Fourier analysis-Fourier Series

The name is very big. This article introduces the physical meanings behind basic mathematical concepts and concepts such as Fourier series and Fourier transformation.

1. Introduction to Fourier Analysis

The basic purpose of expanding to Fourier series is to break down a signal (the function of time variable t) into different frequency components. These basic construction blocks are sine and Cosine functions.

Sin (NT) Cos (NT)

For example, evaluate the following functions: F (t) = sin (t) + 2cos (3 t) + 0.3sin (50 t)

The function has three components. The oscillation frequency is 1 [sin (t) part] 3 [2cos (3 T) part] 50 [0.3sin (50 t) part].

A common problem to be solved in Signal Analysis is: filtering out noise. For example, when playing a recording tape, the special background sound is a high-frequency (sound) noise, and multiple devices can partially filter it out. 0.3sin (50 t) causes the jitter of the f curve in Figure 1, so that the coefficient 0.3 is equal to 0. The function f (t) = sin (t) is obtained) + 2cos (3 TB)

Except for high-frequency jitter, the image is almost the same as that in Figure 1.

This example shows a method for filtering out noise. This method is to expand the signal f (t) using the sine and cosine signals:

Then, the coefficients an and BN corresponding to the filtering frequency are equal to 0. In this example, the signal F has been expressed as the sum of sine and cosine, so the processing process is very easy. However, most signals are not expressed in this way. One of the purposes of studying Fourier series is to study how to effectively break down a function into the sum of sine and cosine, so as to implement various filtering algorithms.

Ii. Fourier transform Classification

Based on the different types of the original signal, we can divide the Fourier transform into four types:

1	Non-periodic continuous signal	Fourier transform)
2	Periodic continuous signal	Fourier Series)
3	Non-periodic Discrete Signal	Discrete Time Domain Fourier Transformation (Discrete Time Fourier Transform)
4	Periodic Discrete Signal	Discrete Fourier Transform (Discrete Fourier Transform)

Is the legend of four original signals:

Iii. Fourier series Calculation

3.1.1 In the interval

This theorem can be equivalent to a set.

It is the Orthogonal Function System on L2 ([-π, π.

The coefficient of Fourier series can be calculated by applying theorem 1.1. The following theorem is obtained:

3.1.2 Other intervals

3.2 cosine and sine Expansion

Below is an interesting part: even continuation and odd Extension

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The Fourier series image shows that it fits the original function very well.

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Another example:

The plural form of 3.3 Fourier Series

Brief description:

3.4 convergence theorem of Fourier Series

Some theorem is simply listed. In practical application, the convergence conditions are basically met.

This theorem indicates that, as K increases, the Fourier coefficient AK and BK converge to 0.

Convergence Theorem at continuous points:

Convergence Theorem of <but left and right limits exist> at discontinuous points:

Consistent convergence:

Mean convergence:

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References:

1. Basis of wavelet and Fourier Analysis

<This book is well written and focuses on Intuitive understanding, but lacks a physical layer explanation>

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