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ch3_dtft

# ch3_dtft - MIT OpenCourseWare http/ocw.mit.edu HST.582J...

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Unformatted text preview: MIT OpenCourseWare http://ocw.mit.edu 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: http://ocw.mit.edu/terms . 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 3 - FOURIER REPRESENTATION OF SIGNALS AND SYSTEMS c Bertrand Delgutte and Julie Greenberg, 1999 Introduction In Chapter 2, we studied the operation of linear, time-invariant filters in the time domain. Further insights and greater simplicity in the analysis are provided when filters are studied in the frequency domain, that is, when sinusoidal signals (or equivalently, complex exponentials) are used as inputs. The key to this simplicity is twofold: (1) The response of a linear, time- invariant system to a sum of sine waves is always a sum of sine waves at the same frequencies. (2) Fourier’s theorem expresses any reasonably well-behaved signal as an infinite sum of sine waves. Therefore understanding the response to sine waves suﬃces to predict responses to arbitrary signals. Frequency-domain analysis is particularly useful for separating signal from noise when the signal and the noise occupy different frequency bands, or at least when the signal-to-noise ratio differs for different frequency regions. The frequency-domain properties of filters can be introduced with either discrete-time or continuous- time signals. We will first present detailed results for discrete-time signals and systems and then extend these results to continuous-time signals. In Chapter 5, we will consider sampling theo- rems that provide a relation between these two types of signals. The frequency representation of random signals will be studied in Chapter 12. 3.1 Frequency response of LTI systems According to the convolution theorem introduced in Chapter 2, the responses of linear, time- invariant (LTI) systems to arbitrary inputs can be computed if the unit-sample response is known. Indeed, convolution is usually the simplest method to compute the output of a digital filter if either the input or the impulse response is only a few samples in duration. There is another condition for which the responses of LTI systems can be readily computed: when the inputs are complex exponentials of the form e j 2 πfn . Specifically, let us consider the response of a system with unit-sample response h [ n ] to a complex exponential with frequency f : ∞ y [ n ] = h [ n ] ∗ e j 2 πfn = h [ m ] e j 2 πf ( n − m ) m = −∞ This can be written as: y [ n 2 πfn ∞ ] = e j h [ m ] e − j 2 πfm = H ( f ) x [ n ] m = −∞ 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 (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. with...
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ch3_dtft - MIT OpenCourseWare http/ocw.mit.edu HST.582J...

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