1.1 A Fourier series decomposes periodic
functions or periodic signals into the sum of a
(possibly infinite) set of simple oscillating
functions, namely sines and cosines (or
complex exponentials).
1.1.1 It is very useful in Physics because it allows us to
detect and analyse periodicities in apparently
random functions
1.1.2 Useful for solving linear
differential equations -
superposition
1.2 Every function can be
represented as a sum of even
and odd functions
1.3 The Euler-Fourier formulae
(TO REMEBER!) help up
to compute the coefficients
an and bn
1.3.1 If f(x) is odd, no an
coefficients will be present
1.3.2 If f(x) is even, no
bn coefficients will
be present
1.3.3 Dirichlet conditions,
sufficient but not
necessary
1.3.3.1 1. f(x) must be periodic 2. f(x) must be single valued
with a finite number of discontinuities in one period
3.f(x) must have a finite number of discontinuities in one
period 4.It has to be possible to compute the integral of
the absolute value of f(x) over a period
1.4 Convergence of Fourier
series
1.4.1 IMPORTANT: Gibbs
Phenomenon
1.4.1.1 Fourier series overshoot at a jump
and this overshooting is not
eliminated even with a very high
number of elements in the sum
1.5 Fourier/Frequency Space
1.5.1 The frequency components,
spread across the frequency
spectrum, are represented as
peaks in the frequency domain.
(See wikipedia's excellent
animation on the "frequency
domain" page)
1.6 Parseval's theorem
1.6.1 The average value of the square of a function is equal
to the sum of the average values of the square of the
Fourier compontents
2 Fourier Transforms
2.1 Provide a Fourier representation of non-periodic functions
2.1.1 It is employed to transform signals between time (or spatial) domain
and frequency domain. It is reversible, being able to transform from
either domain to the other.
2.2 It could be
useful to
remember
some of the
most important
fourier transform
pairs
2.2.1 sine/cosine - is a combination of
dirac delta functions
2.2.2 delta function - is a constant (if we are absolutely sure about the
position in x or t domain, then we will have an infinite spread in
the frequency domain - very important in quantum mechanics)
2.2.3 gaussian - is a gaussian!
2.2.4 The Fourier transform of a Fourier transform is the original function over 2Pi