MATLAB: Are the height of the peaks different in the “FFT for Spectral Analysis” demo in MATLAB 7.11 (R2010b)

MATLABSignal Processing Toolbox

I am running the fftdemo.m file that ships with MATLAB. The resulting power spectral density plot shows two peaks with different amplitudes. Because the original two sine waves that were generated have the same amplitude, I expect that the peaks of the PSD plot have the same amplitude as well.

Best Answer

The two sine waves do have the same power, but it is not immediately evident just by looking at plot of Pyy. There are two reasons why. The first reason is because of the random noise that is being added. If you were to rerun the code several times, you would see the height and width of the two peaks change. This is because the noise that is being generated will be different on successive runs.
Now let's get the noise out of the picture. Try running this code:
t = 0:.001:.25;
x = sin(2*pi*50*t) + sin(2*pi*120*t);
X = fft(x,251);
Pxx = X.*conj(X)/251;
f = 1000/251*(0:127);
plot(f,Pxx(1:128))
You will see two peaks of different height. Note that the power is not determined by the height of the peak, but by the integral of the peak. If you execute:
sum(Pxx(1:25))
sum(Pxx(26:50))
to integrate each peak, it will return 62.5318 and 62.4499, respectively, which are basically equal to each other.
Now, if you want the amplitude of the two peaks to be the same, the easiest thing to do is to increase the length of the FFT by padding it with zeros, thus increasing the signal resolution. The following code should do the trick by increasing signal resolution by a factor of 10:
t = 0:.001:.25;
x = sin(2*pi*50*t) + sin(2*pi*120*t);
X = fft(x,2510);
Pxx = X.*conj(X)/251;
f = 1000/2510*(0:1270);
plot(f,Pxx(1:1271))
Note that we have not changed anything about the way we analyze our data, we are just using a different resolution. Thus there was nothing wrong with the previous result where the height of the peaks was different. It is just that the resolution used in that case was not fine enough.
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