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 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.

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Subsections

• The Z Transform
• Existence of the Z Transform
• Shift and Convolution Theorems
  • Shift Theorem
  • Convolution Theorem
  
  
• Z Transform of Convolution
• Z Transform of Difference Equations
• Factored Form
• Series and Parallel Transfer Functions
  • Series Case
  • Parallel Case
    • Series Combination is Commutative
  
  
• Partial Fraction Expansion
  • Example
  • Complex Example
  • PFE to Real, Second-Order Sections
  • Inverting the Z Transform
  • FIR Part of a PFE
    • Example: The General Biquad PFE
  • Alternate PFE Methods
  • Repeated Poles
    • Dealing with Repeated Poles Analytically
    • Example
    • Impulse Response of Repeated Poles
    • So What's Up with Repeated Poles?
  • Alternate Stability Criterion
  • Summary of the Partial Fraction Expansion
  • Software for Partial Fraction Expansion
    • Example 2
    • Polynomial Multiplication in Matlab
    • Polynomial Division in Matlab
  
  
• Problems

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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.