Fourier Series and Transforms

Fourier Series and Transforms

• Orthogonal functions

• Fourier Series

• Discrete Fourier Series

• Fourier Transform

• Chebyshev polynomials

Scope: we are trying to approximate

an arbitrary function and obtain

basis functions with appropriate

coefficients.

Fourier Series

The Problem

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

) ( )

( )

(

0

x a

x g

x

f

i

i i

N











... 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



^

a

dx x g x

f ( ) ( ) 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 / 1 2

2

 



 



 



 

a

N

N

x f x g x dx

g x

f

Orthogonal Functions

... where x₀=a and x_N=b, and x_i-x_i-1=x ...

If we interpret f(x_i) and g(x_i) 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 i

i i

b

a

f x g x dx

f x g x x

1

) ( ) ( lim

) ( ) (

f

g

  

 0

i i i

i

g f g

f

Periodic functions

-15 -10 -5 0 5 10 15 20

0 10 20 30 40

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

) ( )

2

( x f x

f   

)

( )

2

( x f x x

f    

... we want to approximate this function with a linear combination of 2

periodic functions:

) sin(

), cos(

),..., 2

sin(

), 2 cos(

), sin(

), cos(

,

1 x x x x nx nx

 

 ^







 f x g

x a

a

cos( kx ) b

sin( kx ) 2

) 1 ( )

(

Orthogonality

... are these functions orthogonal ?

0 ,

0 0

) sin(

) cos(

0 0 ,

, 0

) sin(

) sin(

0 0 2

0 )

cos(

) cos(





















































k j

dx kx jx

k j

k j k j

dx kx jx

k j

k j

k j

dx kx jx



















... 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 )

( )

( x  f x

 or

g x  f x



g

k

leading to

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

 















 



   



 

 



 

 

dx x

f kx

b kx

a a a

x f x

a g

N k

k k

k n

k

2

1 0

2 cos( ) sin( ) ( )

2 ) 1

( )

( 2





 







































N k

dx kx x

f b

N k

dx kx x

f a

kx b

kx a

a x

g

k k

N k

k k

N

,..., 2 , 1 ,

) sin(

) 1 (

,..., 1 , 0 ,

) cos(

) 1 (

with )

sin(

) 2 cos(

) 1 (

1 0

Fourier approximation of |x|

... Example ...

.. and for n<4 g(x) looks like leads to the Fourier Serie







   



 ...

5 ) 5 cos(

3 ) 3 cos(

1 ) cos(

4 2

) 1

( ₂ x ₂ x ₂ x

x

g  



  



 x x

x

f ( ) ,

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

0 1 2 3 4

Fourier approximation of x ²

... another Example ...

 2 0

, )

( x  x

 x  f

.. and for N<11, g(x) looks like leads to the Fourier Serie





 



 ^N

k

N kx

kx k x k

g

1 2

2

) 4 sin(

) 4 cos(

3 ) 4

(  

-10 -5 0 5 10 15

-10 0 10 20 30 40

Fourier - discrete functions

N i x

2 



.. the so-defined Fourier polynomial is the unique interpolating function to the function f(x_j) 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

*

















k kx

x N f

b

k kx

x N f

a

N j

j j

N j

j j

k k

 

2 ) 1 sin(

) 2 cos(

) 1

( ^{1 *}

1

*

*

*

*

0 a kx b kx a kx

a x

g ^m _m

k

m  





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

) ( )

*

(

i i

m

x f x

g 

Fourier - collocation points

... with the important property that ...

... in our previous examples ...

-10 -5 0 5 10

0 0.5 1 1.5 2 2.5 3 3.5

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

Fourier series - convergence

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

-10 -5 0 5 10 15

-10 0 10 20 30 40

50 N = 4

-10 -5 0 5 10 15

-10 0 10 20 30 40

50 N = 8

Fourier series - convergence

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

-10 -5 0 5 10 15

-10 0 10 20 30 40

50 N = 16

-10 -5 0 5 10 15

-10 0 10 20 30 40 50

N = 32

Gibb’s phenomenon

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

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():

1 cos

8 cos

8 )

4 cos(

cos 3

cos 4

) 3 cos(

1 cos

2 )

2 cos(

2 4

3 2































... which leads us to the definition:

Chebyshev polynomials - definition

N n

x x

x T T

n ) 

(cos( )) 

( ),  cos( ),  [  1 , 1 ], 

cos(   

... for the Chebyshev polynomials T_n(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 2

4 4

3 3

2 2

1 0

and ]

1 , 1 [ for

1 )

(

where 1

8 8

) (

3 4

) (

1 2

) (

) (

1 ) (

N n

x x

T

x x

x T

x x

x T

x x

T

x x

T x T

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)

n n k

x

 cos( k  ),  0 , 1 , 2 , 3 ,...,

Chebyshev collocation points

These extrema are not equidistant (like the Fourier extrema)

n n k

x

 cos( k  ),  0 , 1 , 2 , 3 ,...,

k

x(k)

Chebyshev polynomials - orthogonality

... are the Chebyshev polynomials orthogonal?

2 0 1

1

, ,

0 0 2

/ 0 ) 1

( )

( k j N

j k

for

j k

for

j k

for x

x dx T

x

T



 

 



 

 













 



 



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

such that

1 2

/

1 x

... this can be easily verified noting that

) cos(

) ( ),

cos(

) (

sin ,

cos











j x

T k

x T

d dx

x

j

k

 







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 g_n(x)

) ( )

2 ( ) 1

( )

(

1 0

0

T x c T x

c x

g x

f

k

k k

n











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

(minimum)

  ⁰

) 1 ( )

(

1

1

2

 



 





 



 



x

x dx f

x

c

g

Chebyshev polynomials - interpolation

... to obtain ...

n x k

x dx T

x f

c

, 0 , 1 , 2 ,...,

) 1 ( ) 2

(

1 2



  



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

n k

d k f

c

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

0



 _

 ^{  }

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

n k

d k f

c

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



 







 





... which means that the coefficients c_k are the Fourier coefficients a of the periodic function F()=f(cos )!

Chebyshev - discrete functions

N i

x



 cos

... leading to the polynomial ...

in this particular case the coefficients are given by

2 / ,...

2 , 1 , 0 ,

) cos(

) 2 (cos

1

* f k k N

c N ^N

j

j

k 













 ^m

k

k k

m x c T c T x

g

1

*

* 0

* ( )

2 ) 1

( ₀

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

... with the property

N 0,1,2,..., j

j/N) cos(

x at )

( )

( _j

* x  f x   

g_m

Chebyshev - collocation points - |x|

f(x)=|x| => f(x) - blue ; g_n(x) - red; x_i - ‘+’

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

0 0.2 0.4 0.6 0.8

1 N = 8

0 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 ; g_n(x) - red; x_i - ‘+’

32 points

128 points

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

0 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 1

0 0.2 0.4 0.6 0.8

1 N = 128

Chebyshev - collocation points - x ²

f(x)=x² => f(x) - blue ; g_n(x) - red; x_i - ‘+’

8 points

64 points

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

0 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 1

0 0.2 0.4 0.6 0.8 1

1.2 N = 64

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

Chebyshev vs. Fourier - numerical

f(x)=x² => f(x) - blue ; g_N(x) - red; x_i - ‘+’

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

0 0.2 0.4 0.6 0.8 1

0 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 ; g_N(x) - red; x_i - ‘+’

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 ; g_N(x) - red; x_i - ‘+’

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

N i x_i 2

 i

x_i N

 cos

periodic functions limited area [-1,1]

) sin(

),

cos(nx nx



 cos

), cos(

) (



 x

n x

T_n

 

) 2 cos(

1

) sin(

) cos(

2 ) 1 (

* 1 1

*

*

*

*

0

kx a

kx b

kx a

a x

g

m m k m

k k

















 ^m

k

k k

m x c T c T x

g

1

*

* 0

* ( )

2 ) 1

( ₀

collocation points

domain

basis functions

interpolating function

Fourier vs. Chebyshev (cont’d)

Fourier Chebyshev

coefficients

some properties













N j

j j

N j

j j

kx x

N f b

kx x

N f a

k k

1

*

1

*

) sin(

) 2 (

) cos(

) 2 (



 ^N

j

j

j k

N f ck

1

* 2 (cos )cos(  )

• 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