**Fourier ****Transform**

**Chapter 2 Fourier series of a square wave as a bar diagram
**

**Chapter 2.1 Square wave graph**

In the next chapter, we will construct the

**Fourier Transform**of the simplest

**non-periodic**function, which is a

**single rectangular pulse**. It is somehow related to the

**even square wave**that we decomposed into a

**Fourier Series**in the

**“Rotating Fourier Series”**

**chapter 8**. The term often appears in the description of the so-called. a

**bar graph**, which is simply a

**Fourier Series**in graphical form. In this chapter, we will show several

**bar graphs**to approximate a

**square wave**with different periods, as

**S7(t)**sums of several of its first harmonics.

**Fig. 2-1
**

**Square wave**with parameters:

**A=1**-amplitude

**ω0=1/sec**-pulsation corresponding to period

**T=2πsec≈6.28sec**.

**ϕ=0**(phase)

**50%**-square wave

**duty**cycle

The course started theoretically at the beginning of the world for

**t=-∞**in the middle of the

**pulse**when

**A=1**.

So for

**t=0**it’s also going to be in the middle of the

**pulse**. It is an

**even**function. Just like, for example,

**cosine**, which also starts in the “middle”. The left semi-time axis t is not shown.

**Chapter 2.2 The seventh approximation of an even square wave is S7(t)=c0+h1(t)+h3(t)+h5(t)+h7(t).****
**In

**chapter 8**of the article “Rotating Fourier Series” we calculated the approximation of the even square wave

**S7(t)**when

**ω0=1/sec—>Fig. 2-3**. We calculated the coefficients an at

**cos(nω0t)**as

**double**the centroids

**scn**of the trajectory

**F(njω0t)**of the function

**f(t)=even square wave**. If you’re not too familiar with these

**centroids**, we just computed the coefficients

**a(n)**of the

**Fourier Series**. We did not take into account the remaining harmonics, i.e. for

**n>7**, which resulted in a “wavy” approximation

**s7(t)**of a square wave.The

**even**harmonics

**hn(t)**are

**zero**because

**a(2)=a(4)=a(6)=…0**. For

**n=∞**we would get a perfect square wave from animation

**Fig. 2-1**. All Fourier Series coefficients

**b(n)**are

**zero**because

**f(t)**is an

**even**function.

**Fig.2-2
S7(t)=c0+h1(t)+h3(t)+h5(t)+h7(t)
**You can see in the animation

– an

**even square wave**with period

**T=2πsec**, i.e.

**ω=1/sec**

– its

**seventh**approximation

**S7(t)**as incomplete Fourier series.

– constant component

**c0=a0=0.5**

– harmonics

**h1(t), h3(t), h5(t) and h7(t)**.

After substituting the specific parameters

**ω0, a(1), a(3), a(5)**and

**a(7)**from the description above, we obtain the formula for

**S7(t)**as a

**Fourier Series**approximation of the function

**f(t)**.

**Fig.2-3**

The specific formula for the

**Fourier Series**approximation of a square wave in

**Fig. 2-1**.

**Chapter 2.3 The single side band bar diagram of a square wave
**

**Single-band**, because only for

**non-negative**pulsations.

**The most accurate**version of the

**Fourier Series**is, of course, its formula itself. For example,

**n=7**and an

**even square wave**, it is

**Fig. 2-3**

**A less accurate**(because it depends on the age and glasses of the observer), but more intuitive version are time charts, e.g.

**Fig. 2-2**. The rectangular function itself, the constant component and the

**4**harmonics of the

**S7(t)**approximation are visible. We see that the components add up to give

**S7(t)**.

**A bar diagram**as an

**animation**, it is the most intuitive. The pulsations

**ω**and the amplitudes

**a(n)**of the harmonics and the constant component

**a0**are clearly visible. And where is the

**φ**phase? Nowhere. Because the

**bar diagram**is suitable only for the analysis of even functions

**f(t)**. These functions have only a

**real**component where either

**φ=0**(positive bars) or

**φ=180º**(negative bars). So let’s make a

**bar diagram**for

**S7(t)**in

**Fig. 2-3**. That is, for the

**square wave**approximation in

**Fig. 2-1**when

**n=7**.

**Fig. 2-4
**

**Bar diagram**of a

**single**side band of a square wave for

**n=7**and

**ω=1/sec**

More precisely, it is an animation of the formula in

**Fig. 2-3**, and the chart you see in

**Fig. 2-4**(before animation) is the initial state

**S7(t)**, i.e.

**for t=0**. The harmonics

**hn(t)**for

**ω**even pulsations are

**zero**. Otherwise – they do not exist.

**1.**The animation takes about

**3T=3*2π sec≈19sec**

**2.**Harmonic pulsations are placed on the horizontal axis for

**ω=0…+7/sec.**

**3.**

**a0=+0.5**the constant component of a square wave, i.e. the

**bar**for

**ω=0**

**4**. Animation of the

**first**harmonic

**h1(t)**i.e. the bar for

**ω=1/sec**

**…**

**7.**Animation of the

**7th**harmonic

**h7(t)**, i.e. a bar for

**ω=7/sec**

**8.**Sum of

**S7(t)**a0 and

**4**harmonics

**hn(t)**i.e. the left extreme bar. Compare with

**S7(t)**in

**Fig. 2-2**. Notice that it is in phase with

**h1(t)**. If

**n=∞**, you would see perfect “rectangular” motions without jitter, such as in

**Fig. 2-1**. Without the so-called the

**Gibbs effect**– maximum “swing” at the beginning and end of the impulse.

**In**the initial state

**t=0**, you see the amplitudes of the harmonics

**hn(t)**, which is with the Fourier coefficients

**a(n)**of the square wave in

**Fig. 2-2**. Negative amplitudes for

**h3(t)**and

**h7(t)**means that these are the values at the initial moment for

**t=0**, which results from the formula

**Fig. 2-3**. In other words,

**φ=180º**, because in this phase the cosines

**h3(t), h7(t), …**when

**t=0**“start”.

**The**function

**f(t)=S7(t)**on the left side of the graph is at any time the sum of all swinging bars, i.e. the constant component

**a0**and the

**4**harmonics

**h1(t), h3(t), h5(t)**and

**h7(t )**. They “swing” with pulsations

**ω=n*ω0=n*1/sec**, that is with

**ω=1/sec, 3/sec, 5/sec**and

**7/sec**. The dotted line is the constant component

**a0=0.5**.

**I emphasize**that the animation with the swinging bars is the most intuitive version of the

**Fourier Series**! You see the sum

**S7(t)**, the

**4**harmonics hn(t) and the

**DC**component

**a0**. And the Fourier coefficients

**a(n)**themselves are bars at the initial moment. We can formally treat the constant component

**a0**as a harmonic

**h0(t)**with

**zero**pulsation.

**Note:**

If the

**bar graph**is so cool and intuitive, why isn’t it used for all periodic functions

**f(t)**? The answer is simple. Because it applies only to functions of

**even**functions for which the

**Fourier series**has no

**sinusoidal**components, i.e.

**b(n)**.

**Chapter 2.4 The double side band bar diagram of a square wave
**The left “negative” semi-axis

**ω**in

**Fig. 2-4**is, apart from the

**f(t)**line, undeveloped. What a waste!

From trigonometry we know that

**cos(ωt)=cos(-ωt)**. So let’s divide each harmonic

**hn(t)**by

**2**and divide them into

**2**semi-axes of

**ω**.

So

**h+n(t)=h-n(t)=hn(t)/2**

So we created a

**double side**band graph

**of a square wave**.

**Fig. 2-5
**

**Bar diagram**of a

**double**side band of a

**square wave**for

**n=7**and

**ω=1/sec**.

**Each**

**right**harmonic

**h+n(t)**band has its symmetric left

**h-n(t)**counterpart.

For obvious reasons

**1. a0=+0.5**the constant component of a square wave, i.e. the bar for

**ω=0**and is the same as

**a0**for the

**double-band**in

**Fig. 2-4**

**2.**The remaining

**harmonics**, or “swings”, are the halves of

**Fig. 2-4**

**3.**

**f(t)**bar is the same as in

**Fig. 2-4**

What’s all this for? You will find that the introduction of

**negative**pulsations will create a certain elegance in the formulas. Even the animation is more aesthetic than

**Fig. 2-4**. Remember the centroids

**scn**of the trajectory

**F(njω0t)**in the article

**“Rotating Fourier Series”**? I was intrigued as to why

**c(n)=2*scn**. Why the

**complex**amplitude of the

**nth**harmonic

**hn(t)**, i.e.

**c(n)**, is a double

**scn**and not just a (single)

**scn**. That’s what math says. Agreed, but there was a “why?” in the back of my head. Now I know. Because

**harmonics**have to be

**divided**into

**positive**and

**negative**pulsations!

**Chapter 2.5 Influence of square wave pulsations on bar diagram graph.
**

**Chapter 2.5.1 Introduction**

More precisely, the influence on

**double-band bar diagrams**, because we will only be interested in such.

**Fig. 2-5**is a

**double-band bar diagram**of the even square wave in

**Fig. 2-1**with period

**T=2π sec–>ω=1/sec**. If the amplitude were

**A=2**, the graph would be

**2**times

**higher**. It is obvious. And period

**T**?

We will examine

**2**cases, because the first-

**T=2π sec–>ω=1/sec**is as above–>

**Fig. 2-5**.

**– T=4π sec–>ω=0.5/sec**

**– T=1π sec–>ω=2/sec**

**Chapter 2.5.2 Square wave bar diagram when T=4π sek–>ω=0.5/sek
**

**Fig. 2-6
**

**Square wave**with parameters:

**A=1**-amplitude

**ω0=0.5/sec**-pulsation corresponding to period

**T=4πsec≈12.56sec**.

**ϕ=0**(phase)

**50%**-square wave

**duty**cycle

A wave similar to that in

**Fig. 2-1**. Only the smaller pulsation

**ω0=0.5/sec**, i.e.

**T=4π sec**. It will be interesting to see how this affects the

**bar diagram**.

**Fig. 2-7
**

**Bar diagram**of a

**double**side band of a

**square wave**for

**n=7**and

**ω=0.5/sec**.

The

**bars**are

**2**times slower and

**2**times “denser” than in

**Fig. 2-5**.

**Chapter 2.5.3 Square wave bar diagram when T=1π sek–>ω=2/sek
**

**Fig. 2-8
**

**Square wave**with parameters:

**A=1**-amplitude

**ω0=2/sec**-pulsation corresponding to period

**T=1πsec≈3.14sec**.

**ϕ=0**(phase)

**50%**-square wave

**duty**cycle

A wave similar to that in

**Fig. 2-1**. Only the bigger pulsation

**ω0=2/sec**, i.e.

**T=1π sec**. It will be interesting to see how this affects the bar diagram.

**Fig. 2-9
**

**Bar diagram**of a

**double**side band of a

**square wave**for

**n=7**and

**ω=2/sec**.

The

**bars**are

**2**times

**faster**and

**2**times “rarer” than in

**Fig. 2-5**.

**Chapter. 2.6 Conclusions
**

**1. Bigger**signal–>

**Higher**bars

**2. “Wider**signal in time”—>The

**bars**are more densely distributed

Looking ahead and generalizing to the

**Fourier Transform**

**1. Bigger**signal–>

**Higher**transform

**2. “Broader**in time signal”—>

**Narrower**Transform