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Converting Propagation Distance to Delay Length

We may regard the delay-line memory itself as the fixed ``air'' which propagates sound samples at a fixed speed $ c$ ($ c=345$ meters per second at $ 22$ degrees Celsius and 1 atmosphere). The input signal $ x(n)$ can be associated with a sound source, and the output signal $ y(n)$ (see Fig.2.1 on page [*]) can be associated with the listening point. If the listening point is $ d$ meters away from the source, then the delay line length $ M$ needs to be

$\displaystyle M = \frac{d}{cT} \quad{\hbox{samples}},

where $ T$ denotes the discrete-time sampling interval. In other words, the number of samples delay is the propagation distance $ d$ divided by $ cT$ , the distance sound propagates in one sampling interval. In practice, rounding $ M=d/cT$ to the nearest integer causes no audible difference, unless the echo time is so short that the system is not really perceived as an echo (we'll learn about ``comb filters'' in §2.6 below).

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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4
Copyright © 2023-08-20 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University