6.8 Summary

In this chapter we have looked at equalization of both linear phase variation (due to delay error) and polynomial amplitude error across the band of interest. In the latter case, we saw that the amplitude response requiring equalization could be expressed as a sum of ramp functions. The equalizing weights that minimize the weighted mean square error across the signal band are found as the solution of a matrix equation, the components of which are values of the Fourier transforms of the distorted responses. Thus, for the amplitude distortions, we require the transforms of the ramp functions, and these are found to be derivatives of the sinc function. Including the ramp ^r - snc _r pairs in the set of transform pairs, we now have the tools for carrying out effective equalization for a range of problems without having to perform explicitly any integration whatever.

After showing that the method is successful in a single channel, including compensating for both amplitude distortion and delay mismatch, the case of forming sum and difference beams using an array was taken. Very effective equalization indeed was found to be possible, and, as shown in the interpolation study of Chapter 5, quite short equalization filters are adequate for high performance if there is some degree of oversampling. This does not mean sampling at several times the minimum rate, but typically at only 20% to 50% higher. Only a simple array, of 16 elements in a regular linear configuration, was taken,...