Implementation Structures for Recursive Digital Filters

This chapter introduces the four direct-form filter implementations, and discusses implementation of filters as parallel or series combinations of smaller filter sections. A careful study of filter forms can be important when numerical issues arise, such as when implementing a digital filter in a fixed-point processor. (The least expensive ``DSP chips'' use fixed-point numerical processing, sometimes with only a 16-bit word length in low-cost audio applications.) For implementations in floating-point arithmetic, especially at 32-bit word-lengths or greater, the choice of filter implementation structure is usually not critical. In matlab software, for example, one rarely uses anything but the filter function, which is implemented in double-precision floating point (typically 64 bits at the time of this writing, and more bits internally in the floating-point unit).

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Subsections

• The Four Direct Forms
  • Direct-Form I
    • Two's Complement Wrap-Around
  • Direct Form II
    • More about Potential Internal Overflow of DF-II
  • Transposed Direct-Forms
  • Numerical Robustness of TDF-II
  
  
• Series and Parallel Filter Sections
  • Series Second-Order Sections
    • Matlab Example
  • Parallel First and/or Second-Order Sections
    • First-Order Complex Resonators
    • Real Second-Order Sections
    • Implementation of Repeated Poles
  • Formant Filtering Example
  • Butterworth Lowpass Filter Example
  • Summary of Series/Parallel Filter Sections
  
  
• Pole-Zero Analysis 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.