Asking a Signal How Much of One Rhythm It Holds

Last timeOne Signal, Two Descriptions

Multiply the signal by the rhythm you are asking about and add up the result. The answer is the amount of that rhythm, and every other rhythm in the signal contributes exactly nothing.

The previous lesson claimed that a signal is a sum of steady rhythms and left the

obvious question open. Given a recording, how is the amount of any particular rhythm

actually found?

The answer is one line long and is worth stating before it is justified. Choose the

rhythm you want to ask about. Multiply the recording by it, instant by instant. Add

up all the products. The total is large when that rhythm is present and zero when it

is not.

Why multiplying and adding works

Consider what the product of two rhythms looks like. If the probe and the signal

rise and fall together, every product is a square and nothing is ever negative, so

the running total climbs. If they are at different rates, they drift in and out of

step, and the product spends exactly as much time positive as negative.

FIG 1The product of a signal with a matched probe and with a mismatched one

signal times a probe at the same ratethe same signal times a probe at a different rateand times a probe one rate away

The matched product never goes below zero, because it is a square, so adding it up over the record gives a large positive total. Both mismatched products are above and below zero in equal measure, and adding either of them up over a whole number of cycles gives not a small number but exactly zero. The cancellation is not approximate and does not depend on the record being long. It depends only on the record covering whole cycles, which is the condition the seventh lesson is about losing.

The exactness matters. A method that returned the right amount plus a little

contamination from every other rhythm would be a fitting procedure, and the answers

would have to be adjusted against each other. This one returns the amount and

nothing else, so each amount can be computed on its own.

The lesson stops here

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The contents

Asking a Signal How Much of One Rhythm It Holds

Why multiplying and adding works

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