The convolution theorem

The last section already indicated the importance of the convolution operation. Essentially, filtering is equal to convolving a function with a specific filter function and, especially important for our purposes, interpolation is essentially a linear filtering operation, as was pointed out by Schafer and Rabiner [52]. A really interesting property is revealed when we look at what happens in frequency domain when we convolve two functions. This is stated by the convolution theorem.

Theorem 2 (Convolution Theorem)   The spectrum of the convolution of two functions in spatial domain is equivalent to the product of the transforms of both input signals, and vice versa, symbolically,

$$\mathrm{f}_1 \ast \mathrm{f}_2 \Leftrightarrow$$ F₁ F₂ (3.28)

$$\mathrm{F}_1 \ast \mathrm{F}_2 \Leftrightarrow$$ f₁ f₂ (3.29)

So one possibility to convolve two functions could be to transform them to the frequency domain, multiply them there and transform them back to spatial domain. This is especially useful if the circumstances require to transform the functions to the frequency domain anyhow, which results in a quite cheap way for convolution.

Furthermore, the convolution theorem provides an excellent means to perform signal analysis in frequency domain. It allow, for instance, to examine (in a very simple way) the consequences of translating a function along the x-axes. Since translation is convolving the function with a translated impulse (Eq. 3.24) and the frequency response of the impulse is given by Eq. 3.21, we have

$$\mathrm{f}(x-t) = \mathrm{f}(x) \ast \delta(x-t) \rightleftharpoons \mathrm{F}(\omega) e^{-2\pi j\omega t}$$ (3.30)

This property is also known as the Shift Theorem:

Theorem 3 (Shift Theorem)  

$$\rightleftharpoons \mathrm{F}(\omega)e^{2\pi j\omega t}$$ f(x + t) (3.31)

$$\rightleftharpoons \mathrm{F}(\omega)e^{-2\pi j\omega t}$$ f(x - t) (3.32)

That means, translating a function only affects its phase spectrum and not its magnitude spectrum.