Transcript Document

Fourier
Fourier Analysis
Fourier Analysis
• When we analyse a function using Fourier
methods, the function is decomposed into
its frequency components.
• This analysis is used in signal processing,
filtering.
Fourier Analysis
0
0
• Consider the two waveforms above, where the
first waveform is made up of only one pure
wave.
• The second waveform (which could be from a
steel pan or speech) is made up of one
fundamental (pure) wave, with many other
waves of higher frequency.
Fourier Analysis
• If we consider the Fourier of the waveforms
we find the following:
Intensity
Intensity
f0
frequency
f 0 2 f 0 3 f 0 frequency
Fourier Analysis
• Where f 0 is called the fundamental
frequency.
f0 
velocityof sound
0
• f 0 ,2 f 0 ,3 f 0 ,... are the harmonics.
Fourier Series
Fourier Series
• The Fourier series is used to represent other
functions.
• This is achieved by a series of sines and
cosines within the given interval, which can
be used as an approximation to the function.
Fourier Series
• The general formula for the fourier series:
a0  
 nx 
 nx 
f x     an cos
  bn sin

2 n1 
 L 
 L 
• Where the coefficients are given by,
1
 nx 
an   f x cos
dx
L L
 L 
L
1
 nx 
bn   f x sin
dx
L L
 L 
L
Fourier Series
• Writing a Fourier series means finding the
the coefficients an, bn.
• Consider the example.
• Write the function f(x)=x as a Fourier series
on the interval –Π,Π.
Fourier Series
• The Fourier series is used for approximating
or analysing periodic functions.
Fourier Transform
Fourier Transform
• While a Fourier is useful for periodic
functions, the Fourier transform or integral
is used for non-periodic functions.
Fourier Transform
• While a Fourier is useful for periodic
functions, the Fourier transform or integral
is used for non-periodic functions.
Fourier Transform
• The Fourier transform of a function ht  is,

1
 2ift


H f 
h
t
e
dt

2 
• For compactness we use the complex
exponent function.
• The Fourier transform transforms from the t
domain to the f domain.
Fourier Transform
• By convention time t is used as the
functions variable and frequency as the
transform variable.
H   
1
2

it


h
t
e
dt


• However these can be reversed or variable
such as position and momentum used. eg.
position to frequency.
Fourier Transform
• A plot of the square of the modulus of the
Fourier transform ( H   vs  ) is called the
power spectrum.
• It gives the amount the frequency
contributes to the waveform.
Fourier Transform
• The inverse Fourier is,
ht  
1
2

 H  f e

2ift
df
Fourier Transform
• Example: Fourier transform of sin 2k0 x  .
Fourier Transform
• Example: Fourier transform of sin 2k0 x  .
 Fx k  



f x e2ikxdx
Fourier Transform
• Example:Fourier transform of sin 2k0 x  .
 Fx k   f x  e  2ikxdx



Fx sin 2k0 x k    sin 2k0 x e2ikxdx


 e 2ik0 x  e 2ik0 x
  
2i
 
  2ikx
e
dx

Fourier Transform
• Example: Fourier transform of sin 2k0 x  .

 Fx k  

f x  e  2ikxdx


Fx sin 2k0 x k    sin 2k0 x e2ikxdx


 e 2ik0 x  e 2ik0 x
  
2i
 


i
2
 e

 2i  k  k0  x
  2ikx
e
dx


 e 2i k k0 x dx
Fourier Transform
• However the deltafunction is,
 x  
 Fx sin2k0 xk  
i
1
2

1
2
 2ikx
e
dk


i
2
 k  k0    k  k0 
Discrete Fourier Transform
Discrete Fourier Transform
• If ht  or H  f is known analytically, the
integral can be evaluated numerically or
numerically using one of the previous
techniques. If a table of values is know
interpolation can be used to evaluate the
function.
Discrete Fourier Transform
• However we consider the case for directly
Fourier transforming functions that are only
known at sampled points.
• By sampling a function at N times, we
determine N values for the Fourier
transform of the function ( N independent
values).
Discrete Fourier Transform
• If the samples are truly independent, the
DFT produces a function with is periodic
between the sampling period.
Discrete Fourier Transform
• If the samples are truly independent, the
DFT produces a function with is periodic
between the sampling period.
• If the function we are analysing is actually
periodic, the first N points should all be in
one period to guarantee independence.
Discrete Fourier Transform
• The time interval T is the largest time over
which we are sampling our function.
• NB: for a periodic function it is the period
of the function.
Discrete Fourier Transform
• The time interval T is the largest time over
which we are sampling our function.
• NB: for a periodic function it is the period
of the function.
• T therefore determines the lowest frequency.
Discrete Fourier Transform
• Assume that the function ht  we wish to
transform is measured or sampled at a
discrete number of points N+1 times (N
time intervals).
Discrete Fourier Transform
• Assume that the function ht  we wish to
transform is measured or sampled at a
discrete number of points N+1 times (N
time intervals).
• Let  be the time interval between
samples.
h2 h
hN
h1
3
h4
 2 3 4
N
Discrete Fourier Transform
H  f   hi e
2if  k 

Discrete Fourier Transform
• Because we are sampling at discrete times
we have a discrete set of frequencies.
Discrete Fourier Transform
• NB:  = dt.
Discrete Fourier Transform
• NB:  = dt.
• Assuming that the samples are evenly
spaced, hn  hn  n   N ,  N  2 ,..., 0,1,..., N.
2
2
2
Discrete Fourier Transform
• NB:  = dt.
• Assuming that the samples are evenly
spaced, hn  hn  n   N ,  N  2 ,..., 0,1,..., N.
2
2
2
• The inverse of the time interval gives the
sampling frequency.
n
fn 
N
Discrete Fourier Transform
• The maximum sampling frequency is called
the Nyquist frequency (when n = N/2).
fc 
1
2
Discrete Fourier Transform
• The maximum sampling frequency is called
the Nyquist frequency (when n = N/2).
fc 
1
2
• The Discrete Fourier Transform has the
form,
N 1
H n   hk e
k 0
 2ikn / N
Discrete Fourier Transform
• The Discrete Inverse Fourier Transform has
the form,
1 N 1
2ikn/ N
hk 
H ne

N n 0
Discrete Fourier Transform
• Example: Suppose we sample N=8 values.
Then n = -4,-3,-2,-1,0,1,2,3,4
 N  N  2
N
N
n
,
,..., 0,1,...,
n
fn 
2
4
2
2
2
N
1

 fc
N  2 2
• We have 9 point, 1 more than we are
allowed. Thus there is some redundant data.
Discrete Fourier Transform
• This is because the endpoints (-4 and 4)
give the same information, the Nyquist
 N  N  2
N
frequency.
n
,
,..., 0,1,...,
2
2
2
n
f
-4
-3
-2
-1
0
1
2
3
4
 4 1

8 2
3
8
1
4
1
8
0
1
8
1
4
3
8
1
2
f(Hz); Δ=½ s
-1
-¾
-½
-¼
0
¼
½
¾
1
Discrete Fourier Transform
n  1

No new info
n  4
Discrete Fourier Transform
• The DFT can be applied to any complex
values series. The computation time is
proportional to the square of the number of
points.
Fast Fourier Transform
Fast Fourier Transform
• A much faster algorithm was developed by
Cooley and Tukey called the Fast Fourier
Transform (FFT).
Fast Fourier Transform
• A much faster algorithm was developed by
Cooley and Tukey called the Fast Fourier
Transform (FFT).
• The most popular implementation is the
radix-2 FFT. It’s computational time is
proportional to N log2 N  .
Fast Fourier Transform
• The FFT algorithm is based on a divide and
conquer approach. The initial problem is
divided into smaller and smaller problem.
These computations are recombined to give
the final result.
Fast Fourier Transform
• The FFT algorithm is based on a divide and
conquer approach. The initial problem is
divided into smaller and smaller problem.
These computations are recombined to give
the final result.
• The simplest implementation continually
halves the dimensions of the DFT until it
becomes unity.