In Figure P7.25 is a sampler, followed by an ideal lowpass filter, for reconstruction of...

In Figure P7.25 is a sampler, followed by an ideal lowpass filter, for reconstruction of x(t) from its samples . Form the sampling theorem, we know that if is greater than twice the highest frequency present in x(t) and , then the reconstructed signal will exactly equal x(t). If this condition on the bandwidth of x(t) is violated, then will not equal x(t). We seek to show in this problem that if , then for any choice of T, and x(t) will always be equal at the sampling instants; that is,

To obtain this result, consider eq. (7.11), which express in terms of the samples of x(t):

With , this becomes

By considering the values of α for which [sin(α)]/α = 0, shown from eq. (p7.25-1) that, without any restriction on x(t), for any integer value of k.