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The CT complex exponential is NOT periodic

  1. Feb 9, 2012 #1
    I'm taking a signals and systems class and the textbook (Signals and systems by Oppenheim) says the CT complex exponential of the form x(t) = C eat with C and a complex is a periodic signal. I fail to see how.

    Let C = |C| e (exponential form of a complex number)
    and a = r + jω (rectangular form)

    Plugging into x(t) = C eat, and using euler's formula

    x(t) = |C| ert * [cos(α+ωt) + j sin(α+ωt)]

    What's inside the brackets is obviously periodic but the whole function is obviously not because of the ert term...

    A function is periodic if there exists a T such as x(t) = x(t+T) for any t, and obviously there isn't any such T for the CT complex exponential...

    Any clarifications?
     
  2. jcsd
  3. Feb 9, 2012 #2
    I can only suggest you revise Euler's formulae. You have one too many exponentials in your expression.

    eiθ = cosθ + i sinθ

    go well
     
  4. Feb 9, 2012 #3
    If the formula is from the book, I don't see how. If it is not from the book, can you type out the exact formulas given by the book?
     
    Last edited: Feb 9, 2012
  5. Feb 9, 2012 #4
    x(t) = C eat with C and a complex numbers

    If C is a complex number, it can be written as |C| e where |C| is the magnitude of C (nothing fancy here, its just the exponential form of a complex number)

    If a is a complex number, it can be written as r + jω (nothing fancy again, this is the rectangular form of a complex number)

    Plugging back
    x(t) = C eat
    = |C| e * e(r+jω)t
    = |C| e * ert * ejωt
    = |C| ert *ej(α+ωt)
    = |C| ert *[ cos(α+ωt) + j sin(α+ωt) ]

    This is not periodic...
     
  6. Feb 9, 2012 #5
    OK, I miss read [itex]\alpha[/itex] as "a". That's cause the confusion. It looks good to me!!!

    [tex]x(t)=C e^{at}=|C|e^{j\alpha}e^{rt+j\omega t}=|C|e^{j\alpha}e^{rt}e^{j\omega t}=|C|e^{rt} e^{j(\alpha+\omega t)}[/tex]

    [tex] e^{j(\alpha + \omega t)}=\cos (\alpha + \omega t) + j\sin(\alpha + \omega t) [/tex]
    [tex]\Rightarrow \;x(t)=|C|e^{rt}[\cos (\alpha + \omega t) + j\sin(\alpha + \omega t) ][/tex]

    If r is a negative number, the amplitude decrease with time and the signal decay.
    If r is positive, then the signal grow with time and most likely you got a problem!!!
     
    Last edited: Feb 9, 2012
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