Two Numbers for Every Rhythm
Last timeAsking a Signal How Much of One Rhythm It Holds
A cosine probe returns nothing for a rhythm that is fully present but shifted a quarter cycle. The repair is to ask twice, and the pair of answers is a turning arrow with a length and a direction.
The recipe from the previous lesson has a hole in it, and the hole is not small.
Here is a signal it reports nothing at all for.
Take the rhythm the probe is looking for, and shift it by a quarter of its cycle, so
that it begins at zero and rises while the probe begins at one and falls. The two
now agree for a quarter cycle and disagree for the next quarter, over and over, and
the products cancel exactly.
Asking twice
So probe twice at every rate: once with a cosine and once with a sine. The first
answer is how much of the rhythm starts at its peak, the second how much starts at
zero and rising, and between them they cover every possibility.
That last claim is the one that needs checking, because it is what stops this
becoming a search over all possible shifts. A rhythm at a given rate shifted by any
amount whatever can be written as some of the cosine plus some of the sine, with
nothing left over. Two probes have therefore already asked about every shift there
is, and a third probe at that rate would return nothing that is not a combination of
the two answers already held.
The pair is an arrow
Read the two answers as coordinates: the cosine answer across, the sine answer up.
The pair is then an arrow, and the arrow is the useful object.
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 planThe contentsThis 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