One-Dimensional Fourier Transform

First, we will consider the one-dimensional Fourier transform.

 

⊆ →

∞

Defi nition 1 (1-D FT) ]If g(x): R    C is a square integrable function, that is,

2

∫ .g(x). dx < ∞,                                            (2.17)

− ∞ .   .

then the Fourier transform of g(x), gˆ (k) is given by

∞

gˆ (k) =

∫ g(x) exp (− 2π ikx) dx.                                  (2.18)

− ∞

The Fourier transform maps the vector space of square integrable func- tions onto itself.  The inverse Fourier transform of gˆ (k) results in the original function g(x):

 

g(x) =

∫   gˆ (k) exp (2π ikx) dk.                                  (2.19)

∞

− ∞

The Fourier transform can be written in a more compact form if the abbreviation

w = e^{2π i}                                                    (2.20)

is used and the integral is written as an inner product:

∞

.g(x) |h(x). =

∫ g∗ (x)h(x)dx,                               (2.21)

− ∞

where * denotes the conjugate complex. Then

gˆ (k) =.w^kx.g(x). .                                           (2.22)

2.3 Wave Number Space and Fourier Transform                               43

The function wt can be visualized as a vector that rotates anticlockwise on the unit circle in the complex plane. The variable t gives the number of revolutions.

Sometimes, it is also convenient to use an operator notation for the Fourier transform:

gˆ = Fg  and  g = F− 1gˆ.                                           (2.23)

◦ •

A function and its transform, a Fourier transform pair, is simply denoted by g(x)       gˆ (k).

For the discrete Fourier transform (DFT ), the wave number is now an integer number that indicates how many wavelengths fi t into the interval with N elements.

Defi nition 2 (1-D DFT) The DFT maps an ordered N-tuple of complex numbers g_n, the complex-valued vector g,

T

g = Σ g₀, g₁,..., g_{N− 1}Σ  ,                                           (2.24)

onto another vector g ˆ of a vector space with the same dimension N.

Σ

.

N− 1

gˆ _v

1

.

= √ N

g_n

n=0

exp

2π inv

− N

, 0 ≤ v < N.               (2.25)

The back transformation is given by

N− 1

 1

N

g_n = √ N

v.=0

gˆ _v

exp

. 2π inv Σ

, 0 ≤ n< N.                (2.26)

Again it is useful to use a convenient abbreviation for the kernel of the DFT; compare Eq. (2.20):

N

w_N = w^1/N = exp. 2π i Σ                               (2.27)

As the continuous Fourier transform, the DFT can be considered as the inner product of the vector g with a set of N orthonormal basis vectors

1                                                     T

b _v = √  Σ w⁰, w^v, w^2v,..., w^{(N− 1)v}Σ

.                 (2.28)

 

Then

N   N    N    N                  N

 

1 N− 1

n=0

N

v

gˆ _v = √ N . w^{− nv} g_n =. b _v. g .  = b   T g.                                (2.29)

Note the second compact notation of the scalar product between two vectors on the right-hand side of the equation that will be used in the following.

44                                                                                        2 Image Representation

 

a

b

0                                                                     0

 

1                                                                     1

 

2                                                                    2

 

3                                                                    3

 

4                                                                     4

 

5                                                                     5

 

6                                                                     6

 

7                                                                     7

 

8                                                                     8

 

=

Figure 2.13: The fi rst 9 basis functions of the DFT for N                   16; a real part (cosine function), b imaginary part (sine function).

 

Equation (2.29) means that the coeffi cient gˆ _v in the Fourier space is obtained by projecting the vector g onto the basis vector b _v. The N basis vectors b _v are orthogonal to each other:

v   v           v− v

b T b ' = δ      ' =

1 v = v'                                   (2.30)

0

.

otherwise.

 

Consequently, the set b _v forms an orthonormal basis for the vector space, which means that each vector of the vector space can be expressed as a linear combination of the basis vectors of the Fourier space. The DFT calculates the projections of the vector g onto all the basis vectors directly, i. e., the components of g in the direction of the basis vectors. In this sense, the DFT is just a special type of coordinate transformation in an M-dimensional vector space. Mathematically, the DFT diff ers from more familiar coordinate transformations such as rotation in a three- dimensional vector space (Section 7.2.2) only because the vector space is over the fi eld of the complex instead of real numbers and has many more dimensions.

The real and imaginary parts of the basis vectors are sampled sine and cosine functions of diff erent wavelengths (Fig. 2.13). The index v denotes how often the wavelength of the function fi ts into the interval

2.3 Wave Number Space and Fourier Transform                               45

Table 2.1: Comparison of the continuous Fourier transform (FT), the Fourier series (FS), the infi nite discrete Fourier transform (IDFT), and the discrete Fourier transform (DFT) in one dimension ( w = e2π i).

Type                         Forward transform                  Backward transform

∞                                                                           ∞

FT: R ⊆ → R

gˆ (k) =  ∫   g(x)w− kxdx                g(x) =  ∫   gˆ (k)wkxdk

∞

− ∞ ∆ x                                                                                                                                              − ∞

FS: [0, ∆ x] ⊆ → Z

gˆ _v

1

= ∆ x

∫ g(x)w− vx/∆ xdx g(x) =.

gˆ v wvx/∆ x

v

0                                                                        =− ∞

∞

1/∆ x

IDFT:  Z ⊆ →  [0,  1/∆ x] gˆ (k) =  .

gnw− nk∆ x                      gn

= ∆ x∫

gˆ (k)wnk∆ xdk

n=− ∞                                                                       0

.    − √ ˆ g  =        g  wv nN

.√ ˆ g  = g  wn vN

1  N− 1                                                               1 N− 1

DFT: Z_N ⊆ → Z_N

vn N

n=0

vn N

v=0

 

 

[0, M]. The basis vector b ₀ is a constant real vector. The projection onto this vector results in the mean of the elements of the vector g.

Besides the continuous and discrete Fourier transforms there are two other forms you may be familiar with: the Fourier series (FS) that maps a function in a fi nite interval [0, ∆ x] to an infi nite series of coeffi cients and the infi nite discrete Fourier transform (IDFT ) that maps an infi nite series of complex numbers to a fi nite interval [0, 1/∆ x] in the Fourier space. Therefore it is illustrative to compare the DFT with these transforms (Table 2.1).

 

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