ch2_dfilt - MIT OpenCourseWare http:/ HST.582J /...

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MIT OpenCourseWare HST.582J / 6.555J / 16.456J Biomedical Signal and Image Processing Spring 2007 For information about citing these materials or our Terms of Use, visit: .
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Harvard-MIT Division of Health Sciences and Technology HST.582J: Biomedical Signal and Image Processing, Spring 2007 Course Director: Dr. Julie Greenberg HST582J/6.555J/16.456J Biomedical Signal and Image Processing Spring 2007 Chapter 2 - DIGITAL FILTERS ± c Bertrand Delgutte and Julie Greenberg, 1999 Introduction Digital filtering is one of the most useful type of processing for discrete-time signals. Digital filters arise naturally in the modeling of physical systems, and have been used for numerical calculations at least since the invention of calculus in the seventeenth century. They are used for separating signals from noise and for frequency analysis, an operation which often reveals important features in the signal. The theory of digital filters is so well developed that it is usually easy to design filters that meet even complicated sets of specifications. Much of this theory directly parallels the theory of analog filters which, historically, was developed first. Indeed, even in the present time, digital filters are often intended to simulate the operation of continuous-time physical systems. In these notes, we will emphasize digital filters because of their importance in the laboratory exercises, and because the underlying mathematical concepts are easier to understand with discrete-time signals than with continuous-time signals. The name “filter” is used because these signal-processing elements typically “pass” or amplify certain frequency components of the signal, while they “stop” or attenuate others. Such a “frequency domain” analysis of digital filters will be treated in Chapter 3. The goal of this chapter is to define digital filters and describe their basic properties. 2.1 Filters defined by linear diference equations A discrete-time system is any mathematical transformation that maps a discrete-time input signal x [ n ] into an output signal y [ n ]. We will begin by considering discrete-time systems defined by a linear, constant-coefficient difference equation (LCCDE) , which constitute an important class of digital filters: K M y [ n ]= ± a k y [ n k ]+ k =1 ± b m x [ n m ] (2.1) m =0 Eq. 2.1 can be used to compute the output y [ n ]attime n from a finite number of previous values of the input ( x [ n ]) and the output. The maximum of the numbers M and K is called the order of the filter. If the input signal is only defined after a certain time, say for n n 0 , then values of both the input and output for a short time prior to n 0 must be known in order to initialize the diFerence equation. Specifically, y [ n ] must be known for n 0 K n n 0 1, and x [ n ]for n 0 M n n 0 1. In many applications, it is justified to assume that these values are zero Cite as: Julie Greenberg, and Bertrand Delgutte. Course materials for HST.582J / 6.555J / 16.456J, Biomedical Signal and Image Processing, Spring 2007. MIT OpenCourseWare (, Massachusetts Institute of Technology. Downloaded on
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ch2_dfilt - MIT OpenCourseWare http:/ HST.582J /...

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