Would anyone know if there is any relationship between the FFT amplitude vs. data length?

For instance, in Matlab, if signal_30k had 30,000 data points, sampled at 1000 Hz, and I do this:

fft_amplitudes = abs( fft( signal_30k ) );

then dominant frequency's resulting amplitude is some number.

but the same signal with the same frequency components, signal_500k had 500,000 data points, also sampled at 1000 Hz, and had the same operation done on it:

fft_amplitudes = abs( fft( signal_500k ) );

the same dominant frequency's resulting amplitude would be another number but much larger.

Is there any relationship between the number of data points and the resulting amplitude that comes out of the FFT? The goal is to ensure that regardless of the number of samples, the amplitudes have some means of being compared with one another.

Any help would be greatly appreciated.


If you want to have the same amplitudes, you need to scale the FFT result by the length of the FFT:

Fs = 100.0
f = 5;

t1 = np.arange(0, 5, 1/Fs)
t2 = np.arange(0, 50, 1/Fs)

s1 = np.sin(2*np.pi*f*t1)
s2 = np.sin(2*np.pi*f*t2)

plt.plot(t1, s1)

plt.plot(t2, s2)

S1 = np.fft.fft(s1) / len(t1)
f1 = np.linspace(0, Fs, len(t1))
S2 = np.fft.fft(s2) / len(t2)
f2 = np.linspace(0, Fs, len(t2))

plt.plot(f1, abs(S1))

plt.plot(f2, abs(S2))

enter image description here

If you want time and frequency domain signal to have the same power, you need to scale the FFT output by 1/sqrt(length).

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  • $\begingroup$ of course, Max, this all depends on the definition of the FFT. but, using the most common definition, you are correct. $\endgroup$ – robert bristow-johnson Dec 1 '16 at 7:40

Yes there is such a relationship. Consider a real sinusoid first with an amplitude A and $N$ samples. The size $N$ FFT output will be an impulse at the frequency of that sinusoid with a magnitude equal to $N\cdot A/2$.

In case of a complex sinusoid, this number would be $N\cdot A$. For a proof, see the following definition of DFT. \begin{equation*} X[k]=\sum _{n=0}^{N-1} x[n] e^{-j2\pi nk/N} \end{equation*} and consider the frequency equal to 0 (DC component). That is a rectangular sequence of amplitude $A$ and length $N$. Plugging in DFT expression and summing over it gives the value $N\cdot A$. Use Euler's identity and the result of the real sinusoid becomes clear. Since other sinusoids are just a shift in frequency, similar results hold for them as well.

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There are multiple FFT implementations. Different FFT implementations can have different default scale factors (1/N, 1/sqrt(N), etc.)

One common FFT implementation is energy preserving (Parseval's theorem) in the forward direction. Thus, since a stationary sinusoid twice as long in the time domain will have twice as much energy (assuming no windowing artifacts), its FFT result will scale proportionally to double the magnitude. You would thus scale FFT result magnitude by 1/N to get a length invariant sinusoidal amplitude.

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