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It's common to equate the appearance of clipped waveforms with harmonic spectra. Symmetric clipping is, for example, often equated with dominant 3rd-order harmonics. Asymmetric clipping, on the other hand is equated with dominant 2nd-order harmonics. While these relationships are true under certain conditions, general waveform and spectra relationships are more complex.
It's common to equate the appearance of clipped waveforms with harmonic spectra. Symmetric clipping is, for example, often equated with dominant 3rd-order harmonics. Asymmetric clipping, on the other hand is equated with dominant 2nd-order harmonics. While these relationships are true under certain conditions, general waveform and spectra relationships are more complex.
First of all, it is very difficult to distort a sine wave and produce only one harmonic (2nd or 3rd). Usually, a distorted sine wave contains a mix of harmonics that include 2nd, 3rd, and higher-order harmonics as well as intermodulation (IM) products. These IM products are sums and differences of the fundamental(s) and harmonics.
First of all, it is very difficult to distort a sine wave and produce only one harmonic (2nd or 3rd). Usually, a distorted sine wave contains a mix of harmonics that include 2nd, 3rd, and higher-order harmonics as well as intermodulation (IM) products. These IM products are sums and differences of the fundamental(s) and harmonics.
The relationship from wave-shape to spectrum is unique, however, the reverse is not (if we ignore phase in the spectrum). In other words, many different-looking waveforms can contain the same harmonics and sound the same.
The relationship from wave-shape to spectrum is unique, however, the reverse is not (if we ignore phase in the spectrum). In other words, many different-looking waveforms can contain the same harmonics and sound the same.
The following experiments show some properties that shed light on the relationships between wave-shapes and harmonics. Rather than distort a sine wave to produce harmonics, we can mix sine wave harmonics to produce distorted wave-shapes.
The following experiments show some properties that shed light on the relationships between wave-shapes and harmonics. Rather than distort a sine wave to produce harmonics, we can mix sine wave harmonics to produce distorted wave-shapes.
Spice is a perfect tool for this experiment. Three sine sources are used. One is a fundamental (220Hz) and two others produce the second (440Hz) and third (660Hz) harmonics. Mixing these in controlled fashion produces various waves-shapes that both agree with our intuitions and do not.
Spice is a perfect tool for this experiment. Three sine sources are used. One is a fundamental (220Hz) and two others produce the second (440Hz) and third (660Hz) harmonics. Mixing these in controlled fashion produces various waves-shapes that both agree with our intuitions and do not.
One of the points of this experiment is that intuiting the harmonic spectra of distorted wave-forms is unreliable. The other point is that playing with these sources and wave-shapes is pretty easy and fun (with Spice) and you can gain some interesting insights into these matters.
One of the points of this experiment is that intuiting the harmonic spectra of distorted wave-forms is unreliable. The other point is that playing with these sources and wave-shapes is pretty easy and fun (with Spice) and you can gain some interesting insights into these matters.
Experiment 1
Experiment 1
Mix 2nd and 3rd harmonics with a fundamental to produce the classic asymmetric and symmetric clipping waveforms. The sources and wave-forms are shown below. The relative amplitudes and phases or the sources are shown, but may be in a format that's unfamiliar, so they're summarized.
Mix 2nd and 3rd harmonics with a fundamental to produce the classic asymmetric and symmetric clipping waveforms. The sources and wave-forms are shown below. The relative amplitudes and phases or the sources are shown, but may be in a format that's unfamiliar, so they're summarized.
V1 - 2nd harmonic: 440Hz amplitude: 0.25 phase 90 degrees duration: 0-100ms
V2 - 3rd harmonic: 660Hz amplitude: 0.12 phase 0 degrees duration: 0-50ms
V3 - fundamental: 220Hz amplitude: 1 phase 0 degrees duration: 50ms-100ms
The green trace shows the harmonics measured at the + end of V2. The red trace shows the sum of all the signals measured at the + end of V3. Note that for the first 50 ms (the left screen half) only the 3rd harmonic is added, and during the second 50ms (the right screen half) only the 2nd harmonic is added. This makes both mixtures visible in one image.
The green trace shows the harmonics measured at the + end of V2. The red trace shows the sum of all the signals measured at the + end of V3. Note that for the first 50 ms (the left screen half) only the 3rd harmonic is added, and during the second 50ms (the right screen half) only the 2nd harmonic is added. This makes both mixtures visible in one image.
The phase of the 2nd harmonic (V1) may be puzzling, but a 90 degree shift is needed to produce the smoothly clipped waveform on the right side. Other shifts are explored below. The point of this is that these mixtures (amplitude and phase) produce the classic asymmetric and symmetric clipped wave-shapes.
The phase of the 2nd harmonic (V1) may be puzzling, but a 90 degree shift is needed to produce the smoothly clipped waveform on the right side. Other shifts are explored below. The point of this is that these mixtures (amplitude and phase) produce the classic asymmetric and symmetric clipped wave-shapes.
When the amplitudes or phases are varied, the wave-form shapes change and lose their classic shapes. Other mixtures with additional harmonics may also produce similar classic shapes, but if the 2nd and 3rd harmonics alone are used, these mixes (roughly) are the only ones that produce the classic shapes.
When the amplitudes or phases are varied, the wave-form shapes change and lose their classic shapes. Other mixtures with additional harmonics may also produce similar classic shapes, but if the 2nd and 3rd harmonics alone are used, these mixes (roughly) are the only ones that produce the classic shapes.
This is interesting by itself since it means that a waveform distortion function has to produce this exact wave shape by relatively gentle clipping in order to produce ONLY 2nd or ONLY 3rd harmonics.
This is interesting by itself since it means that a waveform distortion function has to produce this exact wave shape by relatively gentle clipping in order to produce ONLY 2nd or ONLY 3rd harmonics.
Since a guitar signal's amplitude varies and clipping is often a static function of amplitude (like a diode clipper), that exact condition rarely occurs and if it does, it's fleeting. That's an insight into why distorted wave-forms invariably contain a series of harmonics rather than just one.
Since a guitar signal's amplitude varies and clipping is often a static function of amplitude (like a diode clipper), that exact condition rarely occurs and if it does, it's fleeting. That's an insight into why distorted wave-forms invariably contain a series of harmonics rather than just one.
Experiment 2
Experiment 2
Alter the phase of the two harmonics by 45 degrees. Keep all amplitudes the same.
Alter the phase of the two harmonics by 45 degrees. Keep all amplitudes the same.
V1 - 2nd harmonic: 440Hz amplitude: 0.25 phase 45 degrees duration: 0-100ms
V2 - 3rd harmonic: 660Hz amplitude: 0.12 phase 45 degrees duration: 0-50ms
V3 - fundamental: 220Hz amplitude: 1 phase 0 degrees duration: 50ms-100ms
Note that the harmonic spectra have not changed at all. Only the phases of the harmonics have change. Since the human ear doesn't perceive phase, you should not hear any difference between the waveforms produced as results of experiments 1 or 2. However, the resulting wave-shapes are clearly no longer the classic asymmetric or symmetric clipped shapes.
Note that the harmonic spectra have not changed at all. Only the phases of the harmonics have change. Since the human ear doesn't perceive phase, you should not hear any difference between the waveforms produced as results of experiments 1 or 2. However, the resulting wave-shapes are clearly no longer the classic asymmetric or symmetric clipped shapes.
The Experiment 1 and 2 signals contain exactly the same harmonic content because they are generated by the same harmonic sources in the same proportions. BUT they look drastically different due to their harmonic phase shifts. Similar phase shifts are common in amplifiers, typically in tone stack circuits.
The Experiment 1 and 2 signals contain exactly the same harmonic content because they are generated by the same harmonic sources in the same proportions. BUT they look drastically different due to their harmonic phase shifts. Similar phase shifts are common in amplifiers, typically in tone stack circuits.
Careful examination shows that the left half of the Experiment 2 signal still shows symmetry, while the right half is more asymmetric. However, these qualities have nothing to do with clipping - neither half of the result waveform is clipped.
Careful examination shows that the left half of the Experiment 2 signal still shows symmetry, while the right half is more asymmetric. However, these qualities have nothing to do with clipping - neither half of the result waveform is clipped.
You can try this yourself, or you can find more experiments and insights into waveshapes and spectra in amplifiers in Guitar Amplifier Design.
You can try this yourself, or you can find more experiments and insights into waveshapes and spectra in amplifiers in Guitar Amplifier Design.
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