High-range resolution for radar by oversampling and LMS pulse compression

High-range resolution for radar by oversampling and LMS pulse compression

W.D.Wirth

Abstract: High range resolution for radar systems can be achieved in combination with pulse compression of polyphase coded signals without an increase in the signal bandwidth of the antenna and receiver system. The received signal is oversampled with an increased clock rate to also use some of the information in the spectral side lobes. The compression filter function is computed by the least mean square error method to concentrate the compression result on one sampling value. Experiments are performed with echo signals measured with the experimental phased array system ELRA. Parallel measurements and evaluation with high and low bandwidth signals allow comparisons. With ELRA parameters, resolution can be improved by a factor of 4. The price for this is a loss of 6-9 dB in the signal-to-noise ratio, greater complexity for processing and a faster ADC.

1 Introduction

A high-range resolution is required for some radar applica- tions, for example for the separation of closely spaced targets or for single target classification. Usually, a higher signal bandwidth is then required. This has the conse- quence that the antenna system with its transmit-receive modules has to be designed for the higher bandwidth. This can result in problems, in particular with phase-steered antenna arrays with their limited bandwidth caused by their modulo 2.n phase shifters. Additional controlled delay lines on the sub-array level are required together with phase shifters of the delay-line type, which show a phase shift proportional to the frequency. The frequency range occu- pied by the radar system extends accordingly, which may be a problem particularly in densely used frequency bands.

There have been suggestions to improve the range resolution by inverse filtering, i.e. by a filter with a frequency transfer fimction inverse to the signal spectrum, to extend the signals’ bandwidth after reception [l] or by applying the methods of super-resolution [2].

For a Doppler-tolerant pulse compression, a linear or nonlinear frequency modulation (LFM or NLFM) has to be applied. From this type of modulation polyphase codes can also be derived, e.g. the Frank, P3 or PNL code [3,4]. Noise-optimal compression at the receiver side is usually performed by a matched transversal filter (MF). If the Doppler frequency of the target is known, e.g. with fixed targets or flying targets during the tracking process, a transversal filter with more taps than the code length (mismatched filter) and therefore more degrees of freedom can be determined; this achieves substantially improved sidelobe suppression. By using the additional degrees of freedom with the least mean square error (LMS) method

0 IEE, 1999 IEE Proceedings online no. 19990127 DOI: 10.1049/ip-rsn:19990127 Paper first received 6th August and in revised form 16th December 1998 The author is with Forschungsinstitut fur Funk und Mathematik, Neue- nahrer Str. 20, D-53343, Wachtberg-Werthhoven, Germany

IEE Proc.-Radar. Sonar Navig., Vol. 146, No. 2, April 1999

between the desired and the attainable compressed signal function, the mismatched compression filter function is determined [4].

In the following, an existing [4] approach is examined: whether and how without an increase of the design bandwidth of the antenna system, and only with an increased sampling rate of the analogue/digital converter, can an improved range resolution be achieved by a transversal compression filter, designed by the LMS method. An accurate knowledge of the received signal function of a point target is assumed.

With realised radar systems, the intermediate frequency and video signal bandpass filters are matched to the length of the subpulse and are not exactly band-limiting. If the signal is sampled as usual with the Nyquist rate, some information will be lost. This additional information in the frequency sidebands outside the design bandwidth may be exploited to a certain extent by oversampling.

Within the experimental phased array radar system ELRA, developed at FFM [5], the matched filters for the rectangular subpulses for the receiving channels are realised with surface acoustic wave (SAW) filters in a convenient way. These filters have a frequency transfer function of the well known form sin XIX with high side lobes in the frequency domain. The bandwidth of this filter is given by

1 B = 1.2- z

This is nearly the usual choice of the bandwidth B= UT. The sampling period for the complex video signal with I and Q-components is chosen as T. We recognise that, by applying this type of a matched filter for the subpulses, there are side lobes outside the nominal bandwidth and this part of the spectrum may be extracted additionally by oversampling.

2 Polyphase codes

A pulse with linear frequency modulation (LFM) is assumed, which will have a frequency deviation given by

95

the nominal receiver bandwidth. The phase according to the LFM is well known with the frequency deviation Af and the total pulse length T:

e =

Af cp(t) = 71-t2 T

- 0 -

0 1 0

- 0 -

With the usual sampling rate equal to Af; according to a subpulse length z = UAf; a polyphase code with the code length N = Tlz develops:

71 cp =-n2 O l n i N - 1 ( 3 )

f , = exp(jcp,) (4)

" N From this, the code sequence results:

A further popular version of the polyphase code is the Frank code. It results from a stepwise rise of the frequency up to Af The code phases follow with the code length N=p2 and withp an integer:

271 cp.=g.- O < i t N

I P

with gi = (i div p)(i mod p ) (5)

3 Calculation of filter function for pulse compression

The matrix C ( K + M - 1, A4) may contain in columns the received signal sequences c I . . . cK resulting from the coded pulses. Each column represents one of the M range bins. The increasing shift downward corresponds to the respective range:

C =

The ideal target signal vector s(M,1) contains in a column the target distribution; thus, a 1 at the respective range position according to the target range, and zeros otherwise. The received signal due to the transmitted code sequence is represented together with the noise vector n by

x = C s + n (7) From the measured signal n, the target distribution shall be estimated with a filter matrix W:

i= W*X (8) W shall be determined to minimise the mean square error F between s and i:

(9) F = E{(s - i)*(s - i)) = truceE{(s - i)(s - i)*}

From eqns. 7 and 8 and with I the identity matrix:

s - i = (I - W*C)s - W*n (10) With E{ss*} = P , E{nn*} = Q and E{sn*} =0, we get

E((s - ;*)(s - S)*) = (I - W*C)P(I - C* W ) + W*QW = P + W*(CPC* + Q) W - W*CP - PC* w (1 1)

96

Via the identity concerning W

W * A W - W * B - B * W = -B*A-'B+(W -A-'B)*A(W - A - ' B ) (12)

we arrive at the expression for F:

F = trace(P - PC*(CPC* + Q)-'CP

+ (W - (CPC* + Q)-'CP)*(CPC* + Q)

x (W - (CPC* + Q)-'CP)) (13)

From this, the required minimum follows if the square expression concerning W becomes equal to 0:

W = (CPC* + Q)-'CP (14) Individual targets are at a certain range, and all ranges are equal in probability. Therefore, we may set P = I .

Thus we get

W = (CC* + Q)-'CI (15) W contains columns of C modified by ( C C + Q ) - I . According to numerical examinations, the elements on the diagonals of (CC* + Q) - ' are almost constant, and so W is simply a shift of a filter vector w (column of W ) relative to the signal sequence. We can thus assume for the computation of the filter vector w, for example, a selection vector e(M,I) with a 1 in the centre, and zeros otherwise:

Thus, the filter vector w for a transversal FIR filter is as follows:

w = (CC* + Q)-'Ce (17) The noise power is represented by Q. For a strong noise Q outweighs CC*, and the filter vector turns into the noise- optimal filter function w = c, which is the usual matched filter.

With 'conventional' pulse compression, the elements of vector s are assigned to the individual subpulses. The centre value 1 represents one subpulse or range bin. Here the value 1 however, for the achievement of a higher resolution, represents, only an individual value of the oversampled signal sequence. The algorithm must thus try to concentrate the compressed signal on only one clock position of the oversampled sequence. All remaining clock places correspond to sidelobes and are minimised in the mean square sense. How well that is possible is examined below with simulated and measured radar echo signals.

The application of the filter vector w in a transversal filter represents a linear processing. The linear superposed echo sequences of neighbouring targets can thus be sepa- rated according to the response function of the compression finction calculated according to eqn. 17.

4 Subpulse filter

The transmitter emits a pulse containing a finite sequence of subpulses, each of length z, with the individual phases

IEE Proc.-Rada,: Sonar Navig., Vol. 146, No. 2, April 1999

of the selected phase code. Each subpulse is filtered at the receiver by a filter matched to the subpulse, i.e. convolved with its filter function in the time domain. For the ELRA system in the long-range search mode, the subpulse length is chosen as lops. A SAW filter with an approximately rectangular filter function in the time domain is applied as a matched filter for the subpulses. From received rectangular subpulses, triangular pulses are caused with length 20ps. The triangle function may be d(t). The sampling points, identified by p , correspond to the oversampling clock rate increased by a factor r compared to the conventional clock rate UT:

L t = - p p r

To prepare the convolution of the code series f with the function d by a matrix multiplication, we form a matrix D with elements dip. Each row i contains the triangle function dip around the index ir:

P db = 1 - i - - with p = (i - 1)r . . . ir . . . (i + 1)r I rl f o r i = l . . . N (19)

Matrix D therefore has the dimension (N,r(N+ 1) + 1). With the code series f according to eqn. 4 written as a (1 ,N)-row matrix, it follows that

c = f D (20)

The received signal c is from a point target, which is produced after matched subpulse filtering with the r-fold oversampling rate. By eqn. 20 each code element J; is multiplied with dip and all contributions from the subpulses are superposed. The signal vector c has to be applied in matrix C from eqn. 6. C is then used in eqn. 17 to compute W.

5 Gain or SNR loss

In addition to the increased sampling rate for the analogue- digital convertor, the price for the LMS compression for higher resolution is a loss of the signal-to-noise ratio (SNR). It is calculated as the ratio of the gain with the high-resolution compression (LMS filter) to that with the matched filter (MF) compression. For the MF compression, the oversampling rate is also assumed for a fair comparison.

For the gain computation, we have to take into account the covariance matrix R of the noise after the subpulse filter and for the oversampling rate. With an optimal compression filter for the rectangular subpulse (ELRA, 10 ps), the correlation is a triangular function:

. r (21)

pt = 0 otherwise

The covariance matrix R then contains diagonals with equal values pt at distance t from the main diagonal. The gain is with both filter types, with ym as maximum output signal:

(22) y”, gw = w*Rw

This gain will be computed for both filter types and compared by their ratio. The resultant loss in gain, expressed in dB, then indicates the necessary increase in

IEE Proc.-Radar. Sonar Navig., Vol. 146, No. 2, April 1999

SNR to achieve the high resolution by the LMS filter compared to the MF.

6 Pulse compression of simulated signals

By its signal variety, the ELRA system offers a possibility for practical testing of the suggested compression proce- dure. For this purpose, pulses for high and low resolution are transmitted in a selected beam direction, from which we receive echoes from a group of neighbouring fixed targets.

Short pulses with a duration of 1 or 2ps: their reception with the matched bandwidth of 1 MHz resolves the targets with corresponding high resolution. (ii) Pulses with the code length 16 with the Frank code and with the subpulse length 10 ps: reception is performed with the optimal subpulse filter (time function l o p s rectangle). The corresponding standard sampling period is 10 ps. These signals are also sampled in the 1 ps period, so that an oversampling with r = 10 results. Compression of these signals in the conventional way and with the LMS filter for high resolution then permits a comparison with the echoes from (i).

(i)

1 .o

0.8

0.6

0.4

0.2

0 280 300 320 340 360 380 400

range samples for oversampling Fig. 1 Target distance v = 15, P3 code, N = 16, oversampling factor r = IO, gain loss 6.3 dB, noise factor q = 0.16 - - - MF ~ LMS

Resolution by MF and LMS compression

280 300 320 340 360 380 400 range samples for oversampling

Fig. 2 Target distance v = 3, P3 code, N = 16, oversampling factor r = IO, gain loss 6.3 dB, noise factor q = 0.16 - - - MF - LMS


91

For an initial investigation, the compression is examined comparatively with simulated signals. The parameters are selected according to the ELRA parameters described above. Two targets with Frank codes are generated. The range distance of the targets can be selected. The subpulse MF is represented first by the modelled function d according to eqn. 19. The calculation of the LMS filter vector w follows eqn. 17. The diagonal noise covariance matrix is set as e=@. The factor q is selected as q=O.O015 trace(CC*). This choice proved to be appropriate with the experiments with measured signals.

Additionally, for comparison, the conventional compression function (MF) is applied to the oversampled signal sequence. The results are presented in Figs. 1 and 2. In Fig. 1 the target distance is selected as v= 15 samples of the oversampling rate. This corresponds to 1.5 conventional range bins. It can be recognised that with MF compression (dashed line) the targets are just resolved. The result from the LMS filter is shown by the solid line. In Fig. 2, with a target distance of v = 3 samples, target resolution is achieved with the LMS filter (solid line) but not with the MF (dashed line). Thus, we take v = 3 as the resolution limit for the LMS filter. The resolution by the LMS filter is thus improved compared to the MF by a factor of 5. The gain loss is calculated to be 6.32 dB from eqns. 21 and 22. Therefore, we have to increase the SNR by this gain loss to achieve the higher resolution with the LMS filter compared to the compression with the MF. Higher sidelobes also

1 . o r

0.8

0.6

0.4

0.2

I \ I \

I I I I I \

- I I I I I I I I

- I I I \ I \ I \

- : " I

- \

range of samples for oversampling Fig. 5 Target distance v= 15, Frank code, N= 16, oversampling factor r= 10, gain loss 9.3 dB, noise factor q=O.16

- LMS


- - - MF

280 300 320 340 360 380 400

range samples for oversampling

Fig. 6 Target distance v = 4, Frank code, N= 16, oversampling factor r = 10, gain loss 6.3 dE%, noise factor q = 0.16 - - - MF - LMS


/ 6 b lb 1; Ib 1 6 118 ;O

Fig. 3 Measured amplitude response of subpulse filter against time for rectangular subpulse

r -1.10

-1.40

-1.60

0 2 4 6 8 10 12 14 16 18 20

Fig. 4 rectangular subpulse

98

Measured phase response of subpulse filter against time for

12 0

120 100 80 60 40 20 0

Doppler variation of target signals up tofdT= 0.3 Fig. 7

develop. The demand for high resolution, low side-lobes and moderate SNR loss cannot be achieved together.

In the next step, the response of the subpulse MF was measured. The amplitude and phase responses are shown in Figs. 3 and 4, respectively. The deviation from the triangular model with zero phase is apparent. With this

IEE Proc.-Radar: Sonar Navig., Vol. 146. No. 2, April I999

16

14

12

10

8

6

4

2

'0 100 200 300 400 500 600

1.0

0.8

0.6

0.4

0.2

Fig. 8 Magnitude of w

-

-

-

-

-

I - .

subpulse response function, the simulation for Fig. 1 and 2 was repeated. The results are shown in Figs. 5 and 6. The target distances are now v = 15 and v = 4, respectively. The resolution improvement is thus given by the factor 3.75. The gain or SNR loss is calculated to be 9.23 dB.

In the simulation program, a Doppler shift is additionally given to the target signals. Fig. 7 shows, with a waterfall representation, that with a rise of the Doppler shift from zero (function in the foreground) up to fdT= 0.3 the high resolution is preserved. According to the ELRA parameters (A = 0,11 m) for this example, a speed interval A v results:

fdT = 0.3

For T= 160ps results, fd= 1875Hz and

AV=-= 1 0 3 ~ ~ (23) 2

The corresponding phase progression within a pulse with N = 16 and a total length of 160 ps is A cp =360" fdT = 108". It will thus be possible to estimate the Doppler frequency with sufficient accuracy to apply the high- resolution LMS compression.

As a check, the magnitude of the filter function w is shown in Fig. 8: the filter hnction values are distributed reasonably across the length of the pulse.

1.4

-~ ~

0 20 40 60 80 100 120 140 160 180

Fig. 9

IEE Proc.-Radar, Sonar Navig., Vol. 146, No. 2, April 1999

Reference signal computed fvom Frank code and measured jilter

1 .o 1 ' 1

a 50 100 150 200 250 300 1.0 I I\' 1

b 100 150 200 250 300 350 1.0 ' 1

" 150 200 250 300 350 400

Fig. 10 Compression of echo signals a Short pulse echoes, transmit pulse 2 p s b Transmit subpulse 10 ps , N= 16, MF compression c Transmit subpulse IO ps , N= 16, oversampling r= 10, LMS high-resolution compression

7 Compression of measured echo signals

As the next step, recorded fixed target echoes from the ELRA system were processed. With the Frank code, the target reference signal was calculated; its magnitude is shown in Fig. 9. The partial decrease of the amplitude results from the effect of the subpulse filter, which converts the originally rectangular subpulses into twice as broad overlapping triangles. The superposition results in an amplitude reduction according to the phase shift between neighbouring code elements.

Figs. 10 shows the echo sequence to the 2ps transmit pulse with corresponding high resolution (Fig. loa): the conventional compression (MF) with subpulses of 10 ps, Frank code of length 16, and sampling rate of lps (Fig. 1 Ob); and the result of high-resolution compression applied to the echo signals from subpulses of lops and Frank code of length 16 (Fig. 1Oc). Comparison between Figs. 10b and 1Oc clearly shows a resolution improvement. The result resembles, to a large extent, Fig. IOU from the wideband measurement.

The reference signal function c was still assumed with a model for the transmit signal. Inaccuracies with the modulation of the transmit signal caused by the effect of the phase shifter for phase modulation, as well as the effect of

o v I I I , I I I I I , 0 20 40 60 80 100 120 140 160 180

Fig. 1 1 Measured reference signal from aircraft

99

1.0,,

50 100 150 200 250 300

b 100 150 200 250 300 350 1 .o I

C 150 200 250 300 350 400

Fig. 12 Compression of echo signals a Short pulse echoes, transmit pulse 1 ps b Transmit subpulse 10 ps, N = 16, MF compression c Transmit subpulse 10 ps, N= 16, oversampling r= IO, LMS high resolution compression

the transmission amplifier, could not thereby be included yet.

The use of a real measured signal could give a firther improvement. This was achieved by the reception of echo pulses from an aircraft. This forms an isolated target with small dimensions in comparison to the resolution in space according to the subpulse length. A target was tracked on its tangential flight route. Recorded echo signal blocks from 32 pulses were evaluated. Possible clutter signals were eliminated by elimination of the components’ average values. The Doppler frequency was determined by a Four- ier transformation, so that flight conditions with a very low Doppler frequency could be selected.

The received signal function is shown with its amplitude in Fig. 11. Deviations from the idealised signal in Fig. 9 are identified. This signal sequence was used in its complex form as the reference signal for the calculation

of w. Fig. 12a shows for comparison the echo signal from a Ips pulse Fig. 12c shows a further improvement of the resolution by the LMS filter in comparison to Fig. 10.

8 Conclusions

A higher range resolution may be achieved without increasing the design bandwidth of the radar system. As there is no exact band-limiting in realised radar systems, by oversampling and corresponding pulse compression (with a filter function derived with an LMS algorithm), the information in the spectral sidelobes of the received signal can be exploited. The price for this additional quality is a higher conversion rate of the ADC, greater processing complexity and a certain loss in S N R . In particular with phased array systems, this may be compen- sated by a longer dwell time if necessary.

9 Acknowledgments

The author would like to thank Mr. W. Sander for the realisation of the measurements with the E L M system and the supply of the signal files derived from it.

10 References

1 HERRMAN, G.E., and KELLY, L.L.: ‘Enhanced resolution in simple radars’, IEEE Trans. Aerospace Electron. Syst., 1989, AES-25, (l), pp. 64-72. GABRIEL, W.F.: ‘Superresolution techniques in the range domain’. Proceedings of IEEE Int. Radar Conf. May 1990, Arlington, USA, pp. 263-267 WIRTH, W.D.: ‘Compression of polyphase-codes with doppler shift.’ Proceedings of IEE Int. Radar Conf. October 1992, Brighton, UK, pp. 469412

4 GROGER, I., SANDER, W., and WIRTH, W.D.: ‘Experimental phased array radar E L M with extended flexibility’. Proceedings of IEEE Int. Radar Conf. May 1990, Arlington, USA, pp. 286-290 COOK, C.E., and BERNFELD, M.: ‘Radar signals’ (Academic Press, London, New York, 1967)

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High-range resolution for radar by oversampling and LMS pulse compression