Fourier Transforms in Radar And Signal Processing_8

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Fourier Transforms in Radar And Signal Processing_8 Equalization 141Figure 6.7 Array response with equalization: (a) five taps, 20% oversampling; and (b) four taps, 50% oversampling.142 Fourier Transforms in Radar and Signal ProcessingFigure 6.8 Sum beam frequency response; effect of bandwidth: (a) 10% bandwidth; and (b) 200% bandwidth. Equalization 143equalization, and the dotted response is that for the case of only integerdelay compensation (in units of the sampling interval). [In Figure 6.8(a),this curve is identical to the dashed curve, as all the delays are within ±0.5sampling intervals, so no integer compensation is feasible.] We note that,with the same set of parameters, the response is essentially independent ofthe fractional bandwidth—the shapes of the responses are virtually identical.In the second case, the sampling rate is much higher, of course; in this case,as the bandwidth is 2f 0 , where f 0 is the center frequency, the sampling rateis 2.4f 0 . To sample at this rate may be impracticable at radar frequencies,but may well be feasible for sonar, where broadband (or even wideband)operation is much more commonly required and the actual signal frequenciesare much lower. The fact that these responses are very similar is not a coincidence, butillustrates the fact that the response is essentially independent of the fractionalbandwidth and only depends on how well the delays are matched. Thisdepends, for a given set of equalization filter parameters (m and q ), on howclose the required delays are to integer multiples of the sampling period.This will vary, in general, from one element to another and will depend onthe beam steered direction and the element separation. In particular cases,the delays required may all be integral, in sampling periods, in which casethe matching will be exact, in principle, and the response will be completelyflat. At the other extreme, the delays required might all be half-integral,which is the worst case for matching. In general, however, there will be aspread of delays and the performance will be intermediate. This is illustratedin Figure 6.9. Here the frequency axis is the frequency offset from the center,normalized to the bandwidth, so the range shown is just the band overwhich equalization is required. For this figure the parameters were chosenin order to include the two extreme cases described above. The delay requiredfor an element at distance d from the array center is given in (6.28), andputting c = f 0 0 and dividing by the sampling period 1/qF , the requireddelay in units of sampling periods is given by = (d / 0)q sin (F /f 0 ) (6.30) The element separation was increased to one wavelength, so that theelement positions are given as (2n + 1)/2 wavelengths (n an integer from−8 to +7 for the 16-element array). The steer direction remained at 50degrees, but q was increased to 1.3054 so that q sin 50° = 1. The delayrequired for element n is then given, from (6.28), by (2n + 1) (F /2f 0 ). Ifwe choose F = f 0 , the 100% bandwidth case, we see that all the delays are144 Fourier Transforms in Radar and Signal ProcessingFigure 6.9 Sum beam response with frequency offset for various bandwidths.half integral, the worst case, while if F = 2f 0 , the 200% case, the delays areintegral and we have the flat response shown. The other curves are for thecases of 10%, 20%, . . . , 90% bandwidths (not giving a monotonic sequenceof peak ripple levels), and because the fractional delays are distributed overthe full range, the results are much the same and intermediate between theextreme cases. These are also the same results as for bandwidths of 110%to 190%, because it can be seen that for this case (where q sin = 1), theresults for a fractional bandwidth F /f 0 of r and 2 − r will give the sameresult. The effect of varying the equalization parameters is shown in Figure 6.10for the 50% bandwidth case and for the array separation of 0.5 wavelengths.In Figure 6.10(a), we fix the sampling rate at 1.2, or 20% above the minimum,and vary the number of taps m in the equalizing filters. We note that evenwith three tap filters, the ripple in the center of the band is quite small (upto 0.25 dB above the fully equalized level), but there is a rather rapid fallin gain at the edges of the band, starting well within the flat-top region(at ±0.4 bandwidths offset). As m increases, the response improves, and atm = 9 the gain only falls off rather sharply outside the flat top of the signal Equalization 145Figure 6.10 Sum beam frequency response; variation with equaliza ...