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Relation to Finite Difference Approximation

The Finite Difference Approximation (FDA) (§7.3.1) is a special case of the matched $ z$ transformation applied to the point $ s=0$ . To see this, simply set $ a=0$ in Eq.$ \,$ (8.5) to obtain

$\displaystyle s \;\to\; 1 - z^{-1} \protect$ (9.7)

which is the FDA definition in the frequency domain given in Eq.$ \,$ (7.3).

Since the FDA equals the match z transformation for the point $ s=0$ , it maps analog dc ($ s=0$ ) to digital dc ($ z=1$ ) exactly. However, that is the only point on the frequency axis that is perfectly mapped, as shown in Fig.7.15.


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``Physical Audio Signal Processing'', by Julius O. Smith III, W3K Publishing, 2010, ISBN 978-0-9745607-2-4.
Copyright © 2014-03-23 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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