# Fourier transform

## 7. Fourier Transform accounting

Fourier Transform Chapter 7. Fourier Transform accounting Chapter 7.1 Introduction My main goal was to understand the very idea of ​​the Fourier Transform and that it is sort of a continuous version of the Fourier Series. The Fourier Series applies to successive harmonics of periodic functions f(t) with pulsation ω0=2π/To, which in the complex version are complex coefficients c(n): –c(0) is a constant component –c(1), …

## 6. The Fourier Transform of the exponential function

Fourier Transform Chapter 6. The Fourier Transform of the exponential functionChapter 6.1 Function descriptionIn chapter 3 and 4, You have already seen the transform of a single square wave, which was an example of an even function. Time for a more general one that doesn’t have to be even.It is, for example, the function f(t).f(t)=0 dla t<0f(t)=exp(-1t) dla t>=0 Fig. 6-1The f(t) functionAt first, f(t) looks like a single pulse of finite duration. But it lasts all …

## 5. Fourier Transform in general

Fourier Transform Chapter 5. Fourier transform in general Chapter 5.1 Introduction Not all functions are served by the Fourier Transform. Fig. 5-1 Functions served by the Fourier Transform They have a finite duration or infinite but with a finite energy/field. Fig. 5-1a Even functions E.g. single pulse A=1, Tp=1sec from chapter 3 and 4. Or a piece of a symmetrical parabola. Their transforms are “easier” real functions than complex  functions. Fig. 5-1b Not necessarily even. Usually …

## 4. Fourier Transform of a single square wave pulse part 2

Fourier Transform Chapter 4 Fourier Transform of a single square wave pulse part 2 Chapter 4.1 Fourier series of rectangular pulse trains A=1 Tp=1sec with increasing period To. Fig. 4-1 Fourier series of sequences of rectangular pulses A=1 Tp=1sec as bar diagrams. This is a graphical summary of the chapter 3, in which the pulses A=1 T=1sec become increasingly rare. The formula for the a(n), the …

## 3. Fourier Transform of a single square wave pulse part 1

Fourier Transform Chapter 3 Fourier Transform of a single square wave pulse part 1 Chapter.3.1 Introduction It is difficult to find a simpler non-periodic function f(t) than a single rectangular pulse in Fig. 3-1a. It is an even function with amplitude A=1 and duration Tp=1sec. The Fourier Transform is generally a complex function. But thanks to the even of f(t), its Fourier Transform F(ω) is also a real function. And this one is easier to analyze than the complex function. …

## 2. Fourier series of a square wave as a bar diagram

Fourier Tranform Chapter 2 Fourier series of a square wave as a bar diagram Chapter 2.1 Time chart of a square wave In the next chapter, we will build the Fourier Transform of the simplest non-periodic function, which is a single rectangular pulse. It is somehow related to the even square wave which we decomposed …

## 1. Introduction

Fourier Transform Chapterł 1 Introduction. We already know that almost every periodic function f(t) can be decomposed into cosine and sine with different amplitudes An and Bn and with pulsations nω0. What about “normal” functions f(t), i.e. aperiodic functions? It is similar, only their decomposition into harmonics is more difficult to imagine. Their amplitudes An …

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