Allpass filters

Figure 3.13 Allpass filter formed by modification of a comb filter (clockwise from top-left) flow diagram, time response, frequency response, and pole-zero diagram.

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

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.

In a digital simulation, a frequency-dependent speed of propagation can be implemented in a lumped fashion using allpass filters which have a non-uniform delay versus frequency. 

That is, each delay element becomes an allpass filter which approximates the required delay versus frequency. A diagram appears in Fig. 10.14, where Ha (z) denotes the allpass filter which provides a rational approximation toz co/c(a>)... 

Because allpass filters are linear and time invariant, they commute like gain factors with other linear, time-invariant components. Fig. 10.15 shows a diagram equivalent to Fig. 10.14 in which the allpass filters have been commuted and consolidated at two points. For computability in all possible contexts (e.g., when looped on itself), a... 

Alternatively, a lumped allpass filter can be designed by minimizing group delay,... 

Section of a stiff string where allpass filters play the role of unit delay... 

Now we discuss algorithms that reproduce late reverberation. The material is presented in roughly chronological order, starting with reverberators based on comb and allpass filters, and proceeding to more general methods based on feedback delay networks. 

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 ... 

The phase response of the allpass filter is a non-linear function of frequency, leading to a smearing of the signal in the time domain. 

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. 

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... 

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. 

This is verified to be identical to equation 3.21. Any series combination of elementary filters - for instance, a series allpass filter - can be expressed as a feedback delay network with a triangular feedback matrix [Jot and Chaigne, 1991],... 

There are many ways to implement allpass filters [Moorer, 1979, Jot, 1992b] two methods are shown in Figures 3.13 and 3.14. 

Lang and Laakso, 1994] Lang, M. and Laakso, T.I. (1994). Simple and robust method for the design of allpass filters using least-squares phase error criterion. IEEE Trans. Circuits and Systems, 41(l) 40-48. 

A notch filter or a second-order allpass filter can be obtained using the circuit shown in Fig. 7.113. It is based on the fact that a notch or second-order allpass can be realized by subtracting a second-order bandpass transferfunction from an appropriate constant. Setting Ci = C2 = CinFig. 7.111andadding Vi by means of Ra and Rg as shown in Fig. 7.113, we obtain, using superposition. 

FIGURE 7.113 Circuit used to obtain a second-order notch or a second order allpass filter. 

---