Finite and Infinite support in time and frequency domain

Question

It's known that signals with a finite support in time has an infinite support in frequency (talking about Fourier Transforms).

A common example that is given for the claim is the triangle function with it's Fourier transform $\mathrm{sinc}^2$ function.

I wanted to understand why the following example is not a good one:

$x(t) = \textrm{rectangular}$

$X^f(\omega)=\mathrm{sinc}(\omega)$

We were told in the class that we can't assume that sinc really have infinite support (and that is what I'm not understanding), so the given signal (rectangular) won't be an example for a signal that has a finite support in time and infinite support in frequency.

Answer 1

The coreof the question relies on how to define for the support of a function with $x$ in a domain $\mathcal{D}$. The most common notion in DSP is the closure of the set where $f$ does not vanish, i.e. $$S=\overline{\{x\in\mathcal{D}:f(x)\ne0\}}\,,$$ hence this as a closed set. Here, clearly, for a cardinal sine, the closure of $\mathbb{R}/ \mathbb{Z^*}$ is $\mathbb{R}$ (hence not finite).

There are general results, some even stronger (and beyond my reach), such as in Michael Benedicks, Fourier transforms of functions supported on sets of finite Lebesgue measure, J. Math. Anal. Appl., 106 (1985), pp. 180-183. In a mundane way: if a function and its Fourier transform both have finite measure, then the function is zero almost everywhere.

Yet there are more involved notions for support, like the essential support, which depend on a measure. And you can have a (somewhat pathological) function with a support of $1$, but a $0$ essential support (like the Dirichlet function). This is not the case for continuous functions, like the triangle, the cardinal sine or its square, where both support notions coincide, see for instance Essential support vs. classical support for a continuous function. So both are good examples to that respect.

So, I suspect that the fact that $\sin(x)/x$ is not Lebesgue-integrable (and possibly the rectangular function is not continuous) played a role in your teacher's "not a good one" assertion.

PS: as evoked by Marcus Müller, note that it is somewhat funny to work in $L_2$ with projection on sine and cosine functions that are not square integrable!

Other sources :

• W.O Amrein, A.M Berthier, On support properties of $L^p$-functions and their Fourier transforms, J. Funct. Anal., 24 (1977), pp. 258-267
• Bandwidth Versus Time Concentration: The Heisenberg–Pauli–Weyl Inequality, SIAM J. Math. Anal., 15(1), 151–165

Read More: https://epubs.siam.org/doi/10.1137/0515012

Answer 2

The sinc is just fine an example as a signal with infinite support:

Support being defined as the smallest interval in which the function has non-zero values, it's trivial to see that $\text{sinc}(x)=\sin(x)/x$ (for all $x$ but a single point) becomes arbitrarily small for large $x$, but never actually constantly $\text{sinc}(x>\xi)=0$ for any finite $\xi$. So, q.e.d., sinc has infinite support.

We use this exact property to illustrate to students all over the world that the perfect filter (i.e. something that looks like a rectangle in frequency domain) can never be implemented as a FIR filter, as that would need to be infinitely long.