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In this section the problem of matching partial cascades of elementary allpass filters to the target phase function will be examined. There are two basic schemes: in the first, a (possibly large) number of identical elementary filters are juxtaposed; in the second, the elementary filters of the cascade have distinct coefficients. For the latter, the computational burden in the design process limits the order of the cascade to about six at the present time.

It is worth emphasising that the numerical determination of the coefficients is specific to one choice of the parameters X, AT (distance and digitisation timestep). The process must be repeated for a set of choices.

5.5.1 A product of identical elementary allpass filters

Only the case of real a* is considered here. The H((o) function for this filter was briefly described in Section 5.3 above. When the number of cascaded identical elementary filters is large, the tangent of the phase angle function appears as in Fig. 5.3. The solid line designates the constructed filter, and the symbols designates the target phase angle function. The similarity between these curves is apparent If it were possible to match the values of CD at which the tangent has infinities with those of the target function, the design would be highly

succesful. Unfortunately this matching appears to be possible only over limited ranges on the 0) axis. This is explained by the fact that the infinities occur whenever the phase angle function crosses kJt/2 fork=l,3,5... The intervals between co points at which these crossings occur are governed locally by the gradient of the phase angle with respect to co. Moreover, the second derivative

governs the rate at which the intervals shorten along the axis. Therefore, it is desirable to match the gradient and the curvature, as well as the value of the phase angle.

With only two parameters to vary, a and N , that is the (real) constant in the elementary allpass filter and the number of such filters in the cascade, it is clear that it will not be possible to match these three values at more than one point on the 0) axis. By the device of bandpass filtering, the effect of the filter away from this point can be suppressed. In the designs of Figs. 5.3 -5.5 only the slope and the curvature were matched (in an exact sense). It may be seen that the fit is locally very good, although the order of the designed cascade had to be as high as 57 in one case. This method holds some promise, especially at large co where other methods have difficulty; but further work is needed to include the third matching criterion into the design algorithm.

5.5.2 Coefficients for a heterogenous cascade

It was seen in the previous section that the cascade of identical allpass filters may give a good fit in the region of large co. For values of co near the origin it is necessary to construct a cascade of allpass filters with unequal coefficients. For reasons explained previously, complex-valued coefficients are used forming

symmetric pairs, yielding a real-valued h(n).

If the number of the unknown coefficients is chosen to be the same as the number of the co-locations where the functions are matched, the problem is equivalent to the solution of a number of simultaneous nonlinear equations. When the problem involves more positions on the co-axis than the number of

the unknown a* coefficients, the design will try to minimize the sum of the squares of certain residuals at chosen locations. Numerical algorithms for both these strategies were implemented in this work.

Since the phase angle tan^q/p) of a single allpass filter is a multiple valued function, see Eq. 5.4, it is not practical to match the quantity

<h . . -ir „ \<72 + tan

for the overall filter. Instead, the routines match the tangent of the phase of the cascade with recursively updated p and q.

For one filter only, one can always find Re{a) and 7m(a) that would give tan(6) the same value as tan(-©2jt/g) at two arbitrary locations. It was found more convenient to align the locations of the zero and of the singularity in tan(G) where sin(co2.x/g) or cos(o9x1 g) is zero.

One elementary filter covers at most two branches of the tan(6) function. When trying to match more branches it was found that a maximum of six branches could be succesfully matched with one cascade of allpass filters, thus covering the range 0-1.84 rad/sec. Other, similarly constructed cascades may be used to cover the rest of the ©-axis. The effect of each cascade on the phase behaviour outside the band of interest is eliminated by bandpass filtering.

In one particular case ( x=50m, T = l s ), the error minimisation exercise resulted in the digital filter presented in Table 5.1. The quality of the matching

achieved by this filter can be seen in Fig. 5.6.

Table 5.1 Coefficients b, for x=50m, T =ls 0 0.4288517 1 -0.9145078 2 1.9277369 3 -2.1324127 4 2.4873923 5 -1.3609384 6 1.0000000

The filter in Table 5.1 has been performance-tested by inputting white noise and analysing the cross-spectrum between input and output The results, shown in Figs. 5.7 and 5.8, indicate that the performance is adequate.