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 RD and let f (x) (RD ! 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)T denotes the frequencies along the axes. The original signal, 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:
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, f (x), can be differentiated by computing the inverse Fourier transformation of —jrnF(m):
2. The Fourier transformation off (x) * g(x), where f (x) and g(x) are real functions, is given by the product of their Fourier transformations:
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 G(a> are the Fourier transformations of f (x) and g(x), respectively.
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 iff (x) = f (—x) and is called antisymmetric iff (x) = —— (—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/symm(x) is the symmetric real function and/anti(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 = (u1, u2,..., uD)T denote a D-vector where all components are integers and are bounded as 0 < us < W — 1 (s = 1,2,... ,D), and let a target real function defined over u be denoted by/(u), where/(u): ZD ! R. Then, the DFT of/(u) is defined as follows:
Here, n = (n1, n2,..., nD)T is a D-vector of which all components are integer, where ns = 0,1,..., W — 1, and the frequency along the s-th axis, a>s, is proportional to ns and is given as a>s = ns/W voxel-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 n = 0,1,..., W — 1. Let a W-vector, f, denote the input function, where f = [f(0) ,f(1) ,..., f(W — 1)]r, and let a W-vector, c, denote the discretized sinusoidal function, where c = [g-2^n0/W , e-2njni/W,..., g-2^n(W-ivW]T. Then F[f] = F(n) = f ? c = fTc.