Can you compute the amplitude/power of original signal from Fourier transform?

Question

After taking the Discrete Fourier Transform of some samples with scipy.fftpack.fft() and plotting the magnitude of these I notice that it doesn't equal the amplitude of the original signal. Is there a relationship between the two?

Is there a way to compute the amplitude of the original signal from the Fourier coefficients without reversing the transform?

Here's an example of sin wave with amplitude 7.0 and fft amplitude 3.5

from numpy import sin, linspace, pi
from pylab import plot, show, title, xlabel, ylabel, subplot
from scipy import fft, arange

def plotSpectrum(y,Fs):
 """
 Plots a Single-Sided Amplitude Spectrum of y(t)
 """
 n = len(y) # length of the signal
 k = arange(n)
 T = n/Fs
 frq = k/T # two sides frequency range
 frq = frq[range(n/2)] # one side frequency range

 Y = fft(y)/n # fft computing and normalization
 Y = Y[range(n/2)]

 plot(frq,abs(Y),'r') # plotting the spectrum
 xlabel('Freq (Hz)')
 ylabel('|Y(freq)|')

Fs = 150.0;  # sampling rate
Ts = 1.0/Fs; # sampling interval
t = arange(0,1,Ts) # time vector

ff = 5;   # frequency of the signal
y = 7.0 * sin(2*pi*ff*t)

subplot(2,1,1)
plot(t,y)
xlabel('Time')
ylabel('Amplitude')
subplot(2,1,2)
plotSpectrum(y,Fs)
show()

I'd try math.stackexchange.com The programming part is really tangential to your question. — Jake
Can you give an example ? There are many reasons not to have the right amplitude : wrong fullscale, forgot to take the numbers of sample into consideration (depending on the method), ... — lucasg
It says in your code's comments that you are only plotting half the frequency range. That's why you only get half the amplitude (for any strictly real input signal). — hotpaw2

Paul R Paul R · Accepted Answer · 2012-11-14T15:06:01

Yes, Parseval's Theorem tells us that the total power in the frequency domain is equal to the total power in the time domain.

What you may be seeing though is the result of a scaling factor in the forward FFT. The size of this scaling factor is a matter of convention, but most commonly it's a factor of N, where N is the number of data points. However it can also be equal to 1 or sqrt(N). Check your FFT documentation for this.

Also note that if you only take the power from half of the frequency domain bins (commonly done when the time domain signal is purely real and you have complex conjugate symmetry in the frequency domain) then there will be a factor of 2 to take care.

Can you compute the amplitude/power of original signal from Fourier transform?

1 Answers