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Phase and amplitude spectrum of signal

  1. Oct 2, 2014 #1

    etf

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    Hi!

    1. The problem statement, all variables and given/known data


    My task is to calculate amplitude and phase spectrum of this signal:
    postavka1.jpg

    2. Relevant equations

    My idea is to calculate complex Fourier series of this signal, $$f(t)=\sum_{n=-\infty}^{n=+\infty}Fne^{j\frac{2n\pi t}{T}},$$ where $$Fn=\frac{1}{T}\int_{0}^{T}f(t)e^{-j\frac{2n\pi t}{T}}. $$Fn will be some complex number, which can be written as $$|Fn|e^{j\Theta n},$$ where $$|Fn|$$ is amplitude spectrum and $$\Theta n$$ is phase spectrum.

    3. The attempt at a solution

    I got $$Fn=\frac{E\tau }{T}\frac{\sin{(nw0\tau /2)}}{nw0\tau /2}e^{-jnw0(t1+\tau /2)}, $$ where w0=2*pi/T. We see that phase spectrum is $$\Theta n=-nw0(t1+\tau /2)$$ and amplitude spectrum is $$|Fn|=\frac{E\tau }{T}\frac{\sin{(nw0\tau /2)}}{nw0\tau /2}.$$ Now for some values of $$\tau, $$ $$E$$ and $$T$$ I can plot amplitude and phase spectrum as function of n?

     
    Last edited: Oct 2, 2014
  2. jcsd
  3. Oct 2, 2014 #2

    collinsmark

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    Yeah, that looks right to me. :)

    The assumption of course is that f(t) is periodic, and that same period repeats itself from -infinity to infinity. (Otherwise you need to use the Fourier transform instead.)

    By the way, notation wise, you forgot your [itex] dt [/itex] when setting up your integral, and I'm also assuming that several times when you wrote [itex] Fn [/itex] you actually meant [itex] F_n [/itex]. Other than stuff like that, it looks good to me.
     
  4. Oct 3, 2014 #3
    What about the value at n=0 ?
     
  5. Oct 3, 2014 #4
    There must be a dc term too.
     
  6. Oct 3, 2014 #5

    collinsmark

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    Yes, that's right. Perhaps one could try to evaluate the limit of [itex] \frac{\sin x}{x} [/itex] as [itex] x \rightarrow 0 [/itex].

    (Hint: l'Hôpital to the rescue)

    [Edit: btw, lazyaditya, recall that n goes from [itex] -\infty [/itex] to [itex] \infty [/itex]. So n = 0 is in the middle there somewhere. :)]
     
    Last edited: Oct 3, 2014
  7. Oct 3, 2014 #6
    L'hospital rule can't be applied to discrete sequence and since Fourier series of a periodic signal is discrete in nature thus dc term need to be calculated by keeping n=0 in the equation used for calculating Fourier series coefficient .
     
  8. Oct 3, 2014 #7

    collinsmark

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    Correct, if you want to be mathematically rigorous about it, one cannot technically use l'Hopital's rule for a discrete value, as you say. This thread was posted in the engineering section though. We're not the most rigorous bunch.

    [Edit, But yes, lazyaditya's method is preferred; it is mathematically better and less likely to get you into trouble in future problems. Treat n = 0 differently than all the other ns, if you would otherwise find yourself in a divide by 0 situation. Evaluate the integral substituting 0 for n in the original integral to produce [itex] F_0 [/itex] specifically. It turns out in this case the answer you get is the same either way (such as using the non-rigorous [itex] \frac{\sin x}{x} =1 [/itex], when x approaches 0) in this particular problem, but the rigorous approach is better in general.]
     
    Last edited: Oct 3, 2014
  9. Oct 3, 2014 #8

    etf

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    @collinsmark
    Yes, I forgot dt and I didn't know how to write "n" in index :)
    @lazyaditya
    I didn't notice that for n=0, Fn is undefined. With substitution n=0 in Fn I got F0=E*tau/T.

     
  10. Oct 3, 2014 #9

    etf

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    And that's general rule when Fn is undefined for some value n, I put n in original formula and calculate?
     
  11. Oct 3, 2014 #10

    etf

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    One more question: If I have $$|Fn|$$ and $$\Theta n$$ of some signal $$f(t),$$ and I want to calculate phase and amplitude spectrum of $$f(t+t1),$$ where t1 is some time shift, can I use phase and amplitude spectrum which I have already found for $$f(t)$$ to calculate phase and amplitude shift for $$f(t+t1)$$, or I have to start with calculation from beginning (Represent f(t+t1) in terms of complex Fourier series, calculate |Fn| etc)?
     
  12. Oct 3, 2014 #11
    Yes, you can use it.
     
  13. Oct 3, 2014 #12

    etf

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    So if $$f(t)=\sum_{n=-\infty }^{n=\infty }Fne^{jnw0t}, $$ $$f(t+t1)$$ will be $$\sum_{n=-\infty }^{n=\infty }Fne^{jnw0(t+t1)}$$ and it will have same amplitude and phase spectrum as $$f(t)$$? I'm sorry to bother you but I have exam very soon and I'm trying to learn it...
     
  14. Oct 3, 2014 #13
    Yes it is correct and you are not bothering anyone .
     
  15. Oct 3, 2014 #14

    collinsmark

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    Changing [itex] f(t) \rightarrow f(t + t_1)[/itex] will cause a difference in the phase response. Specifically, the new result will be the old result multiplied by a complex spiral, sometimes called a corkscrew function, of the form [itex] e^{j n \{ \mathrm{something} \} t_1} [/itex]. In other words, [itex] F_n \rightarrow F_n e^{j n \{ \mathrm{something} \} t_1} [/itex]. I'll let you work out what that something is.
     
    Last edited: Oct 3, 2014
  16. Oct 3, 2014 #15

    etf

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    I think I got it :) I created new thread with some problem which involve time shift...
     
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