# amplitude of (signal) after FFT operation?

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Mary Jon le 15 Nov 2014
I have this code, I am suppose sin of amplitude 10 with frequency 200hz and sampling frequency 20000 hz and do FFT on this signal,
why the Amplitude after FFT is 1000?? where the amplitude must be stay 10
Fs = 20000;
t = 0:1/Fs:0.01;
fc1=200;
x = 10*sin(pi*fc1*t)
x=x';
xFFT = abs(fft(x));
xDFT_psd = abs(fft(x).^2);
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Ait m'barek Soufiane le 2 Août 2017
Hi.. I have the same qst, did you find the answer?

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### Réponse acceptée

Matt le 15 Nov 2014
Modifié(e) : Matt le 17 Nov 2014
Mary,
In general, to return a FFT amplitude equal to the amplitude signal which you input to the FFT, you need to normalize FFTs by the number of sample points you're inputting to the FFT.
Fs = 20000;
t = 0:1/Fs:0.01;
fc1=200;
x = 10*sin(pi*fc1*t)
x=x';
xFFT = abs(fft(x))/length(x);
xDFT_psd = abs(fft(x).^2);
Note that doing this will divide the power between the positive and negative sides, so if you are only going to look at one side of the FFT, you can multiply the xFFT by 2, and you'll get the magnitude of 10 that you're expecting.
The fft documentation has a pretty good example that illustrates this and some other fft best practices.
*Edited for clarity, - see Matt J's comment for the original statement.
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632541 le 21 Avr 2021
Hi Matt J,
That's true if the FFT is being used to compute Fourier Series coefficients. If the idea is to approximate a continuous Fourier Transform integral, the FFT needs to be scaled by the time sampling interval 1/Fs. If the idea is to preserve signal energy (Pareseval's theorem), the FFT needs to be normalized by 1/sqrt(N).
How do I decide which one to use ?
Guanjiang Chen le 11 Mai 2021
Could anyone please tell me what 'a FFT amplitude equal to the amplitude signal which you input to the FFT' mean? What is the amplitude of signal I input? which set will affects the amplitude of input signal? the sample number or the sample frequency?

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### Plus de réponses (3)

Matt J le 16 Nov 2014
Modifié(e) : Matt J le 16 Nov 2014
You also have to be careful about how you design your frequency space sampling. In your current code, the frequency sampling interval is Fs/length(t)=99.5025 Hz. But the frequency you are trying to sample is at 100 Hz, so your Fourier Space sampling will never hit this. And, because the spectrum is sharply peaked, you can get significant errors with this deviation.
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Matt J le 21 Avr 2021
@Kaveri A I think you've answered your own question. If your application expects discrete-space signal energy to be preserved, normalize by 1/sqrt(N).
632541 le 23 Avr 2021
@Matt J My signal is taken from sensor with sampling frequency 50MSPs. And I need fourier coefficients from FFT . Which method do I apply?
1/N?
1/Fs?
1/sqrt(N)?
Which one?

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Brince Babu le 13 Nov 2020
I need to sample a analog real time signal first like a sin wave and then how to do fft in real time such that I can get the magnitude and phase angle of each of the sample
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Guanjiang Chen le 11 Mai 2021

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Frantz Bouchereau le 29 Juil 2021
There are various ways in which you can compute and plot true power spectrum or power spectral density in MATLAB (when I say 'true power spectrum' I mean that the output values correspond to actual power values).
1) If you want to compute the power spectrum without having to specify many parameters and want the function to choose the best parameters for you, you can use pspectrum. Calling the function without outputs will give you a plot with the computed power spectrum.
2) If you want to compute power spectrum or power spectral density and want full control over the window size, window overlap, window type, and number of FFT points, you can use the Welch periodogram pwelch function. Calling the function without outputs will give you a plot with the computed power spectrum.
3) If you want to just visualize the power spectrum, you can use the Signal Analyzer app. The app let's you visualize your signals simultaneously in the time, frequency, and time-frequency domains. You can zoom into signal regions of interest and analyze the spectra at those zoomed regions.
4) If you have split your signals into multiple signal frames you can use the Spectrum Analyzer scope.
Finally, here is a popular MATLAB doc page that explains the relationship between FFT and true power spectra: Power Spectral Density Estimates Using FFT.
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