More on Perfect Reconstruction 2.3 2002/08/06 2003/07/09 14:05:19 GMT-5 Roy Ha rha@rice.edu Justin Romberg jrom@rice.edu Roy Ha rha@rice.edu Richard G. Baraniuk richb@rice.edu Michael Haag mjhaag@rice.edu Justin Romberg jrom@rice.edu Mariyah Poonawala mariyah@rice.edu Prashant Singh prash@ece.rice.edu perfect reconstruct reconstruction sampling sinc This module examines the idea and formula behind perfect reconstruction in more depth.

Introduction In the previous module on reconstruction, we gave an introduction into how reconstruction works and briefly derived an equation used to perform perfect reconstruction. Let us now take a closer look at the perfect reconstruction formula: f t n f s t n t n We are writing f t in terms of shifted and scaled sinc functions. t n t n n is a basis for the space of bandlimited signals. But  wait . . . .

Derive Reconstruction Formulas What is t n t n t k t k ? This inner product can be hard to calculate in the time domain, so let's use Plancharel Theorem · · 1 2 ω ω n ω k

if n k sinc n sinc k 1 2 ω ω n ω k 1 if n k sinc n sinc k 1 2 ω ω n ω n 1 2 ω ω k n 1 2 k n k n 0 In we used the fact that the integral of sinusoid over a complete interval is 0 to simplify our equation. So, t n t n t k t k 1 n k 0 n k Therefore t n t n n is an orthonormal basis (ONB) for the space of bandlimited functions. Sampling is the same as calculating ONB coefficients, which is inner products with sincs

Summary One last time for f t bandlimited Synthesis f t n f s n t n t n Analysis f s n t n f t In order to understand a little more about how we can reconstruct a signal exactly, it will be useful to examine the relationships between the fourier transforms (CTFT and DTFT) in more depth.