C.2 CONTINUOUS-TIME FOURIER TRANSFORM
Definition: Fourier Transform 
The continuous-time Fourier transform of x(t) for 
is
(C.16) 
where ω is radian frequency.
It is obtained from the bilateral Laplace transform X(s) by substituting 
(with real part 
), assuming the ROC includes the imaginary axis (as is shown for the four cases in Figure C.1). The inverse Fourier transform is
(C.17) 
Since radian frequency is related to ordinary frequency as 
, the Fourier transform can also be expressed using the following symmetric transform pair:
(C.18) 
(C.19) 
The units of f are s−1 denoted by hertz (Hz); ...
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.
