[Fourier transform and its application study notes] Course overview

1. Fourier series

(1th Lesson)

Periodic phenomena are expressed mathematically by trigonometric functions.

(2nd Lesson)

Phenomena can become cyclical through periodicity.

There is a periodic function called the Fourier series, which is composed of trigonometric functions of different frequencies and can be further deduced into the form of complex exponents, in which the coefficients are called Fourier coefficients.

The Fourier coefficients can be deduced to be

$C _m = \displaystyle{\int_{0}^{1}}e^{-2\pi imt}f (t) dt$

So is it possible for all periodic functions to be represented as a combination of trigonometric functions of different frequencies? The class professor did not elaborate, here is the derivation process, the result is certainly correct, the condition is that there are infinitely many Fourier coefficients, that is, frequency coverage ($-\infty<k<\infty$).

(3rd lesson)

Convergence question, is the form of the re-composition of the Fourier coefficients exactly the same as the original function? The convergence problem is discussed here, and they are convergent on the $l^2$.

(4th Lesson)

In another direction, the Fourier series is regarded as the orthogonal basis of the $l^2$ space by the complex exponential $e^{2\pi ikt}$ of different frequencies, and the Fourier coefficients are equivalent to the projection of the original function on the orthogonal base.

(4th Lesson, 5th Lesson)

The application of Fourier series in thermal equation and the introduction of convolution

2. Fourier transform

(6th lesson)

From the Fourier series to the Fourier transform, from the periodic function to the non-periodic function, in fact, the cycle is considered infinitely large, but at this time the Fourier coefficient will become 0, so need to find another way (5th). The Fourier transform can be obtained by adjusting the formula slightly.

$\MATHCAL{F} F (s) = \displaystyle{\int_{-\infty}^{\infty} e^{-2\pi ist}f (t) DT}$

And the Fourier transform of the rectangle function and the triangular function is discussed.

(7th Lesson)

The Fourier transform of Gaussian function is discussed, and the duality of Fourier transform is discussed, and duality is very helpful for the calculation of the Fourier transform later.

(8th Lesson)

This paper discusses the time delay, scale transformation and convolution of Fourier transform, which can help to understand the essence of Fourier transform (temporal domain, frequency domain change), and also help to calculate the Fourier transform later.

(9th Lesson)

The convolution on the time domain is the product of the frequency domain, so the convolution is often used for filtering, and the Fourier derivative theorem is introduced by convolution.

$\mathcal{f} (f ') (s) = 2\pi is (\mathcal{f} F) (s) $

Then we discuss the heat equation again.

(10th Lesson)

Solving the central limit theorem by Fourier transform (Gaussian distribution, normal distribution)

3. Fourier transform of the distribution (generalized Fourier transform)

We refer to the above Fourier transform as the traditional Fourier transform, it has some limitations, such as convergence of the discussion, for example, some functions are difficult to calculate, for example, some functions can not be Fourier transform, so in order to solve such problems we introduced a new Fourier transform: Distributed Fourier transform.

(11th lesson)

First you need to know what the best function of the Fourier transform is, and through the understanding of the Fourier transform, we find the fast-descending functions (Schwartz function).

(12th lesson)

We take the $\varphi$ function as the test function in the distributed Fourier transform, which corresponds to the distribution (generalized function) $T $, has the following relationship

$<t,\varphi>$

This is known as matching paring, which is $t$ acting on $\varphi$.

The distribution of Fourier is introduced here

$<\mathcal{f}t,\varphi> = <t,\mathcal{f}\varphi>$

The effect of the Fourier transform of the distributed $t$ on the $\varphi$ of the test function is similar to that of the distribution $t$ to the Fourier transform of the test function $\varphi$, which means that we can get the distribution $t$ Fourier transform by the following equation.

The Pulse function $\delta$ is a typical distribution

(13th lesson)

Some examples of distributed Fourier transforms

(14th lesson)

Distribution Fourier transform derivative theorem, product, convolution; three properties of the $\delta$ function are introduced:

• Sampling characteristics
• Shift characteristics
• Scaling Features

These three features are often used in subsequent calculations

4. Application of Fourier transform

(15th lesson)

The relationship between Fourier transform and diffraction imaging

(16th Lesson, 17th lesson)

The ш function is introduced by the crystal Imaging, the ш function is extended by the $\delta$ function, with all the properties of the $\delta$ function, and the ш function Fourier transform is computed.

(17th, 18th, 19th lessons)

Sampling theorem

5. Discrete Fourier transform

(19th Lesson)

The discrete Fourier transform is the evolution of discrete sampling of continuous functions.

(20th Lesson)

Several important characteristics of discrete Fourier transform

• Input/Output periodicity
• The discrete complex exponent has orthogonality
• Index independence

(21st lesson)

Derivation of discrete Fourier transform DFT and discrete Fourier inverse transform idft matrices

The duality of DFT can be very helpful for computing.

(22nd Lesson)

Fast Fourier transform FFT is a fast algorithm for DFT

6. Linear system

(23rd Lesson)

• A conceptual understanding of linear systems, an understanding of its superposition principle and a positive proportional relationship
• Discrete finite-dimensional linear systems can be expressed by multiplication of matrices
• continuous infinite dimensional linear systems can be expressed with the integral of kernel function

(24th Lesson)

For linear system input pulse function, the result of linear system output is impulse response

Linear time-invariant systems are expressed by convolution

(25th Lesson)

Eigenvalue, eigenvector/feature function of linear time-invariant systems

7. High-dimensional Fourier transform

(26th Lesson)

Two-dimensional Fourier transform is equivalent to the one-dimensional variable $x$, into a two-dimensional variable $\underline{x} (x_1,x_2) $, we regard this two-dimensional variable as a vector.

The original multiplication $st$ also becomes the inner product $\underline{x}\cdot\underline{\xi}=x_1\xi_1+x_2\xi_2$, why this change, please refer to this lesson in-depth understanding.

High-dimensional empathy.

(27th lesson)

It is discussed that some high-dimensional Fourier transforms have simple operation methods.

(28th Lesson)

The shift and scale change of Gauviffli leaf transform are discussed, and these operations have higher degrees of freedom than one dimension. The high-dimensional $\delta$ function has the same characteristics as a one-dimensional $\delta$ function.

(29th Lesson)

High-dimensional ш functions have higher degrees of freedom, and the ш Fourier transform and two-dimensional sampling theorem are computed.

(30th Lesson)

Radon transformation, medical image imaging, Fourier transform has played a full role in it

[Fourier transform and its application study notes] Course overview