Symmetry properties and a few simple theorems play a central role in our thinking about the DFT. These same properties carry over to the Fourier integral transform as the DFT sums go over into Fourier integrals. Since these properties are generally quite easy to prove, (and, hopefully, you are already familiar with them), we will simply list them in this section. (The proofs are exercises for you :-))
The symmetry properties are, of course, crucial for understanding and manipulating Fourier transforms. They can be summarized by
and Equation 8b
The applications of these basic symmetry properties leads to
and for two cases of special interest we can show that
if is real, then is Hermitian,
and if is imaginary, then is anti-Hermitian,
The similarity theorem results from a simple change of variable in the Fourier integral
and likewise for the familiar shift theorem
The power theorem which states that
can be specialized to Rayleigh’s theorem by setting
In more mathematical works, Rayleigh’s theorem is sometimes called Plancherel’s theorem.
Of course, the important and powerful convolution theorem — meaning linear convolution — is valid in continuum theory also:
The many variations of the convolution theorem arising from the symmetry properties of the Fourier transform apply as well. For example, the autocorrelation theorem is as follows:
The function is called the cross-power spectrum. When we set , this equation states the important result that the Fourier transform of the autocorrelation of a function is its power spectrum:
The formal similarity between these continuous-theory properties and those of the DFT makes them easy to remember and to visualize. But, there are essential differences between the two. The DFT with its finite sum has no convergence questions. The Fourier integral transform, on the other hand, has demanded the attention of some of our greatest mathematicians to elucidate its convergence properties. As we know, the absolute integrable condition is only a start; it can be relaxed — quite easily at the heuristic level — to include the sine wave/ function pair. The sine wave’s ill behaviour is characteristic of a wide class of functions of interest in DSP that do not decay sufficiently fast at infinity for them to possess a Fourier Transform in the normal sense.