# Complex Exponential Signals

Consider a continous-time signal

*x*(*t*) = *K e ^{iω0
t}* ,

for some real constants *K *and* ω *_{0}.
Its CTFT is

*X*(*ω*) = *K
* ∫_{
(− ∞
to ∞
)}* e ^{−i}*

^{(}

^{ω-ω0}^{)}

^{ t}*dt*

which is again not easy to evaluate. This integral is mathematically very subtle. The answer is

∀*ω
*∈*
Reals*, *X*(*ω*)
= 2*π* *K δ
*(*ω -* *ω*_{0}
)

where *δ * is the Dirac delta function.
What this says is that a complex exponential in the time domain is concentrated
at one frequency in the frequency domain (which should not be surprising). We
can verify this answer easily by evaluating the inverse CTFT,

*x*(*t*) = (1/2*π*)
∫_{
(− ∞
to ∞
)}* **X*(*ω*
)*e ^{iω t}*

*dt*

= *K* ∫_{
(− ∞
to ∞
)}* **δ
*(*ω -* *ω*_{0}
) *e ^{iω t}*

*dt*

*= K e ^{iω0
t}*

*,*

where the final step follows from the **sifting property** of the Dirac
delta function.