November 2012
Intermediate to advanced
794 pages
22h 9m
English
book
Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications
by John J. Shynk
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C.4 DISCRETE-TIME FOURIER TRANSFORM
Definition: Discrete-Time Fourier Transform 
The DTFT of x[k] for 
is
(C.43) 
where j is included in the argument of 
to distinguish the DTFT from the continuous-time Fourier transform 
(the notation 
is also used for the DTFT).
The DTFT is not the same as the discrete Fourier transform (DFT) which is a function of discrete frequency. It is well known that the DFT is a useful transform because of the computational advantages of a fast Fourier transform (FFT) implementation. We do not directly use the DFT for any of the topics covered in this book, so a discussion of it is not included here, except to mention that the DFT can be derived from the DTFT by uniformly sampling ω on the unit circle.
The DTFT is obtained from the bilateral z-transform by evaluating X(z) on the unit circle where (with r = 1), which implies that the ROC of X(z) includes |z| = 1. ...
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