Chapter List

Overview

Previously we studied the problem of isolating the individual complex exponential components of the signal function s( t), given the data vector d with entries s( m ?) , m = 1, , M, where s( t) is

we assume that ? _n < ?/ ?. The second approach we considered involved calculating the function

for ? < ?/ ?. This sum is an example of a (finite) Fourier series. As we just saw, we can extend the concept of Fourier series to include infinite sums. In fact, we can generalize to summing over a continuous variable, using integrals in place of summation; this is what is done in the definition of the Fourier transform.

The Fourier Transform

In our discussion of linear filtering, we saw that if f is a finite vector f = ( f ₁ , , f _M ) ^T or an infinite sequence , then it is convenient to consider the function F( ?) defined for ? ? ? by the finite or infinite Fourier series expression

If f( x) is a function of the...