0.0 The Sampling and Reconstruction Problem

The idea of sampling is simple -- record the signal amplitude at some particular values in time, preferably uniformly spaced. We want these samples to represent the continuous-time signal. With just this primitive idea of recording uniform samples of the signals, we see that the problem of representing the signal becomes ill-posed.

Reconstruction using the samples (in green) using two methods -- sinc and linear interpolation.

In the above figure, the same samples in green are satisfied by two signals $y_1$which is a bandlimited signal and $y_2$which is generated by the linear B-spline kernel. These are not the only two signals that satisfy the samples, and in theory, there are infinitely many such signals.

The reconstruction problem is to sample a given signal such that the samples completely define the signal. From the example above, this requires some additional information about the signal. Shannon considered bandlimited signals.