Consider the following interpolator 
where is the upsampling of sequence by an integer factor of , and where is the transfer function of an ideal discrete-time LTI lowpass filter with gain and cutoff frequency . What is the relation between the energies of signals and ?
It can be easily shown  that the interpolator’s input-output relationship is . Hence, evaluating the Z-transforms on the unit circle, we obtain
The energy of the interpolator’s output signal is
where we used Parseval’s theorem for discrete-time signals. Thus,
is an ideal lowpass filter with the following frequency response
and the following sinc impulse response
Please do note that this is a non-causal filter with an impulse response of infinite length. We can then write the interpolator’s output signal’s energy as follows
Performing a change of variable, , we obtain
is the energy of the input signal. Since , we can conclude that the interpolator amplifies the input signal’s energy by a factor of , which is somewhat intuitive.
We considered the case where signals are bi-infinite sequences and is a non-causal ideal lowpass filter. It would be interesting to consider the more realistic case where the signals are finite sequences and is a causal FIR filter.
 Alan V. Oppenheim, Ronald W. Schafer, Discrete-Time Signal Processing, 2nd edition, Prentice-Hall, 1999.