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 $f=440$ Hz) and walk around the room with one ear plugged. If the room is reverberant you should be able 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^4.2 (amplitude doubled) due to constructive interference. In a diffuse reverberant soundfield, the distance between nodes is on the order of a wavelength (the ``correlation distance'' within the random soundfield).

Figure: A comb filter with a sinusoidal input.

The way reverberation produces nodes and antinodes for sinusoids in a room is illustrated by the simple comb filter, depicted in Fig. 4.3.^4.3 Figure 4.3 also plots a portion of the unit-amplitude, sinusoidal input signal. Since the comb filter is linear and time-invariant, its response to a sinusoid must be sinusoidal (see previous section). The output signal is plotted for two different choices of delay. The feedforward path has gain $0.5$, and the delay is one period in the first case and half a period in the second. With the delay set to one period, the unit amplitude sinusoid coming out of the delay line constructively interferes with the amplitude $0.5$ sinusoid from the feed-forward path, and the output amplitude is therefore $1+0.5=1.5$. In the other case, with the delay set to half a period, the unit amplitude sinusoid coming out of the delay line destructively interferes with the amplitude $0.5$ sinusoid from the feed-forward path, and the output amplitude therefore drops to $\left\vert-1+0.5\right\vert=0.5$.

Consider a fixed delay of $\tau$ 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., $f\tau=0,1,2,3,\ldots\,$, or $f=n/\tau$, for $n=0,1,2,3,\ldots\,$. On the other hand, destructive interference happens at all frequencies for which number of periods in the delay line is an integer plus a half, i.e., $f\tau = 1.5, 2.5, 3.5,$ etc., or, $f = (n+1/2)/\tau$, for $n=0,1,2,3,\ldots\,$. 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.

Figure: Comb filter amplitude response when delay $\tau =1$ sec.

The amplitude response of a comb filter has a ``comb'' like shape, hence the name.^4.4 Note that if the feedforward gain is increased from $0.5$ to $1$, the comb-filter gain ranges between 0 (complete cancellation) and $2$. Negating the feedforward gain inverts the gain curve, placing a minimum at dc^4.5 instead of a peak.