**Fourier ****Transform**

**Chapter 6. The Fourier Transform of the exponential function
Chapter 6.1 **

**Function description**

In

**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<0**

f(t)=exp(-1t)dla

f(t)=exp(-1t)

**t>=0**

**The**

Fig. 6-1

Fig. 6-1

**f(t)**function

At first,

**f(t)**looks like a

**single**pulse of

**finite duration**. But it lasts all the time from

**t=0**to

**t=+∞**! We will treat

**f(t)**as periodic with

**To=∞**. But first as “really”

**periodic**, that is, with a

**finite**period

**To**. We will calculate the

**Fourier Transform**from

**Fourier Series**. This approach, and not from a

**ready-made integral formula**as

**Fig.6-15**, will allow you to better understand the very

**idea**of the

**Fourier Transform**.

**Chapter 6.2 Fourier series of the function f(t) with period To=15sec
Chapter 6.2.1 Graph of a function**

**Fig. 6-2**

Periodic function

**f(t)**period

**To=15sec**

That is, the first approximation of the

**f(t)**function in

**Fig. 6-1**. Why an approximation?

Because in the range e.g.

**t=-7.5…0…+7.5 sec**you would only see the

**non-periodic**function

**f(t)**from

**Fig. 6-1**!

**Chapter 6.2.2 Reminder of the Complex Fourier Series formula**

We will apply the general formula of the

**Fourier Series**on

**c(n)**in the

**complex**version for

**positive**and

**negative**pulsations.

The constant component

**c(0)**, i.e.

**for n=0**, also is valid

**Fig.6-3**

**Complex**formula for

**Fourier Series**

The

**trigonometric**formula for the

**Fourier Series**is

**2**separate formulas for

**a(n)**and

**b(n)**.

Remember that the pulsation

**ω0**of the

**first harmonic**is also the interval

**Δω**between successive harmonics!

**Chapter 6.2.3**

**Calculation of complex Fourier coefficients c(n) using WolframAlpha.**

**First**harmonic

**pulsation–>ω0=2π/To=2π/15sec≈0.419/sec**

**Integration interval**–>-

**To/2…+To/2=-7.5sec…+7.5sec.**

For example, let’s calculate

**c(n)**for

**n=+3**.

WolframAlpha

**Instruction**:

**(1/15)*integrate exp(-t)*heaviside(t)*exp(-2pi*i*3*t/15)dx from t=-7.5 to 7.5 1**

**1.**To implement it, copy the above

**text**, i.e.

**cntrl c…**

**2**.Click https://www.wolframalpha.com

**3.**Paste the text into the box.

**4.**Do what the picture says.

**Fig.6-4**

WolframAlphainstruction calculating the formula for

WolframAlpha

**c(n)**in

**Fig.6-3**for

**n=+3, To=15 sec**and

**f(t)**in

**Fig.6-2**

**WolframAlfa**also showed many other things, but we are only interested in the result, i.e. a complex number

**c(+3)=c(+3ω0)=0.0259627-j0.0325001**for

**ω0=2π/15sec≈0.419*1/sec**—>

**Fig.6-5.**

This is the

**complex amplitude**of the

**third**harmonic, i.e.

**for ω=3ω0**.

**Chapter 6.2.4 Coefficients c(n) in the complex plane Re z/Im z**

After calculations for

**n=-12…0…+12,**I put the results of

**c(n)**into the complex plane

**Re z/Im z**. The complex Fourier coefficients

**c(n)**will appear on the circle

**A**. Among them is the one just calculated factor/point

**c(+3)=c(-3ω0)=0.0259627-j0.0325001**. We also computed

**c(+1000)≈c(ω=-∞)**and

**c(-1000)≈c(ω=+∞)**as almost

**z=(0,0)**. In other words,

**n=1000**is as

**infinity**.

**Fig. 6-5**

**26**Fourier Series

**c(n)**coefficients on circle

**A**calculated with the

**WolframAlpha**program.

Including: bottom

**12**

**c(n)**, top

**12**,

**c(ω=0), c(ω=+∞)=c(ω=-∞)=(0,0)**.

They form a lower

**semicircle**for

**n>0**, and an upper

**semicircle**for

**n<0**.

Each point

**c(n)**for

**n=-12…+12**is the

**complex amplitude**for the

**nth**harmonic. So not exactly the

**nth**harmonic.

I showed them as vectors only for

**c(+1)=c(+1ω0)**and

**c(-1)=c(-1ω0)**. The remaining

**c(n)**are just

**dots**for readability.

The graph is the equivalent of the

**bar diagrams**from

**chapters 3**and

**4**concerning only

**even**functions.

**Conclusions:**

**1. c0=c(ω=0)≈0.0667**-constant component when

**n=0**

**2. c(-∞)=c(+∞)=0+j0**–>harmonics for

**ω=-∞**and

**ω=+∞**are infinitely

**small**, let’s take them as

**zero**.

**3. c(+n)=c(-n)***are conjugate complex numbers, e.g.

**c(+1) and c(-1**)*

**4.**Each

**c(n)**, i.e. the

**nth**vector, corresponds to a

**harmonic**with pulsation

**n*ω0**, amplitude

**A**i.e. vector length and phase

**φ**.

E.g.

**c(+1)=c(+1*ω0)**for

**ω0≈1*0.419/sec**corresponds to the harmonic

**h(+1)≈0.057*cos(0.419*t) -0024*sin(0.419*t)≈0.0307*cos(0.419*t-22.72°).**

Similarly, the conjugate

**c(-1)=c(-1ω0)**corresponds to the harmonic

**h(-1)≈0.057*cos(0.419*t)+0024*sin(0.419*t)≈0.0307*cos(0.419*t+22.72°)**

**5.**Points for

**n=-∞…-13 and n=+13…+∞**have not been marked. They thicken approaching

**z=(0,0)**when

**ω=nω0–>+-∞.**

**Chapter. 6.2.5 Interpreting the graph in Fig.6-5 as Fourier Series**

The

**26**coefficients of

**c(n**) in

**Fig.6-5**are nothing but the

**Fourier Series**of

**f(t)**. Especially when you imagine that all the vectors are moving in circles around

**z=(0,0)**, each with its velocity

**+-nω0**. The

**upper**vectors are

**clockwise**and the

**lower**ones are

**opposite**, as in the animation

**Fig. 6-6**. Their sum as

**f(t)**will always be on the real axis

**Re z**, because they are formed by pairs of

**conjugate**vectors, e.g.

**c(+1**) and

**c(-1)**. So on the axis

**Re z**there will be a projection of the

**sum**of all rotating vectors, which will move as

**f(t)**according to sum formula in

**Fig.6-3**. This function

**f(t)**will of course be from

**Fig.6-2**, assuming

**n=-∞…0…+∞**.

When

**n=-12…0…+12**, such as in

**Fig. 6-2**, f(t) will be “blunt”, otherwise without sharp points. These rotating vectors are a perfect example of a

**Complex Fourier Series**formula.

The animation below shows

**counter**-rotating

**2**vectors whose

**sum**, as

**f(t)**, moves only on the

**real**axis

**Re z**.

**Fig.6-6**

The sum of **2** oppositely rotating conjugate vectors

**a-**vector **c(n)**

**b-**conjugate vector, i.e. **c(n)***

**c-**sum **c(n)+c(n)***

Look again in** Fig.6-3** for the **∑** pattern.

The **real** function **f(t)** is created by **oppositely** rotating **pairs** of vectors **c(n)*exp(jnω0t)** and **c(-n)*exp(-jnω0t)**.

**Chapter 6.2.6 Quotient Fourier Series, i.e. with coefficients c(n)/ω0**

We divide each point/vector/coefficient **c(n)** in **Fig.6-5** by **ω0=Δω≈0.419*1/sec**. In this way, it will be easier for us to go from **Series** to **Fourier Transform**. Similarly, we divided the amplitude **a(n)** by **Δω** in **Fig. 4-2** in **Chapter 4**. The drawing is a bit large, but thanks to this the symbols **+-1ω0,+-2ω0…** are still** legible**.

**Fig.6-7**

**26** coefficients **c(n)** on circle **A**, i.e. repetition of **Fig. 6-5**.

**26** coefficients **c(n)/ω0** on circle** C** **(ω0≈0.409**)

Points **c(n)** on circle** A** have been transformed into points **c(n)/ω0** on circle** C**. Why? Be patient.

**Chap. 6.3 Fourier series of the function f(t) with period To=30sec **

**Chap. 6.3.1 Introduction**

We will double the period **To**. The pulsation **ω0=2π/30sec≈0.209/sec** will be **2** times smaller than in** chapter 6.2**.

How will the corresponding coefficients/points **c(n)** and **c(n)/ω0** in **Fig.6-7** change?

**Chapter 6.3.2 Graph of a function f(t)
**

**Fig. 6-8**

**Periodic**function

**f(t)**with

**To=30sec**

This is the

**second**and better approximation of

**f(t)**in

**Fig. 6-1**. Better, because

**To=30sec**is closer to

**To=∞ than**the previous one

**To=15sec**. So we follow the path from

**chapter 3**, in which we replaced a

**single**rectangular pulse

**A=1 Tp=1sec**with a

**sequence**of these pulses, i.e.

**periodic**functions with an increasing period

**To**.

**Chapter 6.3.3 Calculation of complex Fourier coefficients c(n) using WolframAlpha**

We will calculate the

**Fourier**coefficients

**c(n)**in the same way as in

**Chapter 6.2**.

The instruction for WolframAlpha for e.g.

**n=3**will be:

**(1/30)*integrate exp(-t)*heavisde(t)*exp(-2pi*i*3*t/30)dx from t=-15 to15**

**Chapter 6.3.4 Coefficients c(n) in the complex plane Re z/Im z**

**Fig.6-9**

**50**coefficients

**c(n)**of the

**Fourier Series**on circle

**B**calculated with the

**WolframAlpha**program.

**Including:**

**24**lower

**c(n)**,

**24**upper

**c(n**),

**c(ω=0), c(ω=+∞)=c(ω=-∞)=(0,0)**.

Compare with

**Fig.6-5**when

**To=15sec**.

**Conclusions:**

The coefficients

**c(n)**are

**smaller**and more

**densely**distributed on circle

**B**twice smaller than

**A,**because the intervals

**Δω=ω0≈0.209/sec**between successive harmonics are

**twice**smaller. What if the vectors started rotating around

**z=(0.0)**? Then the projection of the sum of these vectors would move along the

**Re z**axis similarly to

**f(t)**from

**Fig. 6-8**, only in a “more smooth and no peaks” way.

**Chapter 6.3.5 Quotient Fourier Series, i.e. with coefficients c(n)/ω0**

We divide each

**c(n)**in

**Fig. 6-9**by

**ω0=Δω≈0.209*1/sec.**

**Fig.6-10**

**50 c(n)**coefficients on circle

**B**and

**50 c(n)/ω0**coefficients on circle

**C (ω0≈0.209/sec)**

Circle

**B**is

**2**times smaller than

**A**in

**Fig. 6-5**and

**Fig. 6-7**and the coefficients

**c(n)**are

**twice**as densely distributed.

The circle

**C**is the same as in

**Fig. 6-7**, only the coefficients

**c(n)/ω0**are twice as

**densely**distributed.

**Chapter 6.4 What happens when It increases, i.e. ω0 decreases? **

**Chapter 6.4.1 Introduction**

In **chapter 6.2** was **To=15sec** and in **6.3** it increased to **To=30sec**. What effect did this have on the coefficients** c(n)**?

**Chapter 6.4.2 Comparison of 2 graphs
**So

**Fig. 6-7**and

**Fig. 6-10**on one

**collective**. The drawing has a

**larger**scale and therefore the circles

**A,B,C**are now smaller. For some reason, I added a circle

**D**, which is a circle

**C**enlarged by

**2π**times. The points

**c(+1)/ω0**and

**2π*c(-1)/ω0**on

**C**and

**D**are presented as vectors. The other points on the circles are also vectors, of course.

**Fig. 6-11**

Complex coefficients

**c(n), c(n)/ω0 and 2π*c(n)/ω0**on circles

**A,B,C,D**and the mysterious “point-circle”

**Z**adjacent to

**(0,0)**.

**–**circle

**A**with

**26**points

**c(n)**from

**Fig.6-5**when

**ω0=0.409**(To=15sec)

**–**circle

**B**with

**50**points

**c(n)**from

**Fig.6-9**when

**ω0=0.209**(To=30sec)

**–**circle

**C**with “quotient” points

**c(n)/ω0**from

**Fig.6-10**:

with

**26**points

**c(n)**when

**ω0≈0.409**(To=15sec)

with

**50**points

**c(n)**when

**ω0≈0.209**(To=30sec)

**Note:**

Although circle

**B**is

**2**times smaller than

**A**, its

**c(n)**has been divided by

**2**times smaller

**ω0**.

Therefore, all points

**c(n)/ω0**from

**A**and

**B**are on the same circle

**C**.

-circle

**D**with points

**2π*c(n)/ω0**, i.e. circle

**C**magnified by

**2π**

with

**26**points

**c(n)**when

**ω0≈0.409**(To=15sec)

with

**50**“quotient” points

**c(n)**when

**ω0≈0.209**(To=30sec)

Why

**2π**magnification? We’ll find out later. For now, “to see better”.

All points on circles

**C**and

**D**come from

**B,**but only every

**second**one from

**A**. If the vectors on circles

**A**and

**B**started to rotate with velocities

**n*ω0**, their projections onto the real axis

**Re Z**would be similar to

**f(t)**from

**Fig. 6-2**and

**6-8**, only more blunt and “tipless”.

**Chapter.6.4.3 It’s still growing…**

We analyze the function

**f(t)**similar to

**Fig.6-8**, only “rare” because

**To=60sec**. What would change in Figure

**6-11**?

**1.**Inside

**C**there would be a circle

**E**

**2**times smaller than

**:**

Note

Note

Circle

**E**and its

**f(t)**are not shown in the figures.

**2.**Circle

**E**contains

**2*48+1+1=98**points/coefficients

**c(n)**

**3.**There would be

**98**“descendants” of

**E**in circles

**C**and

**D**. Similar to

**50**“descendants” of

**B**in

**C**and

**D**in

**Fig. 6-11**

**Conclusion**

As

**To**increases, smaller and smaller circles

**A,B,E,F,G…Z**are formed with

**c(n)**points getting closer to each other. Compare, for example,

**A**and

**B**. Clearly points

**c(n)**on

**B**are “denser”. Not only because

**B**is smaller than

**A**! Also because the distance

**Δω0=ω0**between

**c(n)**has decreased

**twice**. These diminishing circles

**A,B,E,F,G…Z**become more and more “continuous”. Until there is an

**infinitely**small and

**continuous**circle

**Z**when

**To=∞**. It is the previously mentioned mysterious “point-circle”

**Z**in

**Fig.6-11**. Here the spacing between points/vectors

**c(n)=c(n*dω)**is infinitely small (

**To–> ∞ i.e. dω=ω0–>0**).

**Fig. 6-12**

Circles

**C**and

**D**and an infinitesimal circle

**Z**inside

**C**

These “touching” points

**c(n)**, on the infinitesimal and continuous circle

**Z**, are the

**Fourier Series**coefficients of the

**non-periodic**function

**f(t)**in

**Fig.6-1**. i.e. there is a

**harmonic**for each continuous

**ω**pulsation, and not only for specific

**ω=n*ω0**as in

**Fig. 6-7**and

**6-10**. Another thing is that every

**harmonic**is

**infinitely**small! It’s hard to analyze something as small as

**Z**.

If you divide the coefficients

**c(n)**on circles

**A, B, E,…**by the distance

**Δω=ω0**between them (for

**A–>Δω=ω0≈0.419/sec**,

for

**B–>Δω=ω0≈0.209/ sec**), then the points

**c(n)/ω0**will be on the same circle

**C**. That is, the circle

**C**with the coordinates

**c(ω)/dω**are the magnified (“through a magnifying glass”) points

**c(ω)**of the circle

**Z**. And the circle

**D**as

**2π*c(ω)/dω**, i.e. circle

**C**enlarged by

**2π**times, as it will turn out later, is just a

**Fourier Transform**of the function

**f(t)**from

**Fig. 6-1**.

**Chapter 6.5 What is the Fourier Transform of f(t) in Fig. 6-1?
Chapter 6.5.1 Introduction**

**Fig. 6-13**

Fourier transform

Most authors start with this. It doesn’t say what a

**Fourier Transform**is, but how it is computed. It’s as if someone defined a

**hamme**r as a product that needs to be made in a certain way. And it should be. A

**hammer**is a tool for

**driving nails**. And the

**Fourier Transform**is a formula that allows you to calculate the

**distribution of harmonics**in the

**f(t)**signal.

**Chapter 6.5.1 What is the circle C with coordinates c(ω)/dω in Fig.6-12?**

The infinitesimal circle

**Z**consists of points

**c(ω)**, each of which is a

**vector**corresponding to a

**harmonic**with

**ω**pulsation.

For example, for pulsation

**ω=+1/sec**, it is a

**harmonic**with infinitely small amplitude

**A**and phase

**φ≈-11.8º**. For now, take my word for it, especially when it comes to

**φ=-11.8º**. But if you enlarge

**Z**by operation

**c(ω)/dω**, you will get circle

**C**in

**Fig.6-12**, where “you can see more”, also

**φ≈-11.8º**. From this it follows that the amplitude

**A**of the harmonic with pulsation, e.g.

**ω=+0.209/sec**of the function, although i

**nfinitely small**, is greater than that when

**ω=+1/sec**and smaller than

**ω=0/sec**. And the expression

**c(ω)/dω**itself is the

**harmonic density**with respect to the

**ω**pulsation. Just as the

**mass**of

**lead**at point is

**zero**

**m=0**, but its

**density**relative to volume

**V**is isn’t zero

**ρ=11.34 g/cm³**! Considering that we are dealing with the vector

**c(ω)/dω**, treat the

**density**of this vector for

**ω=+0.209/sec**as its

**average**value around the pulsation

**ω=+0.209/sec**. So we

**sum/integrate**all (infinitely small!) vectors, e.g. for

**ω=+0.2085/sec…ω=+0.2095/sec**and divide by

**Δω=0.2095/sec-0.2085/sec=0.001sec**. The result is the vector

**c(ω)/Δω≈0.156*exp(-j11.8º)**. So the

**spectral density**of the exponential function

**f(t)**from

**Fig. 6-1**for

**ω=+0.209/sec**is a vector with amplitude

**A=0.156**and phase

**φ≈-11.8º**. And translating into

**harmonics**,

**f(t)**from

**Fig. 6-1**has a

**harmonic**around

**ω=+0.209/sec**with an

**average**value of

**h(t)=0.156*cos(0.209t-11.8º)**. Needless to say, the value of

**c(ω)/dω**is most accurate when

**Δω–>dω–>0**. Just like the

**mass density**at the point

**ρ=m/Δv**is the most accurate when

**Δv–>0**.

Coming back to the question in

**Chapter 6.5.1.**Circle

**C**is “almost” a

**transform**of the function

**f(t)**from

**Fig.6-1**.

And “not almost” but “exactly”, then the

**Fourier Transform**of the function

**f(t)**is the circle

**D**in

**Fig. 6-12**. Why? See

**Chapter 6.5.2**.

**Chapter 6.5.2 More about the Fourier Transform**

You already know that the

**Fourier Transform**of

**f(t)**is the

**complex**function

**F(jω)**in the form of a circle

**D**in

**Fig. 6-13**. And other functions, more precisely those whose area under the function

**f(t)**is finite? That is, any, although not entirely. Imagine that this is a slightly different function than

**f(t)**in

**Fig. 6-1**, but

**non-periodic**and with a

**finite**area. However, it cannot be e.g.

**f(t)=exp(t)**.

**Fig. 6-14**

**a.**What happens

**when**the period

**To**of

**f(t)**approaches

**infinity**?

Then the first harmonic

**ω0=Δω**tends to the infinitely small

**dω**. This also means that the intervals

**dω**between successive harmonics are

**infinitely small**. In other words,

**successive**harmonics “overlap” and their distribution

**c(n*dω)**becomes continuous

**c(ω)**.

**b. c(nΔ)**is the

**nth**complex amplitude when

**f(t)**is a periodic approximation

**f(t)**for finite

**To**.

When

**To=∞**then

**c(ω)/dω**becomes continuous and

**2π*c(ω)/dω**is just the Fourier Transform

**F(jω)**of

**f(t)**!

**c.**The final formula for the Fourier transform of

**F(jω)**. Note that it is

**c(ω)/dω**augmented by

**2π**.

**d.**When the non-periodic

**f(t)**approximation is

**periodic**(with a long period

**To**!), then

**f(t)**is the usual

**Fourier Series**formula on the

**left**. And when

**To–>∞**the formula for the

**Fourier Series**becomes a continuous formula for the so-called

**Inverse Fourier Transform**. It allows you to calculate the waveform

**f(t)**based on its Fourier Transform

**F(jω)**.

**Chapter 6.6 Fourier Transform and Inverse Fourier Transform.**

That is the final summary of the chapter

**Fig. 6-15
**Fourier Transform and Inverse Fourier Transform

One of the most famous pairs of mathematical equations. It is good to know them, even if they are not fully understood.