The Time Limited Band Limited Theorem

We have seen that the behaviour of signals and their Fourier transforms at infinity can be a major concern. Indeed, in any practical digital computing scheme, signals and the Fourier transforms have to be of limited length. Normally then, any general statement that can be made about the length of Fourier transform pairs will have considerable bearing on digital signal processing, both in theory and in practice.

Our discussion addresses time-limited and band-limited signals. A time-limited signal is one that is confined to a finite length of time and is zero outside that interval. For a time-limited signal, its total energy is contained in an interval

2a:

A= \int_{-a}^{a} |f(t)|^{2}dt

Likewise, a band-limited signal has its spectrum completely confined to a finite frequency interval, so that its total energy is

B= \int_{-b}^{b}|F(\omega)|^{2}d\omega

The time-limited band-limited theorem says that no signal can be both time-limited and band-limited, except for the trivial case where f(t) is identically equal to zero. We prove this important theorem by assuming f(t) is both time-limited and band-limited; then, we show that f(t) necessarily must be the null function. First, observe that not only is this signal

f(t)=\int_{-b}^{b}F(\omega)e^{i\omega t }d\omega =0 for |t| \geq a

but all of its derivatives must also be zero at |t| \geq a. Therefore, differentiating wrt to time under the integral n times gives

\int_{-b}^{b}F(\omega)e^{i\omega a}(\omega)^{n}d\omega for n=0,1,2 \ldots

Next, we write the inverse Fourier transform of a band-limited signal in a special form. For such a signal, we can write

f(t)=\int_{-b}^{b}F(\omega)e^{i\omega t}d\omega=\int_{-b}^{b}F(\omega)e^{i\omega (t-a)}e^{i\omega a}d\omega

Then, using the power series expansion for the exponential function allows term by term integration to give

f(t)= \sum_{n=0}^{\infty} \frac {(i(t-a))^{n}}{n!}\int_{-b}^{b}F(\omega)e^{i\omega a }(\omega)^{n}d\omega

as an alternative representation of a band-limited signal in terms of its spectrum. But, we have shown that if the signal is also time limited, each of the integrals in this sum is identically zero. Hence, f(t)=0 is the only function that can be both time-limited and band-limited.

The theorem immediately raises a spectre of a fundamental nature for digital signal processing because it says that every signal must be infinitely long either in time domain or in the frequency domain, or both. We will see the consequences of this requirement later where we develop the relationship  between continuous signals and their sampled counterparts.

So far, our discussion of the continuous Fourier integral transform has been on a general level, giving powerful theorems and properties applicable to a wide variety of continuous functions. Next, to exemplify these theorems and also to form a basis for further discussion, we introduce a repertoire of Fourier transforms particularly important to digital signal processing.

To continue later…

Nalin Pithwa

 

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