The Price of Looking for a Finite Time
Last timeWhat Was Lost Before You Started
A transform computed from a finite record is the transform of the signal multiplied by the record. That spreads every pure rhythm across neighbouring rates, and there are two cures.
Take a recording of a single pure tone, transform it, and look at the result.
It will not be a single spike. It will be a peak at roughly the right place with
a spray of smaller values on either side, reaching all the way across the range.
Try the same thing with a tone at a slightly different rate and the spray will
be much smaller or much larger, apparently at random.
The temptation is to blame the arithmetic. The arithmetic is exact. What has
happened is that the thing transformed was never the tone. It was a finite
stretch of the tone, which is a different signal, and the transform reported
that different signal correctly.
The record is a multiplication
Write the finite record as the signal multiplied by a shape that is one for the
duration of the record and zero everywhere else. That is exactly what taking a
finite record does: it keeps the signal where the record exists and replaces it
with nothing elsewhere.
The previous lesson gives the rest immediately. Multiplying two signals in the
first description is sliding their spectra along each other in the second. So
the spectrum computed from a finite record is the true spectrum with the
record's own spectrum slid along it, and the record's own spectrum is therefore
the thing to look at. It is not a spike either.
Why the ends are what matter
The spray is caused by the two ends of the record and not by its length. Cut a
rhythm off part way through a cycle and the record begins at one value and ends
at another, with an implied jump between them. The transform of a finite record
treats the record as though it repeated, as the sampling lesson's wrapping
already hinted, so that jump is a real feature of what was transformed. A single
rhythm cannot produce a jump. Many rhythms added together can, and that is what
the transform reports.
The exception is worth stating precisely because it explains a great deal of
confusing experience. If the rhythm completes a whole number of cycles within
the record, the last reading hands over to the first reading smoothly, there is
no jump, and all the energy lands at one rate exactly. Test signals are
frequently chosen this way, deliberately or by luck, and then everything looks
perfect until the first real measurement arrives.
The lesson stops here
5 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.
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See the planThe contentsThis is the reading half
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The contents