Comb filter response

Figure 3.12 Comb filter (clockwise from top-left) flow diagram, time response, frequency response, and pole diagram.

Schroeder determined that the comb filter could be easily modified to provide a flat frequency response by mixing the input signal and the comb filter output as shown in figure 3.13. The resulting filter is called an allpass filter because its frequency response has unit magnitude for all frequencies. The z transform of the allpass filter is given by ... 

The poles of the allpass filter are thus the same as for the comb filter, but the allpass filter now has zeros at the conjugate reciprocal locations. The frequency response of the allpass filter can be written ... 

An allpass filter has a flat magnitude response, and we might expect it to solve the problem of timbral coloration attributed to the comb filter. However, the response of an allpass filter sounds quite similar to the comb filter, tending to create a timbral coloration. This is because our ears perform a short-time frequency analysis, whereas the mathematical property of the allpass filter is defined for an infinite time integration. 

By combining two elementary filters in series, we can dramatically increase the echo density, because every echo generated by the first filter will create a set of echoes in the second. Comb filters are not good candidates for series connection, because the only frequencies that will pass are those that correspond to peaks in both comb filter respones. However, any number of allpass filters can be connected in series, and the combined response will still be allpass. Consequently, series allpass filters are useful for increasing echo density without affecting the magnitude response of the system. 

A parallel combination of comb filters with incommensurate delays is also a useful structure, because the resulting frequency response contains peaks contributed by all of the individual comb filters. Moreover, the combined echo density is the sum of the individual densities. Thus, we can theoretically obtain arbitrary density of frequency peaks and time echoes by combining a sufficient number of comb filters in parallel. 

The environmental setting of the reproducing system greatly affects objective (measurable by test instruments) and subjective (discernible to trained listeners) frequency response and imaging location. Comb filtering is especially apparent when studying the diffraction patterns where sound waves from planal sources impinge the object. Not unexpectedly, objects with minimal sharp transitions produce the least serious off-axis colorations. 

Comb filter An electrical filter circuit that passes a series of frequencies and rejects the frequencies in between, producing a frequency response similar to the teeth of a comb. 

Figure 9.10. Fourier comb filter performance, (a) and (b) Possible circuits, (c) Response of the circuit shown in (a) and discussed in the text relative gain is expressed in db. (d) Performance characteristics of the filter shown in (a) upper trace is 5-kHz-input square wave lower trace is filter output for 5-kHz-square-wave input, (e) Filter response under noise conditions random noise has been added to the 5-kHz-input square wave.

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