Why sinusoids¶

Every page that follows describes speech as a sum of sine waves. Before accepting that, it is worth asking, "Why sine waves and not something else?" The answer is not that they are familiar. It is that they are the only shapes that survive what a vocal tract does to a sound.

The problem: your description has to survive a filter¶

Say you want to describe the sound coming out of somebody's mouth, and you want the description to be short.

The obvious move is to describe the waveform. Speech waveforms look like repeating pulses, so you might record the shape of one pulse, say how often it repeats, and be done. Three or four numbers for the shape, one for the rate. That is a genuinely compact description, and for the sound at the vocal folds it is close to right.

Magnify a vowel and the proposal looks sound. The same motif comes round again and again, close enough each time that one shape and one rate feel like a fair description of it.

Then the sound has to get out of the head, and the description falls apart.

Between the vocal folds and the air is a tube: the throat, the mouth, the position of the tongue and jaw and lips. That tube resonates. It boosts some frequencies and suppresses others, and the boosting changes as the tube changes shape, which is what saying different vowels physically is. By the time the sound reaches a microphone it has been through a filter, and the pulse you carefully described is no longer that pulse. It has been smeared into something with ringing tails and a different shape depending on which vowel is being said.

So the description has to be redone for every vowel. Worse, there is no simple rule for redoing it: the filtered shape is not the original shape scaled or shifted, it is a different shape.

That is the real constraint, and it is not about compression at all. Any useful description of speech has to behave predictably when the signal goes through a filter, because a filter is the last thing that happens to every speech sound before you hear it.

What survives a filter unchanged¶

Ask the question the other way round. Instead of picking a description and hoping it survives filtering, ask which signals come out of a filter looking like themselves.

Push a square wave through a filter and you get something that is not a square wave: the corners round off, the flat tops sag or ring. Push a single pulse through and you get a smeared pulse with a tail. Push almost any shape you can name through a filter and the shape changes.

Push in a sine wave and a sine wave comes out. Same frequency. Only two things about it have changed: it may be louder or quieter, and it may be shifted in time.

Try it. Send each of the four shapes through the filter below and move the cutoff around. Then press Rescale and realign, which undoes exactly those two permitted changes and draws whatever is left over.

The residual for the sine stays under one percent wherever you put the cutoff, and what remains is the filter still settling. For the square and the sawtooth it never falls below a third of the input. The triangle is the mildest of the three, because its harmonics are weak to begin with, and even it does not get close. No single gain and no single delay can account for what the filter did to any of them.

That property is not a coincidence and it is not approximate. It holds for every filter that is linear and does not change its behaviour over time, which covers the vocal tract over the tens of milliseconds a coder cares about. Sine waves are the signals that a filter can only scale and delay, never reshape. In the language of linear systems they are the eigenfunctions of the system: the inputs that come back as multiples of themselves.

Nothing else has this property. That is the whole reason the rest of this site is written in sinusoids.

The same argument, on the radio¶

Nothing in the last section mentioned sound. It said "linear" and it said "does not change over time", and everything else followed. Any system with those two properties has sinusoids as its eigenfunctions, whatever it is made of.

A radio is such a system. So is a length of coax, an antenna, a filter in an IF strip, and a path through the atmosphere. They scale a sine wave and delay it, and they cannot reshape it, for the same reason the throat cannot. This is why the same words appear on both sides of a radio conversation. Amplitude, frequency and phase describe a vowel and they describe a carrier. A frequency response describes a mouth and it describes a feedline.

The overlap is not an analogy. It is the same theorem applied twice. The audio being coded here is a pressure wave a few hundred hertz across, and the signal carrying it is an electromagnetic wave a few hundred megahertz across, and the mathematics does not distinguish them. That is convenient for a radio amateur: the Fourier machinery in the next chapter is already familiar equipment pointed somewhere new.

The three numbers¶

A sinusoid is fully specified by three quantities, and every later chapter uses these words in this way.

Amplitude is how large the swing is. Double it and the sound is louder.

Frequency is how many cycles pass per second, in hertz. Higher means higher pitched.

Phase is where in its cycle the wave sits at the moment you start looking, measured as an angle. Two sinusoids of identical amplitude and frequency can still differ by being offset in time, and phase is that offset.

Amplitude and frequency behave the way intuition suggests. Phase does not, and it turns out to matter far less to the ear than the other two. That is convenient to the point of being suspicious, and it is worth a chapter of its own: it is the reason a 2400 bit/s codec can throw away half of what it measures and still be understood. Chapter 4 takes it up.

What follows from it¶

Those three numbers are the whole vocabulary, and the filter argument is what makes them sufficient. A filter can change the first and the third of them. It cannot touch the second, and it cannot introduce a fourth. Once you know that, several things that look like separate design decisions turn out to be the same decision.

A filter becomes a list of numbers. If every sine goes in and comes out as itself with a new size and a new delay, then a filter is fully described by what it does to each frequency: how much it scales that frequency, and how much it delays it. That is a curve rather than a process, and a curve you can measure, transmit, and reason about.

The description stops depending on the consonants or vowels The measurement you make is how much energy sits at each frequency. That is the same kind of measurement whether the speaker is saying "ah" or "ee". The numbers differ; the description does not have to be reinvented.

The tool for measuring the frequency is the Fourier transform, and in a real coder its discrete form, the DFT. It takes a block of samples and reports how much of each frequency is in it. That is the subject of chapter 2.

Here is that measurement, run on real speech, before any of the machinery behind it has been explained. Move through a sentence and change the speaker.

The curves have nothing in common. The axes are identical every time, and that is the property the rest of the site is built on.

Next: The Fourier transform and the DFT. How to find out which sinusoids are actually in a block of samples, and what "frequency resolution" costs. Back to the start. If you would rather skip the mathematics, Multi-band excitation is where the codec itself begins.

Last reviewed: 2026-08-11.