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**Search Introduction to Digital Filters**

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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:

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

Let 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 *:

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.

- The
*Z*Transform - Existence of the
*Z*Transform - Shift and Convolution Theorems

*Z*Transform of Convolution*Z*Transform of Difference Equations- Factored Form
- Series and
Parallel Transfer Functions

- Partial Fraction Expansion
- Example
- Complex Example
- PFE to Real, Second-Order Sections
- Inverting the Z Transform
- FIR Part of a PFE
- Alternate PFE Methods
- Repeated Poles
- Alternate Stability Criterion
- Summary of the Partial Fraction Expansion
- Software for Partial Fraction Expansion

- Problems

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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