Analysis of a Digital Comb Filter

In Chapter 1, we extensively analyzed the simplest lowpass filter, $y(n) = x(n) + x(n - 1)$ 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 simplest lowpass 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 analytical tools we will be learning more about in later chapters. The purpose is to introduce and illustrate the practical utility of these tools before diving into their systematic development.

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 need to see its effect on sounds passing through it--you need to see its frequency response. 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. The purpose of this book is to describe how to do this kind of analysis.

As a preview of things to come, we will analyze and evaluate the above example comb filter rather thoroughly. Don't worry about understanding all 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 you have studied the general theory presented in subsequent chapters, as it provides a concise review of the main topics covered. If you already fully understand the analysis illustrated in this chapter, you might consider skipping ahead to Chapter 10, §10.4.