FIR Digital Filter Design

FFT processors implement long FIR filters more efficiently than any other method, thanks to the speed of the FFT algorithm and the Fourier convolution theorem (§2.3.5). The convolution theorem states that the convolution of an input signal with a filter impulse response is given by the inverse DTFT of the signal's spectrum and the filter's frequency response . Of course, in practice the DTFT is used in sampled form, replacing it with a (zero-padded) FFT. To make the most of FFT processors for FIR filter implementation, we need flexible ways to design all kinds of FIR filters for use in such FFT processors.

This appendix provides a starting point in the area of FIR digital filter design. The so-called ``window method'' for FIR filter design is discussed in some detail, and it is compared with the optimal Chebyshev method. Other methods, such as least-squares, are discussed briefly to provide some perspective. Tools for FIR filter design in both Octave and the Matlab Signal Processing Tool Box are listed where applicable. For further information on digital filter design, see the documentation for the Matlab Signal Processing Toolbox and/or [193,258,212,187,241].

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Subsections

• The Ideal Lowpass Filter
• Least-Squares Impulse Response Design
  • Examples
  
  
• Frequency-Sampling FIR Design
• The Window Method
  • Matlab Support for the Window Method
  • Online Demonstration of the Window Method
  • Bandpass Filter Design Example
    • Matlab code
    • Results
    • In case you don't have the Matlab Signal Processing Toolbox
    • Comparison to the Optimal Chebyshev FIR Bandpass Filter
    • Matlab code
    • Results
  
  
• Hilbert Transform Design Example
  • Choice of Kaiser Window
  • Windowing an ``Ideal'' Impulse Response
  • Hilbert Transformer by the Window Method
  • Comparison to Optimal Chebyshev FIR Filter
    • Conclusions
  • Comparison to use of the hilbert function
  
  
• Postlude on Hilbert Transform Theory
• Generalized Window Method
• Minimum Phase Filter Design
• Minimum Phase and Causal Cepstra
• Optimal FIR Digital Filter Design
  • Filter Specifications
  • Ideal Lowpass Filter Revisited
  • Optimal Least-Squares Filters
  • Optimal Chebyshev Filters
  • Lp norms
    • Special cases
    • Filter Design using Lp Norms
  • Least-Squares Linear-Phase FIR Filter Design
    • Matrix Formulation: Optimal Design, Cont'd
    • Least Squares Optimization
    • Geometrical Interpretation of Least Squares
    • Matlab Support for Least-Squares FIR Filter Design
  • Chebyshev Optimal Linear Phase filter Design
  • More General Real FIR Filters
  • Complex FIR Filter Design
    • Related Paper
  • Arbitrary FIR Magnitude and Phase Specification
  • Second-Order Cone Problems (SOCP)
  • Nonlinear Optimization in Octave

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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 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.