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Spectral density of signal

  1. Nov 13, 2013 #1
    1. The problem statement, all variables and given/known data
    First of all, I have to calculate power spectral density S(jw) of this signal:
    q7i2xd1mawd8yrpvz4kk.png
    It looks something like that:
    patgymqa16sv87b2ajrj.png
    t>=0


    2. Relevant equations



    3. The attempt at a solution
    It seems to me, that I can't use this standart formula:
    9qh6yuco66x2t4dzyqbe.png

    So.. I see that there is signal multiplication- exponent and sinus, but still not clear how to get spectral density S(jw).
    Any ideas?
     
  2. jcsd
  3. Nov 13, 2013 #2

    LCKurtz

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    This problem is definitely not in my comfort zone, but why can't you use$$
    \sin(wt) = \frac{e^{iwt}-e^{-iwt}}{2i}$$and use the standard formula with just exponentials?
     
  4. Nov 13, 2013 #3
    I have got limit from 0 to infinity. I guess, when I use infinity as limit, then I will get e^(inf) which is infinity and then all solution will be infinity. But speaking about the task.. in my opinion there have to use other method, but I don't know which.
     
  5. Nov 13, 2013 #4

    LCKurtz

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    But your ##S(t)## has ##e^{-at}## in it. If ##a## is large enough, won't that help you at your ##\infty## limits? I haven't worked it out because that's your job. Have you actually tried it? And your graphic shows only for ##t>0##.
     
  6. Nov 14, 2013 #5
    Also I noticed, that we can't use the same "w" in exponent and sinus.
    So, when I use this formula, then I get:
    0iqnd8kzk3q2qy6xq.png
    But what about range? It's from 0 to infinity, but I still don't understand some maths there. I guess, that when limit is 0, than all equation is 0. But what about infinity?
    -U*exp(-t*(a+jw)) = -U [when t=inf]
    What is cos(t*w0) and sin(t*w0), when t=inf?
     
  7. Nov 14, 2013 #6

    LCKurtz

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    Why do you guess everything is zero? Did you try it?

    What happens to terms like ##e^{-at}\sin(\omega_0 t)## and terms like ##e^{-at}e^{iwt}## when ##t\to\infty##?
     
  8. Nov 14, 2013 #7
    It's not 0, I tried it, but I found out, that we can't just use standart formula, because a sine wave spectral density has only a delta function at the carrier frequency since the signal contains just
    one spectral component namely the carrier frequency.
    So I need to do this task in other way. I think using this property:
    kn139p2flydp5413iik9.png

    When I try it with standart formula:
    n5mf96e2jza6t4iutbs.png
    But it don't give same signal as it was at the beggining, when I use inverse formula:
    wr63gmt9eetzr3f4m8uj.png
     
  9. Nov 14, 2013 #8

    rude man

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    1. What is the Fourier integral X(f) of this time function x(t)?
    2. What is the energy represented in x(t)? Hint: invoke Parseval's theorem.
    3. What is the formula for power, given X(f)? Recall that power = energy averaged over time.
    4. What then is the power spectrum G(f)? 2G(f), when integrated over all frequencies, gives you the power.

    Note that I'm using only positive frequencies, that's why it's 2G(f) instead of G(f). That factr also impacts your computation of time-averaged energy.
     
  10. Nov 14, 2013 #9

    LCKurtz

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  11. Nov 15, 2013 #10
    Are you saying, that it should be the correct answer?
    n5mf96e2jza6t4iutbs.png
    And here is the way, how I get it, if it's necessary:
    73wyzwuivwffq02hpn_thumb.jpg
    But, when I use this formula:
    wr63gmt9eetzr3f4m8uj.png
    (inverse formula of spectral density or inverse Fourier trasnsform formula)
    I should get back s(t) as it was at the beginng, but it's not. Or maybe I can't use inverse formula in this case?
    If we are speaking about requirement for the integral to converge, I'm not convinced about this method, because I'm not sure wheather this 'infinity' criterie is acceptable or not when we got sinus. I knpw that pure sine in the time domain evaluates to a delta function in the frequency domain, but in my case it is combinated with exponential term. It should leads to a spread of the energy in the frequency domain, as one conversant person said, but still I'm confused.

    1. If it's angular freqvency, than formula is almost the same as PSD forumula:
    aw0janux4rttm34gkb.png
    And then result shuld be the same:
    Result without limits:
    0iqnd8kzk3q2qy6xq.png
    Result with limits (0..inf):
    n5mf96e2jza6t4iutbs.png
    But same understanting, which I mentioned to LCKurtz.
    Maybe I have to use two Fourier function multiplication?
    Can you explain something more to clear my doubts?
    Then I'll continue the task, considering your points. And at the end, as I understand, after I use Parseval's theorem, to get power spectral density, I only need to square the modulus?
     
    Last edited: Nov 15, 2013
  12. Nov 15, 2013 #11

    rude man

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    PSD of x(t) is not simply the Fourier transform of x(t). Power immediately implies some kind of squared function. (For example, power dissipated in a 1 ohm resistor = V2, V is volts). So my hint is to consider squaring X(f), which is energy (Parseval), then finding the total energy divided by the total time which is the average power.

    PSD integrated from f = -∞ to +∞ is the average power in the signal. So two times the integral of PSD over the positive frequency range of interest is also the average power in the signal. That should get you the correct expression for PSD.

    P.S. I like to use f instead of ω.
    So my Fourier transform is X(f) = ∫x(t)exp(-jwt)dt with ω = 2πf and working with f from then on.
     
  13. Nov 15, 2013 #12
    I think, that we are speaking about different things. My fault.
    And I don't have to find PSD, if it's:
    a87ad2211534759928fb1955e31e2d2d.png
    And I don't have to find energy spectral density neither:
    c13c0db8c0bdbbeb25ff1a20f271a010.png
    It was non-understanding with language and translation..
    So.. I still don't know, how exactly I should say, but it's like Fourier transform. I have to calculate function in frequency domain.
    For example:
    041sk4atntuz3liixxi.png
    I have to calculate my task like in this example.
    And after that, also there are few extra tasks linked with S(jw), but now.. it means that it is Fourier transform and this is the answer of my s(t)?
    n5mf96e2jza6t4iutbs.png
     
  14. Nov 15, 2013 #13

    LCKurtz

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    You never responded to my questions, quoted below, in post #4.

    So I don't know if you understand or not why that is the answer.
     
  15. Nov 15, 2013 #14
    In that case it gives 0 and then all part f(inf) gives 0.
    There was doubts in my mind about Dirichlet's condition and sin(inf*w), but of course.. e^(-a*inf)=0, then it not matters.
    Now it's clear. Thanks.
     
  16. Nov 15, 2013 #15

    rude man

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    OK, if all you need is the Fourier transform of your time function then follow the advice in posts 2 and 4.
    The Fourier transform definitely exists.
     
  17. Nov 15, 2013 #16
    Also I have to calculate amplitude spectrum from S(jw).
    jwhar9akwy1dpxtykums.png
    But it's quite complicated s(jw) to seperate real and imaginary part.
    j9cdqesqm6grxogu51f.png
    Are there is no easiest method?
    So here is easy example:
    q566kxv9ia5w5r1kj7z.png
    What's the way in my S(jw) case? Must use this method?
     
  18. Nov 15, 2013 #17

    rude man

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    I agree, the math is messy.

    You should also consider that mutliplication in the time domain = convolution in the frequency domain.
     
  19. Nov 18, 2013 #18
    Task is done. Thanks.
     
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