One Signal, Two Descriptions
A recording can be written as a list of values in time order, or as a list of rhythms with an amount beside each. Both hold everything, and each makes a different question easy.
A microphone does one thing. It reports the air pressure in front of it, over and
over, at regular instants. A recording is that sequence of readings written down in
order, and nothing else. Play the list back through a loudspeaker and the sound
returns, so the list is complete.
It is also mute. A violin and a flute playing the same note produce two different
lists, and no inspection of either list says which instrument it was. The lists do
not come with the ingredients labelled, because the ingredients were added together
before the microphone ever saw them, and addition covers its tracks.
Adding rhythms hides the rhythms
Take three steady rhythms: a slow one, one three times as fast at a third of the
height, and one five times as fast at a fifth. Each one on its own is the simplest
repeating shape there is. Their sum is not.
The temptation is to look for the ingredients by eye, and it fails immediately. Two
quite different recipes can produce curves that look much the same, and one recipe
can produce a curve that looks like nothing at all. This is not a failure of
attention. Addition genuinely destroys the separation, and getting it back requires
an operation, not a closer look.
Where the rhythms come from
A steady rhythm is what a point does when it goes round a circle at a constant rate
and you watch only its height. That is worth taking seriously rather than treating
as a picture, because everything later in this course is easier in terms of turning
arrows than in terms of waves.
Here are two arrows, joined end to end. The first turns slowly. The second is
mounted on the tip of the first and turns three times as fast. The far tip traces a
path, and the height of that tip, plotted as it goes, is a sum of two rhythms.
the two arm lengths, squared and addedunder 0.001% of it gone by the end, at every setting of every slider
- stepped by
- a → cos(0.06)*a - sin(0.06)*b b → sin(0.06)*a + cos(0.06)*b c → cos(0.18)*c - sin(0.18)*d d → sin(0.18)*c + cos(0.18)*d
- each rule reads
- the state as it was when the step began
- started at
- a = 1.0, b = 0.0, c = m, d = 0.0
- held steady
- the two arm lengths, squared and added = a*a + b*b + c*c + d*d
- drift
- under 0.001% over the run, against 0.20% allowed
Everything in the rest of the course is this picture with more arms. A signal is
claimed to be a sum of turning arms of different rates, and the job of the transform
is to report how long each arm is and where it was pointing when the clock started.
The second description holds everything
The reasonable suspicion at this point is that describing a signal by its rhythms
must be a summary, and that summaries lose things. It is worth killing that
suspicion early, because it is wrong in a specific way that matters.
Count. A recording of a thousand readings is a thousand numbers. The description in
terms of rhythms, done properly, is also a thousand numbers, and the thousand
readings can be rebuilt from them exactly, to the last digit. No approximation is
involved and no detail has been dropped. The conversion is a change of description,
not a compression.
- the reading taken at instant n, which is the first description: the list the microphone produced
- how much of the rhythm that completes k whole cycles across the record is present
- where in its cycle that rhythm was when the record began, which the next two lessons show is as necessary as the amount
- how many readings there are, which also fixes how many distinct rhythms there is room for
Which description to ask in
Neither description is the true one. They make different questions easy, and the
split between them is sharp enough to be useful as a rule of thumb.
Ask where in this hour of recording the door slammed. In the first description this
is a matter of looking for the large numbers, and it takes one pass. In the second
it is close to hopeless, because a bang lasts no time at all and therefore contains
a little of every rhythm, so it appears as a faint contribution spread over the
whole spectrum with nothing to say about when it happened.
Now ask which note is being held. In the second description the answer is the
largest entry, read directly. In the first it requires counting peaks and hoping the
signal is clean, which it never is.
The two descriptions are not rivals, then, and the question of which is correct does
not arise. The transform is worth the trouble because real work asks both kinds of
question about the same signal, and converting between the descriptions turns out to
be cheap enough to do constantly.
That cheapness is not obvious and was not always true. It is the subject of the last
lesson, and for the twenty years before that lesson's algorithm was published, the
second description was understood perfectly well and used very little.
Recap
- A signal measured at regular instants is a list of numbers. Nothing in that list says what the signal is made of, because adding steady rhythms together destroys every visible trace of the rhythms that went in.
- The second description lists how much of each steady rhythm the signal contains. It is not a summary and it throws nothing away: it has exactly as many numbers in it as the recording did, and the recording can be rebuilt from it exactly.
- Which description to use is decided by the question. Where something happened is easy in the first and painful in the second; what something is made of, and what a filter will do to it, are easy in the second and painful in the first.
This is the reading half
Starting the course gives you your own copy of it. Every idea on every page has problems standing under it, marked with a reason rather than a tick, and any sentence you do not believe can be opened and argued with. None of that can happen on a page nobody owns.
The contents