A discrete-time realization of the Equation 1 is given by a system that implements the transfer function where m m is the number of samples of delay. When the delay τ τ is an integral multiple of the sampling quantum, m m is an integer number and it is straightforward to implement the Equation 2 by means of a memory buffer.

In fact, an m m-samples delay line can be implemented by means of a circular buffer, that is a set of M M contiguous memory cells accessed by a write pointer IN IN and a read pointer OUT OUT , such that

where the symbol ρ ρ is used for the quotient modulo M M. At each sampling instant, the input is written in the location pointed by IN IN , the output is taken from the location pointed by OUT OUT , and the two pointers are updated with

In words, the pointers are incremented respecting the circularity of the buffer. In some architectures dedicated to sound processing, memory organization is optimized for wavetable synthesis, where a stored waveform is read with variable increments of the reading pointer. In these architectures, a quantity of 2r 2 r memory locations is available, and from these M=2s M 2 s locations (with s<r s r ) are uniformly chosen among the 2r 2 r available cells. In this case the locations of the circular buffer are not contiguous, and the update of the pointers is done with the operations

In practice, since the addresses are r r-bit long, there is no need to compute the modulo explicitly. It is sufficient to do the sum neglecting any possible overflow.

Of course, the Equation 3 is also replaced by

It might be thought that, choosing a sufficiently high sampling rate, it is always possible to use delay lines having an integer number of samples. Actually, there are some good reasons that lead us to state that this is not the case in sound synthesis and processing. In sound synthesis, the models have to be carefully tuned without resorting to very high sample rates. In particular, it is easy to verify that using integer length delays in physical models we get errors in fundamental frequencies that go well beyond the just noticeable difference in pitch.

For instance, for a pressure wave propagating in air at normal temperature conditions, the spatial discretization given by the sampling rate F s =44100Hz F s 44100 Hz gives intervals of 0.0075m 0.0075 m , a distance that can produce well-perceivable pitch differences in a wind instrument. Another reason for using fractional delays is that we often want to vary the delay lengths continuously, in order to reproduce effects such as glissando or vibrato. The adoption of integer-length delays would produce annoying discontinuities. The most widely used techniques for implementing fractional delays are interpolation by FIR filters or by allpass filters. These two techniques are, in some sense, complementary. The choice of one of the two has to be made according to the peculiarities of the system to be simulated or of the architecture chosen for the implementation. In any case, a delay of length m m is obtained by means of a delay line whose length is equal to the integer part of m m, cascaded with a block capable to approximate a constant phase delay equal to the fractional part of m m. We recall that the phase delay at a given frequency ω ω is the delay in time samples experienced by the sinusoidal component at frequency ω ω. For instance, consider a linear filtering block enclosed in a feedback loop (see Section 6): the frequency of the k k-th resonance f k f k of the whole feedback system is found at the points where the phase response equates the multiples of 2π 2 . At these frequencies, the components reappear in phase every round trip in the loop, thus reinforcing their amplitude at the output. The phase delay at frequency f k f k is therefore the effective delay length at that frequency, that is the length of an ideal (linear phase) delay line that gives the same k k-th resonance. Figure 1 shows a phase curve and its crossings with multiples of 2π 2 giving a distribution of resonances.

Sounds, propagating in the air, come into contact with surfaces and objects of various kinds and this interaction produces physical phenomena such as reflection, refraction, and diffraction. A simple and very important phenomenon is the reflection of sound about a planar surface. Due to a reflection such as this, a listener receives two delayed copies of the same signal. If the delay is larger than about a hundred milliseconds, the second copy is perceived as a distinguished echo, while if the delay is smaller than about ten milliseconds, the effect of a single reflection is perceived as a spectral coloration. A simple model of single reflection can be constructed starting from the basic blocks described in this and in the preceding chapters. It is constructed as an m m-samples delay line, with the incidental fractional part of m m obtained by FIR interpolation or allpass filtering, cascaded with an attenuation coefficient g g, possibly replaced by a filter if a frequency-dependent absorption has to be simulated. The output of this lossy delay line is summed to the direct signal. Let us analyze the structure in the case that m m is integer and g g is a positive constant not exceeding 1 1. The difference equation is expressed as and, therefore, the transfer function is

In the case that g=1 g 1 , it is easy to see by using the De Moivre formula that the frequency response of the comb filter has the following magnitude and group delay: and it is straightforward to verify that the frequency band ranging from dc to the Nyquist rate comprises m m zeros (antiresonances), equally spaced by F s mHz F s m Hz The phase response is piecewise linear with discontinuities of π at the odd multiples of F s 2m F s 2 m . If g<1 g 1 , it is easy to see that the amplitude of the resonances is while the amplitude of the points of minimum (halfway between contiguous resonances) is

An important parameter of this filtering structure, called non-recursive comb filter (or FIR comb), is the peak-to-valley ratio

Figure 4 shows the response of a non-recursive comb filter having length m=11 m 11 samples and a reflection attenuation g=0.9 g 0.9 . The shape of the frequency response justifies the name comb given to the filter.

The zeros of the comb filter are evenly distributed along the unit circle at the m m-th roots of -g g as shown in Figure 5.

A simple model of one-dimensional resonator can be constructed using the basic blocks presented in this and in the preceding chapters. It is composed by an m m-samples delay line, with the incidental fractional part of m m obtained by FIR interpolation or allpass filtering, in feedback loop with an attenuation coefficient g g possibly replaced by a filter in order to give different decay times at different frequencies. Let us analyze the whole filtering structure in the case that m m is integer and g g is a positive constant not exceeding 1 1. The difference equation is expressed as and the transfer function is

Whenever g<1 g 1 , the stability is ensured. In the case that g=1 g 1 , the frequency response of the filter has the following magnitude and group delay: and it is easy to verify that the frequency band ranging from dc to the Nyquist rate comprises m m vertical asymptotes (resonances), equally spaced by F s mHz F s m Hz . If g=1 g 1 the filter is at the limit of stability, and this is the only case when the phase response is piecewise linear, starting with the value -π2 2 at dc, with discontinuities of π at the even multiples of F s 2m F s 2 m . If g<1 g 1 , it is easy to verify that the amplitude of the resonances is while the amplitude of the points of minimum (halfway between contiguous resonances) is

An important parameter of this filtering structure, called recursive comb filter (or IIR comb), is the peak-to-valley ratio

Figure 6 shows the frequency response of a recursive comb filter having a delay line of m=11 m 11 samples and feedback attenuation g=0.9 g 0.9 The shape of the magnitude response justifies the name comb given to the filter.

The poles of the comb filter are evenly distributed along the unit circle at the m m-th roots of g g, as shown in Figure 7.

In sound synthesis by physical modeling, a recursive comb filter can be interpreted as a simple model of lossy one-dimensional resonator, like a string, or a tube. This model can be used to simulate several instruments whose resonator is not persistently excited. In fact, if the input is a short burst of filtered noise, we obtain the basic structure of the plucked string synthesis algorithm due to Karplus and Strong [3] .

Many of the effects commonly used in electroacoustic music are obtained by composition of time-varying delay lines, i.e., by lines whose length is modulated by slowly-varying signals. In order to avoid discontinuities in the signals, it is necessary to interpolate the delay lines in some way. The interpolation by means of allpass filters is applicable only for very slow modulations or for narrow-width modulations, since sudden changes in the state of allpass filters give rise to transients that can be perceived as signal distortions [4] . On the other hand, linear (or, more generally, polynomial) interpolation introduces frequency-dependent losses whose magnitude is dependent on the fractional length of the delay line. As the delay length is varied, these variable losses give an amplitude distortion due to amplitude modulation of the various frequency components. Coupled to amplitude modulation, there is also phase modulation due to phase nonlinearity of the interpolator, in both cases of FIR and IIR interpolation. The terminology used for audio effects is not consistent, as terms such as flanger, chorus, and phaser are often associated with a large variety of effects, that can be quite different from each other. A flanger is usually defined as an FIR comb filter whose delay length is sinusoidally modulated between a minimum and a maximum value. This has the effect of expanding and contracting the harmonic series of notches of the frequency response. The name flanger derives from the old practice, used long ago in the analog recording studios, to alternatively slow down the speed of two tape recorders or two turntables playing the same music track by pressing a finger on the flanges. The name phaser is most often reserved for structures similar to the comb FIR filter, with the difference that the notches are not harmonically distributed. Orfanidis [5] proposes to use, instead of the delay line, a bunch of parametric notch filters. Each notch is controllable in its frequency position and width. Smith [6] , instead, proposes to use a large allpass filter instead of the delay line. If this allpass filter is obtained as a cascade of second-order allpass sections, it becomes possible to control and modulate the position of any single pole couple, which represent all the single notches of the overall response. A common feature of flangers and phasers is the relatively large distance between the notches. Vice versa, if the notches are very dense, the term chorus is preferred. Orfanidis [5] , suggests to implement a chorus as a parallel of FIR comb filters, where the delay lengths are randomly modulated around values that are slightly different from each other. This should simulate the deviations in time and height that are found in performances of a choir singing in unison. Vice versa, Dattorro [4] says that a chorus can be obtained by the same structure used for the flanger, with a difference that the delay lengths have to be set to larger values than for the flanger. In this way, the notches are made more dense. For the flanger the suggested nominal delay is 1msec and for the chorus it is 5msec. If the objective is to recreate the effect of a choir singing in unison, the fact of having many notches in the spectrum is generally disliked. Dattorro [4] proposes a partial solution that makes use of a recursive allpass filter, where the delay line is read by two pointers, one is kept fixed and produces the feedback signal, the other is varied to pick up the signal that is fed directly to the output. In this way, when both the pointers are at the nominal position, the structure does not introduce any coloration for stationary signals. A final remark is reserved to the spatialization of these comb-based effects. In general, flanging, phasing, and chorusing effects can be obtained from two different time-varying allpass chains, whose outputs feed different loudspeakers. In this case, sums and subtractions between signals at the different frequencies happen “on air” in a way dependent from position. Therefore, the spatial sensation is largely due to the different spectral coloration found in different points of the listening area.