Sampling, data transmission, and the Nyquist rate

The sampling theorem for bandlimited signals of finite energy can be interpreted in two ways, associated with the names of Nyquist and Shannon. 1) Every signal of finite energy and bandwidth W Hz may be completely recovered, in a simple way, from a knowledge of its samples taken at the rate of 2W per second (Nyquist rate). Moreover, the recovery is stable, in the sense that a small error in reading sample values produces only a correspondingly small error in the recovered signal. 2) Every square-summable sequence of numbers may be transmitted at the rate of 2W per second over an ideal channel of bandwidth W Hz, by being represented as the samples of an easily constructed band-limited signal of finite energy. The practical importance of these results, together with the restrictions implicit in the sampling theorem, make it natural to ask whether the above rates cannot be improved, by passing to differently chosen sampling instants, or to bandpass or multiband (rather than bandlimited) signals, or to more elaborate computations. In this paper we draw a distinction between reconstructing a signal from its samples, and doing so in a stable way, and we argue that only stable sampling is meaningful in practice. We then prove that: 1) stable sampling cannot be performed at a rate lower than the Nyquist, 2) data cannot be transmitted as samples at a rate higher than the Nyquist, regardless of the location of sampling instants, the nature of the set of frequencies which the signals occupy, or the method of construction. These conclusions apply not merely to finite-energy, but also to bounded, signals.