Are Non-Sinusoidal Sound Waves Just Magic?

[sophiecentaur]

When I ask whether or not “real” square waves are made of sine waves,

It is that question that needs to be modified before you can get an answer. Different people have answered from their own points of view and they are not necessarily wrong. What do you 'really' mean by 'real' square waves. A real square wave will have a finite slope and a finite amount of ripples, depending on the characteristics of the source. You could Fourier Analyse that wave and the transform will tell you precisely the amplitudes and phases of sinusoids with which you could build up (synthesise) the same square(ish) wave. In practice, there will be a limit to the accuracy with which you can do the synthesis so the reconstituted square wave may 'look' slightly different. (You pays your money . . . .) The square wave that's produced by a high speed electronic switching circuit would require a large number of harmonics of the fundamental for a good match.
Someone with a large set of sinusoidal synthesisers could produce a waveform that would look just like your square wave; it would BE as square wave, although the guy who made it never used any sort of switch to produce it.

 

[olivermsun]

I think this is what is causing my confusion. I’m getting contradicting answers wherever I look because of what seem to be semantic disagreements. When I ask whether or not “real” square waves are made of sine waves, I seem to get about 50% yes and 50% no.

It's because your question includes the words "real" and "made of."

I understand that all mathematical waves can be created by an infinite number of sine waves via Fourier series. I also know that real square waves are not the perfect mathematical “squares” that they are often shown to be. My question is really just whether or not all real square waves can be composed of a finite number of sine waves, and have the appearance shown in some of the images earlier posted of square waves being created by a Fourier series.

The things that prevent a "real" square wave from actually being square tend to insure that what's left is expressible as a sum of sines. If you then assume that what you have is a truly periodic, almost-square wave train, then sure, you can express the wave, to some arbitrary accuracy, as the sum of a "finite number" of sine waves. However, you could just as easily say that the almost-square wave is the sum of a series of waves with some other, non-sinusoids form.

If you want to be more precise than that, the semantics start to matter a great deal.

 

[FScheuer]

It's because your question includes the words "real" and "made of."The things that prevent a "real" square wave from actually being square tend to insure that what's left is expressible as a sum of sines. If you then assume that what you have is a truly periodic, almost-square wave train, then sure, you can express the wave, to some arbitrary accuracy, as the sum of a "finite number" of sine waves. However, you could just as easily say that the almost-square wave is the sum of a series of waves with some other, non-sinusoids form.

If you want to be more precise than that, the semantics start to matter a great deal.

What exactly do you mean by

“tend to insure that what's left is expressible as a sum of sines”

Are there exceptions?

And when you say that an almost square wave can be the sum of non-sinusoidal waves, isn’t it only the case that an almost square wave can be the sum of a finite number of sine waves. To create a sine wave out of square waves would require an infinite number of square waves for example, and wouldn’t it be the same with creating a square wave out of non-sinusoidal waves?

 

[olivermsun]

What exactly do you mean by

“tend to insure that what's left is expressible as a sum of sines”

Are there exceptions?

I gave some examples earlier in the thread. First is that that lots of real world "signals," like a a voltage or the position of a speaker cone, have to be single-valued, continuous, and have continuous derivatives (up to some order). These things can be represented by a Fourier series of sines and cosines. On the other hand, a surface wave train on water could, in principle, have a vertical (or even overturning) faces. The surface height would not be a function representable by a Fourier series.

And when you say that an almost square wave can be the sum of non-sinusoidal waves, isn’t it only the case that an almost square wave can be the sum of a finite number of sine waves. To create a sine wave out of square waves would require an infinite number of square waves for example, and wouldn’t it be the same with creating a square wave out of non-sinusoidal waves?

If a given wave can be represented by a Fourier series, it can also be written as a sum of some polynomials or some other things which aren't square but aren't sines either. The choice of sines and cosines is arbitrary (if very convenient) from a purely mathematical standpoint.

 

[sophiecentaur]

The surface height would not be a function representable by a Fourier series.

Neither could it (overturning faces) be described by a single valued time function so what is your point?

If a given wave can be represented by a Fourier series, it can also be written as a sum of some polynomials or some other things which aren't square but aren't sines either. The choice of sines and cosines is arbitrary (if very convenient) from a purely mathematical standpoint.

There's no limit to the number of orthogonal function that can be used. Raised Cos is the favourite for MPEG, I believe.

 

[olivermsun]

Neither could it (overturning faces) be described by a single valued time function so what is your point?

I was asked whether there are any exceptions to physical waves tending to be representable by Fourier series. I said that such systems are often single-valued, continuous, and have a continuous derivative, so they can be represented by a Fourier series. However, a water wave with a vertical face "would not be a function representable by a Fourier series (emphasis added)." Maybe I should have said "would not be a function and hence would not be representable..."?

 

[Quandry]

Yes. How does your ear perceive that music? In the frequency or in the time domain? You imply that it's only the time domain / oscilloscope picture world. Why is the temporal waveform somehow more real than the multiple frequency sensations that we get when a chord is played? Is the frequency domain really just some mathematical trick? Is the data carried by the raised Cos coding in MPEG meaningless - even when it's being manipulated in a processor, only to be 'real' when it is projected on the TV display or input to a loudspeaker? In particular, which path through the 2D array of pixels is the 'real' one?
You are implying that the linear, one dimensional waveform down a wire is the only true representation of ta sound. What is so special of that way of expressing the sound?

I wasn't going to comment further, but your habit of telling what I seem to be implying, when I am not, broke my resolve.
Time Domain is the analysis of the mathematical functions of the temporal features of signals with respect to time.
Frequency Domain is the analysis of mathematical functions of spectral features of signals with respect to frequency.
"How does your ear perceive that music?"
My ear does not use mathematical function analysis. It is capable of taking serial (temporal) input and parallel (spectral) input and responding to the energy (temporal) and tonal complexity (spectral). To the best of my knowledge it is not fully understood how the brain processes that and why some chords sound good and some do not - but that is not within my scope.

 

[sophiecentaur]

but your habit of telling what I seem to be implying, when I am not,

You may not mean to but a lot of your statements imply things (to me) that you may not have considered.

My ear does not use mathematical function analysis.

No? Hearing is signal processing. We don't know the algorithms involved but it's either Maths or Magic.

Basically, I have a problem with statements in this thread that 'imply' the time domain description of a signal is somehow the 'right one' and that other descriptions are less fundamental. The idea has run right through the thread and seems to be hard to put to bed.

 

[Quandry]

Hearing is signal processing. We don't know the algorithms involved but it's either Maths or Magic

Then I guess it must be magic.