10. Fourier-Discrete Functions
... what happens if we know our function f(x) only at the points
xi
2
i
N
it turns out that in this particular case the coefficients are given by
ak
*
bk
*
2
N
2
N
N
f ( x j ) cos( kx j ) ,
k 0 ,1, 2 ,...
f ( x j ) sin( kx j ) ,
k 1, 2 , 3 ,...
j 1
N
j 1
.. the so-defined Fourier polynomial is the unique interpolating function to the
function f(xj) with N=2m
g ( x)
*
m
1
2
m 1
a0
*
a
k 1
*
k
cos( kx ) b k sin( kx )
*
1
2
*
a m cos( kx )
11. Fourier Spectrum
F ( ) R ( ) iI ( ) A ( ) e
A ( ) F ( )
R ( ) I ( )
2
( ) arg F ( ) arctan
A ( )
( )
i ( )
2
I ( )
R ( )
Amplitude spectrum
Phase spectrum
In most application it is the amplitude (or the power) spectrum that is of interest.
Remember here that we used the properties of complex numbers.
12. When does the Fourier transform work?
Conditions that the integral transforms work:
f(t) has a finite number of jumps and the limits exist from both
sides
f(t) is integrable, i.e.
f ( t ) dt G
Properties of the Fourier transform for special functions:
Function f(t)
Fouriertransform F()
even
even
odd
odd
real
hermitian
imaginary
antihermitian
hermitian
real
13. Some properties of the Fourier Transform
Defining as the FT:
Linearity
Symmetry
Time shifting
f ( t ) F ( )
af 1 ( t ) bf 2 ( t ) aF1 ( ) bF 2 ( )
f ( t ) 2 F ( )
f (t t ) e
f (t )
i t
F ( )
n
Time differentiation
t
n
( i ) F ( )
n
14. f (t )
n
Time differentiation
t
n
( i ) F ( )
n