Fourier transform need help

Discussion in 'General Electronics Chat' started by vead, Sep 6, 2016.

  1. vead

    Thread Starter Active Member

    Nov 24, 2011
    621
    8
    Hello
    I need help to understand use of fourier transform. Fourier transform means the conversation of time domain into
    frequency domain. I don't understand actually what is happening. Here is example
    _20160812_142920.JPG
    Can someone help me with comparing the answer with given questions. Does it means 20hz signal at 2 second?
     
  2. MrAl

    Well-Known Member

    Jun 17, 2014
    2,439
    492
    Hello,

    Just to note, that last part should be sinc(w) not sin(w).

    You mean the interpretation of the result?
    It is the continuous 'spectrum' of frequencies that are considered present in the original signal.

    The whole idea is converting from the time domain to the complex frequency domain. It's a different way to represent functions that often reduces the math in some more complicated expressions while also giving us a new view on the nature of things.
     
  3. vead

    Thread Starter Active Member

    Nov 24, 2011
    621
    8
    Yes I know what is use of fourier transform . I feel I know only definitions I don't have real idae. Fourier transform used in signal and systems and I can solve example. I don't understand under which conditions only Fourier transform is use.
    Can we create simple example to understand use of fourier transform relates with circuit?
     
    Last edited: Sep 6, 2016
  4. MrAl

    Well-Known Member

    Jun 17, 2014
    2,439
    492
    Hi,

    Ok one use is in circuit analysis of a circuit like an RC circuit, where we have the R in series with C and in series with a voltage source of say 1vdc and we want the voltage response across C, and the circuit is void of energy until t=0 at which time the source is turned on, then later maybe the source is turned off and shorted out to vin=0 then.
    In the time domain this is done using the convolution integral but we are trying to avoid that by using the Fourier version in the frequency domain.

    First we find the impulse response of the circuit in the frequency domain which means we replace C with it's impedance zc=1/jwC where jwC=j*w*C. This is just the voltage divider formula:
    Vout/Vin=zc/(R+zc)=(1/jwC)/(R+1/jwC)=1/(j*w*R*C+1)

    and we might replace RC with T so we have:
    (1/T)/(j*w+1/T)

    and T here will be the constant T=RC.

    Next we would find the Fourier Transform of the input voltage which because it is a step at first it would be a different form than a pulse. Most of the time we would just look this up in a table of Fourier Transforms. In this case we find that it is:
    pi*d(w)+1/jw

    where 'd' there is the curly delta symbol used to represent an impulse.

    Next we multiply the two functions together because that is the equivalent to the time domain convolution:
    (pi*d(w)+1/jw)*((1/T)/(jw+(1/T)), where jw=j*w in the denominators.

    Next we might simplify that a little and try to find the Inverse Fourier Transform. When we find the inverse transform of that expression above, that gives us the complete time domain expression without using the convolution integral. Thus we have turned an integral equation into an algebraic equation which sometimes is easier to solve.
    The trick is to be able to find the Inverse transform which sometimes takes a little extra work like by using the partial fraction expansion of the final expression in the frequency domain.

    Using Laplace Transforms is almost the same for this circuit, where the impulse response is (1/T)/(s+1/T) except now the Laplace Transform of the unit step is just 1/s, so we multiply:
    (1/s)*(1/T)/(s+1/T)

    which simplified is:
    1/(s^2*T+s)

    and using inverse Laplace Transform techniques we get;
    f(t)=1-e^(-t/T), where T=RC again.

    So the real chore is to learn how to find the inverse transforms of either type. This is either done using some algebra and some tables, or use some software that finds the inverse transform for you. Tables and software should be available on the web.
     
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