in 1949. It is also commonly called Shannon's interpolation formula and Whittaker's interpolation formula. E. T. Whittaker, who published it in 1915, called it the Cardinal series.
Definition
Given a sequence of real numbers, x[n], the continuous function
(where "sinc" denotes the
Sampling theorem.) Otherwise, the frequency components above the Nyquist frequency "fold" into the sub-Nyquist region of X(f), resulting in distortion. (See Aliasing
This is equivalent to filtering the impulse train with an ideal (brick-wall) low-pass filter with gain of 1 (or 0 dB) in the passband. If the sample rate is sufficiently high, this means that the baseband image (the original signal before sampling) is passed unchanged and the other images are removed by the brick-wall filter.
this is satisfied if the sequence belongs to any of the spaces with 1 ≤ p < ∞, that is
This condition is sufficient, but not necessary. For example, the sum will generally converge if the sample sequence comes from sampling almost any stationary process, in which case the sample sequence is not square summable, and is not in any space.
Stationary random processes
If x[n] is an infinite sequence of samples of a sample function of a wide-sense stationary process, then it is not a member of any or Lp space, with probability 1; that is, the infinite sum of samples raised to a power p does not have a finite expected value. Nevertheless, the interpolation formula converges with probability 1. Convergence can readily be shown by computing the variances of truncated terms of the summation, and showing that the variance can be made arbitrarily small by choosing a sufficient number of terms. If the process mean is nonzero, then pairs of terms need to be considered to also show that the expected value of the truncated terms converges to zero.
Since a random process does not have a Fourier transform, the condition under which the sum converges to the original function must also be different. A stationary random process does have an
. A suitable condition for convergence to a sample function from the process is that the spectral density of the process be zero at all frequencies equal to and above half the sample rate.