TABLE OF CONTENTS A fundamental concept in the study of signals is the frequency content of a signal. For a large class of signals, the frequency content can be generated by decomposing the signal into frequency components given by sinusoids. Following is the basics about Fourier transforming.
For a periodic signal x( t)
with T = 2 ?/ ? 0, the Fourier expansion expressed as a series is available
where
An aperiodic signal x( t) has the Fourier transform
With
The generalized Fourier transform is defined starting from the Fourier transform of a periodic signal as following. For a periodic signal x( t), its Fourier expansion is
Using common Fourier transform pair
Its Fourier transform, or say the generalized Fourier transform of this periodic signal, is
?( ? k ? 0) is a train of impulses located at ? = k ? 0, k = 0, 1, 2,
The Fourier transform of a discrete-time signal x[ n]( n = 0, 1, 2, ) is called discrete time Fourier transform (DTFT) and has the definition
X( ?) has the property of periodicity. It is completely determined by computing X( ?) only over any 2 ? interval, for example, the interval 0 ? ? ? 2 ?.
