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Fourier Analysis
- Fourier Series
- 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).
- It is very useful in Physics because it allows us to
detect and analyse periodicities in apparently
random functions
- Useful for solving linear
differential equations -
superposition
- It is very useful in Physics because it allows us to
detect and analyse periodicities in apparently
random functions
- Every function can be
represented as a sum of even
and odd functions
- The Euler-Fourier formulae
(TO REMEBER!) help up
to compute the coefficients
an and bn
- If f(x) is odd, no an
coefficients will be present
- If f(x) is even, no
bn coefficients will
be present
- Dirichlet conditions,
sufficient but not
necessary
- 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. 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
- If f(x) is odd, no an
coefficients will be present
- Convergence of Fourier
series
- IMPORTANT: Gibbs
Phenomenon
- Fourier series overshoot at a jump
and this overshooting is not
eliminated even with a very high
number of elements in the sum
- Fourier series overshoot at a jump
and this overshooting is not
eliminated even with a very high
number of elements in the sum
- IMPORTANT: Gibbs
Phenomenon
- Fourier/Frequency Space
- 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)
- 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)
- Parseval's theorem
- 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
- 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
- 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).
- Fourier Transforms
- Provide a Fourier representation of non-periodic functions
- 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.
- 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.
- It could be
useful to
remember
some of the
most important
fourier transform
pairs
- sine/cosine - is a combination of
dirac delta functions
- 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)
- gaussian - is a gaussian!
- The Fourier transform of a Fourier transform is the original function over 2Pi
- sine/cosine - is a combination of
dirac delta functions
- Theorems
- 1. Linearity: F[af(x)+bg(x)] = aF(f(x))+bF(g(x))
2.Shift theorem F(f(x+a))=e^iaω*g(ω) 3.Scaling
F(f(ax))=1/|a|*g(ω/a) 4.Exponential moltiplication
F(e^ax*f(x))=g(ω+ia) 5.Convolution Theorem F[f(x)
convoluted with g(x)] = 2PiF[f(x)]F[g(x)]
- 1. Linearity: F[af(x)+bg(x)] = aF(f(x))+bF(g(x))
2.Shift theorem F(f(x+a))=e^iaω*g(ω) 3.Scaling
F(f(ax))=1/|a|*g(ω/a) 4.Exponential moltiplication
F(e^ax*f(x))=g(ω+ia) 5.Convolution Theorem F[f(x)
convoluted with g(x)] = 2PiF[f(x)]F[g(x)]
- Provide a Fourier representation of non-periodic functions
- Convolution
- Convolution Theorem F[f(x)
convoluted with g(x)] = 2PiF(x)Fg(x)
- Be careful with extrema of integration
- Convolution Theorem F[f(x)
convoluted with g(x)] = 2PiF(x)Fg(x)
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