Is this the right way to view the Fourier/Laplace Transforms?

Discussion in 'Math' started by spaceflame, May 5, 2013.

  1. spaceflame

    Thread Starter New Member

    May 5, 2013
    1
    0
    I have always had doubts about this and I wanted to confirm if my reasoning is right or wrong.

    1) A Fourier Transform basically tells us that any LIMITED signal can be converted into an infinite sum of sinosoids of diferent amplitudes and frequencies: [​IMG]

    2) A Transfer Function using FT tells us 2 things: the amplitude of each e^jwt term and the phase for each.

    3) Since the Fourier Transform only works for limited signals (in amplitude and time) we use the Laplace Transform for ALL signals, which is basicaly the same as a FT, but now s=σ+jω. In other words, it tells us that any signal can be written as a sum of sinosoids TIMES an exponencial term (e^σt)
    http://upload.wikimedia.org/math/3/6/1/36193c7b97db891595cec12dd627b814.png

    4) A LT transfer function is different: it tells us how the amplitude of each term e^st changes depending on the σ we choose to use. In other words, we consider only the first term of the sum, in which ω=0, and thus e^s=e^σ and analyse how the magnitude of this first term changes as we change σ

    I am very sorry if I was hard to understand, but right now this is how I understand these transforms.
    Is this wrong or not?
    Thank you
     
  2. gootee

    Senior Member

    Apr 24, 2007
    447
    50
    Fourier transform is only about steady-state. Laplace has that but also includes the transient response.

    Laplace transform enables us to convert a time-domain differential equation into an algebraic equation (in the frequency domain), solve it, then convert the solution back to the time domain. Or, we can leave it in the frequency domain and have our transfer function.

    Fourier transform, by definition, requires periodic signal, extending infinitely in time in both directions. (But in practice, with FFT, we can use it on a finite-time non-periodic signal by fooling the transform, by pretending the signal is in a "circular buffer".)
     
  3. WBahn

    Moderator

    Mar 31, 2012
    17,788
    4,808
    Your claim that the Fourier transform requires the time domain signal to be limited in both time and amplitude is incorrect. As gootee says, by definition the Fourier transform requires a periodic signal which, by definition, is infinite in duration. Also, there is nothing preventing the function from containing a finite number of impulses within each period, so it doesn't have to be limited in amplitude, either.
     
  4. Walks-In-Storms

    New Member

    Aug 12, 2013
    16
    3
    Has anyone here tried to do Fourier Transforms with a TI-83 Plus calculator? I've just signed on to Matlab, but it's going to take me days to wade through all of the confounded tutorials, and I'm having to hold up my designing and building for calculations. I'm (very) new here, and having to re-learn my engineering math into the bargain (math computations are one hell of a lot cheaper than trial and error with materials at today's prices). That applies to both a model or the full-sized final construction (I've already build one, photos of which I posted here elsewhere). I hope I'm not in the wrong place, but what I want to has to do with design of a Vertical Axis Wind Turbine. The turbine design I'm investigating is that with contra-rotating airfoils, and I want to determine whether putting one turbine inside the other is feasible. I want to model each turbine (one will turn the rotors, while the other turns the stator) mathematically, in order to learn what I need to know. Airflow through the "solidity" (frontal area of the turbine) rises and falls as each blade passes through that area, and will vary even more with one turbine inside the other. In fact, it may be that the inside turbine won't turn at all, the airflow having been made so turbulent by the outside turbine's blades.

    (Building one turbine atop the other isn't feasible for me - or even possible [I have to put it on the roof, take it down, and do maintenance all by my lonesome]).

    I'll almost certainly build a small model, but I had great luck in saving money by use of math when building the first turbine, and I'd like to do the same here (I used up about twenty notebooks and reams of paper that time, the reason I've turned to the calculator).

    I'd also like to learn a method of analyzing or modeling the several forces involved in Faraday's Law, that having to do with the new permanent-magnet generator I'm about to build. It would be very handy if I could enter the several factors into the calculator, then make a graph by which to compare various magnet strengths, flux at turbine and rotor speeds, et cetera (I've some long since experience with the TI-83, but none whatever with Matlab).

    Any comments will be much appreciated. Thanks!

    Anybody ever try Fourier analysis on a TI-83 calculator?
     
    Last edited: Aug 20, 2013
Loading...