Fourier Transformation

Idea

Towards Frequency Domain

The Fourier Transformation is a mathematical technique that transforms a signal from its original domain (often time or space) into a representation in the frequency domain. It allows us to break down complex signals into simpler components—specifically, the different frequencies that make up the signal. In essence, the Fourier Transformation provides a different perspective on the same information, highlighting the underlying frequency components and their amplitudes.

High Level Idea

Fourier transformation can be understood as a transformation of coordinates in a Hilbert space (A infinite dimensional vector space equipped with an inner product). The general idea is that, every function can be seen as a vector in a infinite dimensional space, and the transformation between functions could be the different representations of a function vector in different coordinates.

Hilbert Space

A Hilbert space is a complete vector space equipped with an inner product. It generalizes the notion of Euclidean space to infinite dimensions and provides the framework for discussing concepts like orthogonality, projection, and more in a rigorous manner. Functions can be elements of a Hilbert space, with operations like addition and scalar multiplication defined pointwise.

Function Space

In the context of Fourier transforms, we typically consider the Hilbert space $L^2(\mathbb{R})$, the space of square-integrable functions. This means functions $f$ such that:

$${\int }_{-\infty }^{\infty }|f(x){|}^{2}dx<\infty$$

Coordinate Transformation

The Fourier transform can be viewed as changing the "basis" in this function space from the time (or spatial) domain to the frequency domain.

In $L^2(\mathbb{R})$, functions are often represented in terms of the standard basis, which consists of functions like Dirac delta functions that are localized in space. To simplify this, you could view a function as

$$\mathbf{f}=\sum _{n=-\infty }^{\infty }f(n){\mathbf{e}}_n$$

where $f(n)$ is the value of $f$ at point $n$, and ${\mathbf{e}}_n$ is the corresponding standard basis.

Now, if we apply a coordinate transformation ${\mathbf{e}}_n\to {\mathbf{d}}_n$, to obtain the representation of $\mathbf{f}$ in this new space, we are supposed to calculate its projection

$$f(n)\cdot {\mathbf{d}}_n=⟨\mathbf{f},{\mathbf{d}}_n⟩{\hat{\mathbf{d}}}_n={\int }_{-\infty }^{\infty }f(t)d_n(t)\mathrm{d}t\cdot {\hat{\mathbf{d}}}_n$$

where $⟨\cdot ,\ast ⟩=\int \cdot \ast \mathrm{d}t$ is the inner product, and then we sum them up

$$\mathbf{f}=\sum _{n=-\infty }^{\infty }F(n){\hat{\mathbf{d}}}_n$$

where

$$F(n)=⟨\mathbf{f},{\mathbf{d}}_n⟩={\int }_{-\infty }^{\infty }f(t)d_n(t)\mathrm{d}t$$

In particular, the Fourier transform projects these functions onto an orthogonal basis of complex exponentials $e^{i\omega x}$. These exponentials are eigenfunctions of the Fourier transform operator and form a complete orthonormal set in $L^2(\mathbb{R})$.

Derive from Fourier Series

Fourier Series

See Fourier Series for details, we will directly give its form: The Fourier series expansion of the function $f$ is

$$f(x)\sim \frac{a_0}{2}+\sum _{n=1}^{\infty }\left(a_n\cos \frac{n\pi x}{l}+b_n\sin \frac{n\pi x}{l}\right)$$

where

$$\begin{array}{rl}{a}_n& =\frac{1}{l}{\int }_{-l}^{l}f(x)\cos \frac{n\pi x}{l}\mathrm{d}x,n=0,1,2,\cdots \\ b_n& =\frac{1}{l}{\int }_{-l}^{l}f(x)\sin \frac{n\pi x}{l}\mathrm{d}x,n=1,2,\cdots \end{array}$$

Complex Form of Fourier Series

Using Euler's formula, we can derive

$$\cos \theta =\frac{{\mathrm{e}}^{\mathrm{i}\theta }+{\mathrm{e}}^{-\mathrm{i}\theta }}{2},\sin \theta =-\mathrm{i}\left(\frac{{\mathrm{e}}^{\mathrm{i}\theta }-{\mathrm{e}}^{-\mathrm{i}\theta }}{2}\right)$$

Substituting them into the above series expression, assuming the series converges, let $\omega =\frac{2\pi }{T}=\frac{2\pi }{2l}=\frac{\pi }{l}$, we have

$$\begin{array}{rl}f(x)& =\frac{a_0}{2}+\sum _{n=1}^{\infty }\left(a_n\frac{{\mathrm{e}}^{\mathrm{i}n{\omega }_{0}x}+{\mathrm{e}}^{-\mathrm{i}n{\omega }_{0}x}}{2}-\mathrm{i}{b}_n\frac{{\mathrm{e}}^{\mathrm{i}n{\omega }_{0}x}-{\mathrm{e}}^{-\mathrm{i}n{\omega }_{0}x}}{2}\right)\\ & =\frac{a_0}{2}+\sum _{n=1}^{\infty }\frac{1}{2}(a_n-\mathrm{i}{b}_n){\mathrm{e}}^{\mathrm{i}n{\omega }_{0}x}+\sum _{n=1}^{\infty }\frac{1}{2}(a_n+\mathrm{i}{b}_n){\mathrm{e}}^{\mathrm{i}n{\omega }_{0}x}\\ & =\frac{a_0}{2}+\sum _{n=1}^{\infty }\frac{1}{2}(a_n-\mathrm{i}{b}_n){\mathrm{e}}^{\mathrm{i}n{\omega }_{0}x}+\sum _{n=-\infty }^{-1}\frac{1}{2}(a_{-n}+\mathrm{i}{b}_{-n}){\mathrm{e}}^{\mathrm{i}n{\omega }_{0}x}\\ & =\sum _{-\infty }^{\infty }{d}_n{\mathrm{e}}^{\mathrm{i}n{\omega }_{0}x}\end{array}$$

where

$$d_n=\{\begin{array}{ll}\frac{1}{2}(a_n-\mathrm{i}{b}_{-n}),& n>0\\ \frac{1}{2}{a}_0,& n=0\\ \frac{1}{2}(a_{-n}-\mathrm{i}{b}_{-n}),& n<0\end{array}$$

It can be verified that for any value of $n$, $d_n$ can be expressed as

$$d_n=\frac{1}{2l}{\int }_0^{2l}{\mathrm{e}}^{-in{\omega }_{0}x}f(x)\mathrm{d}x=\frac{1}{T}{\int }_0^T{\mathrm{e}}^{-in{\omega }_{0}x}f(x)\mathrm{d}x$$

where ${\omega }_0=\frac{2\pi }{T}$ is called the fundamental frequency, and thus we obtain the complete expression of the complex exponential form Fourier series expansion

$$f(x)=\sum _{n=-\infty }^{+\infty }\left[\frac{1}{T}{\int }_0^T{\mathrm{e}}^{-in{\omega }_{0}x}f(x)\mathrm{d}x\right]{\mathrm{e}}^{\mathrm{i}n{\omega }_{0}x}$$

Fourier Transform

Fourier series expansion is for periodic functions, but in reality, most signals are non-periodic. Non-periodic functions can be viewed as periodic functions with $T\to +\infty$, and when $T=+\infty$, the fundamental frequency ${\omega }_0$ becomes the differential $\mathrm{d}\omega$, so the summation needs to be converted to an integral

$$\begin{array}{rl}f(x)& =\sum _{n=-\infty }^{+\infty }\left[\frac{1}{T}{\int }_0^T{\mathrm{e}}^{-in{\omega }_{0}x}f(x)\mathrm{d}x\right]{\mathrm{e}}^{\mathrm{i}n{\omega }_{0}x}\\ & =\sum _{n=-\infty }^{+\infty }\left[\frac{{\omega }_0}{2\pi }{\int }_0^T{\mathrm{e}}^{-in{\omega }_{0}x}f(x)\mathrm{d}x\right]{\mathrm{e}}^{\mathrm{i}n{\omega }_{0}x}\\ & =\sum _{n=-\infty }^{+\infty }\left[\omega {\int }_0^T{\mathrm{e}}^{-in{\omega }_{0}x}f(x)\mathrm{d}x\right]{\mathrm{e}}^{\mathrm{i}n{\omega }_{0}x}\end{array}$$

Here we take $\omega =\frac{{\omega }_0}{2\pi }$, let $\omega =\mathrm{d}\omega \to 0$ then we have

$$f(x)=\frac{1}{2\pi }{\int }_{-\infty }^{+\infty }{\int }_{-\infty }^{+\infty }{e}^{-i2\pi \omega x}\cdot f(x)\mathrm{d}x\cdot e^{i2\pi \omega x}\mathrm{d}\omega$$

where

$$F(\omega )={\int }_{-\infty }^{+\infty }{e}^{-i2\pi \omega x}\cdot f(x)\mathrm{d}x$$

is called the Fourier transform, and

$$f(x)=\frac{1}{2\pi }{\int }_{-\infty }^{+\infty }F(\omega )\cdot e^{i2\pi \omega x}\mathrm{d}\omega$$

is called the inverse Fourier transform

Applications of Fourier Transform

Applications of Fourier Transform in Optics Fourier Optics