### 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

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 amplitude of the

**nth**harmonic, and the

**bar diagram**represent the same thing. The upper bars were swinging in

**chapter 3**as animations in

**Fig. 3-7…10**. The largest bars are the middle constant components

**a(0)**. The first right bars from

**a(0)**are

**a(+1)**and the

**first left**ones are

**a(-1)**and refer to the first harmonics for

**+,-**pulsations. The lower

**bars**are already so dense that they look like

**red**envelopes.

**Conclusions:**

The

**“upper left”**harmonic distribution for waves with period

**To=2sec**and pulse

**A=1**,

**Tp=1sec**(i.e.

**dense**pulses), consists of sparsely distributed harmonics with large amplitudes

**a(n)**.

The

**“lower right”**harmonic distribution for waves with period

**To=256sec**and pulse

**A=1,Tp=1sec**(i.e.

**rare**pulses), consists of densely distributed harmonics with small amplitudes

**a(n)**.

And what would a

**square wave**with an

**infinite**period

**To**and an impulse

**A=1**and

**Tp=1sec**, i.e. a

**single impulse**, consist of? On the feel, these will be

**infinitesimal**harmonics

**a(n)**with infinitely little different pulsations

**Δω->dω–>0**. So the discrete formula for the amplitude an of the next harmonic

**an=a(n*Δω)**will become a continuous function

**a=a(ω)**. Each

**ω**pulsation will be assigned some amplitude, and not only for

**ω=n*Δω**pulsations. Another thing is that the amplitudes are then i

**nfinitely**small, but not

**zero**! The closest to such a situation is the bar diagram for

**To=256 sec**. Here It is “almost”

**infinity**. These increasingly flat graphs of

**a(n*Δω)**want to tell us something, but what? The answer to such questions is the

**Fourier Transform**, used to study the harmonics of a

**single pulse**, and generally

**non-periodic functions**.

**Chapter 4.2 How to go from Fourier Series to Fourier Transform?
**For now, the transform only for a specific single pulse.

**Fig. 4-1**shows that the bar diagrams become:

–more dense

-the intervals between successive

**a(n)**are getting smaller

–more “flat”

So divide

**an(ω)**from

**Fig. 4-1**by

**Δω**, you get the

**quotient**version. The “flat” defect will disappear and subsequent bar diagrams will look like this. The value of

**ω**becomes continuous when

**Δω–>dω–>0 ω=n*dω**. Because then for every continuous

**ω**there are

**n**and

**dω**such that

**ω=n*dω**.

**Fig. 4-2a**

The quotient

**an(ω)/Δω**, when

**Δω**is a specific finite value, is a

**discrete**function, i.e. it exists only for specific

**ω=n*Δω**.

The quotient

**an(ω)/dω**, when

**dω**is infinitely small, is a

**continuous**function, i.e. it exists for any

**ω**. For every

**n**and

**dω**there is an arbitrary (continuous)

**ω**such that

**ω=n*dω**.

**Fig. Fig.4-2b**

The results of the next

**4 “quotient”**versions of the

**Fourier Series**of square waves in the bar diagrams.

Quotients

**an(ω)/Δω**for

**f(t)**pulse

**A=1, Tp=1sec**with increasing periods

**To=2, 4, 8 and 16 sec**.

**Fig. 4-2c**

This is actually a repetition of

**Fig. 4-2a**, where the transform

**F(ω)**of a single rectangular pulse

**x(t)**is defined as the amplitude density

**a(ω)**with respect to the pulsation

**ω**(multiplied by 2π). Here

**x(t)**is a single rectangular pulse “starting” in its center–>

**Fig.3-1a**, but this applies to all single pulses

**x(t)**being

**even**functions. In

**chapter 5**we will generalize this formula to all

**x(t)**functions, not necessarily even ones.

**Conclusions**:

**1.**The envelope of each plot is the same! i.e. its maximum

**a0/ω0=1/2π≈0.16**and zeros

**ω=0,+/-2π,+/-4π,+/-6π…**are the same for each series. “Something like that” is easier to analyze than the increasingly flat amplitudes in

**Fig. 4-1**.

**2.**The subsequent

**“quotient” Fourier Series**are “denser”, because the intervals

**Δω=ω0**between successive amplitudes

**a(n)**and

**a(n+1)**decrease.

**3.**Equivalents of the bottom

**4**–>”flat” from

**Fig.4-1**you have to imagine. They will be similar to

**Fig.4-2a**, only more “dense”.

**4.**And now the most important thing. The above

**“quotient”**versions, let’s call them

**an/Δω**, are

**discrete**, otherwise discontinuous functions. So they exist only for

**ω=n*Δω**and not for any

**ω**! And when will they become

**continuous**functions? So the envelope of these bar diagrams? Then when

**Δω**becomes infinitely small, i.e. when

**Δω–>dω**as in

**Fig.4-2a**.

**Chapter 4.3 Interpretation of the Fourier Transform F(ω) for a pulse f(t)=A=1 Tp=1sec**

Because it’s the easiest for this particular **f(t)** function. All its harmonics start in the same phase **φ=0** e.g. for **ω=1π/sec** or **φ=-180º** e.g. for **ω=1π/sec**–>**Fig.4-3c**. Therefore, classical **f(x)** bar diagrams can be used, which cannot be said for any **impulse** whose harmonics have different initial phases **φ**. We will talk about such transforms in **Chapter 5**.

**Fig. 4-3
**Transform

**F(ω)**of a rectangular pulse

**A=1**and

**Tp=1sec**(“starting in the middle”)

**Fig. 4-3a**

Transform

**F(ω)**

**Fig. 4-3b**

Bar diagram

**F(ω)**of this transform

How to treat him? E.g.

**F(ω=1π/sec)≈0.63**. Does this mean that for

**ω=1π/sec**, the amplitude of this harmonic is

**0.63**?

**Nooo**! After all, this impulse consists of

**infinitely**small harmonics

**infinitely**little differing in pulsation. So the harmonic

**A(ω=1π/sec)=0**. But the

**sum/integral**of these harmonics in a

**small**range, e.g.

**Δω=0.9999π…1.0001π**will no longer be zero!

**Chapter 4.4 What is the Fourier Transform?**

**Chapter 4.4.1 Introduction
**

**1.**

**The transform**is information about the

**frequency**distribution in a given

**signal**.

**2. So**it is a function of the pulsation

**F(ω)**.

**3.**It is mostly a

**complex function F(ω)**. This will be discussed in

**Chapter 5**. Fortunately, our impulse

**A=1, Tp=1sec**is an

**even**function, and the transforms of

**even**functions are “easier”, because they are

**real F(ω) functions.**Therefore, we can present it in the form of the formula

**Fig.4-3a**or the bar diagram

**Fig.4-3b**.

**Chapter 4.4.2 Rod and transform**

Analogies will make it easier to understand the idea of the

**transform**.

In particular, the harmonic

**A(ω)**for

**ω**is

**zero**, but the transform

**F(ω)**for

**ω**is

**non-zero**.

**Fig.4-4**

**Rod**specific density

**ϱ(x)**and

**square wave**transform

**F(ω)**analogy.

**Fig.4-4a**

**Rod**specific density

**as a distance function**

**ϱ(x)**

The

**mass**of the rod in any cross-section

**x**is

**zero**because the volume of each cross-section is

**ΔV=0**.

The

**rod**density

**ϱ(x)**in any

**x-section**is not

**zero**. A

**lead-aluminum alloy**rod with a cross-section of

**s=1cm2**and

**a length**of

**l=10m**. The

**density**of the

**rod**is the

**smallest**at the ends of

**-5m +5m**and the

**highest**in the

**middle**. In other words, there is only

**aluminum**at the

**ends**, only

**lead**in the

**middle**, and an

**alloy**with an

**intermediate**composition in the rest. E.g.

**ϱ(x=+2)≈9.5g/cm3**

**Fig.4-4b**

Square pulse

**transform**as a function of pulsation

**F(ω)**

The

**amplitude**of the harmonic

**A(ω)**for any

**ω**pulsation is infinitely small, let us assume that it is

**zero**.

The amplitude

**density**with respect to

**ω**, i.e. “almost” the

**F(ω)**transform, because

**F(ω)=2π*a(ω)/dω**is non-zero. E.g.

**F(1π/sec)≈0.1**. This means that the

**average**value of the amplitude

**a(ω=1π/sec)**in the “tiny” range

**ω=1π/sec-dω…1π/sec+dω**is approximately

**0.1**. The average value of

**something**in a certain “tiny” range

**ω**is the density of

**something**in relation to

**ω**.