The Continuous Fourier Integral Transform Part 2: Properties of the Fourier Integral Transform

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

\mathcal {F}f^{*}(t)=F^{*}(-\omega)  Equation 8a

and \mathcal{F}f(-t)=F(-\omega) Equation 8b

The applications of these basic symmetry properties leads to

\mathcal{F}f^{*}(-t)=F^{8}(\omega)

and for two cases of special interest we can show that

if f(t) is real, then F(\omega) is Hermitian, F^{*}(\omega)=F(\omega)

and if f(t) is imaginary, then F(\omega) is anti-Hermitian, F^{*}(-\omega)=-F(\omega)

The similarity theorem results from a simple change of variable in the Fourier integral

\mathcal{F}f(\omega)=(1/|a|)F((\omega)/a) Equation 9

and likewise for the familiar shift theorem

\mathcal{F}f(t-\tau)=e^{-i\tau \omega}F(\omega) Equation 10

The power theorem which states that

2\pi\int_{-\infty}^{\infty}f(t)g^{*}(t)dt=\int_{-\infty}^{\infty}F(\omega)G^{*}(\omega)d\omega Equation 11

can be specialized to Rayleigh’s theorem by setting g=f

2\pi\int_{-\infty}^{\infty}|f(t)|^{2}dt=\int_{-infty}^{\infty}|F(\omega)|^{2}d\omega Equation 12

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:

\mathcal{F} \int_{-\infty}^{\infty}f(\tau)g(t-\tau)d\tau=F(\omega)G(\omega) Equation 13

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:

\mathcal{F}\int_{-\infty}^{\infty}f(\tau)g(t+\tau)d\tau=F(\omega)G^{*}(\omega) Equation 14

The function FG^{*} is called the cross-power spectrum. When we set g=f, this equation states the important result that the Fourier transform of the autocorrelation of a function is its power spectrum:

\mathcal{F}\int_{-\infty}^{\infty}f(\tau)f(t+\tau)d\tau=|F(\omega)|^{2} Equation 15

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/\delta 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.

More later…

Nalin Pithwa

 

 

 

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