Digital Waveguide (DW) Scheme

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Introduction

Finite Difference Time Domain (FDTD) Scheme

FDTD and DW Equivalence

The displacement of an object describes how far away and in what direction it has been moved, or displaced, from its rest position. Displacement is a commonly chosen wave variable in physical modeling. — Click for http://ccrma.stanford.edu/~jos/Mohonk05/Ideal_Plucked_String_Displacement.html

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The sampling rate of a discrete-time signal is defined as the number of samples per second. Its units are thus in Hertz (Hz). Shannon's Sampling Theorem states that the original continuous-time signal can be recovered exactly from the samples if and only if the sampling rate is higher than twice the highest frequency present in the original signal. Any higher frequencies will alias to frequencies below half the sampling rate. — Click for https://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

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Derivation of the sampling theorem which states that any signal can be perfectly reconstructed, in principle, from uniformly spaced samples of that signal, provided that the sampling rate is higher than twice the highest frequency present in the signal — Click for https://ccrma.stanford.edu/~jos/mdft/Sampling_Theorem.html

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A signal is typically a real-valued function of time. A discrete-time signal is typically a real-valued function of discrete time, and is therefore a time-ordered sequence of real numbers. — Click for http://ccrma.stanford.edu/~jos/filters/Definition_Signal.html

A waveguide restricts wave propagation to a particular subspace, which is usually a line. Vibrating strings, woodwinds, and transmission lines are examples of one-dimensional waveguides. — Click for http://en.wikipedia.org/wiki/Waveguide

Traveling waves propagate (i.e. travel) through space. — Click for http://ccrma.stanford.edu/realsimple/travelingwaves

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Sampling is the process of converting a continuous-time signal into a discrete-time signal. — Click for http://ccrma.stanford.edu/~jos/mdft/Sampling_Theory.html

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Physical Audio Signal Processing

Introduction

Finite Difference Time Domain (FDTD) Scheme

FDTD and DW Equivalence

Digital Waveguide (DW) Scheme

We now derive the digital waveguide formulation by sampling the traveling-wave solution to the wave equation. It is easily checked that the lossless 1D wave equation $Ky''=\epsilon {\ddot y}$ is solved by any string shape which travels to the left or right with speed $c \isdeftext \sqrt{K/\epsilon }$ [100]. Denote right-going traveling waves in general by and left-going traveling waves by , where and are assumed twice-differentiable. Then, as is well known, the general class of solutions to the lossless, one-dimensional, second-order wave equation can be expressed as

$\displaystyle y(t,x) = y_r\left(t-\frac{x}{c}\right) + y_l\left(t+\frac{x}{c}\right). \protect$

(E.4)

Sampling these traveling-wave solutions yields

$\displaystyle y(nT,mX)$	$\displaystyle =$	$\displaystyle y_r(nT-mX/c) + y_l(nT+mX/c)$
	$\displaystyle =$	$\displaystyle y_r[(n-m)T] + y_l[(n+m)T]$
	$\displaystyle \isdef$	$\displaystyle y^{+}(n-m) + y^{-}(n+m) \protect$	(E.5)

where a ``

'' superscript denotes a ``right-going'' traveling-wave component, and ``

'' denotes propagation to the ``left''. This notation is similar to that used for acoustic-tube modeling of speech [299].

**Figure E.1:** Digital simulation of the ideal, lossless waveguide with observation points at and .
$\includegraphics[width=\twidth]{eps/fidealCopy}$

Figure E.1 (initially given as Fig.C.3) shows a signal flow diagram for the computational model of Eq.(E.5), termed a digital waveguide model (developed in detail in Appendix C). Recall that, by the sampling theorem, it is an exact model so long as the initial conditions and any ongoing additive excitations are bandlimited to less than half the temporal sampling rate [454, Appendix G]. Recall also that the position along the string, meters, is laid out from left to right in the diagram, giving a physical interpretation to the horizontal direction in the diagram, even though spatial samples have been eliminated from explicit consideration. (The arguments of $y^{+}$ and $y^{-}$ have physical units of time.)