Fourier Methods Flashcards
Any ‘well-behaved’ function f(x) can be expanded in terms of
an infinite sum of sines and cosines
Fourier coefficients
ao=
1/pi ∫ from -pi to pi f(x) dx
Fourier coefficients
an=
1/pi ∫ from -pi to pi f(x)cos nx dx
Fourier coefficients
bn=
1/pi ∫ from -pi to pi f(x) sin nx dx
∫from -pi to pi of
sin mx sin nx dx =
pi δ_mn
∫from -pi to pi of
cos mx cos nx dx =
pi δ_mn
∫from -pi to pi of
sin mx cos nx dx =
0
the Fourier series can also be written in
complex form
f(x)= Σ An e^inx
e^inx=
cosnm +isinnx
e^-inx=
cosnx - isinnx
The Fourier transform can be thought of simply as extending the idea of a Fourier series from
an infinite sum over discrete, integer Fourier modes to an infinite integral over continuous Fourier modes.
define the Fourier transform as H(f)=
∫ h(t) e^-2piift dt
inverse Fourier transform
h(t)=
same but with positive exponential
the FT of the sum of two functions is equal to
the sum of their FTs
if the time domain function h(t) is a real function then its FT is
complex
even function
h(t)=
h(-t)
odd function
h(t)=
-h(-t)
if h(t) is real then
H(-f)=[H(f)]*
if h(t) is imaginary then
H(-f)=-[H(f)]*
if h(t) is even
H(f) is even
if h(t) is odd
H(f) is odd
convolution theorem
ft (g*h) = G(f)H(f)
the FT of the convolution of the two functions is equal to the product of their individual FTs
correlation theorem
the FT of the first time domain function, multiplied by the complex
conjugate of the FT of the second time domain function, is equal to the
FT of their correlation.
FT(g,h) = G(f)H*(f)
