**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), c(2)…c(n)**are the amplitudes of

**complex**harmonics with pulsations

**1ω0, 2ω0… nω0…**

**Amplitudes**and

**phases**are directly visible in

**c(1), c(2)…c(n)**.

The

**Fourier Transform**is concerned with

**non-periodic**functions

**f(t)**. We treat it as a function with period

**To=∞**. This leads to some strange things. The

**constant**component and all

**harmonics**are

**infinitesimal**. We take them as

**zero,**although they exist! The transition of one

**harmonic**into

**another**is

**continuous**, so instead of a specific

**c(ω)**we have their

**average**in the range

**ω-dω…ω…ω+dω**. This

**average**is the

**density**of complex amplitudes

**c(ω)/dω**. More precisely, the

**Fourier Transform**is

**2π*c(ω)/dω**. In this chapter, we will show you how to compute

**Fourier Transforms**of

**non-periodic**functions by definition. So as it is usually done*, not by

**Fourier Series**when

**To=∞**. We did this earlier in

**Chapters 4**and

**6**using

**Fourier Series**, assuming period

**To=∞**. The results should be identical.

**Fig.7-1**

Fourier Transform formula

Calculation of transforms is facilitated by various formulas derived from the above definition. Some convolutions of functions, residua, etc. Also transforms of integrals, derivatives, functions shifted in time. Accounting matters, however, are not the main focus of this article.

**Chapter 7.2 Fourier Transform of a single square wave pulse
**

**Fig.7-2**

Function

**f(t)**as a single wavepulse

**A=1sec**at time

**t=-0.5sec…+0.5sec**

We will compute the transform

**F(jω)**of

**f(t)**using the formula in

**Fig.7-1**.

**Fig.7-3**

Calculation of the transform

**F(jω)**of the function

**f(t)**from

**Fig.7-2**

**Integration**interval in the range

**t=-0.5…+0.5**. In the remaining range, i.e.

**t=-∞..-0.5**and

**t=+0.5…+∞**, the

**integrand function**is

**zero**and has no effect on the value of the

**integral**. Don’t be surprised that at some point a rather complicated

**quotient**of

**complex numbers**will turn into a decent

**real**function

**sin(0.5ω)**. Well, that’s the math, and if you don’t believe it, check it out with

**WolframAlphA**.

**Fig.7-4**

Fourier Transform

**F(jω)**of the function

**f(t)**from

**Fig.7-3**

How to treat e.g.

**F(1π)≈0.63**? You already know well that although the amplitude for

**ω=1π/sec**exists, it is infinitely small, with some reluctance you can say that it is

**zero**. However, it changes continuously and in the range of e.g.

**1π/sec-dω…+1π/sec…1π/sec+dω**its

**average**value relative to

**ω**is

**F(1π)≈0.63**. The result is, of course, the same as in

**Fig. 4-3**in

**chapter 4**. There we treated

**f(t)**as periodic with

**To=∞**and therefore we could use the

**Fourier Series**formulas. Thenote also applies to

**chapter 7.3.**Recall that

**f(t)**is an even function, and therefore its transform

**F(jω)**is a real function that can be represented as an ordinary graph.

One more thing. The expression

**F(jω)**and

**F(ω)**can be used interchangeably. Either way, the result is a

**complex**function that can also be

**real**, as in

**Fig. 7-4**.

**Chapter 7.3 Fourier transform of the function t=exp(-t) in the interval t=0…+∞**

So like in **chapter 6**

**Fig.7-5**

Function** f(t)** as exponential** exp(-t)** in the interval **t=0…+∞
**

**Fig.7-6**

Calculation of the transform

**F(jω)**of the function

**f(t)**from

**Fig.7-5**

It is not an

**even**function and therefore its

**transform**is a fully

**complex function**.

**Fig.7-7**

Plot of the transform

**F(jω)=1/(1+jω)**

For

**ω=+0.209/sec**, we computed

**F(+0.209)=1/(1+j0.209)≈0.209-j0.200=0.978*exp(-j11.8º)**as the

**red**vector. How? For example, inserting

**1/(1+0.209i)**into the window at

**https://www.wolframalpha.com.**We also calculated

**F(ω)**from the formula for

**ω=-2.412/sec-1/sec,+1/sec and +2.412/sec**, these complex numbers are marked with

**red**dots. And

**ω=+∞,-∞**? Here we assumed

**ω=+1000,-1000**as “almost”

**infinity**. As expected,

**F(jω)≈0**. Needless to say, all values of

**F(jω)**for

**ω=-∞…+∞**form a

**circle**.