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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 . HST582J/6.555J/16.456J Biomedical Signal and Image Processing Spring 2007 Chapter 5  SAMPLING IN TIME AND FREQUENCY c Julie Greenberg and Bertrand Delgutte, 1999 Introduction In previous chapters, we studied how discretetime signals can be obtained by sampling continuous time signals, and then considered the design and analysis of digital filters in both the time and frequency domains. This chapter considers the operation of sampling in more detail. In par ticular, we will study the operations of both sampling in time and sampling in frequency, and examine the effects of sampling in one domain on the representation of the signal in the other domain. Using the concept of sampling, we will establish the relationships between the different Fourier transforms (CTFT, DTFT, CTFS, DFS, DFT). Finally, we will consider two important applications of sampling; implementing continuoustime LTI systems with digital filters is based on sampling in time, while spectral analysis is based on sampling in frequency. 5.1 Sampling in time 5.1.1 Discretetime signals as special continuoustime signals Before discussing sampling in time, it is useful to establish an additional relationship between discretetime signals and continuoustime signals. In particular, discretetime signals can be considered as a special case of continuoustime signals, specifically, a weighted sum of impulses spaced at regular intervals. To see this, consider a discretetime signal x [ n ] with DTFT X ( f ), and define the continuoustime signal x s ( t ) as a sum of shifted impulses weighted by the values of x [ n ]: x s ( t ) = x [ n ] ( t nT s ) (5.1) n = The CTFT of this signal is X s ( F ) = x [ n ] ( t nT s ) e j 2 Ft dt n = = x [ n ] ( t nT s ) e j 2 Ft dt n = = x [ n ] e j 2 FnTs = X ( f ) n =  f = FT s = F HarvardMIT Division of Health Sciences and Technology HST.582J: Biomedical Signal and Image Processing, Spring 2007 Course Director: Dr. Julie Greenberg . Fs Thus, when x s ( t ) and x [ n ] are related as in (5.1), the CTFT of x s ( t ) is identical to the DTFT of x [ n ]. Because the relation between a signal and its transform is unique, this means that x [ n ] 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]. and x s ( t ) can be considered as different representations of the same signal. In other words, one interpretation of discretetime signals is as the weighted sum of continuoustime impulses, ( t nT s ), spaced at regular intervals equal to the sampling period.), spaced at regular intervals equal to the sampling period....
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