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Physical Audio Signal Processing
    Passive Impedances
       Passive Reflectances

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Passive Reflectances

From (L.19), we have that the reflectance seen at a continuous-time impedance $ R(s)$ is given for force waves by

$\displaystyle S(s) \isdef \frac{F^{-}(s)}{F^{+}(s)} = \frac{R(s)-R_0}{R(s)+R_0} $

where $ R_0$ is the wave impedance connected to the impedance $ R(s)$. As discussed earlier, all passive impedances are positive real. It can be easily verified that $ R(s)$ positive real implies that $ S(s)$ is stable and has magnitude less than or equal to $ 1$ on the $ j\omega $ axis (and hence over the entire left-half plane, by the maximum modulus theorem), i.e.,

$\displaystyle \left\vert S(s)\right\vert \leq 1,$   re$\displaystyle \left\{s\right\} \leq 0 $

Any stable $ S(s)$ satisfying (M.1) is called a passive reflectance.

Solving for $ R(s)$, we can characterize every passive impedance in terms of its corresponding reflectance:

$\displaystyle R(s) = R_0\frac{1+S(s)}{1-S(s)} $

The reflectance is always defined relative to an impedance $ R_0$ which is the impedance attached to $ R(s)$ to create an impedance discontinuity and thereby generate (frequency-dependent) reflections.

In the discrete-time case, we have the same basic relations, but in the $ z$ plane:

$\displaystyle S(z)$ $\displaystyle \isdef$ $\displaystyle \frac{F^{-}(z)}{F^{+}(z)} = \frac{R(z)-R_0}{R(z)+R_0}$ (M.1)
$\displaystyle R(z)$ $\displaystyle =$ $\displaystyle R_0\frac{1+S(z)}{1-S(z)}$ (M.2)
$\displaystyle \Gamma(z)$ $\displaystyle =$ $\displaystyle \Gamma _0\frac{1-S(z)}{1+S(z)}$ (M.3)

and

$\displaystyle \left\vert S(z)\right\vert \leq 1, \quad \left\vert z\right\vert \leq 1 $

Stable functions $ S(z)$ satisfying (M.1) are also known in the mathematics literature as Schur functions. In the limit as damping goes to zero (all poles of $ R(z)$ converge to the unit circle, with interlacing zeros as required to remain positive real), the reflectance approaches the transfer function of an allpass filter. Thus, the Schur function is a generalization of an allpass transfer function to allow for loss. Recalling that a lossless impedance is called a reactance, we can say that every reactance gives rise to an allpass reflectance. Thus, for example, waves reflecting off a mass at the end of the string will be allpass filtered. It is intuitively very straightforward that the reflection magnitude cannot exceed $ 1$ at any frequency when reflecting from a passive impedance back into a passive medium at impedance $ R_0$. It is also intuitively satisfying that lossless reflection involves only a phase shift and no amplification or attenuation at any frequency.

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Next: Passive String Terminations
written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.


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