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Back to Physics 434, 2012: Information Processing in Biology.
Here we introduce the idea of Fourier series and Fourier transforms. We have discussed in the previous lecture why we need them.
Main Lecture
- Consider a function
periodic on
.
- We would like to approximate this function as
, takin
at some point.
- From this expression, we can find the coefficients
self-consistently. Indeed, let's multiply the equation by
and integrate from
to
.
- All terms containing products
are zero.
- For the
terms, we have
.
- Completing the integrals, we have:
.
- We can do similar to find
: multiply by
, and integrate.
- This gives:
,
,
.