A High Speed Target Detection Approach Based on STFrFT

Pang Cunsuo,Ran Tao The Department of Electronic Engineering,

Beijing Institute of Technology Beijing, China

10801153@bit.edu.cn,rantao@bit.edu.cn

Pang Cunsuo National Key Laboratory for Electronic Measurement,

North University of China Taiyuan, Shanxi, China 10801153@bit.edu.cn

Abstract—The fractional Fourier transform (FrFT) is a potent tool to analyze the chirp signal. However, it fails in some applications, such as the echo is NFLM signal when the radar detects the high speed target. The short-time fractional Fourier transform (STFrFT) is proposed to solve this problem. It displays the time and FrFD-frequency information jointly in the short-time fractional Fourier domain. According to characteristic of high speed target, we establish the echo signal model with high-order phase, then analyze the target detection performance by numerical simulation to get the relationship among the optimal number of pulses, signal-to-noise ratio and acceleration changing rate. Finally, the simulations verify that the principle of the method is correct and can provide guidance for practical applications.

Keywords-high order phase; long time integration; high speed target; STFrFT

I. INTRODUCTION

Along with the technology development, the velocity of high speed target can reach 3000m/s or more, the traditional radar cannot meet demand detection. The echo signal of the high velocity target has phase characteristics of secondary and above. The conventional FFT technique cannot effectively accumulate signal’s energy. For missile testing, the short time Fourier transform and Wigner transform was proposed in [1], but the STFT may lead to decreased energy when it is used for a long time accumulation about non-stationary series. Moreover, time-frequency representations of Wigner are bilinear; it is difficult to detect multiple targets for the cross-fuzzy. In [2], according to the characteristics of FrFT, the detection of the anti-radiation missiles was research in the Gaussian noise. In [3], according to frequency characteristics of anti-radiation missiles, the local methods of RAT were proposed, however, it does not apply in low SNR. In [4], a pre-processing method was proposed to remove the Doppler ambiguity. The above study focuses on the second phase of echo signal; in fact, the high speed targets in flight will produce more than the second order phase. According to the characteristics of high speed target, in this paper, we will analyze high order phase characteristic of echo and the detection methods.

II. ECHO SIGNAL MODEL OF HIGH SPEED TARGET

It is a challenging task that radar and sonar detect high speed weak target in noise and clutter. Detecting a high speed weak target requires a long accumulation time. However, the phase of target echo cannot be modeled with a

few parameters during the long time accumulation. This problem is actually a non-parametric detection problem. Here, the echo signal of high speed target is assumed to be non-linear FM signal, and the noise is additive white complex Gaussian noise. So the target echo signal can be expressed as:

2 3( ) exp[ 2 ( ( ) ( ) )] ( )s s sx n j fnT nT nT w n= + + +

Where denotes the rate of change of speed.denotes the rate of change of acceleration. sT denotes the pulse repetition time, for simplicity, we assume that the most high order phase of target echo is 3, and set sT = 1.

According to maximum likelihood algorithm [5], we have:

2 3

2 3

( , , ) ( ) ( )

( ) exp[ 2 ( )]

exp[ 2 (( ) ( ) ( ) )]

n N

n Nn N

n Nn N

n N

H f x n x n

x n j fn n n

j f f n n n

=

==

==

=

=

= + +

= + +

By defining:

, ,f f f= = =

We can obtain:

2 3

2 3

1

2 3 2

1

2 3 2 1/2

1

( , , ) exp[ 2 ( )]

1 2 exp( 2 )cos[2 ( )]

[{1 2 cos(2 )cos[2 ( )]}

{2 sin(2 )cos[2 ( )]} ]

n N

n NN

nN

nN

n

H f j fn n n

j n fn n

n fn n

n fn n

=

=

=

=

=

= + +

= + +

= + +

+ +

From“(3)”, we know that H is not entirely even along of its coordinates. Such as:

2011 International Conference on Instrumentation, Measurement, Computer, Communication and Control

978-0-7695-4519-6/11 $26.00 © 2011 IEEE

DOI 10.1109/IMCCC.2011.189

744

( , , ) ( , , )H f H f=( , , ) ( , , )H f H f=( , , ) ( , , )H f H f

( , , ) ( , , )H f H f

By defining 0= ,we can obtain:

3 1/2

1

( , 0, ) {1 2 cos[2 ( )]}N

n

H f fn n=

= + +

Now when we consider only three order phase of target echo, the result of maximum likelihood estimator is showed in Fig. 1. Fig. 1(a) is three-dimensional map of target echo to verify “(4)” and “(5)” is correct. Fig. 1(b) shows that the result of maximum likelihood estimator is not symmetrical and the estimate results will affect target detection with the increase of acceleration changing rate .

(a)

-0.2 -0.1 0 0.1 0.20

20

40

60

80

100

120

140

f

gamma=0

gamma=1e-6

gamma=1e-5

(b)

Figure1 The result of maximum likelihood estimator (a) Three dimensional map of target echo

(b) Frequency dimensional projecting map of target echo

A. Data Model As the targets high velocity motion, amplitude, time

delay and phase of the echo signal are changed, and echo signal has the characteristics of polynomial phase [6].thus, the n-th range-compressed echo can be written as:

2

0 0 0

0 0

2 3

, sinc exp

exp exp 2

exp exp( )

d d

dd d

d d

f fx t n A D B t nT j

fj f t nT j f nT

j nT j nT

Where 0D BT denotes the time-bandwidth product. Bdenotes the bandwidth of transmitted signal, 0 denotes the initial time delay, 0 02 /v c denotes the time delay rate,

02 /df v denotes the Doppler frequency, 2 /d adenotes the Doppler frequency rate, 2 / 3d denotes the Doppler frequency rate of change and denotes the wavelength ,T denotes the pulse repetition time.

B. Effect of Doppler Frequency Changing Rate In order to consider the impact of high-order phase, we

assume 0 0, 0d .In addition, from “(6)”, we can see that the maximum of Doppler frequency X f can be written as:

31

00

1max exp

2d

N

dn

NX f A D j n

So we have the output SNR as follows:

2

, 2

( )d

out

MAX X fSNR

N

For a given d , we can calculate the corresponding N

by “(8)”.If we plot points ,d N in the logarithmic coordinate as shown in Fig. 2 , we find that these points are formed into a straight line approximately. So using the linear least squares (LLS) fit, we can get:

1/3, round 1.9919 /opt a dN

III. PERFORMANCE ANALYSIS OF STFRFT

According to the stability of high speed target motion in a short time, a fast STFrFT method is presented to detect the echo signal of high speed target in Low SNR.

745

Figure2 Relationships of 10log N with 10log

A. Algorithm of The STFrFt

, ( , ) ( ) ( ) ( , )x p pSTFrFT t u x g t K u d

Where ( )x is the input signal, g(t) is the Gaussian window function, The FrFT of the Gaussian function has Gaussian support in all fractional domains. ( , )pK u is a kernel function, Detailed definition can be seen in [7][8].

B. Window Length In STFrFT analysis, it is desirable to use the longest

window possible that does not extend to include regions with substantially different frequency content.

According to“(6)”, the largest window length of STFT can be expressed as:

'TPT

=

Where

3

1 1 1' max{ , }d

Ta a

= =

The largest window length of STFrFT can be expressed as:

3

1

d

QT

=

C. Transform Order The traditional transform order estimation is a two-

dimensional search, in the search process, the method in [2] needs a large amount of computation. In this paper, we consider the actual background of the radar target detection, the transform order can be coarse search by STFT, on this

basis, the semi-Newton method is used to fine search, and the iterative process can be expressed as:

'

1

( )[ , ]n n n

S up a b

p+ =

where n is the result of n times iterations , n is the n-th search step size , the main operation of semi-Newton method in each iteration is a one-dimensional search and a first-order partial derivative calculation.

At last, the detecting scheme of high speed target is summarized by the following steps.

Step 1: STFT is used to estimate the order ( )opt P of sub-signals.

Where ( )opt P defines the equivalent order for the short-time window with P samples.

Step 2: On the basis of step 1 we use globally optimized STFrFT to get the optimal order of sub-signals [7].

1( ) ( )

2' tan tan( )2opt Q opt N

NM

1( ) ( ) ( )1

1 [ ], int( / )i i

L

opt N opt P opt Pi

L N PT

Where, ( )opt N defines the transform order for the complete N-sample signal and ( )'opt Q defines the equivalent order for the short-time window with Q samples.

Step 3: On the basis of step 2, semi-Newton method is used to get the optimal order ( )''opt Q of sub-signals.

Step 4: On the basis of step 3, we get the third order phase of target echo.

1

'

( ) ( )1

1 [cot '' cot '' ], ' int( / )j j

L

d opt Q opt Qi

L N QT

Step 5: On the basis of step 4, we use FrFT to get the optimal detection performance of target.

D. Numerical Example The outputs of STFrFT for a NLFM signal are

contrasted below with the FT and FrFT for the same signal.

The signal, nominally 2 32 (20000 100 20 )( ) j t t ts t e ,is sampled in

the range { [0,1]}t at a sampling rate of 1000Hz. Globally optimized STFRFT outputs for the entire signal using a 128-point sliding Gaussian window. Fig. 3 shows the results of FT, Fig. 4 shows the results of FrFT, Fig. 5 shows the results of the STFrFT. Fig. 6 shows detection probability for the two algorithms at different SNR. Simulation results demonstrate that the proposed algorithm to improve signal

-5 -4.8 -4.6 -4.4 -4.21.7

1.75

1.8

1.85

1.9

1.95

2

log10( )

log

10

(N)

Computation

LLS fit

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to noise ratio than the FrFT algorithm about 2dB when the detection probability is 80%.

200 400 600 800 1000

20

40

60

80

Am

plit

ude

frequency

Figure 3 The detection results of FT

200 400 600 800 1000

100

200

300

400

500

Am

plit

ude

frequency(u)

Figure4 The detection results of FrFT

200 400 600 800 1000

200

400

600

800

Am

plit

ude

frequency(u)

Figure5 The detection results of STFrFT

-20 -18 -16 -14 -12 -10 -8 -60.2

0.4

0.6

0.8

1

Pd

SNR

FrFT

The proposed algorithm

Figure6 Comparison of the proposed algorithm and FrFT

IV. CONCLUSIONS

According to the characteristics of high speed target, We analyzed the relationship radar radial velocity, acceleration, acceleration changing rate with time, and established echo signal model to analyze the high order phase how to impact on target detection performance, Finally, by numerical simulation, we get the relation between optimal number of pulses and acceleration changing rate. Some results provide theoretical guidance for the application.

ACKNOWLEDGMENT

This work was partially supported by the National Natural Science Foundation of china for Distinguished Yong Scholars (Grant No.60625104).

REFERENCES

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[3] Fang Qian-xue, Wang Yong-Liang , Wang Shou-Yong, “ARM detection technique based on local Radon-Ambiguity transform”, Systems Engineering and Electronics,China,vol.30,pp.2151-2154, November 2008

[4] Chen JianChun etc, “A study on anti-Radiation Missile Launching Warning Technique”, International Symposium on Noise and Clutter Rejection in Radars and Imaging Sensors,Japan,pp.485-490,1994.

[5] THEAGENIS J.ABATZOGLOU, “Fast maximum likelihood joint estimation of frequency and frequency rate ”, IEEE Transactions on Signal Processing,American,vol.22,pp708-715,Six 1986

[6] Zhang Nan, Tao Ran, Wang Yue, “A Target Detection Algorithm Based on Scaling Processing and Fractional Fourier Transform”, Acta Electronica Sinica, American,vol.38,pp683-688,March 2010,

[7] Chris Capus, Keith Brown, “Short-time fractional Fourier methods for the time-frequency representation of chirp signals”, J.Acoust.Soc.Am,vol.113 ,pp. 3253-3263,Six 2003.

[8] Ran Tao, Yan-Lei Li, and Yue Wang, “Short-Time Fractional Fourier Transform and it’s Applications”, IEEE Transactions on Signal Processing,American,vol.58,pp.2568-2580,May 2010

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