Digital Sound & Music: Concepts, Applications, & Science, Chapter 4, last updated 6/25/2013
represents the PAG, the maximum amount by which the original sound can be boosted
This is Equation 4.14 originally discussed in Section 126.96.36.199.
4.3.5 The Mathematics of Delays, Comb Filtering, and
In Section 188.8.131.52, 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.
Filtering in C++