# Fourier transform of general complex signal

I am trying to understand a sinusoid and its fourier transform.

Given an example sinusoid

$$Ze^{iwt}$$, how do i calculate the fourier transform of it?

How do i even represent this?

In euler form?

like $$Zcost(wt) + iZsin(wt)$$ ?

Been looking at online materials but still cannot understand

• What is it that you don't understand, exactly? Any online resource would tell you, at least from the engineering/physics perspective where you do some hand-waving, that it's a shifted impulse (scaled by some constant). Commented Jan 23, 2019 at 3:33
• @Metric like do i integrate the euler form? Commented Jan 23, 2019 at 3:35
• No, you integrate it directly, but this is probably from an engineering/physics class, so there's going to be some handwaving. Commented Jan 23, 2019 at 3:38
• To offer some assitance towards the classical approach of finding it, I suggest you look at the inverse fourier transform of the dirac delta. Commented Jan 23, 2019 at 3:39

Since I think I know what you're asking for, I'm going to hand-wave a little bit here.

Suppose the fourier transform $$F$$ of a function $$f$$ is defined as

$$F(f)(\omega) = \int_{-\infty}^\infty f(t) e^{-i \omega t} dt$$

and the inverse-fourier transform $$F^{-1}$$ of a function $$g$$ is defined as

$$F^{-1}(g)(t) =\frac{1}{2\pi}\int_{-\infty}^\infty g(\omega) e^{i \omega t} d \omega$$

Since

$$F(\delta)(\omega) = \int_{-\infty}^\infty \delta(t)e^{-i\omega t} d t = \int_{-\infty}^\infty \delta(t)e^{0} d t = \int_{-\infty}^\infty \delta(t) d t = 1$$

we have that

$$F^{-1}(1)(t) = \frac{1}{2\pi}\int_{-\infty}^\infty e^{i\omega t} d\omega = \delta(t)$$

That is,

$$\int_{-\infty}^\infty e^{i \omega t} d \omega = 2 \pi \delta(t) \tag 1$$

Now, if your function is $$f(t) = ze^{i \omega_0 t}$$, then using $$(1)$$ we have that its fourier transform is

$$F(f)(\omega) = \int_{-\infty}^\infty f(t) e^{-i \omega t} d t = z\int_{-\infty}^\infty e^{i (\omega_0-\omega) t} dt = 2 \pi z\delta(\omega_0 - \omega) = 2 \pi z\delta(\omega - \omega_0)$$