FFT Amplitude and FFT Normalization

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

[Soufiane Atouani]

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)

Answer 1

[Chunru]

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).

[Paul]

@Chunru

Do you have a reference or two that defines power as in the second equation above?

Thanks.

[Chunru]

You can google it.

https://www.gaussianwaves.com/2013/12/power-and-energy-of-a-signal/

[Paul]

The linked page shows that (average) power, P_x, is defined as a limit as the number of samples used in the summation goes to infinity, not a summation over a fixed number of samples as in the equations above.

[Chunru]

For strict definition of power, one needs infinite number of samples. For practical definition of power, one have to to use a summation over a finite number to approximate.

Then for random process, finite average can be an estimate of power.

For periodic signal (we have often implicitly assumed that in DFT), the average over a period is just the true power.

The definition (with variations) can be found in almost every text book in DSP, for example, [S.K. Mitra, Digital Signal Processing, 2nd Ed, McGraw-Hill, 2001,Sec 2.1.3].

Definition in matlab:

https://www.mathworks.com/help/signal/ug/measure-the-power-of-a-signal.html#:~:text=The%20power%20of%20a%20signal%20is%20the%20sum%20of%20the,signal%20power%20in%20one%20step.

[Paul]

Thanks for the reference. I'll try to find a copy of Mitra.

FWIW, neither Oppenheim and Schafer, nor Proakis and Manolakis define power as above for a finite duration sequence (periodic and random signals are not finite duration) or a finite window of an infinite duration sequence, so based on my very small sample I'm not sure that almost every textbook includes those definitions.

All but one of the on-line accessible sources I've seen, such as the one at gaussianwaves.com and various university course lecture notes, also do not define power that way for a finite duration sequence, though for sure I haven't done an exhaustive search.

Too bad that doc page doesn't include references ....

[Chunru]

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