# 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.](https://1719691385-files.gitbook.io/~/files/v0/b/gitbook-legacy-files/o/assets%2F-MetmKF4Nky-_P8tzlc8%2F-Mig1mXj1J9cXTWC6wXg%2F-Migx04SqYTQOg1CGHyn%2Fimage.png?alt=media\&token=a5b6c6fc-9090-4ddb-9eea-4c9fbfead585)

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.
