CYBK-073

The impulse response function for the horizontal propagation of the wave motion, Eq. (3.10), (Dommermuth 1986) is the target time domain sequence used in this recursive filter design. As mentioned previously, a finite portion of this infinite length sequence is used. A direct truncation of h(t) at some length ttot=KT, where T is the digitization interval, would create ripples in the magnitude characteristics of the designed filter. In addition to that it was found that a function with sharp decay is more likely to create stability problems when approximated by a recursive digital filter. Windowing the function h(t) in the time domain would avoid the instability, but, due to the special frequency content distribution of h(t) along the time axis (Fig. 3.2), this procedure would suppress the higher frequencies present in h(t) in an uncontrolled manner, therefore the preferred solution is to window in the frequency domain in order to include all frequency components of interest and then inverse Fourier transform and digitize in the time domain. The analytical form of this function was presented in Eq. 3.11. The digitization interval T and the

maximum (Nyquist) frequency 1/2T Hz it implies have to agree with the cutoff frequency used in the frequency domain windowing.

The filter designs contained in this study were obtained with the output error method (Section 4.2.2.2). An initial design was carried out by minimizing the equation error of the recursive filter under consideration using the non-iterative methods described in Section 4.2.2.1. The equation error obtained from this

method is accompanied by an output error (see Eq.4.29), usually higher in value. If this output error was acceptable, the design process could be stopped; otherwise it would proceed to a second phase in order to further minimize the output error.

Both linear programming and generalized inverses were used in the preliminary task. In general linear programming would yield the smaller final equation error but would very often lead to an unstable solution, particularly for higher orders of N, M. The generalized inverse method found a rather less accurate but usually stable solution. Sometimes ill-conditioning was encountered. In mild cases the measures discussed in Section 4.2.2.1 overcame the problem, but in some cases the method had to be abandoned and the LP method was resorted to.

The non-iterative approach to minimize the final output error, discussed in Section 4.2.2.2 was found impractical due to conditioning problems of the generally large matrices [B]. Steepest descent would converge at a very slow rate whereas the modified Gauss method was found to lead the solution to the minumum after fewer iterations. However this method is not appropriate in certain cases because the solution of the normal equations for the determination of a search direction is meaningless if the matrix [TtT] in Eq.4.42 is severely ill-conditioned. In these cases steepest descent was the only option left. Instability was encountered during minimization and treating it by inverting the unstable poles was not found to disturb the process. The iterations would stop when both the value of the objective function J(a,b), and the slope of the gradient vector [V«/(a, b), V*/ (a, b)]T fell bellow a certain threshold, in this case lfi3.

Choosing the order of the recursive filter was seen to be very much dependent on the particular problem considered. An initial choice N=M=K/4 was usually sufficient, but in some cases lower order filters could be designed whereas in others the necessary order could be as high as N=M=K/2.

Considering Eq. 3.10, the instantaneous frequency of the impulse response function h(t) is gt/4x, showing that higher frequencies occur at farther points

along the t axis. Therefore, if it is desired to capture frequencies up to q)c, the larger the distance x it is desired to transfer wave motion, the larger is the time portion of h(t) to be retained. A practical implication of this observation is that higher frequencies cannot be easily propagated to large distances. These difficulties are encountered in the design of the corresponding recursive filter, where an inconvenient case would result in a higher order filter. When the cutoff frequency was chosen 271 recursive filters of reasonable order were designed for distances ranging from 3 to 20 m; reducing the frequency requirements to n it was possible to design filters for distances up to 50m. The designed recursive filters are listed in Appendix B, Table B.l for coc=7i and selected examples for the case (Oc=2n are listed in Table B.2.

The quality of the filters is demonstrated for certain cases in Figures 4.1-4.10. The almost perfect agreement with the magnitude and phase characteristics of the ideal filter is apparent. The error in the phase angle and in the magnitude between the designed and the ideal filters has been calculated as follows:

In Figures 4.11-4.12 these are shown for both the recursive and the corresponding FIR filter (Eq. 3.11) of the same length (K=M+N), for all the designed cases. It can be seen that the error in the digital filter is in most cases more that an order of magnitude smaller than that of the corresponding FIR having the same length; the latter would have to be of a much higher length than N+M to achieve the same error levels. £ ^ = 2(1 / W o ) I - 1 t o ) I)2 I

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