{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

113_1_chapter17

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

Info icon This preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
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
Image of page 1

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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.
Image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern