Digital Waveguide Models

The term $y^{+}[(n - m)T_s] = y^{+}(n - m)$ can be interpreted as the output of an -sample delay line of input $y^{+}(n)$ . Similarly, the term $y^{-}[(n + m)T_s] = y^{-}(n + m)$ can be interpreted as the input to an -sample delay line with output $y^{-}(n)$ .
Physical wave variables are given by the superposition of traveling waves. In a one-dimensional system, we can use two systems of unit delay elements to model left- and right-going traveling waves and sum delay-line values at corresponding “spatial” locations to obtain physical outputs, as depicted below.

Figure 5: Discrete-time simulation of ideal, lossless wave propagation with observation points at and .
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Any ideal, lossless, one-dimensional waveguide can be simulated in this way. The model is exact at the sampling instants to within the numerical precision of the processing system.
To avoid aliasing, the traveling waveshapes must be initially bandlimited to less than half the sampling frequency.
In many modeling contexts, the calculation of physical output values can be limited to just one or two discrete spatial locations. Individual unit delay elements are more typically combined and represented by digital delay lines, as shown below.

Figure 6: Digital waveguide simulation of ideal, lossless wave propagation using delay lines.
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The delay lines can be initialized with displacement data corresponding to any bandlimited, arbitrary waveshape.