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. ...
Get Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications now with the O’Reilly learning platform.
O’Reilly members experience books, live events, courses curated by job role, and more from O’Reilly and nearly 200 top publishers.
