# The CT complex exponential is NOT periodic

1. Feb 9, 2012

### fishingspree2

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. Feb 9, 2012

### Studiot

I can only suggest you revise Euler's formulae. You have one too many exponentials in your expression.

eiθ = cosθ + i sinθ

go well

3. Feb 9, 2012

### yungman

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
4. Feb 9, 2012

### fishingspree2

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...

5. Feb 9, 2012

### yungman

OK, I miss read $\alpha$ as "a". That's cause the confusion. It looks good to me!!!

$$x(t)=C e^{at}=|C|e^{j\alpha}e^{rt+j\omega t}=|C|e^{j\alpha}e^{rt}e^{j\omega t}=|C|e^{rt} e^{j(\alpha+\omega t)}$$

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

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