The AIs Weigh in on the question: Are Harmonics Real or just an artifact of the FFT?
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In Part 1 of this article I noted that a sine wave, perhaps created by a mass on a spring bouncing up and down, results in a single peak in the spectrum. If the spring gets more rigid as it gets compressed, the mass will move less in one direction than the other. We can describe the form of the wave this makes as distorted or clipped. When we pass this clipped wave through the FFT, we get harmonics or multiples of the fundamental frequency.
For example, if the mass bounced up and down 30 times per second, we would see a peak in the spectrum at 30 Hz. If it was clipped, we would also see peaks at 60, 90, 120 and 150 Hz etc. These are called harmonics.
Are Harmonics Real?
The question I posed is: Are the harmonics real? In other words, although the mass is still only bouncing up and down 30 times per second, is there actually real vibration occurring at 60, 90 and 120 Hz Etc? OR are these just artifacts of the FFT?
In Part 2 of the article I suggested an easy way to test the example I just presented. One could attach a spring or a tuning fork with a resonant frequency of 60 Hz to the spring system mentioned above. If there is no “real” 60 Hz vibration then the tuning fork will not make a sound.
In Part 2 of the article I also suggested that a single hit or impulse can be thought of the same way. If a tuning fork has a natural frequency of 60 Hz, it will not make a sound unless it is excited at 60 Hz. It does however make a sound when you strike it against your hip. This means that the single strike or single impulse must contain “real” vibration at 60 Hz. Although I did not hit the tuning fork 60 times per second, the single impulse “contains” that frequency. It is not just a by product of the FFT.
In Part 3 of the article I offered some examples of resonance where the natural frequencies of a system are also harmonics. This is not the norm for most objects, but it is was worth mentioning to avoid confusion.
Non Linearity
Why are there frequencies in the output that do not exist in the input? If I bounce the mass on the spring up and down at 30 Hz, why do I also get frequencies at 60, 90, 120 Hz Etc? The answer is non-linearity. In a non linear system you get things in the output that were not present in the input. You can read more about that here: https://zencovibrations.com/linear-and-non-linear-vibrations/
What Does LinkedIN Think?
I asked the LinkedIN community what they think about this question, both before and after writing my articles. It seems I did not do a very good job of changing minds! The poll below still has 5 days to run, but the sentiment is clear. That said, this is in no way a scientific poll. I did not ask the same people for their opinions before and after reading the articles. Many thousands read the articles and only 114 answered the poll. I also have no way of knowing if the people who responded to the poll actually read any of the articles.
I argued for many years that harmonics are NOT real, so I certainly can see why intelligent people might still think that (even though I proposed some simple experiments in these articles to prove that they are in fact real!)
What Do Our AI Overlords Have to Say About It?
They say the AIs are going to replace us, so I thought I would survey a few LLMs to see what they had to say on the subject. I asked them “Are Harmonics Real?”
Question 1:
Question: A clipped wave will produce harmonics in the FFT. Are the harmonics “real” or are they just a product of the FFT algorithm?
CoPilot
The harmonics observed in the FFT of a clipped wave are indeed “real” in a mathematical and physical sense. When a wave is clipped, the process introduces sharp transitions or discontinuities, effectively altering the waveform. These abrupt changes result in the addition of high-frequency components that were not present in the original signal. The Fourier Transform (FFT) mathematically represents the wave as a sum of sinusoids, so it picks up on these newly introduced frequency components—these are the harmonics you see.
In other words, the harmonics are not artifacts of the FFT algorithm itself; they are genuinely present due to the clipping process. The FFT just reveals them. Clipping is a type of non-linear distortion, and harmonics are a natural byproduct of such distortions in signal processing.
Gemini
The harmonics produced in the FFT of a clipped wave are real, not just a product of the FFT algorithm. Here’s why:
- Clipping introduces non-linear distortion: When a waveform is clipped, its shape is fundamentally altered. This abrupt change in the time domain signal introduces new frequency components that were not present in the original, unclipped signal.
- Harmonics are inherent to non-sinusoidal periodic waveforms: Any periodic waveform that isnt a pure sine wave is composed of a fundamental frequency and its harmonics (integer multiples of the fundamental frequency). Clipping a sine wave forces it to become a non-sinusoidal, albeit distorted, periodic waveform.
- The FFT reveals existing frequency content: The Fast Fourier Transform (FFT) is a mathematical tool that analyzes a time-domain signal and decomposes it into its constituent frequencies. It doesn’t invent frequencies; instead, it reveals the frequencies that are already present in the signal.
- Experimental verification: You can observe these harmonics using a spectrum analyzer (which essentially performs an FFT in hardware). The presence of harmonics after clipping is a physically measurable phenomenon, not an artifact of the FFT process itself.
- Think of it this way: Clipping adds energy to the signal at specific harmonic frequencies due to the waveform distortion. The FFT accurately identifies and quantifies these newly generated frequency components.While the FFT algorithm itself has certain characteristics and potential artifacts (like spectral leakage or aliasing if not used correctly), the harmonics seen in the FFT of a clipped wave are a genuine representation of the altered frequency content of the signal caused by the clipping process.
DeepSeek
When a waveform is clipped (either hard or soft clipping), the resulting distortion introduces real harmonics into the signal. These harmonics are not just artifacts of the FFT algorithm but are physically present in the time-domain signal due to the nonlinear distortion caused by clipping.
Why Are the Harmonics Real?
- Mathematical Basis (Fourier Series/Transform)
- Nonlinear Distortion Mechanism
- FFT as a Measurement Tool
Practical Confirmation
- If you observe a clipped sine wave on an oscilloscope, you can clearly see the flattening (hard clipping) or smoothing (soft clipping).
- A spectrum analyzer (which doesn’t use an FFT) would still show the harmonics.
- Analog systems (e.g., guitar distortion pedals) produce these harmonics without any digital processing.
Conclusion
The harmonics produced by clipping are real and not an artifact of the FFT. The FFT merely reveals them as part of the signal’s true frequency content.
Question 2
Question: As a follow-up to the last question. Let’s say the fundamental frequency of the clipped wave is 30 Hz. The clipped wave is measured on a vibrating table. If I attach a spring with a natural frequency of 60 Hz to the table, will it resonate?
CoPilot
Resonance occurs when the frequency of an external driving force matches the natural frequency of the system—in this case, the spring’s natural frequency of 60 Hz. The fundamental frequency of the clipped wave is 30 Hz, which is half the spring’s natural frequency. However, the harmonics generated by the clipping process could include a frequency component at 60 Hz, depending on how the waveform is distorted.
If the 60 Hz harmonic is present in the vibration caused by the clipped wave, the spring could potentially resonate. This is because resonance doesn’t solely depend on the fundamental frequency—it can also be driven by harmonic components of the waveform that match the spring’s natural frequency.
To test this, you would need to analyze the FFT of the clipped wave to confirm the presence of a 60 Hz harmonic. If it exists and has enough energy to excite the spring, resonance is a plausible outcome.
Gemini
Yes, the spring with a natural frequency of 60 Hz will likely resonate, although the strength of the resonance will depend on several factors. Here’s the breakdown:
- Harmonics are present: As we discussed, clipping a 30 Hz wave introduces harmonics, which are integer multiples of the fundamental frequency. One of these harmonics will be at 60 Hz (the second harmonic).
- Resonance occurs at the natural frequency: Resonance is the phenomenon where a system (in this case, the spring) vibrates with maximum amplitude when driven by an external force at or near its natural frequency.
- The 60 Hz harmonic acts as a driving force: The vibrating table, driven by the clipped 30 Hz wave, will contain a significant component at 60 Hz due to the clipping. This 60 Hz component will act as a periodic driving force on the attached spring.
- Matching frequencies lead to energy transfer: When the driving frequency (the 60 Hz harmonic from the table) matches the natural frequency of the spring (60 Hz), energy will be efficiently transferred to the spring, causing it to oscillate with a larger amplitude – this is resonance.
In conclusion: Because clipping introduces a harmonic at 60 Hz, which matches the natural frequency of the spring, you can expect the spring to resonate.
DeepSeek
Yes, the spring with a natural frequency of 60 Hz will resonate if the clipped wave contains sufficient energy at that frequency (the 2nd harmonic). Here’s why:
1. Harmonics in a Clipped Wave
- A clipped wave (hard or soft clipping) generates real harmonics at integer multiples of the fundamental frequency (30 Hz).
- The strongest harmonics are typically the odd-order (90 Hz, 150 Hz, etc.), but even-order harmonics (60 Hz, 120 Hz, etc.) are also present, especially in asymmetrical clipping.
2. Resonance Condition
- Resonance occurs when a system’s natural frequency matches an excitation frequency.
- If the clipped wave has a 60 Hz component (2nd harmonic), and the spring has a natural frequency of 60 Hz, it will resonate if:
3. Will the Spring Resonate?
- Yes, if: The clipping is severe enough to produce a measurable 60 Hz harmonic. The system has low damping (e.g., a lightly damped spring).
- No or weakly, if: The clipping is mild (weak harmonics). The system is heavily damped. The wave is symmetrically clipped (which suppresses even harmonics, leaving mostly odd ones like 90 Hz).
Practical Consideration
- If the clipped wave is symmetrical (e.g., centered around zero), the even harmonics (60 Hz, 120 Hz, etc.) will be weaker or absent, meaning resonance at 60 Hz may not occur.
- If the clipping is asymmetrical, even harmonics will be stronger, increasing the chance of resonance at 60 Hz.
- You could verify this by running an FFT on the vibrating table’s motion to see if 60 Hz is present.
Conclusion
Yes, the spring will likely resonate at 60 Hz if the clipped wave contains enough energy at that frequency. The presence of even harmonics depends on the clipping symmetry, but in many real-world cases, some 60 Hz content will exist, leading to resonance.
Conclusion
LinkedIN poll answering community Vs LLMs? I’m going to side with the LLMs on this one! I want to reiterate that I spent a number of years arguing that harmonics are NOT real before I changed my mind. I also want to note that I wrote the articles on this topic BEFORE I asked the LLMs to weigh in.
Still not convinced? Why not try the experiments I proposed in the article and see for yourself? If you have an 1800 RPM motor, put a spring on it that is tuned to 60 Hz (2x) and see if it vibrates. If the 2x harmonic is not real, the spring will not vibrate. Or, hit a bell with a mallet and see if it rings. If the hit does not contain the natural frequencies of the bell, the bell will not ring.
You might also query the LLMs yourself and look at the sources they cited to come to their conclusions.
While you are pondering these deep questions, How About Registering for a Vibration Course?
