The CT complex exponential is NOT periodic

[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 e^at with C and a complex is a periodic signal. I fail to see how.

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

Plugging into x(t) = C e^at, and using euler's formula

x(t) = |C| e^rt * [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 e^rt 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?

 

[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

 

[yungman]

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 e^at with C and a complex is a periodic signal. I fail to see how.

Let C = |C| e^jα (exponential form of a complex number)
and a = r + jω (rectangular form)
Are these formulas from the book? It does not make sense as
$e^{ja}=e^{jr}e^{-ω}\;$ according to the given formula.

Plugging into x(t) = C e^at, and using euler's formula
x(t) = |C| e^rt * [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 e^rt 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?

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?

 

[fishingspree2]

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

eⁱθ = cosθ + i sinθ

go well

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?

x(t) = C e^at with C and a complex numbers

If C is a complex number, it can be written as |C| e^jα 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 e^at
= |C| e^jα * e^(r+jω)t
= |C| e^jα * e^rt * e^jωt
= |C| e^rt *e^j(α+ωt)
= |C| e^rt *[ cos(α+ωt) + j sin(α+ωt) ]

This is not periodic...

 

[yungman]

x(t) = C e^at with C and a complex numbers

If C is a complex number, it can be written as |C| e^jα 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 e^at
= |C| e^jα * e^(r+jω)t
= |C| e^jα * e^rt * e^jωt
= |C| e^rt *e^j(α+ωt)
= |C| e^rt *[ cos(α+ωt) + j sin(α+ωt) ]

This is not periodic...

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!