Sliding Becomes Multiplying

Last timeThe Transform of a Finite List

Every filter, blur and echo slides one signal along another and adds up the overlap. In the second description that whole operation becomes one multiplication per rhythm.

This is the lesson that justifies the course. Everything so far has been a change of

description with no obvious payment for the trouble. Here is the payment.

The operation that is everywhere

Take a short list of weights. Lay it over the signal, multiply each weight by the

reading beneath it, add the products, and write the total down as the output at that

position. Slide the weights along by one reading and do it again.

FIG 1A three point average slid along a short record

step	position of the weights	what sits under them	output written down
1	readings one to three	two, eight, two, each multiplied by a third	four
2	readings two to four	eight, two, two	four
3	readings three to five	two, two, two	two
4	readings four to six	two, two, fourteen	six
5	readings five to seven	two, fourteen, two	six

5 steps

The record was two, eight, two, two, two, fourteen, two. The two spikes, one of eight and one of fourteen, have each been spread over three outputs and reduced in height, which is what smoothing is. Note the cost: three multiplications for every output, and if the weights were a thousand long rather than three it would be a thousand. The same operation with weights that sum to zero finds edges instead, with two weights far apart it is an echo, and over a photograph with a disc of weights it is a lens out of focus.

What a filter does to a single rhythm

Now feed the sliding sum a steady rhythm and watch what comes out. Every window the

weights see is the same rhythm viewed from a different point in its cycle, so the

output is that same rhythm again, at some other size and some other starting point.

It is never a different rate. No energy moves from one rate to another, ever, in any

operation of this kind.

The lesson stops here

4 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

Sliding Becomes Multiplying

The operation that is everywhere

What a filter does to a single rhythm

4 more paragraphs to go

This is the reading half

Read alongside