Problem 3

In the figure below, we have a sampler, followed by an ideal low pass filter, for reconstruction of from its samples . From the sampling theorem, we know that if is greater than twice the highest frequency present in and , then the reconstructed signal will exactly equal . If this condition on the bandwidth of is violated, then will not equal . We seek to show in this problem that if , then for any choice of T, and will always be equal at the sampling instants;

that is,

To obtain this result, consider the following equation which expresses in terms of the samples of :

With , this becomes

By considering the value of for which , show that, without any restrictions on , for any integer value of k.