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Why does a digital square wave contain harmonics?
I understand the basic premise of fourier series, where sign waves make up all waveforms.
But it seems weird to think that a digitally represented square wave is made up of sign waves, if it is created by what seems to be just a change in voltage level.
So, how and/or when do the sign waves make there appearance in digital square wave?
I can imagine the sign waves arising into existence when waveform voltage is used to push air with a speaker. There must be imperfection when the instantaneous voltage transition of a square wave is applied to a speaker cone that can't physically respond due to accelerating mass.
Are the sign waves a distortion phenomenon?
Also, I think it seems weird to have a multitude of sign waves of a greater frequency being generated by a square wave that is only making a voltage shift two times per cycle.
This is what happens when I try to "visualize" sound science in a way that makes sense. It reminds me of quantum mechanic waves being both particles and waves depending on how they are observed.
I don't doubt it's true, but I don't see where the sign waves come from in a square wave described predominately by using just two voltage levels?
Comments
Square waves are the sum of only the odd harmonics... the ones that share the zero crossing and rise and fall in alignment with the base sine wave.
Square waves at low frequencies can burn out electronics:
Here's why... that square wave intends to drive a speaker coil to it's maximum excursion.
Either all the way forward or back. And then it intends to hold the speaker in that position for 1/2 of the frequency. Holding a speaker means it' not moving and the resistance of the
magnetic field is what most sound waves experience. Holding a speaker still means there is effectively no resistance so it appears to the electronics as a resistance-less circuit path which is like a solid piece of wire... this effectively is a "short" without anything to hold excess current back. Hold a speaker at 10 volts with no resistance and the wires in the coil get really hot fast and melt.
This state where the electronics is pumping out current into a no-resistance load is just
running the electronics at is max capacity to provide electrons. It's like a rhythmic "short"
for each 1/2 cycle.
The resistance in this case of a speaker coil is actually magnetic field impedance but it's a load never the less). Creating and collapsing a magnetic field is a type of work for the circuit but a square wave intends to expand instaneously and collapse just as fast. It's those
"table top" segments that create the "short" events.
Now in reality the speaker really can't go from zero to max in nano-seconds so square waves get rounded edges by the realities of the physics of the speaker mass. But really low
frequency square waves are still something to use with caution. With a powerful enough amp you can throw the speaker coil right out of it's frame. Speakers can clip... clipping by the way generates unintended square waves. Creating more "short" events.
The section on Additive Synthesis in Synth Secrets by Gordon Reid touches on this:
https://www.soundonsound.com/techniques/introduction-additive-synthesis
You can see in one chart that the wave composed of just the first three squarewave harmonics is already pretty square.
But realize that he's talking about "synthesis" not reality, and FFT is a technique for decomposition and approximation.
Who's to say if a square wave is "really" composed of sine waves?
In reality, sound waves exist in four dimensions, and sound exists in our perceptions.
This may seem a little weird, but the first thing you've got to do is ask yourself what's a square wave. Related to that question is the point that you can't make a square wave in digital by simply doing the voltage flip. If you do, then the thing will alias all over the place and sound horrible. You have to do band limited synthesis and not generate any frequencies that go over Nyquist. (So, you can't generate any of the higher harmonics.)
Here's an experiment you can do in AUM. Take a square wave generator as your source and then put in an all-pass filter as an effect. AUM has an all-pass builtin. Pull the frequency of the all-pass to be fairly low. Now bypass the all-pass filter and record the output from the square wave. About halfway through the recording, push the all-pass back into the path. The audio output won't change with the all-pass in or out. But, if you now go and take the audio file and look at the wave form that was recorded, the parts with the all-pass in won't look square at all. So, is a square wave the way it looks or the way it sounds? If it's the way it sounds, then I'd say you can say the harmonics are there in the square wave. There isn't any distortion or anything of that sort. Our ears and brains perceive a certain set of overtones with the correct amplitudes as a square wave. Essentially, our ears and brain do the Fourier analysis.
One way to generate a digital square wave that is band limited is to generate all of the needed sinusoidal components. I'd say that doing it this way would have to be taken as the harmonics are always there. If you use some other synthesis technique to generate the square, then maybe you could say the harmonics aren't there until our ears perceive them. But, that would be the same thing as if you had generated the square in analog electronically or maybe with an acoustic source.
@horsetrainer : I think your confusion stems from a subtly wrong way that we talk about Fourier analysis. When one says "a waveform is MADE from sine waves, it would be more accurate to say that a waveform could be re-created by summing sine waves". A Fourier analysis provides the sine wave series that approximates the original waveform.
Additionally, you never hear digital sound. The digitally generated audio is converted at some stage (by speakers or headphones which are vibrating membranes) to the acoustic signal we hear.
Since you mentioned quantum mechanics, I thought maybe there might be a different way to think about this that would make it more intuitive. If this makes it more confusing, then ignore what I say here. It's only a different way to think about it.
One of the ways to think about Fourier series is that functions (periodic signals) that are being described form an infinite dimensional vector space. The complex sinusoidal functions of the Fourier series are a set of basis vectors for this vector space. So, any periodic signal can be described as a sum of the complex sinusoids times an amplitude. The amplitudes are the projection of the signal onto the basis vector just like the amplitudes in a position vector in 3D (e.g. P = x i + y j + z k) are the projection of the position vector onto the i, j, k basis vectors.
If all that makes sense, then the question about when (or where) the sinusoids come into the square wave is the same as asking when the x, y, z come into describing a position vector.
Just like in 3D space, there are multiple possible sets of basis vectors to describe periodic functions. But, here's where it gets a bit different. In 3D space we might see something as a rectangular coordinate system in one setting and in a different setting it's easier to visualize in spherical coordinates. For audio signals, our direct perception is in the basis vectors that are the Fourier series. Our ears pick up the signal that way and our brain processes it that way.
Let me try to explain the phenomenon from the time domain which is probably easier to understand.
You have worked with Drambo already, and the Wavetable Oscillator has a nice ultra-steep filter module that can subtract harmonic by harmonic from the rectangle waveform. To demonstrate what the sine waves actually do, I've started muting them beginning from the lowest frequency.
If you want to build the same in Drambo, make sure you set the filter slopes to infinite.
The original square wave:
Minus the fundamental frequency: Every harmonic is there but the original oscillator frequency is missing in the waveform.
You can see how a sinewave with the oscillator frequency has been subtracted from the square wave. (If I had set a sinewave in the oscillator, the output would be completely muted now):
Going further, I have now muted the first five harmonics:
The same, now zoomed in:
You can see two things here:
First, there are still a lot of higher frequencies contained in the square waveform.
Second, while the square lobes have gone, the pulses from the positive-to-negative and negative-to-positive switching are still there and almost at full level.
If you increase the HPF cutoff more and more, removing more and more lower harmonics, you'll see how many sinewaves it takes to build such a complex signal as a square wave. To build a perfectly shaped square wave, you would need an infinite amount of sinewaves. No analog synth can do that but they're close enough for the human ear 😉
Now let's extend this experiment by adding another filter with infinite slope, now a LPF to filter out the upper harmonics:
By narrowing down the window of passed through frequencies, you can extract individual sinewaves and listen to each separately.
I can highly recommend to try that for yourself, turn the HPF and LPF frequency knobs and "get a feel" for now a square wave is "constructed".
I realise this is somewhat of an off-topic aside, but it blew my mind when I realised you could modulate the DC Offset module in Bitwig to make sound...
(might be interesting to some of the people reading this thread)
A(ny) perfect waveform doesn‘t generate any harmonics at all.
But „perfect“ isn‘t possible in real world due to physical limitations of the generator.
Sinewaves are the least demanding because they rise and fall in a smooth way.
Triangles are next, but there‘s a turning of polarity at half the period.
Saws have a rapid rise or fall at either begin or end.
Finally Square waves have a rapid rise and fall at both ends.
These rises and falls (and the polarity change of triangles) can‘t be generated within zero amount of time (as theory requires), hence the waveform gets distorted and that distortion generates harmonics.
The different spectra result from number and position of the transitions within a single cycle.
If you look at oscilloscope images of waveforms from analog synths they only roughly approach the theoretical form of the wave - and these variations generate the specific character of the instrument.
@Telefunky
I disagree with your first statement about 'any' perfect waveform not producing harmonics.
A perfect sinewave will not generate any other harmonics, but the other standard waveforms are compound signals that were constructed mathematically so they produce a series of even and/or odd harmonics, which is either pleasing to the ear or approximates the timbre of various real-world sounds and instruments.
As you rightly say, in practice all generated waveforms are slightly less than perfect, or a lot less. Therefore even a digital sinewave can register small amounts of other harmonics.
Interestingly, the fact these other waveforms (triangle, saw, square) look visually pleasing is coincidental. they were constructed for their harmonic properties.
There’s no disagreement as you wrote „a perfect sinewave“
My statement was more general with any perfect waveform.
While a sine can be approached very, very close to ideal, it‘s still impossible with our current technology to generate a clone of the mathematically perfect waveform.
A tiny bit of distortion remains, though not perceivable by our ears... it‘s still there.
But given an analog generator with zero reaction time, I don‘t see any reason for a waveform to generate harmonics at all.
In practical (real world application and measurement) they of course do, but as a consequence of the imperfect generation (and reproduction), not the shape itself... imho.
In air, waveforms do not generate harmonics, it’s the harmonics that generate the waveform. A waveform is simply the changing sum over time of the sine wave shaped oscillations in pressure at a particular point in space.
Thanks everyone for all your in-depth explanations and perspectives.
Next I'll have to do these experiments and find a way of understanding this that I can apply functionally to sound design.
@NeonSilicon
That all-pass filter experiment was eye opening. I couldn't believe that there were so many very non-square wave shapes that sounded virtually the same as the original square wave. I had to draw them myself to prove they actually sounded as they did.
This is making me think about how I understood string theory in quantum mechanics...
A multitude of vibrating sub-atomic-energy-strings creating patterns of interacting waves that ultimately become expressed at our perceivable macro level as having form.
Our ears don't hear the shapes of waves. The cochlea is arranged in a logarithmic spiral of sensors. A "pure tone" (a sinusoidal wave) stimulates only one small region that the brain can then interpret as a sine wave of a given pitch. Any complex wave form is picked up by the stimulation of multiple of the sensors that correspond to the Fourier series that mathematically describes the wave form. Our brain processes this information to interpret it as a square wave or whatever the waveform is. If the waveform didn't have the upper harmonics in it, then our ears' sensor wouldn't pickup any difference between any waveforms.
An interesting feature of this system is that the sensors don't directly differentiate phase of the wave components. That's why the experiment I described above works. Putting an all-pass filter shifts some of the frequencies but not others. So, the added up waveform doesn't look square anymore. But, it still sounds square because the ear still senses the same harmonic series.
It doesn't matter how the waveform is generated, via superposition of sinusoids directly or a direct acoustical square wave generator. The resulting sound wave directly stimulates the sensors in the ear that match the frequencies in the harmonics. That can only happen because there is energy in the waveform at those frequencies and not at others.
A good book on this area is "The Physics and Psychophysics of Music" by Juan G. Roederer.
Yeah, it's a trip to see and hear. It also explains part of why very different looking waves forms can sound like a square wave or a saw when the come out of oscillators or clarinets for example. The waves coming from those systems are always coupled to some sort of filter (the clarinet body for example). These filters alter the phase of the signals and thus the waveform's shape. But, the thing will still sound mostly square. These types of filters do remove some frequencies too because they are subtractive filters, but the waveform changes are interesting to look at this way. It helped me to think about why certain settings on a Big Muff sound pretty square wave to me (or saw at some settings) but don't ever look like it on a scope. The output of a Big Muff is a tone control -- a filter.
Not to sidetrack the thread too much, but if you want to read a book that might alter your thoughts about quantum mechanics, I really liked the book "The Undivided Universe" by Bohm and Hiley.
That's an interesting perspective.
My question on this "sound wave/harmonic" subject..
Do particular waveforms in air, such as a square wave, have a greater capacity to energize a resonance vibration in materials?
It might seem that the twice per wave cycle "air pressure spike" of the square wave, might impact materials and shake them up similar to physically striking them?
But when those 'waves' are used as modulation sources at slow speeds they act totally different
The might 'sound' the same but when used to modulate a parameter they are definitely not the same...
Exactly. When you are designing a band limited waveform for a digital synth, you'd need to take the possibility of using it as an LFO source into account.
I'm also in the camp that thinks the right way to think of a square way is as the sum of a bunch of harmonics (and a gnarly sum at that). Most physical things vibrate with a fundamental frequency, and then multiples of that frequency -- and our ears and brains have evolved to understand things that operate that way. Our vocal cords, guitar strings, etc; it's the way physics works. The square wave is not something that naturally occurs.
IMO, one of the reasons a synth can sound "otherworldly" is because... the wave forms they can generate are not of this world. If you want a synth to sound natural, you need to emulate the harmonics that occur in nature.
I've been putting together videos for the user manual of my next app; this might help. The app has an AUv3 plug-in, that displays the Fourier transform in a spiral, so that you can see the chromatic pitches align from the center, and it's easier to see the harmonic frequencies, and how things line up in chords.
Wow Audiobus forum delivers! Great question and inspiring answers. Lots of knowledge to be gained here, some of it might take me a while to get my head around. Tried Neosilicon’s experiment and made a video for who might be interested. Fascinating stuff, thanks!
WARNING! VIDEO SOUND MIGHT BE LOUD AND OBNOXIOUS! TURN YOUR VOLUME DOWN!
Nice one, when you drive it thru a filter the sound changes as some of the 'peaks' drive the filter harder
Kinda makes me wonder why some synths have multiple pulse/saw wave samples from 'different synths' when in practice they could just feed the regular oscillator thru an all pass filter before feeding them thru a lowpass filter etc. etc....
Now there’s another idea to try out! Thanks!
Resonant frequencies depend on material, mass and dimensions of the resonant object. This means that the best waveform to energise a resonance vibration in an object is one that contains the most energy (usually its fundamental sinewave) at the resonant frequency of the object. Objects can have multiple resonant frequencies, so if prominent harmonics of the energising waveform match those other frequencies, then more vibrational energy will be imparted to the object, but those frequencies, (and therefore the best energising waveform) depend on the properties of the object.
The only way I’ve found of shaping the (very rough) square waves my homemade oscillators produce is using filters (low pass resistor capacitor networks for the most part) - I don’t understand enough about electronics yet to know if that’s the only way to do it. I guess some oscillators just produce sine waves though - Wien bridge ones apparently do but u can’t for the life of me get to grips with how they actually work yet...
Watch this, you will understand everything and probably more
True... it you hit the object with impulses at it's resonant frequency you can drive it into
physical distortion and break glasses for example. You add reinforcing energy like pushing
a child in a swing. Push hard at the right frequencies and you can wrap the swing around the beam. NOTE: Do not try this with your own kid.
To determine the resonances of a "circuit" or a physical object, you hit it with an "impulse". The impulse is a short burst of energy that does not assume any frequency so it works universally for any circuit or object. Hitting something with a soft mallet generates oscillations at resonant frequencies of the circuit or the object.
Those IR files we buy to simulate a speaker, a microphone, a room are "Impulse Response" recordings. They hit the target with an energy impulse and record the trailing resonances.
There's a math process where you input an infinite impulse into a network (a circuit or a
series of equations) and determine the resulting output waveform. Fourier series plays prominently in the math, as I recall. I'm sure there's a YouTube video demonstrating he math and that most of us would fail to have the basic concepts needed to follow along.
I'll bet there's a Khan Academy math path that gets you there for anyone wanting to get
deeper into the science of acoustics and sound engineering.
Square waves don't play all that well with wave shapers and similar circuits. They tend to just turn them in to shorter or taller square waves. If I remember right (and it's been a long time, so I could easily be wrong) the problem with Wien bridge oscillators for musical usage is that they are hard to tune quickly and dynamically. They make really really low distortion sin waves though.
Lots of synth oscillators are triangle core oscillators, so if you can get your square wave outputs to a triangle, then you can apply all of those circuits. Lots of audio/musical sine wave oscillators are really triangle core with diode wave shaping. A good way to convert a square into a triangle is an integrator. Search for op-amp integrator circuit and you should find some to play with. There are issues to overcome with these circuits, but there's plenty of documentation sources and just a simple circuit could get you a good starting point to experiment with.
Cool!
That would be an interesting thing to try. I'm not sure without doing some math or experimentation, but I think one of the things you might have to overcome is that the waveform changes for a given oscillator at a fixed all-pass corner frequency would be oscillator frequency dependent. I'm pretty sure you could tune this to make it work, but it might be difficult.
Another thing to consider is that although our ear can't hear phase directly, there are higher order effects that phase recognition impacts. One of those, which is related to your idea for using this to do wave shaping is that frequency dependent phase alters transients. It tends to smear them. That could be good or bad depending. Another is that frequency dependent phase manipulations alters our perception of location. There's some research I recently came across that was about how our perception for location for higher frequency comes from the stereo inter-arrival times of the signal, whereas the bass frequency location perception was determined by the stereo perception of the phase shift between the bass frequencies. So, heavily manipulated audio signals can lead to confused perception of location in the sound field.
If you see square waves in the ocean... run for your lives.
Math meets practical reality:
So much great knowledge being dropped here