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:

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
*:

Since multiplying the input transform by the transfer function gives the output transform , we see that embodies the

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

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