose, the mean squared error (MSE) of the

estimates of AR model coefficients is considered. Fig. 11, obtained by simulation results in different conditions, shows the

MSE of estimates versus the number of primary data for two situations; the first is noise-only, and the second considers signal

with 25 SNR dB . As be seen, after an initial transition (almost 3 steps for signal absent), the MSE is monotonically reduced

by increasing the number of data in all detectors. This shows the convergence of all of the algorithms. In the first situation, all

methods have a relatively small error, although the MSE of the ARUKF is a little more than the MSE of the other two

detectors. In the second situation with 25 SNR dB , the MSEs and transient times of the ARKD and specially the AREKF

have significantly increased in comparison with the first situation while the MSE of the ARUKF has a slight increase and its

transient time has even been reduced to 2. The increasing of the initial error is due to the Kalman filter transient state and, after

passing through this transient state, the error decreases with increasing the number of data and the estimate of each method

converges to a final value. The interesting advantage of the ARUKF is that the transient state error of the filter is relatively low

and this results in a better performance of this method. In the following of the convergence study, an analysis with one million

iterations in different conditions shows that the error of each method in each iteration always is less than one and no instability

was seen during the estimation.

Finally, the effect of Doppler shift frequency error of the target on the performance of detectors is investigated. It is common in

radar signal detection to divide the target velocity (or Doppler frequency) domain into small cells and test each cell for the

presence of target [27]. For simulation, like before, the nominal is assumed to be   1 nom rad in the detectors while the data

vector y is simulated with the actual with a difference of 5% compared to the nominal   1 nom rad . Fig. 12 shows the

12

ROC curve of the mentioned detectors for 30N ,   1 nom rad and 10 SNR dB . Comparing the curves Figs. 6 and 12

shows that although the presence of an error in Doppler shift causes a loss in the performance of the detectors, the performance

of the ARUKF is also better than that of other detectors.

5. Conclusion

In this paper, we have concerned adaptive detection of a radar target signal embedded in Gaussian interference based on

only the primary data where the interference has been modelled as an AR process. Herein, ARGLR detector relying on ML

estimation of the AR model parameters and also ARKD detector relying on Kalman estimation of them have been developed.

In this regard, we proposed two new detectors; AREKF based on EKF and ARUKF based on UKF. We demonstrated that the

ARUKF detector significantly has better performance than that of other detectors. However with the increase in the number of

primary data, parameters estimation error of the AR model is reduced and hence the performance of the other detectors

especially ARGLR approaches to ARUKF. Also a considerable advantage of the ARUKF detector compared with both other

KF-based detectors is that the detection probability of ARUKF will not be corrupted by increasing SNR even for small

numbers of primary data and therefore this detector outperforms its counterparts especially in low sample support. At the end,

in order to the generalization of the proposed method it is suggested that this method be developed using secondary data.

6. References

[1] Kelly, E.J. “An adaptive detection algorithm”, IEEE Trans. Aerosp. Electron. Syst., 22(2), pp. 115–12 (1986).

[2] Robey, F.C., Fuhrman, D.L., Kelly, E.J., et al. “A CFAR adaptive matched filter detector”, IEEE Trans. Aerosp. Electron.

Syst., 29(1), pp. 208–216 (1992).

[3] Rafie, A.H. and Taban, M.R. “Two dimensional optimal linear detector for slowly fluctuating radar signals in compound

gaussian clutter”, Scientia Iranica, 21(6), pp. 2213-2223 (2014).

[4] Conte, E., De Maio, A. and Ricci, G. “GLRT-based adaptive detection algorithms for range-spread targets”, IEEE Trans.

Signal Process., 49(7), pp. 1336–1348 (2001).

[5] Himed, B. and Melvin, W.L. “Analyzing space-time adaptive processors using measured data”, Thirty-first Asilomar Conf.

on Signals, systems & computers, Pacific Grove, CA, USA, pp. 930–935 (1997).

[6] Bandiera, F., Besson, O., Coluccia, A., et al. “ABORT-like detectors: A Bayesian approach”, IEEE Trans. Signal Process.,

63(19), pp. 5274-5284 (2015).

[7] Aubry, A., De Maio, A., Foglia, G., et al. “Diffuse multipath exploitation for adaptive radar detection”, IEEE Trans. Signal

Process., 63(5), pp. 1268-1281 (2015).

[8] Said, S., Hajri, H., Bombrun, L., et al. “Gaussian distributions on riemannian symmetric spaces: statistical kerning with

structured covariance matrices”, IEEE Trans. Information Theory, 64(2), pp. 752– 772 (2018).

[9] Kay, S.M. “Asymptotically optimal detection in unknown colored noise via autoregressive modeling”, IEEE Trans. on

Acoustics, Speech, and Signal Processing, 31(4), pp. 927-940 (1983).

[10] Qureshi, T. R., Rangaswamy, M. and Bell, K. L. “Parametric Adaptive Matched Filter for Multistatic MIMO Radar”,

IEEE Trans. Aerosp. Electron. Syst., 54(5), pp. 2202– 2219 (2018).

[11] Haykin, S. and Steinhardt, A. “Adaptive radar detection and estimation”, Wiley, New York (1992).

13

[12] Sheikhi, A., Nayebi, M.M. and Aref, M.R. “Adaptive detection algorithm for radar signals in autoregressive interference”,

IEE Proc., Radar Sonar Navig., 145(5), pp. 309–314 (1998).

[13] Sheikhi, A., Nayebi, M.M. and Aref, M.R. “A powerful practical coherent adaptive radar detector”, 2001 CIE Conf. on

Radar, Beijing, China, pp. 405–409 (2001).

[14] Moniri, A.R., Nayebi, M.M., and Sheikhi, A. “A multichannel auto-regressive GLR detector for airborne phased array

radar applications”, 2003 IEEE Conf. on Radar, Adelaide, SA, Australia, pp. 206-211 (2003).

[15] Sheikhi, A., Zamani, A. and Norouzi, Y. “Model-based Adaptive Target Detection in Clutter Using MIMO Radar”, 2006

CIE Conf. on Radar, Shanghai, China, pp. 1-4 (2006).

[16] Alfano, G., De Maio, A. and Farina, A. “Model-based adaptive detection of range-spread targets”, IEE Proc. Radar Sonar

Navig., 151(1), pp. 2–10 (2004).

[17] Sheikhi, A., Zamani, A., Hatam, M., et al. “Model based adaptive detection algorithm with low secondary data support”.

10th

Int. Conf. on Information Science, Signal Processing and their Applications, pp. 177-180 (2010).

[18] Wang, P., Li, H. and Himed, B. “A parametric moving target detector for distributed MIMO radar in non-homogeneous

environment”, IEEE Trans. Signal Process., 61(9), pp. 2282-2294 (2013).

[19] Li, H., Wang, Z., Liu, J., et al. “Moving target detection in distributed MIMO radar on moving platforms”, IEEE J. Select.

Topics Signal Process., 9(8), pp. 1524-1535 (2015).

[20] Fan, Y.F., Luo, F., Li, M., et al. “Fractal properties of autoregressive spectrum and its application on weak target detection

in sea clutter background”, IET Radar Sonar Navigat., 9(8), Kuala Lumpur, Malaysia, pp. 1070-1077 (2015).

[21] Sohn, K.J., Li, H. and Himed, B. “Parametric GLRT for multichannel adaptive signal detection”, IEEE Trans. Signal

Process., 55(11), pp. 5351–5360 (2007).

[22] Wang, P., Li, H. and Himed, B. “A new parametric GLRT for multichannel adaptive signal detection”, IEEE Trans.

Signal Process., 58(1), pp. 317–325 (2010).

[23] Crouse, D.F. “Basic tracking using nonlinear continuous-time dynamic models”, IEEE M. Aerosp. Electron. Syst., 30(2),

pp. 4-41 (2015).

[24] Jahanbakhshi, S., Pishvaie, M.R. and Boozarjomehry, R.B. “Characterization of three-phase flow in porous media using

the ensemble Kalman filter”, Scientia Iranica, 24(3), pp. 1281-1301 (2017).

[25] Kiani, M. and Pourtakdoust, S.H. “Spacecraft attitude and system identification via marginal modified unscented Kalman

filter utilizing the sun and calibrated three-axis-magnetometer sensors”, Scientia Iranica, 21(4), pp. 1451-1460 (2014).

[26] Taban, M.R. and Sheikh Mozaffari, A. “Adaptive radar signal detection in Gaussian clutter with autoregressive model

using Kalman filter”, Journal of Radar, 2(3), pp. 1-12 (2014).

[27] Skolnik, M. I. “Introduction to Radar Systems”, 3rd ed., McGraw-Hill, New York (2001).

[28] Aubry, A., De Maio, A., Pallotta, L., et al. “Radar detection of distributed targets in homogeneous interference whose

inverse covariance structure is defined via unitary invariant functions”, IEEE Trans. Signal Process., 61(20), pp. 4949–4961

(2013).

[29] Haykin, S. “Kalman filtering and neural networks”, Wiley, New York (2001).

[30] Kay, S.M. “Fundamentals of statistical signal processing”, vol. 2, Prentice Hall, New Jersey (1993).

14

Figure Captions:

Fig. 1. Pd aginst Pfa for AREKF with SNR = 10dB, =1rad and four values of N

Fig. 2. Pd aginst SNR for AREKF with Pfa = 10 -2

, =1rad and four values of N

Fig. 3. Pd aginst Pfa for ARUKF with SNR = 10dB, =1rad and four values of N

Fig. 4. Pd aginst SNR for ARUKF with Pfa = 10 -2

, =1rad and four values of N

Fig. 5. Pd aginst Pfa for AREKF and ARUKF compre with ARGLR and ARKD for SNR =10dB, =1rad and N =10

Fig. 6. Pd aginst Pfa for AREKF and ARUKF compre with ARGLR and ARKD for SNR =10dB, =1rad and N =30

Fig. 7. Pd aginst Pfa for AREKF and ARUKF compre with ARGLR and ARKD for SNR =10dB, =1rad and N =50

Fig. 8. Pd aginst SNR for AREKF and ARUKF compre with ARGLR and ARKD for Pfa =10 -2

, =1rad and N =10

Fig. 9. Pd aginst SNR for AREKF and ARUKF compre with ARGLR and ARKD for Pfa =10 -2

, =1rad and N =30

Fig. 10. Pd aginst SNR for AREKF and ARUKF compre with ARGLR and ARKD for Pfa =10 -2

, =1rad and N =50

Fig. 11. MSE versus N for noise only and signal with SNR =25dB

Fig. 12. ROC curve of the detectors for SNR =10dB, nom =1rad and N =30

15

Figures:

Fig. 1.

Fig. 2.

Fig. 3.

16

Fig. 6.

Fig. 4.

Fig. 5.

17

Fig. 7.

Fig. 8.

Fig. 9.

18

Fig. 20.

Fig. 11.

Fig. 12.

19

Biographies:

Mehdi Dorostgan received the M.Sc. degree in Communication Engineering from Isfahan University of

Technology (IUT), Iran, in 2004. Currently, he is working towards the Ph.D. degree in Communication

Engineering at the department of Electrical Engineering, Yazd University, Iran. His research interests include

detection, estimation and array signal processing.

Mohammad Reza Taban was born in Isfahan, Iran, in 1968. He received the B.Sc. degree in 1991 from Isfahan

University of Technology (IUT), Isfahan, Iran, the M.Sc. degree in 1993 from Tarbiat Modarres University,

Tehran, Iran, and the Ph.D. degree in 1998 from Isfahan University of Technology, all in Electrical Engineering.

He joined the Yazd University faculty, Yazd, Iran, in 1999. Since 2016, he has been a Professor of Electrical and

Computer Engineering at the Isfahan University of Technology, Isfahan, Iran. His main research interests are

statistical signal processing, detection, and estimation. He has been in the scientific committee of the 10th

international ISC conference (ISCISC2013) and the 2nd Iranian Conference on Avionics System (ICAS2015). Dr.

Taban has published more than 90 technical papers in international and national journals and conferences