What Was Lost Before You Started

Last timeSliding Becomes Multiplying

Measuring at regular instants makes whole families of rhythms identical. Which ones collapse together follows from one line of arithmetic, and no later processing can separate them again.

A camera records a wheel turning. In the recording the wheel turns slowly

backwards. Nobody doubts which direction the real wheel went, and nobody thinks

the camera is broken. Something happened between the wheel and the recording

that cannot be repaired by looking harder at the recording.

The same thing happens to every signal that is measured at regular instants,

which is every signal that reaches a computer. It is not a defect of the

equipment and it does not get better with a more expensive measuring device. It

is a consequence of taking readings at instants rather than continuously, and

the only way to deal with it is to understand exactly which rhythms it confuses,

and then arrange for the confusing ones not to be present.

What a list of readings leaves open

A record of a signal is a list of numbers and the times they were taken at. The

list makes a claim about those instants and no claim at all about anything in

between. That is worth sitting with for a moment, because it is easy to look at

a plot of readings joined up by straight lines and forget that the lines were

drawn by the plotting program and were never measured.

Infinitely many continuous signals pass through any given set of readings. Among

them is the smooth one a reader would draw by hand, and also one that jumps

violently up and down between every pair of readings and happens to be back in

place at each one. Both are consistent with the data. Deciding which to believe

is not something the data can do for you.

FIG 1Two rhythms that agree at every reading

one tenth of a cycle between readingsone and one tenth cycles between readings

The two curves are wildly different signals, and at every whole number along the bottom they have exactly the same value. A record taken at those instants contains one list of numbers, and that list belongs to both of them equally. There is no sense in which the slow one is the true answer: the record simply does not contain the question.

An entire family collapses to one list

The arithmetic behind that picture is short enough to do in a line. A rhythm at

a rate of $\omega$ radians per reading is at angle $\omega n$ at reading $n$ .

Now add $2\pi$ to the rate. At reading $n$ the angle is $\omega n + 2\pi n$ ,

which is the original angle plus a whole number of full turns, and a full turn

is nothing. The two readings are identical, at every $n$ , exactly.

So the rates come in families. Starting from any rate, adding or subtracting

whole cycles per reading gives another rate that produces the identical record.

Each family is infinite and spaced one full cycle per reading apart. The next

figure is that fact with a handle on it: two arms turning at rates that differ

by exactly one full turn per step, and the heights they are measured at.

The lesson stops here

3 more paragraphs to go

You have read the opening. The rest of the argument, the problems that check whether it landed, and the lines worth keeping at the end all come with a plan.

The first lesson of every course in the library reads the whole way through, free, so you can see exactly what the rest of them are.

See the plan The contents

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

What Was Lost Before You Started

What a list of readings leaves open

An entire family collapses to one list

3 more paragraphs to go

This is the reading half

Read alongside