Fourier transform power dependent on frequency

[neil.thompson]

Homework Statement

this is something i noticed doing homework rather than homework itself. I plot fft output from different frequency signals, i am not sure why power changes with increasing frequency?

Homework Equations

if i take (with MATLAB notation):

time = 0:0.01:10
y = fft(sin(2*pi*f*time))

with f = 5
then the maximum amplitude of the fft output is about 498.

with f = 10
the maximum amplitude of fft output is 492.

3. My attempt at a solution

I understand the amplitude is 'halved' in both cases because this fft is ambiguous so the energy is spread over two peaks. But why is the energy less when the frequency increases? I have more cycles in the case with more frequency, but I suppose this means I have less samples. Is it usual to normalise this in some way? It seems like this is something you wouldn't want if you were dealing were plotting energy return from doppler shifts.

 

[mighty2000]

Hello,

Thank you for bringing up this observation. The change in power with increasing frequency in your FFT output is due to the sampling rate and the Nyquist-Shannon sampling theorem. This theorem states that in order to accurately represent a signal in the frequency domain, the sampling rate must be at least twice the highest frequency component in the signal. In your example, when the frequency increases from 5 to 10, the sampling rate (0.01) remains the same, but the highest frequency component in the signal (10) is now equal to the sampling rate. This leads to aliasing, where the higher frequency component is "folded" back into the lower frequency component, resulting in a decrease in power. To avoid this, you can increase the sampling rate or decrease the frequency of the signal.

In terms of normalizing the FFT output, it is not necessary if you are only interested in the relative power between different frequency components in the same signal. However, if you want to compare the power between different signals, it is recommended to normalize the FFT output by dividing by the number of samples.

I hope this helps clarify the observed change in power with increasing frequency in your FFT output. If you have any further questions, please don't hesitate to ask.