# Amplitude calculation of the original signal from its FFT

I have a signal whose FFT has a one prominent harmonic ($f_1$) and few other harmonics with a lower amplitude. In order to calculate the amplitude of the original signal I can use a well known theorem: a value of $f_1$ divided by half number of data points in the original signal.

But what if I have harmonics $f_2, f_3, \ldots$? Should I take their amplitudes into account for such calculation?

• Where are your harmonics come from? Aug 17 '16 at 14:33
• To get the accurate amplitude measurement, you must take into account all frequencies and their phases. Taking the amplitude of a single frequency, or even all of them is not enough. Think of the interference of all the waves - phase is crucial.
– jojek
Aug 17 '16 at 14:42

Whoa, clarification is needed. The peak amplitude ($A_p$) of an $N$-length $f_1$ input sinusoid can only be found using

$A_p = 2M/N$

(where $M$ is the the magnitude of the $f_1$ spectral component) if $N$ samples of that $f_1$ input sinusoid represent an exact integer number of cycles.

If that condition holds then you can measure $A_p$ for the $f_1$ sinusoid as well as the peak amplitudes of the harmonics. But those peak amplitude values apply only to the individual sinusoids that comprise your FFT input signal. You CANNOT use the individual measured peak amplitudes to measure the peak amplitude of your nonsinusoidal FFT input. That was the point made in jojek's comment.

• Thank you; I should like to clarify my question with regard to the origin of the harmonics. The signal that I measure is the response signal to a single frequency sinusoid (f1). It goes like that: I have a nanoparticle that vibrates in a certain liquid as a response to the applied single frequency sinusoidal wave. When I analyse the vibration of the nanoparticle by applying fft I see not only the original frequency of the initiating signal, but also its harmonics. My aim is to compute the vibration amplitude of the nanoparticle.
– Eli
Aug 18 '16 at 10:20
• Thus my question is whether to take into account only the Ap of f1 or the Ap-s of other harmonics to calculate the vibration amplitude of the nanoparticle.
– Eli
Aug 18 '16 at 10:20
• To characterize the time-domain properties of a signal, you must take into account ALL of its spectral information (that includes the harmonics). You have an x(n) time-domain sequence that is NOT a pure sinusoid (pure sinusoids have no harmonics). You compute the FFT of x(n). And from the FFT results you want to measure something you call the "amplitude" of your time signal. Karl, perhaps an analysis of the x(n) sequence in the time domain would make it easier to estimate whatever it is you are calling "amplitude." Aug 19 '16 at 18:57