Can You Compute The Amplitude/power Of Original Signal From Fourier Transform?

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 origina

Solution 1:

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.