Fourier Transform Methods


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FOURIER TRANSFORM METHODS David Sandwell, January, 2013
1. Fourier Transforms
Fourier analysis is a fundamental tool used in all areas of science and engineering. The fast fourier transform (FFT) algorithm is remarkably efficient for solving large problems. Nearly every computing platform has a library of highly-optimized FFT routines. In the field of Earth science, fourier analysis is used in the following areas:

Solving linear partial differential equations (PDE’s):

Gravity/magnetics

Laplace

∇2Φ = 0

Elasticity (flexure)

Biharmonic

∇4Φ = 0

Heat Conduction

Diffusion

∇2Φ - δ Φ/ δt = 0

Wave Propagation

Wave

∇2Φ - δ2Φ/ δt2 = 0

Designing and using antennas: Seismic arrays and streamers Multibeam echo sounder and side scan sonar Interferometers – VLBI – GPS Synthetic Aperture Radar (SAR) and Interferometric SAR (InSAR)

Image Processing and filters: Transformation, representation, and encoding Smoothing and sharpening Restoration, blur removal, and Wiener filter

Data Processing and Analysis: High-pass, low-pass, and band-pass filters Cross correlation – transfer functions – coherence Signal and noise estimation – encoding time series

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Figure 1.1 Cartesian coordinate system used throughout the book with z positive up. The z = 0 plane is the surface of the earth. Fourier transforms deal with infinite domains while the fourier series (section 1.6) has finite domains. For our numerical examples we will select an area of length L and with W consisting of uniform cells of size Δx and Δy . This can be represented as a 2-D array of numbers with J = L / Δx columns and I = W / Δy rows.
1.1 Definitions of fourier transforms
The 1-dimensional forward and inverse fourier transforms are defined as:



F (k) = ∫ f ( )x e−i2πkxdx or F (k) = ℑ%& f (x)'(

(1.1)

−∞



f (x) = ∫ F (k)ei2πkxdk or f (x) = ℑ−1%&F (k)'(

(1.2)

−∞

where x is distance and k is wavenumber where k = 1 / λ and λ is wavelength. We also use the
shorthand notation introduced by Bracewell [1978]. These equations are more commonly
written in terms of time t and frequency ν where ν = 1 / T and T is the period. The 2-
dimensional forward and inverse fourier transforms are defined as:

∞∞
F (k) = ∫ ∫ f ( )x e−i2πk•xd2x or F (k) = ℑ2 %& f (x)'(
−∞ −∞

(1.3)

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∞∞
f (x) = ∫ ∫ F (k)ei2πk•xd2k or f (x) = ℑ−21%&F (k)'(
−∞ −∞

(1.4)

where x = (x, y) is the position vector, k = (kx, ky ) is the wavenumber vector, and
k • x = kx x + kyy . The magnitude of the spatial wavenumber β = (kx2 + ky2 )1/2 is used often in later
chapters. For several of the derivations, we’ll also take the fourier transform in the z-direction (i.e. 3-D transform) using the following notation.


F (kz ) = ∫ f ( )z e−i2πkzzdz
−∞

f (z) = ∫ F (kz )ei2πkzzdkz
−∞

(1.5) (1.6)

Fourier transformation with respect to time is also sometimes used to form a 4-D transform.


F (ν ) = ∫ f ( )t e−i2πνtdt
−∞

f (t) = ∫ F (ν )ei2πνtdν
−∞

(1.7) (1.8)

While algebraic manipulation of equations in 4-D is sometimes challenging and error prone, we’ll use computers to help us in two ways. We’ll use the tools of computer algebra to solve the most challenging algebraic manipulations associated with the 3-D and 4-D problems in chapter 4. In addition we’ll provide a FORTRAN subroutine called fourt() which can perform numerical fourier transforms in N-dimensions although we’ll use only 1-, 2-, and 3-D numerical transforms in this book.

1.2 Fourier sine and cosine transforms Here we introduce the sine and cosine transforms to illustrate the transforms of odd and even functions. Also in later chapters we’ll use sine and cosine transforms to match asymmetric and symmetric boundary conditions for particular models.

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Any function f (x) can be decomposed into odd O(x) and even E (x) functions such that

f (x) = E(x)+O(x)

(1.9)

where E (x) = 1 "# f (x) + f (−x)$% and O(x) = 1 "# f (x) − f (−x)$%.

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2

function can be written as

Note the complex exponential

eiθ = cos(θ ) + isin(θ )

(1.10)

( ) ( ) Exercise 1.1 Use equation (1.10) to show that cos(θ ) = 1 eiθ + e−iθ and sin(θ ) = 1 eiθ − e−iθ .

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2i

Using this expression (1.10) we can write the forward 1-D transform as the sum of two parts





F (k) = ∫ f (x)cos(2πkx)dx − i ∫ f (x)sin(2πkx)dx

−∞

−∞

(1.11)

After inserting equation (1.9) into this expression and noting that the integral of an odd function times an even function is zero, we arrive at the expressions for the cosine and sine transforms.





F (k) = 2 ∫ E (x)cos(2πkx)dx − 2i ∫ O(x)sin(2πkx)dx

0

0

(1.12)

Throughout this book we’ll be dealing with real-valued functions. From equation (1.12) it is evident that the cosine transform of a real, even function is also real and even. Also the sine transform of a real odd function is imaginary and odd. In other words when a function in the
space domain is real valued, its fourier transform F (k) has a special Hermitian property F (k) = F (−k)* so one can reconstruct the transform of the function with negative wavenumbers
from the transform with positive wavenumbers. Later when we perform numerical examples

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using real valued functions such as topography, we can use this Hermitian property to reduce the memory allocation for the fourier transformed array by a factor of 2. This is important for large 2-D and 3-D transforms.

1.3 Examples of Fourier Transforms Throughout the book we will work with only linear partial differential equations so all the problems are separable and the order of differentiation and integration is irrelevant. For example the 2-D fourier transform of is given by

( ) ∫ ∫ ( ) ∫ ∫ ( ) F kx, ky

∞$∞ =&f
% −∞ −∞

−i2πk x ' −i2πk y

∞$∞

x, y e x dx)e y dy = & f

(

% −∞ −∞

x, y e−i2πkyy dy')e−i2πkxxdx (

(1.13)

Note that this 2-D transform consists of a sequence of 1-D transforms. This property can be extended to 3-D, 4-D and even N-D; each transform can be performed separately and independently of the transforms in the other dimensions. In the following analysis we’ll only show examples of 1-D transforms but the extension to higher dimensions is trivial.

Delta function – By definition the delta function has the following property;


∫ f (x)δ (x − a)dx ≡ f (a)
−∞

(1.14)

under integration it extracts the value of f (x) at the position x = a . One can describe the delta
function as having infinite height at zero argument and zero height elsewhere. The area under the delta function is 1. In terms of pure mathematics the delta function is not a function and only has meaning when integrated against another function. In this book we use the delta function as a powerful tool, provided to us by the mathematicians, so we trust all the mathematical theory behind it. What is the fourier transform of a delta function? By definition if one performs a forward transform of a function followed by an inverse transform, or vice versa, one will arrive back with the original function. Lets try this using the delta function. By definition, the inverse transform of a delta function is

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∫ δ (k − ko )ei2πkxdk = ei2πkox
−∞

(1.15)

Next lets take the forward transform of equation (1.15). The left hand side will be the delta function because we have performed an inverse and forward transform. The right hand side is given by

( ) ∫ ∫ ∞



δ k − ko = ei2πkoxe−i2πkxdx = e−i2π(k−ko)x dx

−∞

−∞

(1.16)

This result shows that the fourier basis functions are orthonormal. If we consider the special
case of ko = 0 we see that the fourier transform of a delta function is ℑ"#δ (k)$% = 1 . Since fourier transformation is reciprocal in distance x and wavenumber k, it is also true that ℑ"#δ (x)$% = 1 . The
delta function and its fourier transform provide an amazingly powerful tool for solving linear PDEs.

Cosine and sine functions – Lets use the delta function tool and the expressions from Exercise 1.1 to calculate the fourier transform of a cosine function having a single wavenumber
cos(2π k0x) .

∫ ∫ ( ) ( ) ( ) ( ) ∞ cos

2π k

x

e−i2πkxdx =

1



o

ei2πkox + e−i2πkox + e−i2πkxdx = 1 $%δ

k − ko



k + ko &'

−∞

2 −∞

2

(1.17)

So the fourier transform of a cosine function is simply two delta functions located at ±ko .

Exercise 1.2 Show that the Fourier transform of sin(2π k0x) is 1 "#δ (k − ko ) −δ (k + ko )$% .
2i

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Gaussian function – The Gaussian e−πx2 function also plays a fundamental role in solutions to several types of PDEs. Its fourier transform is

F(k) =




e−π x2 e−i2πkxdx

=




e− π ( x 2 +i 2 kx) dx

−∞

−∞

(1.18)

Note that (x + ik)2 = (x2 + i2kx) − k2 . Using this we can rewrite equation (1.18) as

F(k)

=

e−πk2




e−π(x+ik)2 dx

=

e−πk2




e−π(x+ik)2 d ( x

+ ik)

=

e−πk2

−∞

−∞

(1.19)

where we have used the result that the infinite integral of e−πx2 is 1. This is a remarkable and powerful result that the fourier transform of a Gaussian is simply a Gaussian.

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Figure 1.2 Schematic plots of 1-D fourier transform pairs. Solid line is real-valued function while dashed line is imaginary valued function (figure from Bracewell [1978]. 1.4 Properties of fourier transforms There are several properties of fourier transforms that can be used as tools for solving PDEs. The first property called the similarity property says that if you scale a function by a factor of a along the x-axis, its fourier transform will scaled by a−1 along the k-axis and the amplitude will be scaled by a −1 .

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Exercise 1.3 Use the definition of the fourier transform equation (1.1) and a change of variable to show the following. Try positive and negative values for a to understand why the absolute value is needed in the amplitude scaling.

ℑ"# f

(ax)$% =

1

&k) F( +

a 'a*

(1.20)

The shift property says that the fourier transform of a function that is shifted by a along the xaxis equals the original fourier transform scaled by a phase factor. This property is especially useful for numerically shifting a function a non-integer amount of the data spacing along the axis.

Exercise 1.4 Use the definition of the fourier transform and a change of variable to show the following

ℑ#$ f (x − a)%& = e−i2πkaF (k)

(1.21)

The derivative property of the fourier transform is the essential tool used in this book to transform linear PDEs into algebraic equations that are easily solved. It states that the fourier transform of the derivative of a function is the fourier transform of the original function scaled by the imaginary wavenumber.

ℑ#%∂f

& (

=

i2π

kF

(k

)

$∂x '

(1.22)

To show this is true we start with the inverse transform of equation (1.22)

∫ ∂f

=


i2π kF (k) ei2πkxdk

∂x −∞

(1.23)

Next take the forward transform of equation (1.23) and rearrange terms

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∫ ∫ ( ) ∫ ( ) ∫ ℑ#%∂f &( = ∞


i2π koF

ko


ei2πkoxdkoe−i2πkxdx = i2π koF

ko

,- ∞ e−i2π(k−ko)xdx/0 dko

$∂x ' −∞ −∞

−∞

.−∞

1

(1.24)

The term in the curly brackets is the delta function δ (k − ko ) given in equation (1.16). The result
is

∫ ℑ#%∂f

& (=



i2π koF (ko )δ (k

− ko ) dko

=

i2πkF (k)

$∂x ' −∞

(1.25)

The final property considered here is the convolution theorem which states that the fourier transform of the convolution of two functions is equal to the product of the fourier transforms of the original functions.

%∞

(

ℑ' ∫ f (u)g(x − u)du* = F (k)G(k)

&−∞

)

(1.26)

To show this is true one can perform the fourier integration on the left side of equation (1.26) and rearrange the order of the integrations

∫ ∫ ∫ ∫ ∫ % ∞

( ∞%∞

(



+∞

.

ℑ' f (u) g(x − u) du* = ' f (u) g(x − u) du*e−i2πkxdx = f (u), g(x − u)e−i2πkx dx/du (1.27)

&−∞

) & −∞ −∞

)

−∞

-−∞

0

Next use the shift property of the fourier transform to note that the function in the curly brackets
on the right side of equation (1.27) is e−i2πkuG (k) . The result becomes

∫ ∫ % ∞

(



ℑ' f (u) g(x − u) du* = G (k) f ( )u e−i2πkudu = F (k)G (k)

&−∞

)

−∞

(1.28)

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Fourier Transform Methods