Hello Forum, everyone is familiar with the Fourier transform and its importance. Its independent variable, the angular frequency w, is a measurable physical quantity. What about the Laplace transform? What is, conceptually, the advantage of taking the Laplace transform of a signal instead of the Fourier transform? I know that by looking at its zeros and poles we can assess the stability of the system for example.... is that not possible with the FT alone? Does the Laplace transform show information on the system that the FT does not? thanks fisico32
Laplace Transform vs Fourier transform
Started by ●October 27, 2009
Reply by ●October 27, 20092009-10-27
On Oct 27, 8:39=A0pm, "fisico32" <marcoscipio...@gmail.com> wrote:> Hello Forum, > > everyone is familiar with the Fourier transform and its importance. > Its independent variable, the angular frequency w, is a measurable > physical quantity. > > What about the Laplace transform? What is, conceptually, the advantage of > taking the Laplace transform of a signal instead of the Fourier transform=?> I know that by looking at its zeros and poles we can assess the stability > of the system for example.... is that not possible with the FT alone? > > Does the Laplace transform show information on the system that the FT doe=s> not? > > thanks > fisico32The fourier transform is a special case of the laplace transform (as I understand it) The Laplace transform correlates a given waveform with every possible (exponential x sinusoidal) wave. The Fourier transform correlates it with every possible sinusoidal wave. Here is the one point of the Laplace transrom: If you correlate a signal with a "decaying" exponential, and the resulting correlation is unbounded, then you have a stability problem. (if you have pole in right hand plane it means that your correlation became infinite with an exponentially decaying sinusoid) It is OK to correlate your system with an exponentially "increasing" sinusoidal wave and have an unbounded result. So you take your signal and correlate it with every possible exponentially increasing and exponentially decreasing sinusoidal wave (sinusoidal means both exponential cosines and exponential sines) and see what happens. One thing to note, when you get a pole in the S plane, that really sets the boundary of instability. There will be an infinite number of poles to the left of the leftmost "official" pole. The fourier transform only correlates the f(t) with pure (non exponential) sinusoids. This is what we typically want to know about when thinking of the frequency response of a circuit (or system) The fourier transform is especially useful because of the convolution theorem, that says multiplication in the frequency domain is equiv to convolution in the time domain. The response of the circuit is the impulse response convolved with the sinusoidal input signal. This is a piece of cake when you take the fourier transform of the system to immediately get the frequency response. In conclusion Fourier transform is good tool for frequency response Laplace transform is good tool for stability analysis. PS Laplace also is a good tool for solving differential equations becuase the S-plane is a mapping of every possible solution to an ordinary differential equation. caveat: this may all be BS so do your own due diligence :-)
Reply by ●October 27, 20092009-10-27
fisico32 wrote:> Hello Forum, > > everyone is familiar with the Fourier transform and its importance. > Its independent variable, the angular frequency w, is a measurable > physical quantity.Not only don't I need review, your statement is too particular to be accurate. The independent variable of a Fourier transform can be angular or spatial frequency, or any physical quantity. The near and far fields of electromagnetic radiation are related by a Fourier transform.> What about the Laplace transform? What is, conceptually, the advantage of > taking the Laplace transform of a signal instead of the Fourier transform? > I know that by looking at its zeros and poles we can assess the stability > of the system for example.... is that not possible with the FT alone?Sure, but why use a sledge hammer when a tack hammer serves?> Does the Laplace transform show information on the system that the FT does > not?Depending on the information, it can show the information more easily. Think of the Laplace transform as a one-sided Fourier transform, eminently suitable for causal functions. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●October 28, 20092009-10-28
On Oct 27, 6:39�pm, "fisico32" <marcoscipio...@gmail.com> wrote:> Hello Forum, > > everyone is familiar with the Fourier transform and its importance. > Its independent variable, the angular frequency w, is a measurable > physical quantity. > > What about the Laplace transform? What is, conceptually, the advantage of > taking the Laplace transform of a signal instead of the Fourier transform? > I know that by looking at its zeros and poles we can assess the stability > of the system for example.... is that not possible with the FT alone? > > Does the Laplace transform show information on the system that the FT does > not? > > thanks > fisico32Fourier transform only holds for s=jw. The Laplace TF holds for all complex frequency sigma+jw. There are some functions which have no FT of meaning. The Laplace TF is usually used for solving ODEs and for transfer functions. Hardy
Reply by ●October 29, 20092009-10-29
just to add to what brent and Jerry and Hardy said... On Oct 27, 9:39�pm, "fisico32" <marcoscipio...@gmail.com> wrote:> > everyone is familiar with the Fourier transform and its importance. > Its independent variable, the angular frequency w, is a measurable > physical quantity.there's a part of that angular frequency that is mathematical, but not physical. how do you physically measure negative frequency? and negative frequency is in the F.T.> > What about the Laplace transform? What is, conceptually, the advantage of > taking the Laplace transform of a signal instead of the Fourier transform?it's hard to derive the FT of the unit step function without a little bit of hand-waving.> I know that by looking at its zeros and poles we can assess the stability > of the system for example.... is that not possible with the FT alone? > > Does the Laplace transform show information on the system that the FT does > not?there is a mathematical concept relating the two called "analytical extension". if a function of a complex variable is "analytic", it turns out that if you know the values of the function on a certain boundary (like the jw axis, where the LT is equal to the FT), then that gives you sufficient information to define the function at all points away from the boundary. so, for decently-behaved complex H (jw), knowing the behavior of H(jw) for all real w is sufficient to know H(s) for s having a non-zero real part. so, for decently-behaved H(s), the LT has no additional information than the FT can tell you. r b-j
Reply by ●October 29, 20092009-10-29
robert bristow-johnson wrote: ...> there's a part of that angular frequency that is mathematical, but not > physical. how do you physically measure negative frequency? and > negative frequency is in the F.T.Who said? You can just as well apply the negative sign in exp(-jwt) to time. :-) ... Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●October 29, 20092009-10-29
On Oct 29, 10:52�am, Jerry Avins <j...@ieee.org> wrote:> robert bristow-johnson wrote: > > � �... > > > there's a part of that angular frequency that is mathematical, but not > > physical. �how do you physically measure negative frequency? �and > > negative frequency is in the F.T. > > Who said? You can just as well apply the negative sign in exp(-jwt) to > time. :-) > > � �... >we've been here before. for the record, i actually think that the term "imaginary" for imaginary numbers is apt, appropriate, and descriptive. it used to be that i could use Google Groups to find old threads and posts, and there used to be one where we all had a free-for-all arguing whether or not the semantic differentiation between "real numbers" and "imaginary numbers" is good (with some folks claiming that they're all just as real as another). i still have the same position i had before in that debate Jerry, i was down in NYC a couple weeks ago (AES convention). and i was driving (had a horrible car tow experience, the bastards). i wish i would have made an arrangement to drive out to Edison (i can't remember exactly the town you're in) and visit. i'm sorry it didn't occur to me then. L8r, r b-j
Reply by ●October 29, 20092009-10-29
On 2009-10-29 14:20:36 -0300, robert bristow-johnson <rbj@audioimagination.com> said:> On Oct 29, 10:52�am, Jerry Avins <j...@ieee.org> wrote: >> robert bristow-johnson wrote: >> >> � �... >> >>> there's a part of that angular frequency that is mathematical, but not >>> physical. �how do you physically measure negative frequency? �and >>> negative frequency is in the F.T. >> >> Who said? You can just as well apply the negative sign in exp(-jwt) to >> time. :-) >> >> � �... >> > > we've been here before. for the record, i actually think that the > term "imaginary" for imaginary numbers is apt, appropriate, and > descriptive.Like surds, irrational, etc. Mathematical terminology from the middle ages tended to be very unkind to all sorts of new fangled things as told by any history of mathematics book. That was back when solving a quadratic equation was a big deal and solution to quartics was a closely guarded trade secret. It is too bad that 500 year old attitudes are taken seriously by some current folks.> it used to be that i could use Google Groups to find old threads and > posts, and there used to be one where we all had a free-for-all > arguing whether or not the semantic differentiation between "real > numbers" and "imaginary numbers" is good (with some folks claiming > that they're all just as real as another). i still have the same > position i had before in that debate > > Jerry, i was down in NYC a couple weeks ago (AES convention). and i > was driving (had a horrible car tow experience, the bastards). i wish > i would have made an arrangement to drive out to Edison (i can't > remember exactly the town you're in) and visit. i'm sorry it didn't > occur to me then. > > L8r, > > r b-j
Reply by ●October 29, 20092009-10-29
On Oct 29, 1:38�pm, Gordon Sande <g.sa...@worldnet.att.net> wrote:> On 2009-10-29 14:20:36 -0300, robert bristow-johnson > <r...@audioimagination.com> said: > > > > > On Oct 29, 10:52�am, Jerry Avins <j...@ieee.org> wrote: > >> robert bristow-johnson wrote: > > >> � �... > > >>> there's a part of that angular frequency that is mathematical, but not > >>> physical. �how do you physically measure negative frequency? �and > >>> negative frequency is in the F.T. > > >> Who said? You can just as well apply the negative sign in exp(-jwt) to > >> time. :-) > > >> � �... > > > we've been here before. �for the record, i actually think that the > > term "imaginary" for imaginary numbers is apt, appropriate, and > > descriptive. > > Like surds, irrational, etc. Mathematical terminology from the middle ages > tended to be very unkind to all sorts of new fangled things as told by > any history of mathematics book. That was back when solving a quadratic > equation was a big deal and solution to quartics was a closely guarded > trade secret. It is too bad that 500 year old attitudes are taken seriously > by some current folks.am i one of them current folk, Gordon? r b-j
Reply by ●October 29, 20092009-10-29
On Thu, 29 Oct 2009 17:38:31 GMT, Gordon Sande <g.sande@worldnet.att.net> wrote:>On 2009-10-29 14:20:36 -0300, robert bristow-johnson ><rbj@audioimagination.com> said: > >> On Oct 29, 10:52�am, Jerry Avins <j...@ieee.org> wrote: >>> robert bristow-johnson wrote: >>> >>> � �... >>> >>>> there's a part of that angular frequency that is mathematical, but not >>>> physical. �how do you physically measure negative frequency? �and >>>> negative frequency is in the F.T. >>> >>> Who said? You can just as well apply the negative sign in exp(-jwt) to >>> time. :-) >>> >>> � �... >>> >> >> we've been here before. for the record, i actually think that the >> term "imaginary" for imaginary numbers is apt, appropriate, and >> descriptive. > >Like surds, irrational, etc. Mathematical terminology from the middle ages >tended to be very unkind to all sorts of new fangled things as told by >any history of mathematics book. That was back when solving a quadratic >equation was a big deal and solution to quartics was a closely guarded >trade secret. It is too bad that 500 year old attitudes are taken seriously >by some current folks.Middle Ages? It wasn't that long ago that simple negative numbers were not universally accepted by mathematicians. A brief discussion at: http://nrich.maths.org/5961 -- Rich Webb Norfolk, VA
