As we all know, your dead right @bogosort,

I was trying to talk with @Jennifer Solomon

They , as they have said, are not like us , scientists and/ or engineers,
They seem to be more into the meta physical / philosophical sciences.

I really want to help them , and they at the start have asked us to keep this mathematical, and implied not to go off topic.

I don't know abut you @bogosort , but when I teach students I start with the basics, and get that agreed, so we can move on

For @Jennifer Solomon

The equation I quoted is the basic of a sine wave, giving the instantaneous amplitude at and time t

V = A sin( wt )


As @bogosort has said, that can be taken further.

They have V = A sin( wt + p ) + c

Where c is a DC offset, and p is a phase offset.

What this means is the "perfect" / base equation is for a sin wave, of an amplitude, which starts at zero at time t=0 , and is symmetrical around zero, going +- A in amplitude.

Thats all very true,
and thank you @bogosort for reminding us all of that,
but not what I would have covered in the first few lesson unless a student asked about DC offset or phase shift .


@Jennifer Solomon , if you want to engage in discussion about the topic and your expressed desire to keep it mathematical, I am more than willing to work with you via the PM,

I see the math for a sine wave.

I see how FFT math can get you the fourier sinusoidal components.

What I do not see is how the math is describing the co-located presence of n-modulated waves in the exact same physical space, each or them with lossless, preserved, 3D-informed information seemingly seemingly stored in a database of sorts (a wave). I see you can get at individual bandwidths, but I do not see how the math is reflecting what is physically observed in physical space.
This is not philosophical. I want to know physically where the data is stored and how the math is describing two or more “objects“ occupying the same space.

This is the heart of my question and is fully on topic.

 

Thank you @Jennifer Solomon

We know then basic equation of a sine wave V = A sin(wt)

Then if we have two independent basic sine waves at the same time, have the amplitude at a given instance in time of.

V1= A1 sin(w1 t)
V1 = A2 sin(w2 t)


If these two are added we have the instantaneous amplitude at time t of

Vo = V1 + V2.

or
Vo = A1 sin(w1 t) + A2 sin(w2 t)

You ok with that ?

 

Can I suggest @Jennifer Solomon

that if you want to talk deeper on the physics,
then what is basically a circuits / engineering forum like this ,
may not be as of use to you as say a physics forum
a suggestion

https://www.physicsforums.com/
https://www.newscientist.com/section/forum/
https://www.thenakedscientists.com/forum/index.php

Perhaps. Or maybe a mod should move this Off-Topic, because the thinkers on this board have contemplated these questions in the framework of actual implementation and Off-Topic is fine with me.

 

Thank you @Jennifer Solomon

We know then basic equation of a sine wave V = A sin(wt)

Then if we have two independent basic sine waves at the same time, have the amplitude at a given instance in time of.

V1= A1 sin(w1 t)
V1 = A2 sin(w2 t)


If these two are added we have the instantaneous amplitude at time t of

Vo = V1 + V2.

or
Vo = A1 sin(w1 t) + A2 sin(w2 t)

You ok with that ?

Yep.

 

Thank you @Jennifer Solomon

Next step,
is to follow that any signal can be "generated" ,
by adding together other signals.

Lets keep to the time domain for now, so we have amplitude , against time

So a simple example first
a square wave can be made by adding together a number of odd harmonics of the orriginal,
A harmonic just being an integer multiple of the base,


So a 1 KHz square wave could be made by adding together a
1 Khz sine of amplitude A1
3 KHz sine of amplitude A3
5 KHz sine of amplitude A5
7 KHz sine of amplitude A7

etc etc.

https://www.sonoma.edu/esee/courses/ee442/lectures/lect04_fourier.pdf

ignore Fourier for now,
just look at the demo of adding odd harmonics of the fundamental to make square wave,

Are we ok with that ?

 

Thank you @Jennifer Solomon

Next step,
is to follow that any signal can be "generated" ,
by adding together other signals.

Lets keep to the time domain for now, so we have amplitude , against time

So a simple example first
a square wave can be made by adding together a number of odd harmonics of the orriginal,
A harmonic just being an integer multiple of the base,


So a 1 KHz square wave could be made by adding together a
1 Khz sine of amplitude A1
3 KHz sine of amplitude A3
5 KHz sine of amplitude A5
7 KHz sine of amplitude A7

etc etc.

https://www.sonoma.edu/esee/courses/ee442/lectures/lect04_fourier.pdf

ignore Fourier for now,
just look at the demo of adding odd harmonics of the fundamental to make square wave,

Are we ok with that ?

Yep.

 

Yep.

@Jennifer Solomon

so a little bit of revision / home work...

what sine wave equation would I have to use to make a triangle wave ,
again a simple triangle wave, based around zero with fixed amplitude.

Question 2
assuming equation V = A sin (wt)

if we wanted to add a phase shift to that sine wave so it did not start at zero at time t=0,
what would the equation be

 

Thinking more deeply about it, I've finally crystallized the issue I'm having: the physical representation of time-based superposition.

The way I think about the physical aspect is in terms of information capacity, which really boils down to how many different ways we can reliably change something.

1D fluctuations over 2 seconds, for example — manifested as a pressure wave, we're dealing only with pressure changes. Where is the immense timbre qualia encoded, "muxed" into 1D voltage oscillations manifested as pressure waves over time?

It's encoded in the way the 1D voltage changes over time. The microphone simply responds to the instantaneous pressure at its location in the room. In a small room with bare walls, a sound source will produce many acoustic reflections, slighly delayed in time, that interfere with the original sound wave. The resulting air pressure at the microphone will be different than if the mic were in an anechoic chamber, or in a cathedral, or in an open field.

I'm not seeing how 1D voltage fluctuations over 1-2 seconds can communicate all of the timbre information embedded in the wave, much less how 88,000 16-bit values correspond to that data.

A few months ago, while experimenting with dither and quantization noise, I decided to quantize an audio signal to 1 bit (no dither) and listen to the result. The most significant bit in an audio sample is a sign bit, i.e., it only encodes the polarity of that particular voltage sample: positive or negative. No amplitude information is preserved, just the sign of the amplitude. My intuition was that this isn't nearly enough information to encode anything meaningful about audio, so I expected that I would hear a horrible hash of square wave-like noise. Upon listening, the sound was indeed severely noisy, but much to my surprise I could still hear the audio (a person counting numbers). In fact, not only I could tell what the words were saying, I could actually hear the vaguely British accent of the person counting.

Even such subtle timbral qualities as the details of the accent of a speaker can be encoded in just the polarity changes of a signal. This is the equivalent of a microphone that only has two output values: pressure is positive or pressure is negative. The resulting signal is a pulse-width modulation representation of the original signal, which is used all the time in other contexts. The surprising thing is that a lowly PWM representation -- which by design ignores almost all the details of the original signal -- nevertheless encodes sufficient detail to hear and understand human language.

From an information capacity standpoint, the signal represented 44,100 bits of information per second. But from an information quality perspective, we threw away almost all of the actual information in the original signal. And this points right at the key concept: meaning is derived from how something changes over time, not the values themselves, or even the nature of the values. The sign of the pressure at the microphone will change in different ways if a person with a British accent is speaking versus someone with an American accent. All of that exquisite detail is encoded in the history of these changes. And to record this history, we only need a 1D format.

 

I want to know physically where the preserved data are stored and how the math is describing two or more “objects“ occupying the same space that a mux is “getting at.”

The math has to "cheat" because a math equation is static, and the information in the physical waves comes from change over time. Since we can't literally show time on a static page, we abstract it with the symbol t and claim that it stands for all time.

As andrewmm said, we represent the physical situation mathematically as a sum of sinusoids. In our model, the microphone sees a time-varying pressure akin to \[\rho(t) = \sum_{i=1}^N A_i \sin(\omega_i t + \phi_i) \] i.e, a sum of \(N\) sine waves of various amplitudes, frequencies, and phases. But of course what the microphone actually sees is the instantaneous value of this sum, which mathematically is just a single number. You rightfully balk at the notion of a single number being able to encode the vast amount of detail in, say, the sound of a piano.

Hopefully, though, you've come to intuit that the single number doesn't encode anything -- it's the history of changes in the procession of those single numbers that contains all the details. Mathematically, we represent this history with expressions such as \[v(t) = A \sin(\omega_0 t + \phi) + B \sin(\omega_1 t + \theta) \] which innocently belies the vast amount of informational compression it represents, a veritable stream of infinite values.

That waves combine in linear superposition is a fundamental property of the universe. We compactly represent this property with algebraic sums. Once humans realized that we could modulate certain characteristics of waves -- amplitude, frequency, and phase -- their informational capacity became available to us and human civilization was inexorably changed.

 

@Jennifer Solomon

so a little bit of revision / home work...

what sine wave equation would I have to use to make a triangle wave ,
again a simple triangle wave, based around zero with fixed amplitude.

Mathematically speaking, in order to get a pure triangle wave, you would need an infinite number of harmonic sinusoidal waves, none of which exist in nature. (This is one of my issues with the true nature of information seeming to have mathematical infinite resolution and lossless conservation, but finite extent when reasoning about it). A finite series of sinusoids cannot a sharp corner make, so you would need infinite terms to truly model it.

Nonetheless, model it within our attenuated perceptions we do. I understand basic wave mechanics, so let’s fast forward to something more relevant to the question to spare any viewers here. My issue is not with the synthesis of complex wave forms, it’s about the harmonic data conservation around multiple fundamentals when combining many of them into a co-located “supra-wave.”

Here’s the equation of a simulated clarinet sound, \( s(t) \), where w1 = frequency of fundamental (in Hz) times 2π:

\( s(t) = sin(w_{1}t) + 0.75*sin(3*w_{1}t) + 0.5*sin(5*w_{1}t) + 0.14*sin(7*w_{1}t) + 0.5*sin(9*w_{1}t) + 0.12*sin(11*w_{1}t) + 0.17*sin(13*w_{1}t) \)​

...that is itself a complex waveform. A real clarinet sound has hella more harmonics, of course.

Let’s just assume a guitar sound looks similar with tweaked number of variables.

So we can take a carrier fundamental frequency, and these two harmonically complex tones, say 440Hz for the clarinet, and 8000Hz for the guitar and mux both those complex waves and their associated harmonics into the same carrier wave at different fundamentals, yes?

The resulting aggregate waveform has a specific shape, and we send that wave down a wire, to a demuxer.

The demuxer can split the 440Hz fundamental somewhere, and the 8000Hz fundamental somewhere else.

Now:
1) How are the associated overtones‘ relative placement for the 440’s fundamental vs. the 8000’s fundamental conserved after aggregating into a carrier wave? This is a physics question. Superposition would seem to intimate 3 waves are colocating in space, but they’re not. There is one wave, so superposition is yet another sh*tty term. It’s an aggregation.

2) What precise circuitry is doing this and how?



.

 

Mathematically speaking, in order to get a pure triangle wave, you would need an infinite number of harmonic sinusoidal waves, none of which exist in nature. (This is one of my issues with the true nature of information seeming to have mathematical infinite resolution and lossless conservation, but finite extent when reasoning about it). A finite series of sinusoids cannot a sharp corner make, so you would need infinite terms to truly model it.

Nonetheless, model it within our attenuated perceptions we do. I understand basic wave mechanics, so let’s fast forward to something more relevant to the question to spare any viewers here. My issue is not with the synthesis of complex wave forms, it’s about the harmonic data conservation around multiple fundamentals when combining many of them into a co-located “supra-wave.”

Here’s the equation of a simulated clarinet sound, \( s(t) \), where \( w_{1} \),p = frequency of fundamental (in Hz) times 2π:

\( s(t) = sin(w_{1}t) + 0.75*sin(3*w_{1}t) + 0.5*sin(5*w_{1}t) + 0.14*sin(7*w_{1}t) + 0.5*sin(9*w_{1}t) + 0.12*sin(11*w_{1}t) + 0.17*sin(13*w_{1}t) \)​

...that is itself a complex waveform. A real clarinet sound has hella more harmonics, of course.

Let’s just assume a guitar sound looks similar with tweaked number of variables.

So we can take a carrier fundamental frequency, and these two harmonically complex tones, say 440Hz for the clarinet, and 8000Hz for the guitar and mux both those complex waves and their associated harmonics into the same carrier wave at different fundamentals, yes?

The resulting aggregate waveform has a specific shape, and we send that wave down a wire, to a demuxer.

The demuxer can split the 440Hz fundamental somewhere, and the 8000Hz fundamental somewhere else.

Now:
1) How are the associated overtones‘ relative placement for the 440’s fundamental vs. the 8000’s fundamental conserved after aggregating into a carrier wave? This is a physics question. Superposition would seem to intimate 3 waves are colocating in space, but they’re not. There is one wave, so superposition is yet another sh*tty term. It’s an aggregation.

2) What precise circuitry is doing this and how?



.

@Jennifer Solomon

Thanks,
Appreciate your chasing at the bit,

But you have not answered the second question that show us that were following each other, and you have no given us the equation for the first question, but some words.

Question 1
what sine wave equation would I have to use to make a triangle wave ,
again a simple triangle wave, based around zero with fixed amplitude.

Question 2
assuming equation V = A sin (wt)

if we wanted to add a phase shift to that sine wave so it did not start at zero at time t=0,
what would the equation be

 

A few months ago, while experimenting with dither and quantization noise, I decided to quantize an audio signal to 1 bit (no dither) and listen to the result. The most significant bit in an audio sample is a sign bit, i.e., it only encodes the polarity of that particular voltage sample: positive or negative. No amplitude information is preserved, just the sign of the amplitude. My intuition was that this isn't nearly enough information to encode anything meaningful about audio, so I expected that I would hear a horrible hash of square wave-like noise. Upon listening, the sound was indeed severely noisy, but much to my surprise I could still hear the audio (a person counting numbers). In fact, not only I could tell what the words were saying, I could actually hear the vaguely British accent of the person counting.

Even such subtle timbral qualities as the details of the accent of a speaker can be encoded in just the polarity changes of a signal. This is the equivalent of a microphone that only has two output values: pressure is positive or pressure is negative. The resulting signal is a pulse-width modulation representation of the original signal, which is used all the time in other contexts. The surprising thing is that a lowly PWM representation -- which by design ignores almost all the details of the original signal -- nevertheless encodes sufficient detail to hear and understand human language.

There is a better way to do this.

Many moons ago I did a similar experiment. I wanted to find out what is the lowest data rate required (bits x sampling rate) in order to digitize and reproduce both voice and music to a "reasonable" degree of quality. Here are my observations.

Single-bit delta modulation (DM) is a well known and utilized technique. Your 1-bit encoding scheme is digitizing amplitude (or polarity). Instead, 1-bit DM is digitizing the polarity of differences from one sample to the next. The encoder is fairly straight forward.



Here is a schematic of a 4-bit tracking ADC. The output of the analog comparator sends an UP/DN signal to the up/down counter. This signal is your 1-bit DM signal. (The digital output of the binary counter reveals the ADC conversions. It is not required for DM encoding.)

The decoder is even simpler. The encoded signal (1-bit DM) sends information to either increase or decrease the voltage. By feeding this signal into an integrator circuit one is able to reconstruct the original analog signal. The quality of the audio is limited only by the bit rate of the DM signal. Typically, a clock speed in excess of 100kHz is required.

There is a trade-off between the number of bits and clock speed. One can reduce the clock speed by increasing the number of bits transmitted. 4-bit DM is a substantial improvement, capable of providing 16 levels of slew voltages.

The final improvement is to encode 16 levels in exponential voltages. Each level presents an exponentially higher voltage to be integrated at the decoder.

In summary, the best performance is achieved with 4-bit delta modulation with nonlinear encoding.

 

There is a better way to do this.

Many moons ago I did a similar experiment. I wanted to find out what is the lowest data rate required (bits x sampling rate) in order to digitize and reproduce both voice and music to a "reasonable" degree of quality. Here are my observations.

Single-bit delta modulation (DM) is a well known and utilized technique. Your 1-bit encoding scheme is digitizing amplitude (or polarity). Instead, 1-bit DM is digitizing the polarity of differences from one sample to the next. The encoder is fairly straight forward.

View attachment 239927

Here is a schematic of a 4-bit tracking ADC. The output of the analog comparator sends an UP/DN signal to the up/down counter. This signal is your 1-bit DM signal. (The digital output of the binary counter reveals the ADC conversions. It is not required for DM encoding.)

The decoder is even simpler. The encoded signal (1-bit DM) sends information to either increase or decrease the voltage. By feeding this signal into an integrator circuit one is able to reconstruct the original analog signal. The quality of the audio is limited only by the bit rate of the DM signal. Typically, a clock speed in excess of 100kHz is required.

There is a trade-off between the number of bits and clock speed. One can reduce the clock speed by increasing the number of bits transmitted. 4-bit DM is a substantial improvement, capable of providing 16 levels of slew voltages.

The final improvement is to encode 16 levels in exponential voltages. Each level presents an exponentially higher voltage to be integrated at the decoder.

In summary, the best performance is achieved with 4-bit delta modulation with nonlinear encoding.

Remember the OP explicitly stated at the beginning and mid way through, that they wanted to stay on topic and do this mathematically, I think as there last post went way off at a tangent, got into how the mind perceives, and does time exist, etc and was closed.
and they dont want to have a second strike,

Im working with them to take them through the class,
I have offered to do this with them on a PM, so as not to block the forum
and they have said no,
but it does seem to be coming on and were staying on topic. frequency mux and the mathematics,
anything else could be accused of being off topic which none of us want.

 

Javier, this is toward you too, of course (and anyone else):

Mathematically speaking, in order to get a pure triangle wave, you would need an infinite number of harmonic sinusoidal waves, none of which exist in nature. (This is one of my issues with the true nature of information seeming to have mathematical infinite resolution and lossless conservation, but finite extent when reasoning about it). A finite series of sinusoids cannot a sharp corner make, so you would need infinite terms to truly model it.

Nonetheless, model it within our attenuated perceptions we do. I understand basic wave mechanics, so let’s fast forward to something more relevant to the question to spare any viewers here. My issue is not with the synthesis of complex wave forms, it’s about the harmonic data conservation around multiple fundamentals when combining many of them into a co-located “supra-wave.”

Here’s the equation of a simulated clarinet sound, \( s(t) \), where \( w_{1} \),p = frequency of fundamental (in Hz) times 2π:

\( s(t) = sin(w_{1}t) + 0.75*sin(3*w_{1}t) + 0.5*sin(5*w_{1}t) + 0.14*sin(7*w_{1}t) + 0.5*sin(9*w_{1}t) + 0.12*sin(11*w_{1}t) + 0.17*sin(13*w_{1}t) \)​

...that is itself a complex waveform. A real clarinet sound has hella more harmonics, of course.

Let’s just assume a guitar sound looks similar with tweaked number of variables.

So we can take a carrier fundamental frequency, and these two harmonically complex tones, say 440Hz for the clarinet, and 8000Hz for the guitar and mux both those complex waves and their associated harmonics into the same carrier wave at different fundamentals, yes?

The resulting aggregate waveform has a specific shape, and we send that wave down a wire, to a demuxer.

The demuxer can split the 440Hz fundamental somewhere, and the 8000Hz fundamental somewhere else.

Now:
1) How are the associated overtones‘ relative placement for the 440’s fundamental vs. the 8000’s fundamental conserved after aggregating into a carrier wave? This is a physics question. Superposition would seem to intimate 3 waves are colocating in space, but they’re not. There is one wave, so superposition is yet another sh*tty term. It’s an aggregation.

2) What precise circuitry is doing this and how?



.

 

There is a better way to do this.

You misunderstand me. I wasn't trying to find a better encoding, I was trying to see what would happen with the worst possible form of PCM encoding. As I said, I was exploring dither, which is used to decorrelate quantization noise from the signal, and to truly understand its effects I needed to understand the worst possible case.

In summary, the best performance is achieved with 4-bit delta modulation with nonlinear encoding.

Perhaps if you're optimizing for minimum bandwidth and/or dynamic range, but I'm interested in applications that optimize for maximum possible accuracy, i.e., 44.1/48 kHz 24-bit audio. For this, the optimal choice is linear PCM generated from highly oversampled delta-sigma ADCs.

 

You misunderstand me. I wasn't trying to find a better encoding, I was trying to see what would happen with the worst possible form of PCM encoding. As I said, I was exploring dither, which is used to decorrelate quantization noise from the signal, and to truly understand its effects I needed to understand the worst possible case.


Perhaps if you're optimizing for minimum bandwidth and/or dynamic range, but I'm interested in applications that optimize for maximum possible accuracy, i.e., 44.1/48 kHz 24-bit audio. For this, the optimal choice is linear PCM generated from highly oversampled delta-sigma ADCs.

Carefull,
getting off topic of the OP question.

 

Javier, this is toward you too, of course (and anyone else):

@Jennifer Solomon

you had chance to look at the two questions yet ?

 

1) How are the associated overtones‘ relative placement for the 440’s fundamental vs. the 8000’s fundamental conserved after aggregating into a carrier wave?

This is a fundamental property of waves. Bricks are stackable, waves are "nestable". One way to think about it is that, physically, bricks respect addition, while waves respect both addition and subtraction. This allows waves to be mixed. And as with any mixture, with the right kind of work we can separate the parts.

Superposition would seem to intimate 3 waves are colocating in space, but they’re not. There is one wave, so superposition is yet another sh*tty term. It’s an aggregation.

The one wave is a mixture of waves. When we mix salt and water, the mixture is colocated in space in a way that the separated salt and water are not.

2) What precise circuitry is doing this and how?

Filters, like RC circuits (the simplest type). Filters contain reactive elements (capacitors and/or inductors) that store energy, which gives them the property of being time (and hence frequency) dependent. By tuning the right kinds of filter to the right set of frequencies, the desired bandwidth can literally be extracted from the composite signal.

So, the goal of the mux is to encode the signals (and all their overtones) into chunks of bandwidth that are far enough apart from each other that the demux can easily separate them by filtering. As an analogy, we can mix chalk and clay in water to form a superposition of three components. Since the size of the chalk and clay particles are different enough, we can use two mechanical filters to separate the three components from being in superposition. However, if the size of the two solutes are similar in size, then we lose the ability to filter them distinctly -- this is the equivalent of overlapping bandwidths.

 

Carefull,
getting off topic of the OP question.

I was responding to MrChips.

 

Carefull,
getting off topic of the OP question.

Lol... it’s ok, bro...information in the same general department of the department store can be helpful.