Synthetic high range resolution radar

Synthetic high range resolution radar achieved by using pulse to pulse stepped frequency signals. Radar target range profile Course 3 4 hours

Synthetic high range resolution radar. Radar target range profile • • • • •

Frequency-Domain Target Signatures Pulse to pulse stepped frequency signals Synthetic Range Profile Generation Effect of Target Velocity Range- Profile Distortion Produced by Frequency Error • Examples (on synthetic data and real data)

Frequency-Domain Target Signatures • Any signal can be described as either a function of time or a function of frequency. • The echo signal from a range-extended target illuminated by a short RF pulse usually is observed in the time domain. Its amplitude and phase versus frequency is the echo signal spectrum, which is a frequencydomain description of the signal. • Measurements of a target’s echo signal in the time and frequency domains provide equivalent data for determining target reflectivity. • The target’s reflectivity profile in range delay can be defined as its echo signal amplitude and phase versus delay measured with respect to the carrier signal of the transmitted pulse.

Frequency-Domain Target Signatures • A continuous series of short RF pulses transmitted at a fixed pulse repetition frequency can be defined as a Fourier series of steady-state frequency components with a frequency spacing equal to the radar’s PRF. • Reflectivity equivalent to that measured from the train of short pulses could be obtained from measurements of the amplitude and phase of the received Fourier series frequency components relative to the respective transmitted component. • This set of frequency-domain measurements of reflectivity is the spectrum of the time-domain echo pulse train. • In practice, what we want is the HRR reflectivity profile of a target, not the periodic echo response. • Frequency spacing can be the reciprocal of the target's range-delay extent, instead of the reciprocal of the radar's PRI.

Frequency-Domain Target Signatures • The time duration of each transmitted frequency component need only be sufficient to produce an approximation to the steady-state echo response. • The pulse duration have to be greater than the target range-delay extent. • If a series of RF pulses were transmitted stepped in frequency from pulse to pulse over a bandwidth β, the set of echo amplitude and phase measurements made relative to each transmitted pulse can be transformed by using the DFT into the range-profile equivalent of echo amplitude and phase measurements obtained relative to a short RF pulse of bandwidth β.

Pulse to pulse stepped frequency signals x(t ) 

Ne 1

X k 0

0

(t  k .Tr ).cos[2 .( f 0  k .f ).(t  k .Tr )]

X 0 (t )  A if t   kTr , kTr  ti  and

Fig. 3.1

X 0 (t )  0A otherwise

Signal SFS

(3.1)

Pulse to pulse stepped frequency signals N 1 t X     Ati  sin c     0  k   i exp  j    0  k    k Tr  2 k 0 N 1 t Ati  sin c     0  k   i exp  j  +  0  k    kTr 2 k 0





Fig. 3.2 SFS Spectrum.





(3.2)

Pulse to pulse stepped frequency signals •

The ambiguity function module of the signal SFS is given by:  ( , F ) 

 1 sin[ .N .( F .Tr  f . )]  1   .sin c [ .F .(ti   ]. N ti  sin[ .( F .Tr  f . )]

Fig.3.3 The SFS signal ambiguity function

(3.3)

Pulse to pulse stepped frequency signals •

The section of the ambiguity function are:

 ( , 0) 

  sin[ .N .f . ] 1 1    N ti  sin[ .f . ]

(3.4)

 0, F  

sin NFTr  1 sin c Fti  N sin FTr 

 

(3.5)

1 N f

F 

Fig. 3.4 Section of the SFS signal ambiguity function for F=0

1 NTr

(3.6)

(3.7)

Pulse to pulse stepped frequency signals

Fig. 3.5 SFS Matched filter

Q

ti  ti B  N 

(3.8)

Synthetic Range Profile Generation - Transmit a series of N exploring pulses with carrier frequency forms: fk = f0 + k·Δf, k=0, 1, …, N-1 - Set a range-delayed sampling gate to collect I and Q samples of the target’s base band echo response for each transmitted pulse. - Store the quadrate components for each of the N pulses. - Apply frequency weighting to each data and corrections for target velocity, phase and amplitude ripple of echo signals, etc. - Take a inverse discrete Fourier transform (DFT-1) of each record to obtain the synthetic range-profile with N elements.

Synthetic Range Profile Generation RECEIVED SIGNAL

yk(t)

MIXER 1 mk(t) SAMPLING CIRCUIT 1

REFERENCE SIGNAL GENERATOR mk(t)

MIXER 2

RANGE SELECTION CIRCUIT

SAMPLING CIRCUIT 2

ADC 1

A pulse from an N pulses series is described by the following relation:

 U k cos  2 f k t  k  , 

xk  t    ADC 2

for kTr  t  kTr  ti

 0 , otherwise 

(3.9) STORAGE

The reference signal zk(t) is:

H(k) DFT-1 PROCESSOR

zk  t   U 0 cos  2 f k  t  k 

h(t)

(3.10)

Fig.3.6 Functional block diagram of signal processor

The received signal from an ideal scattering point target assumed at range R, with radial velocity toward radar vr, is:  U k r cos  2 f k [t   (t )]   k  , for 

yk  t    



0

kTr    t   t  kTr    t   ti

, otherwise

where

  t  2

R  vt t c

(3.11)

Synthetic Range Profile Generation  U 0 k cos  2 f k  t  , for kTr    t   t  kTr    t   ti

The signal at the mixer’s output is:



mk  t    

0 

(3.12)

, otherwise

The difference between the phases of the transmitted signal and the received one is:  2 R 2vt t  c c 

 k  t   2 f k  

(3.13)

The signal at mixer’s output is sampled at: S k  kTr 

ti 2 R  2 c

(3.14)

where k = 0, 1, ..., N1

Replacing time with Sk results the expression for total sampled difference of phases:  2 R 2vt  t 2 R    kTr  i   c  2 c   c

(3.15)

 k  2 f k 

Therefore, after sampling the signal at mixer’s output is: mk  Sk   U 0 k cos k

(3.16)

After mixing the quadrature components results: H  k   U 0 k  cos k  j sin k   U 0 k exp( j  k )

(3.17)

If applying inverse discrete Fourier transform for all N complex samples it will be obtained the target synthetic range-profile which approximates the target weighting function h(n): h  n 

1 N

N 1



k 0



 H  k  exp 

  2  kn , for 0  n  N  1  N 

j

(3.18)

Synthetic Range Profile Generation For a single point scattering target (vr=0), and assuming that H(k) is normalized (U0k=1) the synthetic range-profile target becomes: 1 N 1 2R 2 h  n    exp( j 2 f k ) exp( j kn), for 0  n  N  1 (3.19) N k 0 c N Since fk = f0 + Δf results: h  n 

1 N

N 1

 exp(-j2 f k 0

0

2R 2 2 NRf ) exp[ j k (n  )], for 0  n  N  1 c N c y  n

Using substitution: results:

h ( n) 

(3.20)

2NRf c

1 2 R N 1 2 exp(  j 2 f 0 ) exp( j yk ) N c k 0 N

(3.21)

In the above relation it is recognized the sum of a geometric progression elements, therefore this relation can be rewritten as: h  n 

1 2 1  exp  j 2 y   exp   j 2 f 0  N c  2 y  1  exp  j   N

h ( n) 

or

1 sin  y  N  1   2 R exp  j  y exp  j 2 f0  N sin  y  N   c N

The magnitude of the synthetic range-profile is: h  n 

sin  y y N sin N

(3.23)

(3.22)

Synthetic Range Profile Generation

Fig. 3.7 Range profile for an ideal scattering point Responses from a point target will be maximized when y=0, ±N, ±2N, and so on. The range index nearest each of these peak responses will be referred to as n=n 0. Range positions corresponding to range index n0 are given by: R

n0 c , 2 N f

 n0  N  c ,  n0 2 N f

2 N  c

2 N f

,....

(3.24)

Range maximum values at which detection is unambiguous (range frame) is obtained for y=0: Rmax 

c 2f

(3.25)

Range resolution can be defined as the range increment between any two adjacent discrete range positions. A set of N frequency steps produces N equally spaced range increments so that: R 

c 2 N f

(3.26)

which is equivalent to general expression of range resolution: R  c  2B

c 2 N f

Synthetic Range Profile Generation f [ MHz] Rmax [m]

N = 16 rs[m]

B [ MHz]

N=32 rs[m]

B [ MHz]

N =256 rs[m]

B [ MHz]

0,5

1500

93,75

1,6

46,87

3,2

5,85

25,6

0,2

750

46,87

3,2

23,48

6,4

2,92

51,2

0,3

500

31,25

4,8

15,62

9,6

1,95

76,8

0,4

375

23,43

6,4

11,71

12,8

1,46

102,4

0,5

300

18,75

8

9,37

16

1,17

128

0,6

250

15,62

9,6

7,81

19,2

0,97

153,6

0,7

214,28

13,39

11,2

6,69

22,4

0,83

179,2

0,8

187,50

11,71

12,8

5,85

25,6

0,73

204,8

0,9

166,66

10,41

14,4

5,20

28,8

0,65

230,4

1

150

9,37

16

4,68

32

0,58

256

2

75

4,68

32

2,34

64

0,29

512

3

50

3,125

48

1,56

96

0,19

768

4

37,5

2,34

64

1,1

128

0,14

1024

5

30

1,87

80

0,93

160

0,11

1280

6

25

1,56

96

0,78

192

0,097

1536

Synthetic Range Profile Generation f [ MHz]

Rmax [m]

B [ MHz] N = 16 rs[m]

N=32 rs[m]

B[ MHz]

B [ MHz] N = 256 rs[m]

7

21,42

1,33

112

0,66

224

0,083

1792

8

18,75

1,17

128

0,58

256

0,073

2048

9

16,66

1,04

144

0,5

228

0,065

2304

10

15

0,93

160

0,46

320

0,058

2560

11

13,63

0,85

176

0,42

352

0,053

2816

12

12,5

0,78

198

0,39

384

0,048

3072

13

11,53

0,72

208

0,36

416

0,045

3328

14

10,71

0,66

224

0,33

448

0,041

3584

15

10

0,62

240

0,31

480

0,039

3840

16

9,37

0,58

256

0,29

512

0,036

4096

17

8,82

0,55

272

0,27

544

0,034

4352

18

8,33

0,52

288

0,26

576

0,032

4608

19

7,89

0,49

304

0,24

608

0,030

4864

20

7,5

0,46

320

0,23

640

0,029

5120

Synthetic Range Profile Generation

By analyzing the previous table, one can see that the wider the signal band is, respectively the larger stepped frequency and pulses number are, the better range resolution is. However by increasing the frequency step value the unambiguous range length becomes smaller. From general expression of range resolution one can deduce that signal duration at signal procesor output is Ta=1/NΔf. If input signal duration is ti it results that the compression ratio for one pulse inside the package is given by : mc 

ti N Ta

(3.27)

This ratio has the same value as if obtained by analogical processing. Therefore we can deduce that the synthetic range-profile generation diagram is a digital matched filter for stepped frequency from pulse to pulse signal.

Synthetic Range Profile. Effect Of Target Velocity The synthetic range-profile of the moving target is given by: h  n 

1 N

N 1

 exp k 0

 2  2 R 2vt kn  2 f k   N c c  

j

ti 2 R    kTr  2  c      

The Doppler frequency for a moving target is: F 

2vt





2vt f 0 c

(3.29)

Fig. 3.8 The synthetic range-profile depending on two variables vt and n.

(3.28)

Synthetic Range Profile Generation

Fig 3.9 Synthetic range-profile’s cross sections at constant vt The maximum response shifts as the ratio signal/noise decreases and distortions of the response emerge, for velocities higher than 30m/s (-2.62dB at 20m/s, and -10.84dB at 100m/s). The target velocity produces range-profile attenuation and distortions and it is necessary to compensate the velocity’s radial component to prevent loosing information about position of the scattering points related to viewing angle.

Synthetic Range Profile Generation Frequency Fluctuations Influence. The complex samples, in frequency domain, for a ideal target are: H  k   e j k

 k  2 f k 2 R / c

where

(3.30)

A random error xk in k frequency step gives : j 2 f k 2 R / c  x k  H  k , xk   e  , 0  k  N 1

(3.31)

By applying the Inverse Fourier Transform, results: 2ky e  j 2 f 0 2 R / c N 1 j N  jx k h n, xk   e , 0  n  N 1 e N k 0

where

y n

2 Nf R c

(3.32)

Considering that the errors are independent and distributed according to a normal law of null average value and standard variance , the expectation of h(n,xk) is:



or:

j 2 ky 

1  j 2 f 0 2 R / c N  1 N M h n, x   e e N k 0



j 1 M h n, x   e  j 2 f 0 2 R / c e N





e



 N  1 2y 2N

 j x k

p x k  d x k , 0  n  N  1

sin  y   2  2 / 2 e , 0  n  N 1 y sin N

(3.33)

(3.34)

Synthetic Range Profile Generation Frequency Fluctuations Influence. The position of the detected scattering point corresponds well to the delay  associated to the target. The average value of the amplitude of the principal peak is given by:





2 2 M h n, x   e   / 2

(3.35)

If we consider that the only source of error is the frequency synthesizer results:   2

2R c

(3.36)

One can consider as acceptable value for .