Bandlimiting

Spectrum of a **bandlimited signal** as a function of frequency

A bandlimited signal is a deterministic or stochastic signal whose Fourier transform, or power spectrum, is zero above a certain finite frequency. In other words, if the Fourier transform, or power spectrum has finite support then the signal is said to be bandlimited. This has the consequence that the signal can be fully reconstructed from its samples, provided that the sampling rate is at least twice the maximum frequency in the bandlimited signal. This critical frequency is also referred to as the Nyquist frequency, and the minimum sampling frequency is called the Nyquist rate. This result, usually attributed to Nyquist and Shannon, is known as the Nyquist-Shannon sampling theorem, or simply the sampling theorem.

An example of a simple deterministic bandlimited signal is a sinusoid of the form $x(t)=\sin(2\pi ft+\theta )$ . If this signal is sampled at a rate faster than $f_{s}>2f$ so that we have the samples $x(n/f_{s})$ , where $n$ is an integer, we can recover x(t) completely from these samples. Similarly sums of sinusoids with different frequencies and phases are also bandlimited.

The signal whose Fourier transform is shown in the figure is also bandlimited. Suppose $x(t)$ is the inverse Fourier transform of $X(f)$ shown in the figure. The highest frequency component in X(f) is f_H. As a result, the Nyquist rate is

f_{N}=2f_{H}\,

or twice the highest frequency component in the signal, as shown in the figure. According to Nyquist-Shannon, it is possible to reconstruct $x(t)$ completely and exactly using the samples

x[n]=x(nT_{s})=x\left({n \over f_{s}}\right)

for integer

n\,

as long as

f_{s}>f_{N}\,

, where

T_{s}=1/f_{s}

The reconstruction of a signal from its samples can be accomplished using the Nyquist-Shannon interpolation formula.

Bandlimited versus timelimited

A bandlimited signal cannot be also timelimited. No signal can be arbitrarily narrow in both the time and frequency domains. More precisely, a function and its Fourier transform cannot both have finite support. This fact can be proved by using the sampling theorem.

One important consequence of this result is that it is impossible to generate a truly bandlimited signal in any real-world situation, because a bandlimited signal would require infinite time to transmit. All real-world signals are, by necessity, timelimited, which means that they cannot be bandlimited. Nevertheless, the concept of a bandlimited signal is a useful idealization for theoretical and analytical purposes. Furthermore, it is possible to approximate a bandlimited signal in the real-world to any arbitrary level of accuracy desired.

This relationship between duration in time and bandwidth in frequency also forms the mathematical basis for the uncertainty principle in quantum mechanics.

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Bandlimited versus timelimited

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See also