Laplace transform and fourier transform

[LvW]

Negative resistance can be generated without clear positive feedback.
...........
.

Yes, of course. I didn't argue against it.
However, the question was if you can realize a transfer function by replacing the "R" by "-R" in a first order RC lowpass function.
Please, can you show me a corresponding circuit diagram that has a stable bias point?

LvW

 

[ninju]

s=sigma + jw. s is a complex number. Fourier transform deals only in the case where sigma is 0. They may look similar due to similar formulae, but they are very different.

Consider the eigen function exp(s),
exp(s) is exp(sigma + jw) which is exp(sigma) * exp(jw) , exp(sigma) is just a real number, but exp(jw) consists of a pain of sinusoids. So, Laplace transform is not the same as Fourier transform.

And, 1/(1-jw) can never exist as the corresponding time domain version is exp(t), which keeps increasing with increase in time. To give an idea of how much it increases,
see this:

Abstruse Goose » y=e^x

This function is clearly divergent, and hence we can not find out its frequency components, which is the basic purpose of using fourier transform.

 

[LvW]

.......................
And, 1/(1-jw) can never exist as the corresponding time domain version is exp(t), which keeps increasing with increase in time. .
..........

Ninju, I completely agree with you.
One interesting aspect is the following:

A circuit that realizes (no - better let's say: models or simulates) the function 1/(1-s) or similar functions with a pole in the RHP can be formulated in simulation programs - and an ac analysis gives some results that makes sense (although not usable for real circuits).
Contrary to stable functions such a function exhibits a rising phase. And the maximum slope of the phase function is a good measure for the "distance" of the pole to the imaginary axis (in the RHP). This is an interesting analogon to stable phase functions with a minimum of the negative slope that also is a measure to this distance (in the LHP).
These features can be used to determine the phase margin of a system with feedback in a closed-loop condition (i.e without opening the loop). Probably you know about the problems by opening the loop in some systems.

Regards
LvW

 

[pancho_hideboo]

s=sigma + jw. s is a complex number.
Fourier transform deals only in the case where sigma is 0. They may look similar due to similar formulae, but they are very different.

No.
They are mathematically same.
w is complex.

Your mathematical knowledge is too elementary.

Consider more high level mathematics.

And exp(s) is not eigen function.

eigen function is exp(s*t) or exp(j*w*t). These are kernel of Integral Transformation or expansion basis of continuous space.
Here both "s" and "w" are complex.
Again consider "Analytic Continuation" of Complex Function Theory.

1/(1-s) or 1/(1-j*w) can exist.
But these transfer function will change form due to nonlinearity.

 

[ninju]

w is complex? exp(jw) is complex with cos and j sin.

If w is complex, cos(complex) + jsin(complex)

we'll end up getting hyperbolic functions. Till today, i haven't used hyperbolic functions in fourier transform.

So,Explain what you mean by 'w is complex'.

Read Feynman's lectures in physics, volume 1 to really understand what complex numbers are and why we even use them.

Else, try implementing FFT using only 74xx series of ICs. :grin:

 

[pancho_hideboo]

w is complex? exp(jw) is complex with cos and j sin.
Explain what you mean by 'w is complex'.

Again consider "Analytic Continuation" of Complex Function Theory.
Analytic continuation - Wikipedia, the free encyclopedia

Read Feynman's lectures in physics,

In physics, we don't use "s", instead we use j*w, here w is complex.
Since they are same.
Surely read "Feynman's lectures in physics".
In electric magnetic theory, maxwell equation is solved at frequency domain beased on exp(j*w*t).
But we extend this w as complex number without introducing new number "s".

Surely learn mathematics and physics.

 

[ninju]

"And exp(s) is not eigen function."

please read oppenheim's signals and systems. I'm sure i'd trust the book (in the 3rd reprint of its second edition), written by an MIT professor with 40 years experience in this field.

I suppose you'd call that book "elementary" too.

 

[pancho_hideboo]

"And exp(s) is not eigen function."
please read oppenheim's signals and systems.
I'm sure i'd trust the book (in the 3rd reprint of its second edition),
written by an MIT professor with 40 years experience in this field.

Show me the page where exp(s) is described as eigen function.

Can you undestand a meaning of "eigen function" or "eigen mode" ?

Again exp(s) is not eigen function.
eigen function is exp(s*t) or exp(j*w*t).

See Eigenfunction - Wikipedia, the free encyclopedia
Eigen function which expand continuous space is exp(k*t), here k is continuous eigen value.
In this case, k=j*w.
.

 

[ninju]

@pancho_hideboo: you're turning this into an argument. I disagree that we "don't use s in physics" but thats a different topic. I meant you to read and understand what complex numbers mean in real world. Mathematics is, but a tool to understand physics, so no point if you know 2+2=4 unless you know what it actually means.

What exactly did you mean to say by giving the wikipedia link?

---------- Post added at 22:34 ---------- Previous post was at 22:32 ----------

yeah sorry about leaving t out, felt lazy to type it. Didn't expect people to point out things which are meant to be implicitly understood. Please accept my apologies for that.

 

[pancho_hideboo]

@pancho_hideboo: you're turning this into an argument.
I disagree that we "don't use s in physics" but thats a different topic.

This is because you insist that Fourier Transform and Laplace Transform are different.

Again they are same.

I meant you to read and understand what complex numbers mean in real world.

Of course, I know.
Consider relation between H(z) and H(z=exp(j*w*T)). Here H(z) is Z-Transformation.
This relation is same as H(s) and H(j*w).


Transfer function is a Green Function which are expanded by eigen mode which is expressed as inverse fourier integral.
1/(1-s) and 1/(1-j*w) have same eigen mode, exp(t).
These function can exist in startup or initial state of circuit.
The Designer's Guide Community Forum - Harmonic balance no guarantee for start-up?

Even for exp(t), we can define both Fourier Integral and Laplace Integral.

 

[ninju]

I suppose most people's knowledge is "elementary" too.

Let me ask you a simple question. Why on earth would there be the necessity to learn two different transforms when according to you, they serve the same purpose.

I'll leave you to ponder over this as i'm going to sleep now.


i never said 1/(1-s) can't exist in a system. It can exist.

 

[LvW]

I do not want to interfere in this interesting discussion - although i could!
Only one remark to pancho_hideboo:

Quote: 1/(1-s) and 1/(1-j*w) have same eigen mode, exp(t).
These fuction can exist in startup or initial state of circuit.

*Are you aware that functions in the frequency domain are based on steady-state conditions?

*And, please, to avoid misunderstandings by using the term "exist", make clear if you mean on paper, as an abstract simulated function (see my last contribution) or as a function that can be measured on a real device. Thank you.

LvW

 

[pancho_hideboo]

I *And, please, to avoid misunderstandings by using the term "exist",
make clear if you mean on paper,
as an abstract simulated function (see my last contribution) or as a function that can be measured on a real device.

I can understand your opinion.

But we consider initial state of oscillator as 1/(1-s) or 1/(1-j*w), that is, former is RHP(Right Half Plane) Pole System and latter is Lower Half Plane Pole System.

 

[LvW]

..........
But we consider initial state of oscillator as 1/(1-s) or 1/(1-j*w), that is, former is RHP(Right Half Plane) Pole System and latter is Lower Half Plane Pole System.

I suppose, you mean "Left Half Plane (LHP)" rather than "Lower...."?
Then my question: 1/(1-j*w) has a pole - according to you - in the LHP? This I cannot understand. Please clarify.

 

[ninju]

I think pancho_hideboo means w = real + complex, instead of s. i think he's just using a different variable in laplace transform and confusing it with fourier transform, because, if w in s=sigma + jw is complex, it'd lead to hyperbolic functions.

and he does mean lower half plane.

Upper half-plane - Wikipedia, the free encyclopedia

 

[pancho_hideboo]

I suppose, you mean "Left Half Plane (LHP)" rather than "Lower...."?

No.

Pole of 1/(1-s) is s=1 which is located at Right Half Plane of s-plane.
Pole of 1/(1-j*w) is w=-j which is located at Lower Half Plane of w-plane.

I think pancho_hideboo means w = real + complex, instead of s.

No.

i think he's just using a different variable in laplace transform and confusing it with fourier transform,
because, if w in s=sigma + jw is complex, it'd lead to hyperbolic functions.

Very wrong.
Append comments after learning a little advanced mathematics.

In H(s), s is sigma+j*w. Here both sigma and w are real numbers.
In H(j*w), w is wreal+j*wimag. Here this w does not mean w in s. This w is a complex number as "Analytic Continuation".
Instead both wreal an wimag are real numbers. Their relations between sigma and w of s are followings.
wreal=w of s, wimag=-1*sigma of s

So Right Half Plane of s-plane is mapped to Lower Half Plane of w-plane, and vice versa.

Again surely learn Complex Function Theory(=Analytic Function Theory), especially "Analytic Continulation".

Again Fourier Integral and Laplace Integral are mathematically same.

 

[LvW]

Pancho_hideboo,

please, do me a favour. In what textbook I can learn something about the w-plane?
In the context of frequency domain I never have heard about it.
Thank you.
LvW

 

[pancho_hideboo]

"And exp(s) is not eigen function."
please read oppenheim's signals and systems.
I'm sure i'd trust the book (in the 3rd reprint of its second edition),
written by an MIT professor with 40 years experience in this field.

Show me the page where exp(s) is described as eigen function.

please, do me a favour.
In what textbook I can learn something about the w-plane?

See books on pure mathematics or physics.
Don't see book on engineering.

I think this is a common knowlege.

I think you might be able to see this in the books of following authors.

S.A.Schelkunoff
J.A.Stratton
A.Sommerfeld
W.K.H.Panofsky, M.Phillips

Again Fourier Integral and Laplace Integral are mathematically same.

 

[ninju]

No.


In H(s), s is sigma+j*w. Here both sigma and w are real numbers.
In H(j*w), w is wreal+j*wimag. Here this w does not mean w in s. This w is a complex number as "Analytic Continuation".
Instead both wreal an wimag are real numbers.

Thats what i told. You seem to have used w as the variable instead of s.


please refer to my previous posts where i've clarified that i meant e^st is the eigenfunction. and please stop making edits to your posts continously.


And i'm sure i get what you mean to say, but please answer my previous question, why would there be two transforms with two different names, being learnt by people all over the world , in two different chapters.

I don't think you are wrong, but you are just expressing yourself differently. w in jw of s=sigma + jw is NOT complex. You yourself have stated so. This is the w under discussion in fourier transform, please avoid bringing in unwanted mathematics here, 'your w' and the 'w we've been discussing' are different.

Your w is not fourier transform w.

Please think logically instead of adding confusion to engineering discussion by bringing in mathematics which have no reason to be here. BTW, I too have read two of the books in the list that you've posted.

You just don't seem to understand the meaning of complex. I believe you are just considering it as a pure mathematical tool to solve equations.

 

[pancho_hideboo]

Thats what i told. You seem to have used w as the variable instead of s.

No.
You can't understand my w.

Again reconsider your following post.

I think pancho_hideboo means w = real + complex, instead of s.

please refer to my previous posts where i've clarified that i meant e^st is the eigenfunction.

There is no post which you clarified exp(s*t) is a eigen fuction.
And I don't think you can understand what is an eigen function in early stage of this thread.

why would there be two transforms with two different names,
being learnt by people all over the world , in two different chapters.

This is due to two reasons.

(1) Historical Reason.
Laplace Transform is used as instead of "Heviside Operator Method" with mathematical back ground.

(2) For students or engineers who don't have advanced mathematic knowledges like you.
If we use H(s), we don't have to know much about Complex Function Theory in many cases.
But if we use H(j*w) instead of H(s), we always have to know much about Complex Function Theory correctly.

I don't think you are wrong,

I think you are wrong.

w in jw of s=sigma + jw is NOT complex.
You yourself have stated so.
This is the w under discussion in fourier transform,
please avoid bringing in unwanted mathematics here, 'your w' and the 'w we've been discussing' are different.

You can not still understand Laplaced Transform and Fourier Transform from point of view of high level.

Your w is not fourier transform w.

No.
My "wreal" is a w of Fourier transform in classical or elementary meaning.
My "w" is a natural extension of Fourier Integral as a result of "Analytic Continuation".

Your "w" is same as my "wreal".

Re-learn a derivation process of "Laplace Transform" from "Fourier Transform".