Fourier Analysis

Euler’s formula

• Euler’s formula is a formula that shows that the Taylor series of a complex exponential function can be expressed as a trigonometric function.

  \[\begin{aligned} \exp{i\theta} &=\sum_{n=0}^{\infty}{\frac{(i\theta)^{n}}{n!}}\\ &=\sum_{k=0}^{\infty}{\frac{(i\theta)^{2k}}{(2k)!}}+\sum_{k=0}^{\infty}{\frac{(i\theta)^{2k+1}}{(2k+1)!}}\\ &=\sum_{k=0}^{\infty}{\frac{(-1)^{k}\theta^{2k}}{(2k)!}}+\sum_{k=0}^{\infty}{\frac{\left(i(-1)^{k}\theta\right)^{2k+1}}{(2k+1)!}}\\ &=\cos{\theta}+i\sin{\theta} \end{aligned}\]

• complex plane is a coordinate plane that expresses complex numbers \(Z=x+i \cdot y\) as vectors on the vector space \(\mathbb{R}^{2}:=\mathrm{span}\left\{\mathbf{e}_{X},\mathbf{e}_{Y}\right\}\), which consists of the real axis basis \(\mathbf{e}_{X}=\begin{bmatrix}1\\0\end{bmatrix}\) and the imaginary axis basis \(\mathbf{e}_{Y}=\begin{bmatrix}0\\1\end{bmatrix}\).

  \[\mathbf{v}_{Z}:=\mathbf{e}_{X}x+\mathbf{e}_{Y}y \in \mathbb{R}^{2}\]

• rotation transformation is a linear transformation that rotates a vector on a coordinate plane counterclockwise by $\theta$ around the $X$ axis.

  \[R:=\begin{bmatrix} \cos{\theta}&-\sin{\theta}\\ \sin{\theta}&\cos{\theta} \end{bmatrix}\]

• Therefore, Euler’s formula suggests that the complex exponential function $\exp{i\theta}$ rotates the complex number $Z$ by $\theta$ in the complex plane counterclockwise around the real axis.

  \[\begin{aligned} R\left(\mathbf{e}_{X};\theta\right) &=1\cdot\begin{bmatrix}\cos{\theta}\\\sin{\theta}\end{bmatrix} + 0\cdot\begin{bmatrix}-\sin{\theta}\\\cos{\theta}\end{bmatrix}\\ &\leftrightarrow \cos{\theta}+i\sin{\theta} \end{aligned}\]

Fourier series

• Fourier series is a technique for mapping a periodic signal $X_{T}(t)$, whose pattern repeats with a period of $T$ defined in the time domain, to a frequency domain composed of discrete sinusoid $\phi_{k}(t)$ by approximating it as a linear combination of discrete sinusoid.

  \[X_{T}(t)\approx\sum_{k=-\infty}^{\infty}{a_{k}\phi_{k}(t)}\]

• sinusoid is a unit vector that rotates at a unique speed in the complex plane, and corresponds to the basis vector of a vector space.

  \[\begin{aligned} \phi_{k}(t) :=\exp{i\omega_{k}t} =\cos{\omega_{k}t}+i\sin{\omega_{k}t} \end{aligned}\]
• frequency is the unique rotational speed of a sinusoid, which corresponds to the direction of the basis vector in vector space.
  • $\nu_{0}=1/T$(Hz): The number of vibrations per second as the fundamental frequency
  • $\omega_{0}=\nu_{0}\cdot 2\pi$(rad/s): Angular velocity in revolutions per second as the fundamental angular frequency
  • $\nu_{k}=k\cdot\nu_{0},\omega_{k}=k\cdot\omega_{0}$: Integer multiple of the fundamental frequency as the harmonic

• Sinusoids that share a period $T$ are orthogonal to each other.

  \[\begin{aligned} \langle\phi_{m}(t),\phi_{n}(t)\rangle &=\begin{cases} T\quad &(m = n)\\ 0\quad &(m \ne n) \end{cases} \end{aligned}\]

• Fourier coefficients represent the average amplitude of a sinusoids per unit time, i.e., the frequency components. In vector space, they correspond to the magnitude (coordinate value) of the result of an evaluation operation in the basis direction.

  \[\begin{aligned} \delta_{n}\left[X_{T}(t)\right] &=\langle X_{T}(t),\phi_{n}(t)\rangle\\ &\approx \left\langle \sum_{k=-\infty}^{\infty}{a_{k}\phi_{k}(t)},\phi_{n}(t) \right\rangle\\ &=\sum_{k=-\infty}^{\infty}{a_{k}\left\langle\phi_{k}(t),\phi_{n}(t)\right\rangle}\\ &=a_{n}T + \sum_{k \ne n}{a_{k} \cdot 0}\\ &=a_{n}T \end{aligned}\]

Fourier transform

• Fourier series maps a periodic signal into frequency components in a countable infinite-dimensional space consisting of discrete basis.

  \[\begin{aligned} X_{T}(t) \mapsto \left\{a_{k}\mid k\in\mathbb{Z}\right\} \in \mathrm{span}\left\{\phi_{k}(t)=\exp{ik\omega_{0}t}\mid k\in\mathbb{Z}\right\} \end{aligned}\]

• Non-periodic signal can be viewed as a periodic signal with $T\to\infty$, which results in the generalization of the discrete harmonic ${k\omega_{0}\mid k\in\mathbb{Z}}$ into the continuous harmonic $\omega \in \mathbb{R}$.

  \[T \to \infty \Leftrightarrow \Delta\omega\to 0 \quad (\because\omega_{0}=2\pi/T)\]

• $a_{k}T$, total amplitude per period of a discrete sinusoids, generalizes to the $\mathcal{X}(\omega)$, amplitude of a continuous sinusoids, as the period diverges to infinity ($T\to\infty$) and the fundamental converges to $0$ ($\Delta\omega\to 0$).

  \[\begin{aligned} \mathcal{X}(\omega) &= \lim_{\Delta\omega\to 0}{a_{k}T}\\ &= \lim_{\Delta\omega\to 0}{\left\langle X_{T}(t),\phi_{k}(t)\right\rangle}\\ &= \left\langle \lim_{\Delta\omega\to 0}{X_{T}(t)},\lim_{\Delta\omega\to 0}{\phi_{k}(t)}\right\rangle\\ &= \left\langle X(t),\phi(t)\right\rangle \end{aligned}\]

• In short, the Fourier transform is a generalization of the Fourier series, which maps a non-periodic signal into a spectrum on a non-countable infinite-dimensional space consisting of a continuous basis.

  \[\begin{aligned} X(t) \mapsto \mathcal{X}(\omega) \in \mathrm{span}\left\{\phi(t;\omega)=\exp{i \omega t}\mid\omega\in\mathbb{R}\right\} \end{aligned}\]

Inverse Fourier transform

• Inverse Fourier transform is an operation that reduces the spectrum $\mathcal{X}(\omega)$ to an aperiodic signal $X(t)$, and is performed as an inner product with the dual basis.

  \[\langle\cdot,\phi^{*}(t)\rangle_{\omega}: \mathcal{X}(\omega) \mapsto X(t)\]

• The dual basis of an orthonormal basis is simplified to a basis whose inner product with the original basis is $1$.

  \[\langle\phi_{k}(t),\phi_{k}^{*}(t)\rangle=1\]

• $m=n$:

  \[\begin{aligned} \langle\phi_{m}(t),\phi_{n}(t)\rangle &=\int_{t \in T}{\exp{i(m-n)\omega_{0}t}\mathrm{d}t}\\ &=\int_{t \in T}{1\mathrm{d}t}\\ &=T \end{aligned}\]

• $m \ne n$:

  \[\begin{aligned} \langle\phi_{m}(t),\phi_{n}(t)\rangle &=\int_{t \in T}{\exp{i(m-n)\omega_{0}t}\mathrm{d}t}\\ &=\frac{1}{i(m-n)\omega_{0}}\left(\exp{i(m-n)\omega_{0}T}-1\right)\\ &=0\\ \because\exp{i(m-n)\omega_{0}T} &=\underbrace{\cos{(m-n)2\pi}}_{=1}+i\underbrace{\sin{(m-n)2\pi}}_{=0} \end{aligned}\]

• Therefore, the dual basis of a discrete basis is defined as follows:

  \[\phi_{k}^{*}(t)=\frac{1}{T}\exp{-ik\omega_{0}t}\]

• When extending the dual basis of a discrete basis to a continuous domain, the normalization constant is replaced by $1/T \to 1/2\pi$.

  \[\begin{aligned} \frac{1}{T}\sum_{k}{(\cdot)} =\frac{1}{2\pi}\sum_{k}{(\cdot)\Delta\omega} \xrightarrow{\Delta\omega\to 0} \frac{1}{2\pi}\int{(\cdot)\mathrm{d}\omega} \end{aligned}\]

• Therefore, the dual basis of a continuous basis is defined as follows:

  \[\phi^{*}(t)=\frac{1}{2\pi}\exp{-i\omega t}\]