Analysis of a Digital Comb Filter

In Chapter 1, we extensively analyzed the simplest lowpass filter, from a variety of points of view. This served to introduce many important concepts necessary for understanding digital filters. In Chapter 2, we analyzed the same filter using the matlab programming language. This chapter takes the next step by analyzing a more practical example, the digital comb filter, from start to finish using the analytical tools developed in later chapters. Consider this a ``practical motivation'' chapter--i.e., its purpose is to introduce and illustrate the practical utility of tools for filter analysis before diving into a systematic development of those tools.

Suppose you look up the documentation for a ``comb filter'' in a software package you are using, and you find it described as follows:

    out(n) = input(n) + feedforwardgain * input(n-delay1) 
                      - feedbackgain * out(n-delay2)

Does this tell you everything you need to know? Well, it does tell you exactly what is implemented, but to fully understand it, you must be able to predict its audible effects on sounds passing through it. One helpful tool for this purpose is a plot of the frequency response of the filter. Moreover, if delay1 or delay2 correspond to more than a a few milliseconds of time delay, you probably want to see its impulse response as well. In some situations, a pole-zero diagram can give valuable insights.

As a preview of things to come, we will analyze and evaluate the above example comb filter rather thoroughly. Don't worry about understanding the details at this point--just follow how the analysis goes and try to intuit the results. It will also be good to revisit this chapter later, after studying subsequent chapters, as it provides a concise review of the main topics covered. If you already fully understand the analyses illustrated in this chapter, you might consider skipping ahead to the discussion of specific audio filters in Appendix B, §B.4.

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Subsections

• Difference Equation
• Signal Flow Graph
• Software Implementation in Matlab
  • Sample-Level Implementation in Matlab
  
  
• Software Implementation in C++
• Software Implementation in Faust
• Impulse Response
• Transfer Function
• Frequency Response
• Amplitude Response
• Phase Response
• Pole-Zero Analysis
• Alternative Realizations
  • First-Order Parallel Sections
  • Parallel, Real, Second-Order Sections
  • Parallel Second-Order Signal Flow Graph
  • Series, Real, Second-Order Sections
  
  
• Summary

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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.