Band-Limited-Signals

Band Limited Signals - A bandlimited signal is a deterministic or stochastic signal whose Fourier transform or power spectral density is zero above a certain finite frequency. In other words, if the Fourier transform or power spectral density has finite support then the signal is said to be bandlimited.

A bandlimited signal can be fully reconstructed from its samples, provided that the sampling rate exceeds twice the maximum frequency in the bandlimited signal. This 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 . If this signal is sampled at a rate so that we have the samples , for all integers n, we can recover 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 is a signal whose Fourier transform is , the magnitude of which is shown in the figure. The highest frequency component in is . As a result, the Nyquist rate is

or twice the highest frequency component in the signal, as shown in the figure. According to the sampling theorem, it is possible to reconstruct completely and exactly using the samples

for integer and

as long as

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

A bandlimited signal cannot be also timelimited. More precisely, a function and its Fourier transform cannot both have finite support. This fact can be proved by using the sampling theorem.