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Sine-wave speech audio encoding

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44 pointspero_p20 comments

I love this demo. It’s a good example of how much our perception is affected by our priors.

This is a other great example, where speech is synthesized with an acoustic piano (though the keys are controlled digitally, the sound is just from the hammers and strings). Without the subtitles you probably couldn’t understand, but it sounds pretty clear when you know what it’s saying already.

https://youtu.be/muCPjK4nGY4

you hit it on the nose here. This is a stellar example of priming.

Have you also done your fair share of psychophysics? :-D

Source: https://www.mrc-cbu.cam.ac.uk/people/matt.davis/sine-wave-sp...

???

All sound is a sum of sine waves. The term "sine wave speech" makes as much sense as "wet water."

The demo is cool, but pick a name that makes sense.

> All sound is a sum of sine waves.

No, it's not, though that's a popular way of encoding sound data. A square or triangular wave will still be a sound, with a distinct character to a sine wave of the same frequency.

A square or triangular wave is a sum of sine waves.

As far as I know, any arbitrary waveform can be represented by an _infinite_ sum of sine waves, and approximated by a finite sum of sine waves. However, I don't know if you can assert that a sine wave is the only fundamental building block for sound that exists. Is there a physical reason for that statement?

There's an infinite number of different sets of orthogonal bases that you can construct any signal from. Sine waves (with phase) are a quite natural and convenient choice though, for mathematical reasons (for example, the efficiency of the Fourier transform).

Sure, but there may be a reason why non-sinusoidal sound waves cannot exist in practice. For example, if I try to generate a square wave by applying +5V and -5V to a speaker, the diaphragm does not move instantaneously so the resulting pressure wave is not discrete. However, a triangle waveform may be physically realizable. But I'm not sure about that either, because a speaker is ultimately an inductor and hence the current flowing through it is non-linear.

I did a deep dive into this, but this just seems to be the case about sinusoids. technically any well behaved periodic functions can be the basis function for all fourier-related stuff (there actually is a FT where a square wave is the basis function, but I can't seem to find it) but there seems to be something special about sin(x) and cos(x) that makes it more convenient to analyze our physical world. They seem to be the result (and the only result) for some class of very simple linear differential equations, which because of its simplicity, happens very often in nature. That's why FT uses sin(x) and cos(x).

>there actually is a FT where a square wave is the basis function, but I can't seem to find it

Haar transform? Technically that's a wavelet transform, but that's a distinction without tremendous difference.

i meant the walsh transform, but that's also neat

Any linear time-invariant system bounded in reality has a frequency response that will drop off in magnitude before reaching infinity. An ideal triangular wave is an infinite sum of sine harmonics because of the discontinuity in its derivative. More generally, the order of the derivative in which the discontinuity happens relates to decay rate of the harmonics. A square wave decays as sinc(f) whereas a triangular wave decays as sinc^2(f).

This is true conceptually, in practice, not so much. It's true you can express any periodic waveform as a discrete infinite sum of sine waves, but if you wanted to express any non periodic waveforms, you have to integrate real and imaginary extended "sinusoids" (exp(jwt)) over a continuum of frequencies. A little fucky if I do say so myself.

Nitpick but the finite sum of sines is not an approximation ( except for quantization effects when transforming between time/frequency)

A defining characteristic of finite length orthogonal transforms is that there exists an inverse transform that preserves energy. What that means is for any particular orthogonal transform, such as the DFT which maps a time domain vector to frequency domain vector, the transformed vector is not an approximation or the time domain. It's the same energy, arranged differently. Nothing is lost in the transform, and it can be transformed back.

This is actually why lossy codecs employ lossless transforms - they decorrelate the energy of a signal over a window of time into a representation that has the same energy but compacted into fewer elements of the vector. Some information arranged differently, in a context that makes it more useful.

There is no fundamental building block of a signal other than energy. Whatever domain you choose to view that energy on depends on the context for which the signal is meaningful. For our ears that's time. For our brains it's frequency. For any given algorithm or encoding, it could be something totally different (eg the DCT domain, which is like frequency, but isn't the same).

Another way to think of it is that frequency and time are two compositions of information, there are infinite such compositions, and infinite ways to map between them. None is any more fundamental than the other, only more meaningful or useful in specific contexts.

Sine waves form one of an infinite number of basis functions which you can map a continuous function onto. They’re super popular because the F in FFT means Fast and it’s not a joke.

By way of your friend and mine the laplace transform, you can roughly show that the fourier transform is a special case of a whole family of maps.

so yes, it’s sine waves all the way down, but it’s one of any number of other handy friends all the way down, too.

The physical reason is that the hairs in your ear (evidently) vibrate sinusoidally, which means that’s the decomposition of sound that your brain will hear.

Square and triangle waves do not exist in nature. What you get is always squiggly.

> A square or triangular wave will still be a sound, with a distinct character to a sine wave of the same frequency.

You're arguing with well known facts. Take some time to understand Fourier's theories.

The name is fine. Virtually all digital art involves using pixels, yet “pixel art” is a sensible name for a type of art where pixels are manipulated directly.

The name makes sense because it refers to encoding speech with a very small number (eg. 5) of sine waves, as opposed to the roughly one thousand you'd expect for a FFT window of e.g. 2048 samples http://www.scholarpedia.org/article/Sine-wave_speech

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