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Transfer Function Analysis

This chapter discusses filter transfer functions and associated analysis. The transfer function provides an algebraic representation of a linear, time-invariant (LTI) filter in the frequency domain:

$\textstyle \parbox{0.8\textwidth}{The \emph{transfer function}\index{transfer f...
..., and $X(z)$\ denotes the {\it z} transform of the filter input
signal $x(n)$.}$

The transfer function is also called the system function [60].

Let $ h(n)$ denote the impulse response of the filter. It turns out (as we will show) that the transfer function is equal to the z transform of the impulse response $ h(n)$:

$\displaystyle \zbox {H(z) = \frac{Y(z)}{X(z)}}

Since multiplying the input transform $ X(z)$ by the transfer function $ H(z)$ gives the output transform $ Y(z)$, we see that $ H(z)$ embodies the transfer characteristics of the filter--hence the name.

It remains to define ``z transform'', and to prove that the z transform of the impulse response always gives the transfer function, which we will do by proving the convolution theorem for z transforms.

Previous: Time Domain Representation Problems
Next: The Z Transform

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About the Author: 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 (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 for details.


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