My goal is to understand a derivation in an academic article that I am having trouble wrapping my head around.
A voltage term is defined as a linear superposition of steady periodic harmonics $V(\tilde{i}, \tilde{\omega}, t) = \frac{1}{2}\sum\limits_{k=0}^\infty V_k(\tilde{i}, \tilde{\omega} ) *exp(jk\tilde{\omega} t) + V_{-k}(\tilde{i}, \tilde{\omega})*exp(-jk \tilde{\omega} t)$
where $\tilde{\omega}$ is frequency and $\tilde{i}$ is amplitude of the current waveform. $t$ is time. $j$ is the imaginary number $\sqrt{-1}$.
The voltage coefficient are complex Fourier coefficients corresponding to each harmonic $V_{\pm k} = V_k' \pm j V_k''$
The prime denotes the real part and the double prime denotes the imaginary part.
$V(\tilde{i}, \tilde{\omega}, t)$ is multiplied by a Gaussian windowing function, $ W(t) = exp(-(\frac{\tilde{\omega} t}{2 \pi b})^2) $
Then the combined term, $V(\tilde{i}, \tilde{\omega}, t)*W(t) $ has its Fourier Transform taken
The result, according to the paper, is
$V(\tilde{i}, \tilde{\omega}, \omega) = \frac{1}{2}\sum\limits_{k=0}^\infty V_k'(G_k(\omega) + G_{-k}(\omega)) + V_k''(D_k(\omega) - D_{-k}(\omega)) + \frac{j}{2}\sum\limits_{k=0}^\infty V_k''(G_k(\omega)-G_{-k}(\omega)) + V_k'(-D_k(\omega)-D_{-k}(\omega)) $
where $ G_k(\omega) = \frac{b\sqrt{\pi}}{\tilde\omega}exp(\frac{-(\omega - k\tilde\omega)^2\pi^2b^2}{\tilde\omega^2}) $ and
$ D_k(\omega) = \frac{b\sqrt{\pi}}{\tilde\omega}exp(\frac{-(\omega - k\tilde\omega)^2\pi^2b^2}{\tilde\omega^2})*erfi(\frac{(\omega-k\tilde\omega)\pi b}{\tilde\omega}) $
here is where I get stuck
Let's just start with the first term, $V_k'(\tilde{i},\tilde\omega)*exp(jk\tilde\omega t)* exp(-(\frac{\tilde{\omega} t}{2 \pi b})^2)$ and take the Fourier Transform of that.
$F[W(t)] = \sqrt2\pi*exp(-(\pi^2b^2\omega^2)/\tilde\omega^2)*\frac{b}{\tilde\omega} $
then $F[V_k'(\tilde{i},\tilde\omega)*exp(jk\tilde\omega t)] $ = $V_k'(\tilde{i},\tilde\omega)\sqrt{2\pi}*\delta(k\tilde\omega+\omega) $ when just taking the Fourier cosine transform for the even part.
I can't get how to translate those delta functions to the resulting derivation in the paper. Any help on what I'm not understanding?