Spectral analysis: foundationsComputational Geophysics and Data Analysis 1 Spectral analysis: Foundations Orthogonal functions Fourier Series Discrete

Computational Geophysics and Data Analysis 1Spectral analysis: foundations

Spectral analysis: Foundations

Orthogonal functions Fourier Series Discrete Fourier Series Fourier Transform: properties Chebyshev polynomials Convolution DFT and FFT

Scope: Understanding where the Fourier Transform comes from. Moving from the continuous to the discrete world. (Almost) everything we need to understand for filtering.


Fourier Series: one way to derive them

The Problem

we are trying to approximate a function f(x) by another function gn(x) which consists of a sum over N orthogonal functions (x) weighted by some coefficients an.

)()()(

0

xaxgxfN

iiiN


... and we are looking for optimal functions in a least squares (l2) sense ...

... a good choice for the basis functions (x) are orthogonal functions. What are orthogonal functions? Two functions f and g are said to be

orthogonal in the interval [a,b] if

b

a

dxxgxf 0)()(

How is this related to the more conceivable concept of orthogonal vectors? Let us look at the original definition of integrals:

The Problem

!Min)()()()(

2/12

2

b

a

NN dxxgxfxgxf


Orthogonal Functions

... where x0=a and xN=b, and xi-xi-1=x ...If we interpret f(xi) and g(xi) as the ith components of an N component

vector, then this sum corresponds directly to a scalar product of vectors. The vanishing of the scalar product is the condition for orthogonality of

vectors (or functions).

N

iii

b

aN

xxgxfdxxgxf1

)()(lim)()(

figi

0 ii

iii gfgf


Periodic functions

-15 -10 -5 0 5 10 15 200

10

20

30

40

Let us assume we have a piecewise continuous function of the form

)()2( xfxf

2)()2( xxfxf

... we want to approximate this function with a linear combination of 2 periodic functions:

)sin(),cos(),...,2sin(),2cos(),sin(),cos(,1 nxnxxxxx

N

kkkN kxbkxaaxgxf

10 )sin()cos(

2

1)()(


Orthogonality

... are these functions orthogonal ?

0,00)sin()cos(

0

0,,0

)sin()sin(

0

02

0

)cos()cos(

kjdxkxjx

kj

kjkj

dxkxjx

kj

kj

kj

dxkxjx

... YES, and these relations are valid for any interval of length 2.Now we know that this is an orthogonal basis, but how can we obtain the

coefficients for the basis functions?

from minimising f(x)-g(x)


Fourier coefficients

optimal functions g(x) are given if

0)()(!Min)()(22

xfxgorxfxg nan k

leading to

... with the definition of g(x) we get ...

dxxfkxbkxaaa

xfxga

N

kkk

kn

k

2

10

2)()sin()cos(

2

1)()(

2

Nkdxkxxfb

Nkdxkxxfa

kxbkxaaxg

k

k

N

kkkN

,...,2,1,)sin()(1

,...,1,0,)cos()(1

with)sin()cos(2

1)(

10


Fourier approximation of |x|

... Example ...

.. and for n<4 g(x) looks like

leads to the Fourier Serie

...

5

)5cos(

3

)3cos(

1

)cos(4

2

1)(

222

xxxxg

xxxf ,)(

-20 -15 -10 -5 0 5 10 15 200

1

2

3

4


Fourier approximation of x2

... another Example ...

20,)( 2 xxxf

.. and for N<11, g(x) looks like

leads to the Fourier Serie

N

kN kx

kkx

kxg

12

2

)sin(4

)cos(4

3

4)(

-10 -5 0 5 10 15-10

0

10

20

30

40


Fourier - discrete functions

iN

xi2

.. the so-defined Fourier polynomial is the unique interpolating function to the function f(xj) with N=2m

it turns out that in this particular case the coefficients are given by

,...3,2,1,)sin()(2

,...2,1,0,)cos()(2

1

*

1

*

kkxxfN

b

kkxxfN

a

N

jjj

N

jjj

k

k

)cos(2

1)sin()cos(

2

1)( *

1

1

****0 kxakxbkxaaxg m

m

km kk

... what happens if we know our function f(x) only at the points


)()(*iim xfxg

Fourier - collocation points

... with the important property that ...

... in our previous examples ...

-10 -5 0 5 100

0.5

1

1.5

2

2.5

3

3.5

f(x)=|x| => f(x) - blue ; g(x) - red; xi - ‘+’


Fourier series - convergence

f(x)=x2 => f(x) - blue ; g(x) - red; xi - ‘+’


Fourier series - convergence



Gibb’s phenomenon


0 0.5 1 1.5-6

-4

-2

0

2

4

6

N = 32

0 0.5 1 1.5-6

-4

-2

0

2

4

6

N = 16

0 0.5 1 1.5-6

-4

-2

0

2

4

6

N = 64

0 0.5 1 1.5-6

-4

-2

0

2

4

6

N = 128

0 0.5 1 1.5-6

-4

-2

0

2

4

6

N = 256

The overshoot for equi-spaced Fourier

interpolations is 14% of the step height.


Chebyshev polynomials

We have seen that Fourier series are excellent for interpolating (and differentiating) periodic functions defined on a regularly spaced grid. In many circumstances physical phenomena which are not periodic (in space) and occur in a limited area. This quest leads to the use of Chebyshev polynomials.

We depart by observing that cos(n) can be expressed by a polynomial in cos():

1cos8cos8)4cos(

cos3cos4)3cos(

1cos2)2cos(

24

3

2

... which leads us to the definition:


Chebyshev polynomials - definition

NnxxxTTn nn ],1,1[),cos(),())(cos()cos(

... for the Chebyshev polynomials Tn(x). Note that because of x=cos() they are defined in the interval [-1,1] (which - however - can be extended to ). The first polynomials are

0

244

33

22

1

0

and]1,1[for1)(

where188)(

34)(

12)(

)(

1)(

NnxxT

xxxT

xxxT

xxT

xxT

xT

n


Chebyshev polynomials - Graphical

The first ten polynomials look like [0, -1]

The n-th polynomial has extrema with values 1 or -1 at

0 0.2 0.4 0.6 0.8 1-1

-0.5

0

0.5

1

x

T_

n(x)

nkn

kx extk ,...,3,2,1,0),cos()(


Chebyshev collocation points

These extrema are not equidistant (like the Fourier extrema)

nkn

kx extk ,...,3,2,1,0),cos()(

k

x(k)


Chebyshev polynomials - orthogonality

... are the Chebyshev polynomials orthogonal?

02

1

1

,,

0

02/

0

1)()( Njk

jkfor

jkfor

jkfor

x

dxxTxT jk

Chebyshev polynomials are an orthogonal set of functions in the interval [-1,1] with respect to the weight functionsuch that

21/1 x

... this can be easily verified noting that

)cos()(),cos()(

sin,cos

jxTkxT

ddxx

jk


Chebyshev polynomials - interpolation

... we are now faced with the same problem as with the Fourier series. We want to approximate a function f(x), this time not a

periodical function but a function which is defined between [-1,1]. We are looking for gn(x)

)()(2

1)()(

100 xTcxTcxgxf

n

kkkn

... and we are faced with the problem, how we can determine the coefficients ck. Again we obtain this by finding the extremum

(minimum)

01

)()(2

1

1

2

x

dxxfxg

c nk


Chebyshev polynomials - interpolation

... to obtain ...

nkx

dxxTxfc kk ,...,2,1,0,

1)()(

2 1

12

... surprisingly these coefficients can be calculated with FFT techniques, noting that

nkdkfck ,...,2,1,0,cos)(cos2

0

... and the fact that f(cos) is a 2-periodic function ...

nkdkfck ,...,2,1,0,cos)(cos1

... which means that the coefficients ck are the Fourier coefficients ak of the periodic function F()=f(cos )!


Chebyshev - discrete functions

iN

xi

cos

... leading to the polynomial ...

in this particular case the coefficients are given by

2/,...2,1,0,)cos()(cos2

1

* NkkfN

cN

jjjk

m

kkkm xTcTcxg

1

**0

* )(2

1)( 0

... what happens if we know our function f(x) only at the points

... with the property

N0,1,2,...,jj/N)cos(xat)()( j* xfxgm


Chebyshev - collocation points - |x|

f(x)=|x| => f(x) - blue ; gn(x) - red; xi - ‘+’

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1 N = 8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1 N = 16

8 points

16 points


Chebyshev - collocation points - |x|

f(x)=|x| => f(x) - blue ; gn(x) - red; xi - ‘+’

32 points

128 points

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1 N = 32

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1 N = 128


Chebyshev - collocation points - x2

f(x)=x2 => f(x) - blue ; gn(x) - red; xi - ‘+’

8 points

64 points

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2 N = 8

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2 N = 64

The interpolating function gn(x) was shifted by a small amount to be visible at all!


Chebyshev vs. Fourier - numerical

f(x)=x2 => f(x) - blue ; gN(x) - red; xi - ‘+’

This graph speaks for itself ! Gibb’s phenomenon with Chebyshev?

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1 N = 16

0 2 4 6-5

0

5

10

15

20

25

30

35

N = 16

Chebyshev Fourier


Chebyshev vs. Fourier - Gibb’s

f(x)=sign(x-) => f(x) - blue ; gN(x) - red; xi - ‘+’

Gibb’s phenomenon with Chebyshev? YES!

Chebyshev Fourier

-1 -0.5 0 0.5 1-1.5

-1

-0.5

0

0.5

1

1.5 N = 16

0 2 4 6-1.5

-1

-0.5

0

0.5

1

1.5 N = 16


Chebyshev vs. Fourier - Gibb’s

f(x)=sign(x-) => f(x) - blue ; gN(x) - red; xi - ‘+’

Chebyshev Fourier

-1 -0.5 0 0.5 1-1.5

-1

-0.5

0

0.5

1

1.5 N = 64

0 2 4 6 8-1.5

-1

-0.5

0

0.5

1

1.5 N = 64


Fourier vs. Chebyshev

Fourier Chebyshev

iN

xi2

iN

xi

cos

periodic functions limited area [-1,1]

)sin(),cos( nxnx

cos

),cos()(

x

nxTn

)cos(2

1

)sin()cos(

2

1)(

*

1

1

**

**0

kxa

kxbkxa

axg

m

m

k

m

kk

m

kkkm xTcTcxg

1

**0

* )(2

1)( 0

collocation points

domain

basis functions

interpolating function


Fourier vs. Chebyshev (cont’d)

Fourier Chebyshev

coefficients

some properties

N

jjj

N

jjj

kxxfN

b

kxxfN

a

k

k

1

*

1

*

)sin()(2

)cos()(2

N

jjj kf

Nck

1

* )cos()(cos2

• Gibb’s phenomenon for discontinuous functions

• Efficient calculation via FFT

• infinite domain through periodicity

• limited area calculations

• grid densification at boundaries

• coefficients via FFT

• excellent convergence at boundaries

• Gibb’s phenomenon


The Fourier Transform Pair

deFtf

dtetfF

ti

ti

)()(

)(2

1)(

Forward transform

Inverse transform

Note the conventions concerning the sign of the exponents and the factor.


The Fourier Transform Pair

)(

)(arctan)(arg)(

)()()()(

)()()()(

22

)(

R

IF

IRFA

eAiIRF i

)(

)(

A Amplitude spectrum

Phase spectrum

In most application it is the amplitude (or the power) spectrum that is of interest.


The Fourier Transform: when does it work?

Gdttf )(

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.

Properties of the Fourier transform for special functions:

Function f(t) Fouriertransform F()

even even

odd odd

real hermitian

imaginary antihermitian

hermitian real


… graphically …


Some properties of the Fourier Transform

Defining as the FT: )()( Ftf

Linearity

Symmetry

Time shifting

Time differentiation

)()()()( 2121 bFaFtbftaf

)(2)( Ftf

)()( Fettf ti

)()()( Fi

t

tf nn

n


Differentiation theorem

Time differentiation )()()( Fi

t

tf nn

n


Convolution

')'()'(')'()'()()( dttgttfdtttgtftgtf

The convolution operation is at the heart of linear systems.

Definition:

Properties:)()()()( tftgtgtf

)()()( tfttf

dttftHtf )()()(

H(t) is the Heaviside function:


The convolution theorem

A convolution in the time domain corresponds to a multiplication in the frequency domain.

… and vice versa …

a convolution in the frequency domain corresponds to a multiplication in the time domain

)()()()( GFtgtf

)()()()( GFtgtf

The first relation is of tremendous practical implication!


The convolution theorem

From Bracewell (Fourier transforms)


Discrete Convolution

Convolution is the mathematical description of the change of waveform shape after passage through a filter (system).

There is a special mathematical symbol for convolution (*):

Here the impulse response function g is convolved with the input signal f. g is also named the „Green‘s function“

)()()( tftgty

nmk

fgym

iikik

,,2,1,00

migi ,....,2,1,0

njf j ,....,2,1,0


Convolution Example(Matlab)

>> x

x =

0 0 1 0

>> y

y =

1 2 1

>> conv(x,y)

ans =

0 0 1 2 1 0

>> x

x =

0 0 1 0

>> y

y =

1 2 1

>> conv(x,y)

ans =

0 0 1 2 1 0

Impulse response

System input

System output


Convolution Example (pictorial)

x y„Faltung“

0 1 0 0

1 2 1

0 1 0 0

1 2 1

0 1 0 0

1 2 1

0 1 0 0

1 2 1

0 1 0 0

1 2 1

0 1 0 0

1 2 1

0

0

1

2

1

0

y x*y


The digital world


The digital world

j

s jdtttgtg )()()(

gs is the digitized version of g and the sum is called the comb function. Defining the Nyquist frequency fNy as

dtfNy 2

1

after a few operations the spectrum can be written as

1

)2()2()(1

)(n

NyNys nffGnffGfGdt

fG

… with very important consequences …


The sampling theorem

dtfNy 2

1

The implications are that for the calculation of the spectrum at frequency f there are also contributions of frequencies f±2nfNy, n=1,2,3,…

That means dt has to be chosen such that fN is the largest frequency contained in the signal.


The Fast Fourier Transform FFT

... spectral analysis became interesting for computing with the introduction of the Fast Fourier Transform (FFT). What’s so fast about it ?

The FFT originates from a paper by Cooley and Tukey (1965, Math. Comp. vol 19 297-301) which revolutionised all fields where Fourier transforms where essential to progress.

The discrete Fourier Transform can be written as

1,...,1,0,1

1,...,1,0,

/21

0

/21

0

NkeFN

f

NkefF

NikjN

jjk

NikjN

jjk


The Fast Fourier Transform FFT

... this can be written as matrix-vector products ...for example the inverse transform yields ...

1

2

1

0

1

2

1

0

)1(1

22642

132

2

1

1

1

11111

NNNN

N

N

f

f

f

f

F

F

F

F

.. where ...

Nie /2


FFT

... the FAST bit is recognising that the full matrix - vector multiplicationcan be written as a few sparse matrix - vector multiplications

(for details see for example Bracewell, the Fourier Transform and its applications, MacGraw-Hill) with the effect that:

Number of multiplicationsNumber of multiplications

full matrix FFT

N2 2Nlog2N

this has enormous implications for large scale problems.Note: the factorisation becomes particularly simple and effective

when N is a highly composite number (power of 2).


FFT

.. the right column can be regarded as the speedup of an algorithm when the FFT is used instead of the full system.

Number of multiplicationsNumber of multiplications

Problem full matrix FFT Ratio full/FFT

1D (nx=512) 2.6x105 9.2x103 28.4 1D (nx=2096) 94.98 1D (nx=8384) 312.6


Summary

The Fourier Transform can be derived from the problem of approximating

an arbitrary function.

A regular set of points allows exact interpolation (or derivation) of arbitrary

functions

There are other basis functions (e.g., Chebyshev polynomials) with similar

properties

The discretization of signals has tremendous impact on the estimation of

spectra: aliasing effect

The FFT is at the heart of spectral analysis

Documents

Spectral analysis: foundationsComputational Geophysics and Data Analysis 1 Spectral analysis: Foundations Orthogonal functions Fourier Series Discrete