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date: 2024.12.25

Fourier Transforms

Sturm-Liouville Theory

Our matters of interest are:

(1) The conditions under which an ODE can be written as an eigenvalue equation with a self-adjoint (hermitian) operator.
(2) Methods for solving ODEs subject to boundary conditions.
(3) Properties of the solutions in (2).

The general eigenproblem characteristics of second-order differential equations are referred to as Sturm-Liouville theory, where the Hermitian operator $\mathcal{L}$ is expressed as:

\[\begin{aligned} \mathcal{L}(x) &= \frac{d}{dx}\left(p(x)\frac{d}{dx} \right)+(\lambda w(x)-q(x)) \\ \mathcal{L}y &= \lambda y \end{aligned}\]

Here, $p(x), q(x), w(x)$ are given functions, and $p(x) > 0$, $w(x) > 0$. $\lambda$ is a real discrete eigenvalue, and $y(x)$ is the eigenfunction, which must satisfy specific boundary conditions on the domain $[a,b]$:

Dirichlet boundary conditions: $y(a)$ and $y(b)$ are fixed to given values.
Neumann boundary conditions: The boundary terms vanish at the boundaries (e.g., $y’(a) + \alpha y(a) = 0$, $y’(b) + \beta y(b) = 0$).

Eigenfunctions in a Sturm-Liouville problem can be normalized by an appropriate constant, and eigenfunctions corresponding to different eigenvalues, $y_n(x)$ and $y_m(x)$, are orthogonal with respect to the weight function $w(x)$: $\int_a^b y_n(x)y_m(x)w(x)dx=0 \quad (\lambda_n \neq \lambda_m)$.

Model of a Vibrating String

The motion of a vibrating string is described by the wave equation:

\[\frac{\partial^2 u(x,t)}{\partial t^2}=c^2\frac{\partial^2 u(x,t)}{\partial x^2}, \quad c=\sqrt{\frac{T}{\mu}}\]

Using separation of variables, $u(x,t) = X(x)T(t)$, the equation splits into two ODEs:

$\frac{d^2T(t)}{dt^2} + \omega^2T(t) = 0$ (temporal equation)
$\frac{d^2X(x)}{dx^2} + \lambda X(x) = 0, \quad \lambda = \frac{\omega^2}{c^2}$ (spatial equation, eigenvalue problem)

If the string is fixed at both ends, the boundary conditions are $u(0,t) = u(L,t) = 0$. Interpreted as a Sturm-Liouville problem, $X(0) = X(L) = 0$. This determines the normal modes and eigenfrequencies of the string. Solving the spatial equation yields eigenvalues $\lambda_n = \left(\frac{n\pi}{L}\right)^2, \quad n=1,2,3,\dots$, with corresponding eigenfrequencies $\omega_n = c\sqrt{\lambda_n} = \frac{n\pi c}{L}$. The eigenfunctions (standing waves) are $X_n(x) = \sin{\left(\frac{n\pi x}{L} \right)}$. Eigenfunctions corresponding to different modes are orthogonal: $\int_0^L X_m(x)X_n(x)dx = 0 \quad (m \neq n)$.

From a physical perspective, the string forms standing wave patterns at specific eigenfrequencies, which depend on the string length $L$, tension $T$, and mass density $\mu$. At specific nodes, the displacement is always zero.

Fourier Series

The Fourier series is defined as a representation of a function or signal as a series of sines and cosines:

\[\begin{aligned} f(x) &= \frac{a_0}{2} + \sum_{n=1}^\infty \left[a_n\cos{nx} + b_n\sin{nx}\right] \\ a_n &= \frac{1}{\pi} \int_0^{2\pi} f(s)\cos{ns}ds, \quad n=0,1,2,\dots \\ b_n &= \frac{1}{\pi} \int_0^{2\pi} f(s)\sin{ns}ds, \quad n=0,1,2,\dots \end{aligned}\]

From the form of the Fourier series, the imposed conditions are that $f(x)$ must have only a finite number of finite discontinuities and only a finite number of extreme values (maxima and minima) in $[0, 2\pi]$ (Dirichlet conditions). Such functions are called piecewise regular.

We can rewrite the equation as:

\[\begin{aligned} f(x) &= \sum_{n=1}^\infty c_ne^{inx}, \\ c_n &= \frac{1}{2}(a_n-ib_n), \quad c_{-n}=\frac{1}{2}(a_n+ib_n), \quad n>0, \quad c_0 = \frac{1}{2}a_0 \end{aligned}\]

Typically, $f(x)$ is a periodic continuous function, decomposed into frequency components via the Fourier series.

The Sturm-Liouville theory forms an orthogonal basis in function space. Thus, any function $f(x)$ can be represented similarly to a Fourier series using the eigenfunctions $y_n(x)$ of a Sturm-Liouville problem:

\[f(x)=\sum_{n=1}^\infty c_ny_n(x), \quad \text{where} \quad c_n=\frac{\int_a^b f(x)y_n(x)w(x)dx}{\int_a^b y_n(x)^2w(x)dx}\]

For instance, in the vibration of a string, the solution $u(x,t)$ is expressed as a linear combination of eigenfunctions $X_n(x)$:

\[u(x,t)=\sum_{n=1}^\infty \left[ A_n\cos{\omega_nt}+B_n\sin{\omega_nt}\right]X_n(x)\]

Here, $A_n$ and $B_n$ are Fourier coefficients.

(Continuous) Fourier Transform

The eigenvalue problems described above generate discrete eigenvalues under finite boundary conditions, which can be represented by Fourier series. However, when a string is infinitely long or boundary conditions are absent, the solution to the wave equation has a continuous frequency spectrum. In such cases, the Fourier transform is used.
The FT is an analysis process, decomposing a complex-valued function $f(x)$ defined in time/space domain into its constituent frequencies function defined in k-space codomain. It is a linear map of a continuous input function ($\mathbb{R} \rightarrow \mathbb{R}$ or $\mathbb{R} \rightarrow \mathbb{C}$) to continuous output function ($\mathbb{R} \rightarrow \mathbb{C}$). The inverse process is synthesis. With an analogy, the Fourier series actually is the synthesis formula which maps a continuous domain (real numbers) to discrete frequency components (integer indices).

\[\begin{aligned} c_n = \frac{1}{P}\int_{-P/2}^{P/2}f(x)e^{-i2\pi \frac{n}{P}x}dx \quad &\text{: analysis formula} \\ f(x) = \sum_{n=-\infty}^{\infty} c_ne^{i2\pi \frac{n}{P}x}, \quad x \in [-P/2,P/2] \quad &\text{: synthesis formula} \end{aligned}\]

$P \rightarrow \infty$, the FT is typically defined as:

\[\hat{f}(\omega) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty f(t)e^{i\omega t}dt\]

For example, the wave equation describing the vibration of an infinitely long string is:

\[u(x,t) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty \hat{u}(k,0)e^{i(kx-\omega t)}dk\]

Here, $\hat{u}(k,0)$ is the Fourier transform of the initial condition $u(x,0)$, and $k = \frac{2\pi}{\lambda}$ is the wavenumber, satisfying the relation $\omega = c|k|$. The Fourier transform converts oscillations occurring in space and time into the frequency domain, allowing analysis of each component.

In the large-$n$ limit, the delta function $\delta_n(t) = \frac{1}{2\pi} \int_{-n}^n e^{i\omega t}d\omega$ is especially useful in Fourier analysis and helps derive results for Fourier integrals:

\[f(x) = \frac{1}{2\pi} \int_{-\infty}^\infty e^{-i\omega x}d\omega \int_{-\infty}^\infty f(t)e^{i\omega t}dt\]

Inverse Fourier Transform

The inverse Fourier transform is given by:

\[\begin{aligned} \hat{f}(\omega) &= \frac{1}{2\pi} \int_{-\infty}^\infty f(t)e^{i\omega t}dt, \\ f(t) &= \frac{1}{2\pi} \int_{-\infty}^\infty \hat{f}(\omega)e^{-i\omega t}d\omega \end{aligned}\]

Transforms in 3D Space

In three-dimensional space, the Fourier transform and its inverse are expressed as:

\[\begin{aligned} \hat{f}(\mathbf{k}) &= \frac{1}{(2\pi)^{3/2}} \int f(\mathbf{r})e^{i\mathbf{k \cdot r}}d^3r, \\ f(\mathbf{r}) &= \frac{1}{(2\pi)^{3/2}} \int \hat{f}(\mathbf{k})e^{-i\mathbf{k \cdot r}}d^3k \end{aligned}\]

Properties of Fourier Transform

Some key properties of the Fourier transform are:

\[\begin{aligned} \left[f(\mathbf{r-R})\right]^T(\mathbf{k}) = e^{i\mathbf{k \cdot R}}\hat{f}(\mathbf{k}) \quad &\text{: translation}, \\ \left[f(\alpha \mathbf{r})\right]^T(\mathbf{k}) = \frac{1}{\alpha^3}\hat{f}(\alpha^{-1}\mathbf{k}) \quad &\text{: change of scale}, \\ \left[f(-\mathbf{r})\right]^T(\mathbf{k}) = \hat{f}(-\mathbf{k}) \quad &\text{: sign change}, \\ \left[f^*(-\mathbf{r})\right]^T(\mathbf{k}) = \hat{f}^*(\mathbf{k}) \quad &\text{: complex conjugation}, \\ \left[\nabla f(\mathbf{r})\right]^T(\mathbf{k}) = -i\mathbf{k}\hat{f}(\mathbf{k}) \quad &\text{: gradient}, \\ \left[\nabla^2 f(\mathbf{r})\right]^T(\mathbf{k}) = -k^2\hat{f}(\mathbf{k}) \quad &\text{: Laplacian} \end{aligned}\]

FT may be defined in some cases for non-integrable functions. $f$ need not be integrable. For example, the FT of the rectangular function is the sinc function, which is not Lebesgue integrable.

It is not generally possible to write the IFT as a Lebesque integral, but when both $f$ and $\hat{f}$ are integrable, the inverse equality holds for almost every $x$, so the FT is injective on $L^1(\mathbf{R})$.

Fourier Convolution Theorem

The operation of combining two functions $f(x)$ and $g(x)$ to produce a new function is called convolution, defined as:

\[\begin{aligned} h(x)=(f*g)(x) &= \frac{1}{2\pi} \int_{-\infty}^\infty g(y)f(x-y)dy, \\ (f*g)(\mathbf{r}) &= \frac{1}{(2\pi)^{3/2}} \int_{-\infty}^\infty g(\mathbf{r'})f(\mathbf{r-r'})d^3r' \quad \text{(in 3D)} \end{aligned}\]

Cross-Correlation Theorem

\[h(x)=(f \star g)(x)=\int_{-\infty}^\infty \overline{f(y)}g(x+y)dy\]

Parseval Relation (Rayleigh’s Energy Theorem)

Let $f(x)$ and $g(x)$ be integrable, and let $\hat{f}(\omega)$ and $\hat{g}(\omega)$ be their FT. If $f, g$ are square-integrable, then

\[<f,g>_{L^2}=\int_{-\infty}^\infty f(x)\overline{g(x)}dx=\int_{-\infty}^\infty \hat{f}(\omega)\overline{\hat{g}(\omega)}d\omega\]

Plancherel Theorem

\[||f||^2_{L^2}=\int_{-\infty}^\infty |f(x)|^2dx=\int_{-\infty}^\infty |\hat{f}(\omega)|^2d\omega\]

Convolutional Neural Network

Signal-Processing Applications

\[\begin{aligned} f(t) &=\frac{1}{2\pi}\int_{-\infty}^\infty F(\omega)e^{i\omega t}d\omega \\ F(\omega) &= \int_{-\infty}^\infty f(t)e^{-i\omega t}dt \end{aligned}\]

Here $f(t)$ is a resolution of the pulse into its angular frequency components and $F(\omega)$ a synthesis of the pulse from its components.
Now assuming a linear response (but for your information, there are other requirements than linearity that must be satisfied by the response function on physical grounds: reality and causality), let us consider a servomechanism for an input of $f_\omega(t)=F(\omega)e^{i\omega t}$ and output of $g_\omega(t)=\phi(\omega)f_\omega(t)$, where $\phi_\omega(t)$ is called a transfer function.

Kramers-Kronig Relation

Discrete Fourier Transform

In real world, we use digital computers with numerically determined FTs, which consist of values given at a discrete set of points. Integrations are therefore converted into finite summations, and the number of the points can be finite or infinite. The Fourier series deals with periodic continuous functions and consists of infinite frequency components. The Discrete Fourier Transform (DFT), on the other hand, transforms finite discrete sampled data into its finite discrete frequency components. For a data length of $N$, only $N$ frequency components are calculated, making it widely used in digital signal processing.

Laplace Transform

The Laplace Transformation is a linear map of a continuous input function ($[0,\infty) \rightarrow \mathbb{R}$ or $[0,\infty) \rightarrow \mathbb{C}$) to a continuous output function ($\mathbb{C} \rightarrow \mathbb{C}$).

It is defined as:

\[F(s) = \mathcal{L}[f(t)]= \int_0^\infty f(t)e^{-st}dt \quad \text{where} \quad s = \sigma + i\omega\]

The Fourier transform is a special case of the Laplace transform with $\sigma = 0$. The LT is a mathematical technique used for solving differential equations which transform them into algebraic equations.