INVERSE FILTERS: DESIGX AND PERFORMANCE USING SURFACE ACOUSTIC WAVES

J.D. Maines, G.R. Rich and J.B.G. Roberts Royal Radar Establishment, Great Malvern, Worcs., U.K.

ABSTRACT. istics to be realized and has encouraged considerable activity in recent years related to band-pass filters having constant group delay and also to matched filters for a range of coded waveforms. paper an inverse filter is described which is designed to minimise interference between received signals which overlap in time. The criteria for the filter design and the SAW implementation of this design is discussed and illustrated by experimental results for two types of device.

The design flexibility of SAW devices permits a wide range of filter character-

Other optimum filters can be made using surface wave techniques and in this

Inverse filters have potential application where overlapping returns lead to garbling of signals and their operation is described by reference to a possible application in an air traffic control system

Introduction

The matched filter has long held a pre-eminent position in the theory and practice of radar signal reception as it is known to be optimum for the threshold detection against white noise of the echo from a point target. We wish to point out that the matched filter is not optimum for some applications, for example in a clutter dominated situation, and here the inverse filter1 optimises signal reception. The inverse filter can be realised using SAW techniques and a specific design is reported here.

The inverse filter is defined to have a frequency response which is the inverse of the signal spectrum it is designed to process. In practice, filters can be designed to approximate to this condition over a finite bandwidth B, so that the spectrum of the signal which emerges from the inverse filter should be nominally flat over this bandwidth. the filter has the appropriate phase response, the output signal can, in the time domain, have a duration 1 1 -which is considerably shorter than 6’ where b is the B 3 dB bandwidth of the input signal. It is the ability to produce this improved resolution in time, poten- tially for any waveform, which enables the wanted signal to be recognised in the presence of other similar, but unwanted returns.

Providing that

Comunications engineers have long been familiar with the use of this technique to reduce intersymbol interference and they provide us with the necessary body of theory to draw upon.

Design and Simulation of the Filter

The expression for the optimum impulse response, when its length is unrestricted, and white noise is present was given by Wiener as

for a sampled d ta system in z-transform notation as used by Di Toro . I(z) is the transmitted waveform, I its complex conjugate and no the rms noise amplitude. The filter is optimum in responding to I plus noise with the closest possible approximation to a delta function, i.e it exhibits residual sidelobes with minimum total energy. When nz dominates the denom- inator we obtain an impulse response proportional to 7 (the time reverse of I since becomes the matched filter. We are more interested in the other extreme case, when n2 is small and

1

= l/z), that is H

H - 1/I

the so-called inverse, clutter or Urkowitz filter. Ideally of course we should adjust H with variations of SIN ratio, for example with target range; this unwel- come complication is not pursued here. For impulse responses of finite length, Di Toro gives an iteration which generates successively larger arrays, each representing a sampled data impulse response which is the best possible approximation to 1/I for its length. The iteration improves H(k) to H(k+l) according to

where R is the residual in the output, IH-1, the sum of squares whose terms are minimised, IH being the filter output from the input I. functions of R (k) . g0 and gl are complicated

The array variables I, H and R represent sampled data functions, or the coefficients of polynomials in

z , and it is an interesting illustration of the power of the Algol 68 computer language that the iteration can be programmed as it stands. For instance the key line in the routine is

-1

r:= output:- i*(h:- h*(gO - gl*r)) - 1.0 where the definition of the multiplying operator * is extended to imply convolution when it finds itself between two arrays, and the arrays h, r and output are flexible, adjusting their bounds as required at each iteration.

A s the iteration proceeds, the impulse response H grows in length and this forces a compromise between the suppression of sidelobe energy and using filters of practicable lengths. For instance, taking I to be a rectangular pulse of duration T, 3 iterations produce an impulse response of length 3T + 2T - 5T. However, by observing that the criterion of minimum energy residual is not perhaps always the best one for design- ing a radar filter we can reduce this demand. concern here is to narrow the main response while keeping other peak responses below some tolerable level. Truncating the 5T impulse response above to a duration 3T raises the far-out sidelobes but makes them compar- able with the near-in peaks. Figures 5 and 6 illus- trate the impulse response and output for this case which was adopted as the design for our first attempt to make a SAW inverse filter and will be referred to as type 1. For all designs discussed here the input pulse is assumed to have a rectangular amplitude envelope.

Our

437

Although this initial design has the required sharp response to the rectangular input, it also exhibits -9 dB sidelobes at AT. fairly empirical approach can be adopted: a new impulse response is formed by adding suitably attenuated self replicas to H, with shifts of T, to cancel the peak sidelobes. This throws up smaller peaks at 2 2T to which the same technique is applicable. The resultant peaks at 2 4T are small enough to be neglected amongst other sidelobes. A final trimming of the output is possible by adding delta functions which add delayed replicas of the input pulse to the output where this is appropriate to make the resultant sidelobes closer to zero. Figures 7 and 8 show just over half of the symmetrical impulse response and the output for this type 2 filter design which was truncated to a length - 7T. This shows a worst sidelobe peak of -18 dB and the - 3 dB width of the main response is 0.07T, compared with 0.6T for the matched filter.

To improve this, a

Doppler Sensitivity

The foregoing design work ignored the possibility that the input waveform may be subject to doppler shifts but it is important for radar applications to predict how this will affect performance. calculations were made of the cross-ambiguity function . The modulus of this complex function of time and frequency is plotted as a surface, sections through which,parallel to the time axis, represent the amplit- ude of the filter output as a function of time for a chosen doppler frequency. Such sections are shown in Fig 1 while Fig 2 shows cuts parallel to the frequency axis, both for the type 1 filter design. The modulus

1 To this end

of the previously calculated output is t he time-axis section through this surface. It will i e seen that large doppler shifts completely ruin the required properties but the perturbation is small when the shift is smaller than 2 1/250 of the frequency range shown. To give a specific example: taking T = 0.45 IJS and specifying time functions at 8.5 ns intervals (to correspond with the finger spacing for a 60 MHz IF on quartz), the frequency scale is 2 59 MHz and the doppler tolerance 20.5 MHz. For a secondary radar operating at 1 GHz this allows for radial velocities up to 3 lo5 knots! In this type of system of course, doppler shifts can be swamped by a frequency mismatch between the radar and the transponder. The second filter design has very similar frequency shift characteristics.

Having established two designs which approximate to the inverse filter for a rectangular input pulse and understood their theoretical behaviour when the signal is doppler shifted, we now examine the problems assoc- iated with the SAW implementation of the device.

SAW Implementation of the Inverse Filter

The wide freedom available to the filter designer using SAW techniques is well known4; the close relation between the device geometry and its impulse response enables a great range of responses to be realised in a predictable way. Standard techniques (with polarity reversal to control phase) were used to produce a type 1 filter. The interdigital comb structure is illus- trated in Fig 3. Although shown with overlap weighting each transducer is in fact "finger-break" weighted. The operating frequency is 60 MHz. The maximum overlap

Figure 1. Time slices through type 1 filter ambiguity surf ace.

Figure 2. Frequency slices through type 1 filter ambiguity surface.

Figure 3 . Simplified diagram of type 1 S A W filter (drawn here with overlap weighting for clarity). Transducer A has 27 finger pairs and the inverse transducer B has 87. C, D, E and F are test transducers having a range of bandwidths.

is 340 wavelengths, this number being chosen to bring the average overlap of the apodised transducer to 53 wavelengths. Conventional photolithography was used in the fabrication with a single stage reduction of 25:l from the artwork.

The four phase-reversals in the input-response were obtained by inverting the regular alternations of finger connections to the two bus-bars. Where this happens, two adjacent fingers are connected to the same pad and the absence of field at this point must cause a gap in the response. In the impulse response of

h38

Fig 7 there are eight such transitions and six negative going delta functions occurring in positive regions of the response. Conventional apodisation using inver- sions of the finger sequence might have lead to sig- nificant departures from the wanted response as the result of a large number of 'dead' fingers, so the stratagem was adopted of separating the generation of the positive and negative portions of the impulse response, electrically feeding two input transducers in parallel and combining their acoustic outputs at a single transducer.

The device is shown in Fig 4, the overall schematic view being supplemented by a simplified diagram showing part of the structure. The delta-

*#-, F \ IO

I I ' I

. . \

C

Figure 4, a) Sketch of apodised transducer of type 2 filter. The shaded sections represent acoustic sources achieved by finger-break weighting. b) Detail of finger pattern, simplified as before.

functions are 3-finger transducers and the corres- ponding gaps in the response of the positive half are achieved by means of "dmy" fingers. The blank spaces in each half are filled with a "grid-iron" of shorted fingers to avoid variations in propagation velocity. The maximum overlap was about half that of the first design because of restriction on the size of available substrates. Further details m y be found in the figure legend. The device could doubtless be improved by more careful normalisation of the finger overlaps, by paying more attention to detail in the configuration of the fingers (there are still some possible gaps in the impulse response) and by the incorporation of diffrac- tion correction.

Experimental Results

Figure 5. type 1 device. requirements indicated on the theoretical plot.

Experimental and theoretical response of the Note the positive and negative phase

Figure 6. Response of inverse filter to input pulse at 60 MHz duration 0.45 useconds - theory and experiment.

The impulse response of the type 1 device is shown in figure 5 and is seen to be close to the theoretical requirement. design input pulse is shown in figure 6 and compares very favourably with the theoretical curve - both theory and experiment show 9 dB for the peak sidelobe level. The insertion loss measured to the peak of the output pulse is 57 dB.

The inverse filter response to its

Detailed measurements of variation of output pulse with the frequency of the input signal shows the ambiguity surface to be in very good agreement with the theoretical prediction.

The results for the type 2 device are shown in figure 7 and here we display the amplitude of the outputs of the positive and negative phase channels separately for comparison with theory; the agreement is good. However, detailed examination of the phase of the output of the two channels shows that they are not in exact antiphase, the error varying by between 5 and 30 degrees. This phase error produces a depart- ure from the theoretical prediction for the response to the design input pulse as shown in figure 8(a). The close in sidelobes are -14 dB and the far-out sidelobes -18; theoretically sidelobes are below -18 di.

h39

\ \ \ \

Figure 7. illustrating the positive and negative phase require- ments (theory) and the performance of the 2 channel approach (experiment).

Impulse response of the type 2 device and

4 \ \ I b ) I

Figure 8. Response of type 2 filter to input pulse at 60 MHz, duration 0.45 weconds, (a) using device with no phase compensation, (b) simulated result, (c) using device with correction circuits.

The phase error arises by the addition of signals due to reflections produced where the input pulse propagates under the many fingers of the structure (410 fingers). mismatch produced by the loading of the quartz surface by the metal electrodes and the effect could be minim- ised by using the split-finger technique5. improved performance shown in figure 8(c) has been achieved by introducing a differential phase correction and a small amplitude correction of the output from each channel by means of external circuitry. improvement cannot be affected simultaneously for all sidelobes because the phase error varies with time. The result illustrated here shows no sidelobe greater than -16 dB. The insertion loss of the type 2 device is 64 dB.

These reflections originate from the

The

The

Conclusions and Discussion

It has been demnstrated that inverse filters can be implemented using SAW techniques. design theory is understood and the SAW device design can be automatically synthesised by computer.

The basic

The

design investigated here, though not fully optimised from the SAW design point of view, theoretically and experimentally achieves an improvement in pulse com- pression ratio over the matched filter of nearly an order of magnitude for simple IF burst waveform. (Note the comparison of the matched filter and the inverse filter in Figure 9.) The technique can be applied for any waveform and is potentially useful where other systems constraints do not permit the transmitted waveform to be chosen for optimum resol- ution and where signal to noise ratio is not a problem.

Figure 9. inverse filters. Ratio of 3 dB widths is 8.4.

Theoretical response of matched filter and

The use of an inverse filter is justified where signal returns which overlap in time require to be resolved in order to effect reliable data processing. The technique could be particularly useful for improving the performance of secondary radars used for Air Traffic Control where there is a "garbling" problem due to overlapping returns. A SAW. solution to this ATC problem has already b en investigated which uses a correlation receiver'; the performance could be improved using the inverse'filter. areas which merit investigation in this respect include mapping radars and radar techniques for detecting underground objects.

Further

There is little doubt that in future other analogue techniques will be available for implementing inverse filter responses but at the present time the capability offered now by SAW devices will prove very effective in establxing the validity of the inverse filter technique in signal processing for a range of systems .

Acknowledgements

The authors wish to express their thanks to various colleagues at RRE: to Mr D J Snell for com- putational services, Mr A S-Young for photoreduction, Mr A Letchford and team for photolithography, and Mr G L Moule for measurement. uced by permission of the Controller, HM Stationery Off ice.

This paper is reprod-

References

1 1953, 1024-1031.

2 1653-1678, (particularly equation (133)).

3 Radar", McGraw Hill 1969, p 153.

4 Tancrell, R H, and Holland, M G, "Acoustic Surface Wave Filters", Proc IEEE, 1971, Vol 59, pp 393-409.

5 Bristol, T U, Jones, W R, Snow, P B and Smith, W R, IEEE Ultrasonics Symposium Proceedings, Boston, (1972).

6 Moule, G L, Proceedings of SAW Components, Devices and Applications Conference, Aviemore, Scotland 1973.

Urkowitz, H, J App Physics, Vol 24, No 8, Aug

Di Toro, M J, Proc IEEE, Vol 56, No 10, Oct 1968,

Rhiaczek, A W, "Principles of High-Resolution