# Fourier Series Expansion

The Fourier series expansion is used for analyzing the frequencies of given signals. Let *x* denote a D-vector where *x **2* R^{D} and let **f ***(x)* (R^{D} ! R) denote an absolutely integrable real function that satisfies

then, the Fourier transform of the function, * F*[f], is defined as

where ш *= (ш**,a***> _{D}**

*)*

*denotes the frequencies along the axes. The original signal,*

^{T}

**f***(x)*

*can be recovered from*

**,**

**F***(*

**!***)*by the inverse Fourier transformation that is defined as

For simplicity, let us assume here that **D ***=* 1. Then, the Fourier transformation of **f***(***x***)* is denoted by **F ***[***f***] = *F(h), and the transformation is linear because it satisfies (2.2) and (2.3). The Fourier transformation has the following properties.

1. The Fourier transformation of the derivative of a function, * f (x),* is given as follows:

More generally,

As shown in Eq. (2.13), by differentiating a function, the Fourier coefficient, * F* [f], is multiplied by j!, and the components of higher frequencies are more enhanced. A function,

*can be differentiated by computing the inverse Fourier transformation of*

**f (x),**

**—jrnF(m):**

2. The Fourier transformation of* f (x) * g(x),* where

*and g(x) are real functions, is given by the product of their Fourier transformations:*

**f (x)**

Analogously, the Fourier transformation of the multiplication of two functions, * f* (x)g(x), is given by the convolution between the corresponding two Fourier transformations:

where * F(!)* and

*are the Fourier transformations of*

**G(a>***and g(x), respectively.*

**f (x)**3. The Fourier transformation of a shifted function is given by rotating the phase of the Fourier transformation of the original function:

4. The real part of the Fourier transformation corresponds to the symmetric component of an input function, and the imaginary one corresponds to the antisymmetric component of the function. A real function is called symmetric if* f (x) = f (—x)* and is called antisymmetric if

*—— (—x).*

**f (x) =**Let the real part and the imaginary part of the Fourier transformation, F(«), be denoted by Re[F(«)] and Im[F(«)], respectively, where

It should be noted that cos(«x) is symmetric and sin(«x) is antisymmetric and that the inverse Fourier transformations of Re[F(«)] and of Im[F(«)] generate a symmetric real function and an antisymmetric real one, respectively. Let

Then/_{sym}m(x) is the symmetric real function and/_{an}t_{i}(x) is the antisymmetric real function. Following Eq. (2.19) allows a unique decomposition of a target real function, */*(*x*), into its symmetric and antisymmetric components as follows:

The discrete Fourier transformation (DFT) is used when a given target function is discrete and its domain is finite. Let *u = (u _{1},* u

_{2},...,

*u*denote a D-vector where all components are integers and are bounded as 0 <

_{D})^{T}*u*1

_{s}< W —*(s =*1,2,... ,D), and let a target real function defined over

*u*be denoted by

*/*(u), where

*/*(u): Z

^{D}! R. Then, the DFT of

*/*(u) is defined as follows:

Here, *n =* (n_{1}, n_{2},..., *n _{D})^{T}* is a D-vector of which all components are integer, where

*n*0,1,...,

_{s}=*W*— 1, and the frequency along the s-th axis,

*a>*is proportional to

_{s},*n*and is given as

_{s}*a>*voxel

_{s}= n_{s}/W^{-1}. The inverse DFT (IDFT) can reconstruct the input signal from

*F*(

*n*) as follows:

As shown in Eq. (2.22), the DFT is an inner product between a given discrete function and a discretized complex sinusoidal function. For simplicity, assume that

* D =* 1. Then the DFT is written as

where * u =* 0,1,..., W — 1 and

*0,1,..., W — 1. Let a*

**n =***-vector,*

**W***f*, denote the input function, where

*f*

*(0) ,*

**= [f***(1) ,...,*

**f***(W — 1)]*

**f**^{r}, and let a

*-vector, c, denote the discretized sinusoidal function, where*

**W***c*

*[g-2^*

**=**^{n0}/

^{W},

_{e}*-2*

**n***jn*

**i/***,..., g-2^*

^{W}^{n(W}-iv

^{W}

**]***T*. Then

*[f] =*

**F**

**F(n) =***f*

**?***c*

**=***f*

^{T}c.