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I have some questions regarding interharmonics. What I'm going to do is first ask just a few, and then as people answer them I would expand this post or create a new question.

Harmonics are sinusoids that have a frequency which is an integer multiple of the fundamental frequency of the original signal \$x(t)\$ they represent. Interharmonics (or inter-harmonics) are defined as sinusoids that have a frequency which isn't an integer multiple of the fundamental frequency of the signal \$x(t)\$. First question: in that definition of interharmonics, is it assumed the signal \$x(t)\$ is periodic, or not?¹

I mean, yes, we use Fourier series usually for periodic signal, but I haven't read a single text book on math, circuit analysis, electronics, or signals and systems where they talk about interharmonics. Fourier's theorem never talks about interharmonics. The only place I've seen a brief discussion on interharmonics is in textbooks about power quality and harmonics. So this makes me wonder whether interharmonics even make sense, or that's just a term invented without any mathematical proof. So, before I ask further questions, I'd like to know the answer to the first question above.

I have another question. As you know, there're various ways to represent a Fourier series. One is the trigonometric form, other is the amplitude-phase form, and the other is the complex exponential form. Writing the amplitudes (i.e. the maximum values or peak values) of the harmonics in terms of the RMS values, the amplitude-phase form is:

\$x(t) = X_0 + \sqrt{2} \displaystyle \sum_{n=1}^\infty X_{\text{rms,} n} \cos{(2 \pi n f_0 t + \theta_n)} \tag*{}\$

My second question is if when a signal has an interharmonic of frequency \$m f_0\$, where \$m\$ is a non-integer positive number, do we sum it to the previous expression as a new sinusoid \$\sqrt{2} X_{\text{rms,} m} \cos{(2 \pi m f_0 + \theta_m)}\$? If not, then how does the interharmonic analitically contribute to the signal \$x(t)\$?

Note ¹: Fourier series can be used to represent a periodic signal with an expression valid for all time \$t\$, or to represent a non-periodic signal in a time interval \$\Delta t\$.

alejnavab
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    "*... and then as people answer them I would expand this post or create a new question.*" Create a new question. Please don't change the question after answers have been given as it makes them look like half-answers. Only edit to improve the original question. – Transistor Sep 25 '20 at 23:35
  • @Transistor Okay, thanks for the suggestion! – alejnavab Sep 28 '20 at 16:15
  • In the context of optics: [four-wave mixing](https://en.wikipedia.org/wiki/Four-wave_mixing). – Rodrigo de Azevedo Oct 28 '20 at 22:30

2 Answers2

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Let’s say you have an interharmonic of 1.5. If you scale your assumed fundamental frequency by 0.5 then you will have two integer harmonic sinusoids contributing to the signal - a 2nd harmonic and a 3rd harmonic. Equivalent, of course, to the original signal - just an integer way to look at it.

First case: Fundamental = f1

$$ x(t) = \cos{(1*2 \pi f_1 t + \theta_n)} + \cos{(1.5*2 \pi f_1t + \theta_m )} $$

Second case: Fundamental = f2 = 0.5f1

$$ x(t) = \cos{(2*2 \pi f_2 t + \theta_n)} + \cos{(3*2 \pi f_2t + \theta_m)} $$

These are equivalent signals (and periodic), I just chose to assume a different fundamental frequency for my analysis so i would have integer harmonics.

relayman357
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  • Interharmonics are still sinusoids, right? – alejnavab Sep 24 '20 at 00:26
  • Yes sir, that is correct. – relayman357 Sep 24 '20 at 00:29
  • Thanks. Following your answer, if I scale the frequency, wouldn't we get a new, different signal than the previous one? For example, if I scale the frequency of the signal \$\cos{(t)}\$ to \$\cos{(0.2t)}\$, these two signal aren't the same, are they? (This is just a simple example.) – alejnavab Sep 24 '20 at 00:35
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    Not if you do it correctly, like i show above (edited my answer). I am not changing the frequency of each sinusoid. I'm just playing around and using a different fundamental - so that i end up with only integer harmonics. – relayman357 Sep 24 '20 at 00:52
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    That trick of scaling frequency was so neat! I checked the two signals in your answer in [this GeoGebra app](https://www.geogebra.org/m/j5zavqxz) and they are indeed the same. However, this has bursted more questions in my head. For example, in the first expression of \$x(t)\$ in your answer, if we suppose that \$f_1=1 \text{ Hz}\$ and \$\theta_n = \theta_m = 0°\$, you can see in the GeoGebra app (or prove it analitically) that the fundamental frequency of \$x(t)\$ is actually \$1/(2 \text{ s}) = 0.5 \text{ Hz}\$, not \$f_1=1 \text{ Hz}\$. – alejnavab Sep 24 '20 at 02:59
  • Another question that arose: Suppose we have a periodic signal \$x(t)\$ with a Fourier series, additionally with \$k\$ interharmonics. Thus, in general, the number of terms in the expression for \$x(t)\$ would be 1 (the DC component) + \$\infty\$ (the number of harmonics) + k (the number of interharmonics). Can you guarantee that we can _always_ scale such an expression such that there're no interharmonics (the same thing you did in your example, but a more general case)? – alejnavab Sep 24 '20 at 03:19
  • Hi @AlejandroNava - you can declare the fundamental frequency to be whatever you want when you are creating the signals. In 1st case if we declare f1 = 1Hz then the signal is composed of a fundamental component and a 1.5 harmonic component. The fundamental you are seeing (properly) in your GeoGebra plot is the "correct" fundamental in regards to periodicity of the waveform. To find it you just find the [GCD](https://en.wikipedia.org/wiki/Greatest_common_divisor) of all the frequency components. – relayman357 Sep 25 '20 at 21:11
  • For example, the GCD in first case is 0.5Hz as you recognized from your analysis in GeoGebra. [Here is](https://opencourses.emu.edu.tr/pluginfile.php/43038/mod_resource/content/0/Fundamental%20Period%20of%20Discrete%20Time%20Signals.pdf) a good page showing how to compute the GCD (greatest common divisor - also called greatest common factor). – relayman357 Sep 25 '20 at 21:15
  • Re: "Can you guarantee that we can always scale such an expression such that there're no interharmonics" - all analog sinusoids are periodic, so yes for them. Not necessarily true for discrete sinusoids - [read here](https://www.sciencedirect.com/topics/computer-science/discrete-sinusoid) for details. I did not know that myself until read that page. [This post](https://physics.stackexchange.com/questions/544322/fourier-analysis-for-physicists/544546#544546) may be helpful too. – relayman357 Sep 25 '20 at 21:20
  • Yes, the sum of sinusoids whose frequency isn't an irrational number, results in a periodic signal; in [this Reddit post](https://www.reddit.com/r/askmath/comments/508roz/comment/d72agks) someone explains why, in the case of the sum of two sinusoids. And yes, we can compute the fundametal period or frequency using the LCM (least common multiple) of the periods, as shown in the document you shared or in [this video](https://youtu.be/A3Il3OyNoso). [Continues in next comment] – alejnavab Sep 25 '20 at 23:10
  • [Following my previous comment] Thus if we sum the DC component plus the infinite harmonics plus the interharmonics, the result would be a periodic signal (assuming the frequency of all of the interharmonics aren't irrational numbers.) But as you said, in my GeoGebra example for the values I chose, the actual fundamental frequency (or fundamental period) of the resulting signal isn't be the same as the one of the harmonics. Isn't this a contradiction? – alejnavab Sep 25 '20 at 23:17
  • Thanks for the links. Now I'll check your last comment in which you replied my question "Can you guarantee that we can always scale such an expression such that there're no interharmonics?" – alejnavab Sep 25 '20 at 23:21
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The IEEE standard #519, titled IEEE Recommended Practices and Requirements for Harmonic Control in Electrical Power Systems, version of the year 2014 (the latest update), on page 3 defines an interharmonic as follows:

interharmonic (component): A frequency component of a periodic quantity that is not an integer multiple of the frequency at which the supply system is operating (e.g., 50 Hz or 60 Hz).

So, when a signal is said to have interharmonics, the signal must be periodic, at least when using IEEE's definition.

As relayman showed in his answer, the presence of inteharmonics in a periodic signal depends on which period (the fundamental period or a multiple of it) is used to compute the coefficients of the Fourier series. The fundamental period \$T_0\$ of a periodic signal is defined as the smallest positive value of \$T\$ that satisfies \$x(t + T) = x(t)\$ for all \$t\$. When we use the fundamental period to compute the continuous-time Fourier series, no interharmonics are present.

alejnavab
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