Digital Sound & Music: Concepts, Applications, & Science, Chapter 4, last updated 6/25/2013

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represents the PAG, the maximum amount by which the original sound can be boosted

without feedback.

( *

This is Equation 4.14 originally discussed in Section 4.2.2.1.

4.3.5 The Mathematics of Delays, Comb Filtering, and

Room Modes

In Section 4.2.2.4, we showed what happens when two copies of the same sound

arrive at a listener at different times. For each of the frequencies in the sound,

the copy of the frequency coming from speaker B is in a different phase relative

to the copy coming from speaker A (Figure 4.27). In the case of frequencies

that are offset by exactly one half of a cycle, the two copies of the sound are

completely out-of-phase, and those frequencies are lost for the listener in that

location. This is an example of comb filtering caused by delay.

To generalize this mathematically, let’s assume that loudspeaker B is d

feet farther away from a listener than loudspeaker A. The speed of sound is c.

Then the delay t, in seconds, is

Equation 4.20 Delay t for offset d between two loudspeakers

Assume for simplicity that the speed of sound is 1000 ft/s. Thus, for an offset of 20 ft, you get a

delay of 0.020 s.

What if you want to know the frequencies of the sound waves that will be combed out by

a delay of t? The fundamental frequency to be combed, , is the one that is delayed by half of

the period, since this delay will offset the phase of the wave by 180. We know that the period is

the inverse of the frequency, which gives us

Additionally, all integer multiples of will also be combed out, since they also will be 180

offset from the other copy of the sound. Thus, we can this formula for the frequencies combed

out by delay t.

Programming

Exercise:

Creating Comb

Filtering in C++