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113_1_chapter17

# 113_1_chapter17 - Chapter 17 THE SAMPLING THEOREM Copyright...

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Chapter 17 THE SAMPLING THEOREM Copyright c 1996 by Ali H. Sayed. All rights reserved. These notes are distributed only to the students attending the undergraduate DSP course EE113 in the Electrical Engineering Department at UCLA. The notes cannot be reproduced without written consent from the instructor: Prof. A. H. Sayed, Electrical Engineering Department, UCLA, CA 90095, [email protected] Sequences are usually obtained by sampling a continuous-time signal at a specific rate. Specifically, a sequence x ( n ) can be obtained by sampling a continuous-time signal x ( t ), say every T s seconds, x ( n ) = x ( t ) | t = nT s An important question that arises is whether x ( t ) can be recovered from knowledge of its samples alone? In other words, are there conditions on how small or how large T s should be so that one can fully recover x ( t ) from its samples? As we shall see, this question is answered by Nyquist’s sampling theorem. In the process of justifying Nyquist’s theorem, we shall also clarify the connections that exist among the transforms we introduced in this book for the study of discrete-time signals (namely, the DTFT and DFT) and the Fourier transform ( FT), which is used in the study of continuous-time signals. Sampling Let T s denote a sampling period (measured in units of time, say seconds). By sampling a continuous-time signal x ( t ) we mean that we form a discrete-time signal x ( n ) from it as follows: x ( n ) = x ( t ) | t = nT s That is, the n -th term of the sequence x ( n ) is the value of the signal x ( t ) at the time instant t = nT s . We refer to F s = 1 /T s as the sampling frequency (measured in units of frequency, say Hertz or samples/second). Intuition . Given a continuous-time signal x ( t ), the more samples we keep of it the more information we have about the signal. Of course, we would like to reduce the number of samples that we need to keep in order to save both in storage requirements and in computa- tional effort. Intuition suggests that the faster the transitions in x ( t ), the more samples we need to keep of x ( t ) (i.e., less spaced sampling is necessary) in order to keep track of the fast variations. Therefore, the choice of a sampling period T s should be related to the frequency content of x ( t ) since this content determines whether x ( t ) has high frequency components or not. The sampling theorem provides a quantitative measure of how high the sampling period T s can be (or how low the sampling frequency F s can be). In order to establish the result, we need to first recall the definition and some of the properties of the Fourier 184

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185 transform (FT) of continuous-time signals. Fourier Transform The Fourier transform pair relations for a signal x ( t ) (for which the FT exists) are given by: x ( t ) = 1 2 π Z -∞ X ( j Ω) e j Ω t d Ω X ( j Ω) = Z -∞ x ( t ) e - j Ω t dt where Ω is the frequency variable, measured in radians/second.
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