# Fourier transform with periodicity at the harmonic frequency

Let's suppose I have a signal $F(t)$ that is periodic, with two periodicities $P_1$ and $P_2$, with $P_1 > P_2$. Suppose that I know the values of the two periodicities.

Using the Fast Fourier transform I can show the two values as peaks in a power spectrum. Now, let's suppose the second periodicity $P_2$ (the faster one), has exactly the same value as the first harmonic of the fundamental value, or $P_2 = 2\times P_1$. This means that I will be not able to distinguish it by using the power spectrum, at least not by looking at the frequency of the peak.

My question is: is there a way to separate the contributions in such a case? For example, is it possible to predict the power of the first harmonic, so that the difference between the predicted power and the observed power of the harmonic peak gives a result significant enough (i.e., greater than $3\sigma$) to say that the first harmonic also "contains" the contribution from a periodicity?

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Would Signal Processing be a better home for this question? – Qmechanic Mar 20 '14 at 8:21
Simply adding the two equal frequency signals gives a single signal with a different amplitude and phase. The issue at hand plays a role in measuring harmonic distortion in audio amplifiers. Signal processing or Electronics would indeed be a better home. – Urgje Mar 20 '14 at 8:52

## 1 Answer

is there a way to separate the contributions in such a case?

If the second period is exactly twice the first, no. If there is a tiny variation, you might be able to pick up fluctuations as the phase the second beats against only the harmonics (but not the fundamental) of the other, but that strikes me as very difficult.

For example, is it possible to predict the power of the first harmonic, so that the difference between the predicted power and the observed power of the harmonic peak gives a result significant enough (i.e., greater than 3σ) to say that the first harmonic also "contains" the contribution from a periodicity?

Not without knowing more than you've said. Even if we limit our set of functions to "ordinary"1 musical instruments, the amplitude of harmonics generally decay as the harmonic number gets higher, but that's just a generalization. Many instruments have strong first and/or second harmonics. Even an amature singer could be trained to produce tones with large amounts of first and second harmonics, and less fundamental. I don't know about things like motors, but I have had personal experience where harmonics induced into electronics from 60Hz power is much stronger than the fundamental. If you are dealing with a limited input set, you could do some experiments and find out for yourself. If not, then the answer is no, not to my knowledge.

As a side-note, it sometimes happens that notes missing their fundamental frequency and no one notices. This is usually due to signal processing (as in telephones) and is rare in real instruments, but it does happen. See the Missing Fundamental for more information.

1 I am not defining this term, but let's just say I mean pitched instruments with "normal" harmonics. Not, for example, percussion.

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