### Constructive and Destructive Interference

Sinusoidal signals are analogous to monochromatic laser light. You
might have seen ``speckle'' associated with laser light, caused by
destructive interference of multiple reflections of the light beam. In
a room, the same thing happens with sinusoidal sound. For example,
play a simple sinusoidal tone (*e.g.*, ``A-440''--a sinusoid at
frequency Hz) and walk around the room with one ear
plugged. If the room is reverberant you should be able to find places
where the sound goes completely away due to destructive interference.
In between such places (which we call ``nodes'' in the soundfield),
there are ``antinodes'' at which the sound is louder by 6
dB (amplitude doubled--decibels (dB) are reviewed in Appendix F)
due to constructive interference. In a *diffuse* reverberant
soundfield,^{4.3}the distance between nodes is on the order of a wavelength
(the ``correlation distance'' within the random soundfield).

The way reverberation produces nodes and antinodes for sinusoids in a
room is illustrated by the simple *comb filter*, depicted in
Fig.4.3.^{4.4}

Since the comb filter is linear and time-invariant, its response to a
sinusoid must be sinusoidal (see previous section).
The feedforward path has gain , and the delayed signal is scaled by .
With the delay set to one period, the sinusoid coming out of the delay
line *constructively interferes* with the sinusoid from the
feed-forward path, and the output amplitude is therefore
.
In the opposite extreme case, with the delay set to
*half* a period, the unit-amplitude sinusoid coming out of the
delay line *destructively interferes* with the sinusoid from the
feed-forward path, and the output amplitude therefore drops to
.

Consider a fixed delay of seconds for the delay line in
Fig.4.3. Constructive interference happens at all
frequencies for which an *exact integer* number of periods fits
in the delay line, *i.e.*,
, or , for
. On the other hand, destructive interference
happens at all frequencies for which there is an *odd number of
half-periods*, *i.e.*, the number of periods in the
delay line is an integer plus a half:
etc., or,
, for
. It is quick
to verify that frequencies of constructive interference alternate with
frequencies of destructive interference, and therefore the
*amplitude response* of the comb filter (a plot of gain versus
frequency) looks as shown in Fig.4.4.

The amplitude response of a comb filter has a ``comb'' like shape,
hence the name.^{4.5} It looks even more like a comb on a dB
amplitude scale, as shown in Fig.4.5. A dB scale is
more appropriate for audio applications, as discussed in
Appendix F. Since the minimum gain is
, the nulls
in the response reach down to dB; since the maximum gain is
, the maximum in dB is about 6 dB. If the feedforward gain
were increased from to , the nulls would extend, in
principle, to minus infinity, corresponding to a gain of zero
(complete cancellation). Negating the feedforward path would shift
the curve left (or right) by 1/2 Hz, placing a minimum at
dc^{4.6} instead of a peak.

**Next Section:**

Sinusoid Magnitude Spectra

**Previous Section:**

Sinusoids at the Same Frequency