# Signal Power Distribution Upon Resampling

1. Aug 5, 2010

Suppose we have a real 25 Hz signal that is sampled at 100 Hz without any anti-aliasing filters.

Taking the DFT results in a series of spikes 25 Hz, 75 Hz, 125 Hz, 175 Hz etc.

Now suppose the 25 Hz signal after it enters the 100 Hz system now enters a 60 Hz system. If we take the DFT now, we see the same repeating spikes as before, however now they are in sets of three.

For a 60 Hz system, the original 25 Hz signal will look like 25 Hz, but the 75 Hz (first alias) will appear as a 15Hz signal to the 60 Hz system. The 2nd alias at 125 Hz will appear as a 5 Hz signal to the 60 Hz system.

Now if we look at the signal power in the 25 Hz signal from the system that samples at 100 Hz and add up the power of the 5 Hz, 15 Hz, and 25 Hz spikes from the 60 Hz system - we will see they are equal according to Parseval's theorem.

But, how exactly do I figure out a closed form way of expressing what fraction of the power goes to each of the individual frequencies?

I tried to fit the signal sin(2*pi*25*n/100) to
a1*sin(2*pi*5*n/60) + a2*sin(2*pi*15*n/60) + a3*sin(2*pi*25*n/60) but had no luck when trying to fit the signals (yes I know the power and amplitude coefficients will be different).

Am I doing the Fourier transform wrong? Can anyone point me to a reference that discusses this topic? Thanks