Comb filters

COMB FILTER MODULATOR FOR PARTIALLY MATCHING SEVERAL SPECTRAL LINES... 

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

Figure 3.14 Schroeder s reverberator consisting of a parallel comb filter and a series allpass filter [Schroeder, 1962],...

The echo density of the parallel combs is the sum of the echo densities of the individual combs. Each comb filter i outputs one echo per timeT , thus the combined echo density, expressed as the number of echoes per second, is [Jot and Chaigne, 1991] ... 

Figure 3.15 Mixing matrix M used to form uncorrelated outputs from parallel comb filters [Schroeder, 1962], At(z) are allpass filters, and Q (z) are comb filters.

Figure 3.29 Modification of Schroeder s parallel comb filter to maximize echo density [Jot, 1992b],...

Jot s method of incorporating absorptive filters into a lossless prototype yields a system whose poles lie on a curve specified by the reverberation time. An alternative method to obtain the same pole locus is to combine a bank of bandpass filters with a bank of comb filters, such that each comb filter processes a different frequency range. The feedback gain of each comb filter then determines the reverberation time for the corresponding frequency band. 

This approach has been extended to a multirate implementation by embedding the bank of comb filters in the interior of a multirate analysis/synthesis filterbank [Zoelzer et al., 1990], A multirate implementation reduces the memory requirements for the comb filters, and also allows the use of an efficient polyphase analysis/synthesis filterbank [Vaidyanathan, 1993],... 

Several techniques have been proposed for modifying the shape of the short-time spectrum so as to emphasize those portions deemed to be important for speech perception or to reduce the amplitude of those portions assumed to be noise. One approach is adaptive comb filtering[Lim et al., 1978], In this method, the fundamental frequency of voiced speech sounds is estimated, and a comb fdter is then constructed to pass signal power in the regions of the pitch harmonics and to suppress power in the valleys in between. Experimental results with normal-hearing subjects, however, have shown no significant improvement in intelligibility with this type of system [Perlmutter et al., 1977][Lim et al., 1978],... 

Comb filter with lowpass filter in feedback loop 113... 

Modification of Schroeder s parallel comb filter to maximize echo... 

The first artificial reverberators based on discrete-time signal processing were constructed by Schroeder in the early 1960 s [Schroeder, 1962], and most of the important ideas about reverberation algorithms can be traced to his original papers. Schroeder s original proposal was based on comb and allpass filters. The comb filter is shown in figure 3.12 and consists of a delay whose output is recirculated to the input. The z transform of the comb filter is given by ... 

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 ... 

Let us consider attempting to create a reverberator using a single comb or allpass filter. For the case of a comb filter, the reverberation time Tr is given by ... 

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. 

Schroeder proposed a reverberator consisting of parallel comb filters and series allpass filters [Schroeder, 1962], shown in figure 3.14. The delays of the comb filters are chosen such that the ratio of largest to smallest is about 1.5 (Schroeder suggested... 

For each comb filter, the pole moduli are the same, and given by ... 

From equations 3.23 and 3.26, we can derive the number of comb filters required to achieve a given modal density D m and echo density De [Jot and Chaigne, 1991] ... 

Schroeder chose the parameters of his reverberator to have an echo density of 1000 echoes per second, and a frequency density of 0.15 peaks per Hz (one peak per 6.7 Hz). Strictly applying equation 3.27 using these densities would require 12 comb filters with a mean delay of 12 msec. However, this ignores the two series allpass filters, which will increase the echo density by approximately a factor of 10 [Schroeder, 1962], Thus, only 4 comb filters are required with a mean delay of 40 msec. 

Lim et al., 1978] Lim, J., Oppenheim, A., and Braida, L. (1978). Evaluation of an adaptive comb filtering method for enhancing speech degraded by white noise... 

Shields, 1970] Shields, V. (1970). Separation of additive speech signals by digital comb filtering. Master s thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology. 

Because the Fabry-Perot interferometer is a comb filter, an additional narrow bandpass interference filter is necessary to isolate the Brillouin scattering and reject Raman scattering or fluorescence. 

PZ (beta is a number between 0 and 1.0, and L is the total length of the string) causes notches to appear in the spectrum at multiples ofy / p. This can be modeled explicitly by feeding in the appropriate initial condition. We can also cleverly get the same effect by placing an inverse comb filter (all zero, harmonically spaced notches) in the excitation path, and adjusting the length to cause the correct nulls in the spectrum. 

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