Math | Fourier Series & Fourier Transform

Basic knowledge

1.1 How to describe a wave

• [picture from Baidu]
• [formula] $x = Asin( ωt + φ ）$
  • A means amplitude.
  • ω represents angular frequency, which means phase angle radian (相角弧度值) changing per unit time. $\omega = 2 \pi f = {2\pi \above{1pt} T}$
  • φ means primary phase, while $ωt + φ$ means phase.

1.2 Standard orthonormal basis

• The standard orthogonal basis can represent any vector in the vector space.

1.3 Hilbert space

• Hilbert space extends Euclid space, and the base (基底) of Hilbert space is usually function. For example, arbitrary function in Hilbert space can decomposed into sine and cosine function thought Fourier Series, where sine and cosine are orthonormal basis.

1.4 Dirichlet’s convergence theorem

Let $f(x)$ takes $2l$ as a period, if $f(x)$ on $[-l, l]$ satisfies that:

• ①. $f(x)$ continuous or only finite first class discontinuities (第一類間斷點：可去、跳躍)
• ②. only finite extreme point (極值點)

we can say that the Fourier Series of $f(x)$ converge in $[-l, l]$.

1.5 Euler’s formula

$e^{ix} = cosx + i*sinx$

Fourier Series

1.1 Formula

$f(x) = {a_0 \above{1pt} 2} + \displaystyle\sum_{n=1}^\infin (a_n*cos({n\pi x \above{1pt} l}) + b_n * sin({n\pi x \above{1pt} l}))$
where
$a_0 = {1 \above{1pt} l} \int_{-l}^l f(x)dx$
$a_n = {1 \above{1pt} l} \int_{-l}^l f(x)*cos({n\pi x \above{1pt} l})dx$
$b_n = {1 \above{1pt} l} \int_{-l}^l f(x)*sin({n\pi x \above{1pt} l})dx$

1.2 Compact formula

$f(x) = \displaystyle\sum_{k=-\infin}^\infin c_k * e^{ikx}$
where $c_k$ represents the projection of $f(x)$ in $e^{ikx}$ direction can be written as $<f(x), e^{ikx}>$. Actually, $c_k$ can be considered as coefficient.

Fourier Transform

1.1 Formula

• forward (time domain to frequency domain)
  $F(\xi) = \int_{-\infin}^\infin f(t)*e^{-i\omega t} dt$
  where $\xi$ means frequency.
• inverse (frequency domain to time domain)
  $f(t) = \int_{-\infin}^\infin F(\xi)*e^{i\omega t} d\xi$

Diff between Fourier Series & Fourier Transform

• Fourier Series is about period function, while Fourier Transform extend $l$ to infinity.

Discrete Fourier Transform (DFT)

1.1 Tips

• DFT approximating that Fourier series approximation on a finite interval where your function is periodic.

1.2 Transform Formula

$\hat{f_k} = \sum_{j=0}^{n-1} f_j * e^{-i2\pi jk \above{1pt} n}$
${f_k} = {1 \above{1pt} n} \sum_{j=0}^{n-1} \hat{f_j} * e^{i2\pi jk \above{1pt} n}$

let $\omega = e^{-i2\pi \above{1pt} n}$

$\begin{bmatrix} \hat{f_0} \\ \hat{f_1} \\ \hat{f_2} \\ ...\\ \hat{f_n} \\ \end{bmatrix} = \begin{bmatrix} 1 & 1 & 1 & ... & 1\\ 1 & \omega & \omega^2 & ... & \omega^{n-1} \\ 1 & \omega^2 & \omega^4 & ... & \omega^{2(n-1)} \\ 1 & ... & ... & ... & ...\\ 1 & \omega^{n-1} & \omega^{2(n-1)} & ... & \omega^{(n-1)(n-1)}\\ \end{bmatrix} \begin{bmatrix} f_0 \\ f_1 \\ f_2 \\ ...\\ f_n \\ \end{bmatrix}$

Fast Fourier Transform (FFT)

• Efficient measure to compute the matrix of DFT.