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:
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.

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).

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.
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:
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:
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