Spectrum of a bandlimited baseband signal as a function of frequency

Sampling bandlimited signals

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 x(t) = \sin(2 \pi ft + \theta) \ . If this signal is sampled at a rate f_s =\frac{1}{T} > 2B so that we have the samples x(nT) \ , for all integers n, 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 a signal whose Fourier transform is X(f)\ , the magnitude of which is shown in the figure. The highest frequency component in x(t)\ is B \ . As a result, the Nyquist rate is

R_N = 2B \,

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

x[n] \ \stackrel{\mathrm{def}}{=}\ x(nT) = x \left( { n \over f_s } \right) for integer n \, and T \ \stackrel{\mathrm{def}}{=}\ { 1 \over f_s }

as long as

f_s > R_N \,

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 (trigonometric) polynomial can have infinitely many roots 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 roots (values for which the polynomial is equal to 0) than their degree 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:

2 \pi W_B T_D \ge 1

where

W_B is a (suitably chosen) measure of bandwidth (in hertz), and

T_D is a (suitably chosen) measure of time duration (in seconds).

References

See also

• Bandwidth
• Nyquist?Shannon sampling theorem
• Nyquist rate
• Nyquist frequency

de:Bandbegrenzung

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