FFT Amplitude and FFT Normalization

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moose
moose on 27 Dec 2015
Commented: Chunru on 24 May 2022
Hello, I have 2 questions: 1) what's the (physical) meaning of the Y-axis in an FFT vs. Frequency plot? (I mean usually I plot the abs(FFT(signal))
2) How can I normlalize my fft plot by the signal energy (if I want to compare to fft plots, of different signals...)?
THANK YOU
  1 Comment
Soufiane Atouani
Soufiane Atouani on 17 May 2022
Hi,
for exmples if you have a signal x sampled at Ts (the inverse of sampling rate) to plot fft with Nfft point you'll write
mag = abs(fft(x,Nfft))
f = 0:fs/Nfft:fs-fs/Nfft
plot(rn mag)
so to normalise your frequency
put f_noramlised = 0:1/Nfft:1-1/Nfft
then plot(f_normalised, mag)
if you want to normalise amplitude too divide your mag by max(mag)

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Answers (1)

Chunru
Chunru on 18 May 2022
Edited: Chunru on 18 May 2022
There are different ways of interpreting the FT. Here is one way according to Parseval's theorem:
or
The LHS of the first equation is the total signal energy. The LHS of the last equation is the power of the signal.
If one computes FT{x(n)} = X(k), then plot out . The integration of over the freq k is the total signal power. Therefore can be interpreted as Power Spectrum Density of x(t). If you plot out , then it can be interpreted as the magnitude spectrum where magnitude square is the power. If you use dB scale, that is , then you are free to interpret it as power or amplitude in dB.
If you plot , it may be interpreted as enengy spectrum densitity.
If you plot , it is just the magnitude of Fourier Coefficients (FT).
  6 Comments
Chunru
Chunru on 24 May 2022
Some references at my hand:
J. Cartmhour, Digital Signal Processing, Prentice Hall, 2000, Sec 12.2
S.K. Mitra, Digital Signal Processing, 2nd Ed, McGraw-Hill, 2001,Sec 2.1.3
D.G. Manolakis, V K Ingle, S. M Kogon, Statistical and Adaptive Signal Processing, McGraw-Hill, 2000, Sec 2.1.2
T. Giannakopoulos and A. Pikraki, Introductio to Audio Analysis, Elsevier, 2014, Sec. 4.3.1
F. Eyben, Real-time Speecha and Music Classificatio by Large Audio Feature Space Extraction, Springer Theses, 2016, Sec 2.2.2

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