PX275 - H6 - fourier transforms in the time domain

considering a time dependent function, $f (t)$

f (t) = {\begin{cases} 0 & t < 0 \\ e^{i ω t} e^{- γ t} & t \geq 0 \end{cases}

so, it has an oscillating term, and a damping term

a useful function is the heavyside (step) function:

H (t) = {\begin{cases} 1 & t \geq 0 \\ 0 & t < 0 \end{cases}

rewriting the amplitude as:

f (t) = H (t) e^{i ω_{0} t} e^{- γ t}

so, the total power:

I_{t o t} = \int_{- \infty}^{\infty} | f (t) |^{2} d t = \frac{1}{2 π} \int_{- \infty}^{\infty} | \tilde{f} (ω) |^{2} d ω

this is parseval's theorem in time domain: $f (t) \to \tilde{f} (ω)$ instead of $f (x) \to \tilde{f} (k)$
similarly, the fourier transform is:

\tilde{f} (ω) = \int_{- \infty}^{\infty} e^{- i ω t} f (t) d t

so, $x$ [length] $\to k$ [length] $^{- 1}$ , and $t$ [time] $\to ω$ [time] $^{- 1}$ , which is the frequency
for the given case:

\begin{aligned} \tilde{f} (ω) & = \int_{- \infty}^{\infty} e^{- i ω t} H (t) e^{i ω_{0} t} e^{- γ t} d t \\ = \int_{0}^{\infty} e^{i (ω_{0} - ω) t - γ t} d t \\ = {[\frac{e^{i (ω_{0} - ω) t - γ t}}{i (ω_{0} - ω) - γ}]}_{0}^{\infty} \\ = \frac{- 1}{i (ω_{0} - ω) - γ} \end{aligned}

now, for $| \tilde{f} (ω |^{2} = \tilde{f} (ω) {\tilde{f}}^{*} (ω)$

\begin{aligned} \tilde{f} (ω) & = \frac{- 1}{i (ω_{0} - ω) - γ} [\frac{- γ - i (ω_{0} - ω)}{- γ - i (ω_{0} - ω)}] \\ = \frac{γ + i (ω_{0} - ω)}{γ^{2} + (ω_{0} - ω)^{2}} \\ ⟹ {\tilde{f}}^{*} (ω) & = \frac{γ - i (ω_{0} - ω)}{γ^{2} + (ω_{0} - ω)^{2}} \\ ∴ | f (ω) |^{2} & = \frac{γ^{2} + (ω_{0} - ω)^{2}}{[γ^{2} + (ω_{0} - ω)^{2}]^{2}} \\ = \frac{1}{γ^{2} + (ω_{0} - ω)^{2}} \\ I_{t o t a l} & = \frac{1}{2 π} \int_{- \infty}^{\infty} \frac{1}{γ^{2} + (ω_{0} - ω)^{2}} d ω \end{aligned}