The shannon-whitaker sampling theorem

The classical theory behind the encoding analog signals into bit streams and decoding bit streams back into signals, rests on a famous sampling theoremwhich is typically refereed to as the Shannon-Whitaker Sampling Theorem. In this course, this samplingtheory will serve as a benchmark to which we shall compare the new theory of compressed sensing.

To introduce the Shannon-Whitaker theory, we first define the class of bandlimited signals. A bandlimited signal is a signalwhose Fourier transform only has finite support. We shall denote this class as $B_A$ and define it in the following way:

Shannon-whitaker sampling theorem

If $f\in B_A$ , then $f$ can be uniquely determined by the uniformly spaced samples $f\left({\displaystyle \frac{n}{A}}\right)$ and in fact, is given by

It is enough to consider $A=1$ , since all other cases can be reduced to this through a simple change of variables. Because $f\in B_{A=1}$ , the Fourier inversion formula takes the form

Comments:

1. (Good news) The set $\left\{\mathrm{sinc}\right(\pi (\mathrm{t}-\mathrm{n})){\}}_{\mathrm{n}\in \mathbb{Z}}$ is an orthogonal system and therefore, has the property that the $L_2$ norm of the function and its Fourier coefficients are related by,
   $$\parallel f{\parallel }_{L_2}^2=2\pi \sum _{n\in \mathbb{Z}}|f\left(n\right){|}^2$$
2. (Bad news) The representation of $f$ in terms of sinc functions is not a stable representation, i.e.
   $$\sum _{n\in \mathbb{Z}}\left|\mathrm{sinc}\right(\pi (\mathrm{t}-\mathrm{n}\left)\right)|\approx \sum _{\mathrm{n}\in \mathbb{Z}}{\displaystyle \frac{1}{|\mathrm{t}-\mathrm{n}|+1}}\to \text{divergences}$$