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Laplace Transform vs Fourier transform

Started by fisico32 October 27, 2009
Jerry Avins <jya@ieee.org> wrote:
(snip, someone wrote)

>> This is because the magnitude of the imaginary component of a >> complex signal is a real signal.
> Of course. All things measurable are real. "Complex" numbers are merely > a clever and useful bookkeeping scheme for manipulating related pairs of > real quantities.
Or pairs of real numbers are a convenient way of measuring complex quantities. Impedance and index of refraction are both complex, and for similar reasons. We can separately measure resistance and reactance, or the real and imaginary parts of the index of refraction. (The imaginary part comes from absorption.) Both are due to the interaction of electrons with atoms, and with each other. Often the available materials are fairly close to ideal, such that we can separate the quantities. Resistors do have inductance, inductors (except superconductors) do have resistance. -- glen
glen herrmannsfeldt wrote:
> Jerry Avins <jya@ieee.org> wrote: > (snip, someone wrote) > >>> This is because the magnitude of the imaginary component of a >>> complex signal is a real signal. > >> Of course. All things measurable are real. "Complex" numbers are merely >> a clever and useful bookkeeping scheme for manipulating related pairs of >> real quantities. > > Or pairs of real numbers are a convenient way of measuring complex > quantities. > > Impedance and index of refraction are both complex, and for similar > reasons. We can separately measure resistance and reactance, or > the real and imaginary parts of the index of refraction. > (The imaginary part comes from absorption.) Both are due to > the interaction of electrons with atoms, and with each other. > > Often the available materials are fairly close to ideal, such > that we can separate the quantities. Resistors do have inductance, > inductors (except superconductors) do have resistance.
Complex numbers are a clever and useful way to represent those quantities in cases where phase shift has meaning. Compact notations expand out ability to comprehend (same root as in prehensile) complicated things. Vector analysis, quaternions, complex numbers, matrix algebra .. without them, we'd be hard pressed to express, let alone understand, some phenomena as concepts. Nevertheless, being built up from simpler stuff (Shall we go back to Peano's axioms?) thay constitute the HLLs of math. Ultimately, all running code executes assembly language. Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
Jerry Avins <jya@ieee.org> wrote:
(snip)
 
> Complex numbers are a clever and useful way to represent those > quantities in cases where phase shift has meaning.
http://en.wikipedia.org/wiki/Refractive_index
> Compact notations expand out ability to comprehend (same root as in > prehensile) complicated things. Vector analysis, quaternions, complex > numbers, matrix algebra .. without them, we'd be hard pressed to > express, let alone understand, some phenomena as concepts. Nevertheless, > being built up from simpler stuff (Shall we go back to Peano's axioms?) > thay constitute the HLLs of math. Ultimately, all running code executes > assembly language.
In the case of discrete ideal L, R, and C, I would have to agree. Consider, though, an electromagnetic wave going through an imperfect (like the ones we have available) dielectric. The motion of the electrons generates new waves that combine with the original wave in such a way as to make it appear to move slower in the material (for n>1), and in most cases decrease in amplitude (absorption). Yes, it is due to the phase of the motion of the electrons relative to the phase of the incoming wave. The electrons move following some fairly simple differential equations, though not necessarily with simple solutions. The wave comes out of the material phase shifted (due to the real part) and attenuated (due to the imaginary part of the index of refraction). Unlike the case of discrete R, L, and C, it is not so easy to separate the two. In non-magnetic materials, the index of refraction is the square root of the dielectric constant. That is, the complex index of refraction is the square root of the complex dielectric constant, describing the interaction of the material with the electromagnetic wave. Also, see: http://en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relation and especially: http://en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relation#Physica_interpretation_and_alternate_form Which in the end describes the interdependence between the real and imaginary parts of the dielectric constant and/or index of refraction. Yes it is all phase shifts, but just saying that leaves a lot out. -- glen
glen herrmannsfeldt wrote:

   ...

> Which in the end describes the interdependence between the real and > imaginary parts of the dielectric constant and/or index of refraction. > > Yes it is all phase shifts, but just saying that leaves a lot out.
There's no doubt that "real and imaginary" is a good way to describe the phenomena. We gain understanding with the compact, high-level, abstraction, but we lose some understanding by taking the mathematical description to be the actual phenomenon. Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
On 10/30/2009 3:09 PM, Jerry Avins wrote:
> glen herrmannsfeldt wrote: > > ... > >> Which in the end describes the interdependence between the real and >> imaginary parts of the dielectric constant and/or index of refraction. >> >> Yes it is all phase shifts, but just saying that leaves a lot out. > > There's no doubt that "real and imaginary" is a good way to describe the > phenomena. We gain understanding with the compact, high-level, > abstraction, but we lose some understanding by taking the mathematical > description to be the actual phenomenon. > > Jerry
Is the level of abstraction the difficulty? Numbers are a means to abstract quantities, fractions to abstract ratios, real numbers to further abstract quantities, and algebra is a means to abstract numbers. Is it really so much different to abstract complex values? I don't see complex quantities as very much different from other abstractions that people seem to be far more comfortable with. -- Eric Jacobsen Minister of Algorithms Abineau Communications http://www.abineau.com
Eric Jacobsen wrote:
> On 10/30/2009 3:09 PM, Jerry Avins wrote: >> glen herrmannsfeldt wrote: >> >> ... >> >>> Which in the end describes the interdependence between the real and >>> imaginary parts of the dielectric constant and/or index of refraction. >>> >>> Yes it is all phase shifts, but just saying that leaves a lot out. >> >> There's no doubt that "real and imaginary" is a good way to describe the >> phenomena. We gain understanding with the compact, high-level, >> abstraction, but we lose some understanding by taking the mathematical >> description to be the actual phenomenon. >> >> Jerry > > Is the level of abstraction the difficulty? Numbers are a means to > abstract quantities, fractions to abstract ratios, real numbers to > further abstract quantities, and algebra is a means to abstract numbers. > Is it really so much different to abstract complex values? > > I don't see complex quantities as very much different from other > abstractions that people seem to be far more comfortable with.
There's no difficulty. The abstraction simplifies the conception. There is a real danger -- hard to avoid -- that the abstraction replaces in one's mind the abstracted reality. jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
Jerry Avins wrote:
> glen herrmannsfeldt wrote: > > ... > >> Which in the end describes the interdependence between the real and >> imaginary parts of the dielectric constant and/or index of refraction. >> >> Yes it is all phase shifts, but just saying that leaves a lot out. > > There's no doubt that "real and imaginary" is a good way to describe the > phenomena. We gain understanding with the compact, high-level, > abstraction, but we lose some understanding by taking the mathematical > description to be the actual phenomenon. >
Isn't this kind of stating the obvious? Come to it, even "real numbers" represented in a computer are really combinations of integers under the hood (and below that, merely aggregations of binary states), so they are no more real than imaginary numbers. All numbers are ultimately mythological; the rest is simply counting, and very few things in the physical world can be measured ~exactly~. Sometimes that matters - the butterfly effect! Richard Dobson
Jerry Avins <jya@ieee.org> wrote:
(snip, someone wrote)

>> I don't see complex quantities as very much different from other >> abstractions that people seem to be far more comfortable with.
> There's no difficulty. The abstraction simplifies the conception. There > is a real danger -- hard to avoid -- that the abstraction replaces in > one's mind the abstracted reality.
Another place where complex numbers seem more than just an abstraction is the evanescent wave: http://en.wikipedia.org/wiki/Evanescent_wave -- glen
glen herrmannsfeldt wrote:
> Jerry Avins <jya@ieee.org> wrote: > (snip, someone wrote) > >>> I don't see complex quantities as very much different from other >>> abstractions that people seem to be far more comfortable with. > >> There's no difficulty. The abstraction simplifies the conception. There >> is a real danger -- hard to avoid -- that the abstraction replaces in >> one's mind the abstracted reality. > > Another place where complex numbers seem more than just an abstraction > is the evanescent wave: > > http://en.wikipedia.org/wiki/Evanescent_wave
I think you confound the explanation for the explained phenomenon. Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;
Richard Dobson wrote:
> Jerry Avins wrote: >> glen herrmannsfeldt wrote: >> >> ... >> >>> Which in the end describes the interdependence between the real and >>> imaginary parts of the dielectric constant and/or index of refraction. >>> >>> Yes it is all phase shifts, but just saying that leaves a lot out. >> >> There's no doubt that "real and imaginary" is a good way to describe >> the phenomena. We gain understanding with the compact, high-level, >> abstraction, but we lose some understanding by taking the mathematical >> description to be the actual phenomenon. >> > > Isn't this kind of stating the obvious? Come to it, even "real numbers" > represented in a computer are really combinations of integers under the > hood (and below that, merely aggregations of binary states), so they are > no more real than imaginary numbers. All numbers are ultimately > mythological; the rest is simply counting, and very few things in the > physical world can be measured ~exactly~. Sometimes that matters - the > butterfly effect!
Most practitioners recognize that. Where complex numbers arise, they all of a sudden believe they are real! Jerry -- Engineering is the art of making what you want from things you can get. &#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;&#2013266095;