Whittaker-Shannon Sampling Theorem: The Finite Interval Version

To summarize: the reconstruction formula

$$f(t) = \sum^{2N}_{k=0} f(x_k)\frac{2\pi}{2N+1}\delta_N (t -x_k)$$ (217)

highlights the key role of the Dirichlet kernel in representing an arbitrary element of $W_{2N+1}$ in terms of a finite set of sampled data. Start with the normalized Dirichlet kernel $\frac{2\pi}{2N+1}\delta_N (t)$ , a vector in $W_{2N+1}$ . By applying discrete shift operations generate a basis. Finally form the linear combination whose coordinates are the sampled values of the function. The resulting formula, Eq.() is also known as (a special case of) the Whittaker-Shannon sampling theorem and it constitutes the connecting link between the analogue world and the world of digital computers.

Lecture 12