Bandlimited
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.
Encyclopedia
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 . If this signal is sampled at a rate faster than so that we have the samples , where is an integer, 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 the
inverse Fourier transform of 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 Nyquist-Shannon, 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.
Bandlimited versus timelimited
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.
Proof: Assume that a signal which has finite support in both domains exists, and sample it faster than the Nyquist frequency. This finite number of time-domain coefficients should define the entire signal. Equivalently, the entire spectrum of the bandlimited signal should be expressible in terms of the finite number of time-domain coefficients obtained from sampling the signal. Mathematically this is equivalent to requiring that a polynomial can have infinitely many zeros in bounded intervals since the bandlimited signal must be zero on an interval beyond a critical frequency which has infinitely many points. However, it is well-known that polynomials do not have more zeros than their orders due to the fundamental theorem of algebra. This contradiction shows that our original assumption that a time-limited and bandlimited signal exists is incorrect.
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 to any arbitrary level of accuracy desired.
A similar relationship between duration in time and bandwidth in frequency also forms the mathematical basis for the
uncertainty principle in
quantum mechanics. In that setting, the "width" of the time domain and frequency domain functions are evaluated with a variance-like measure. Quantitatively, the uncertainty principle imposes the following condition on any real waveform:
where
is a measure of bandwidth , and
is a measure of time duration .
References
See also
...
