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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 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,4.3the distance between nodes is on the order of a wavelength (the ``correlation distance'' within the random soundfield).

Figure 4.3: A comb filter with input signal $ x(t)$ , output signal $ y(t)$ , and delayed-input coefficient $ 0.99$ . The output signal is given by $ y(t)=x(t)+0.99\,x(t-\tau)$ .
\includegraphics{eps/combfilter}

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 $ 1$ , and the delayed signal is scaled by $ 0.99$ . 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 $ 1+0.99=1.99$ . 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 $ \left\vert 1-0.99\right\vert=0.01$ .

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 there is an odd number of half-periods, i.e., the number of periods in the delay line is an integer plus a half: $ 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 4.4: Comb filter amplitude response when delay $ \tau =1$ sec.
\includegraphics[width=4in,height=2.0in]{eps/combfilterFR}

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 $ 1-0.99=0.01$ , the nulls in the response reach down to $ -40$ dB; since the maximum gain is $ 1+0.99
\approx 2$ , the maximum in dB is about 6 dB. If the feedforward gain were increased from $ 0.99$ to $ 1$ , 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 dc4.6 instead of a peak.

Figure 4.5: Comb filter amplitude response in dB.
\includegraphics[width=4in,height=2.0in]{eps/combfilterFRDB}


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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8.
Copyright © 2014-04-21 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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