# Inverse Fourier Transform for band limited, complex transfer function

I have the following band limited, complex transfer function. It describes the reflection response of a dielectric slab in the frequency domain.

$$H(\omega)=\mathrm{rect}\biggl(\frac{\omega-\omega_{c}}{\Delta\omega}\biggl) \biggl(\frac{r(1-e^{j2\beta d})}{1-r^2 e^{j2\beta d}}\biggl)$$

in which $\omega_c=2\pi f_c$ is center frequency, $\Delta\omega=2\pi(f_{max}-f_{min})$is the bandwidth, $d$ is the thickness of the slab, $\beta=kn=\frac{2\pi}{\lambda}n_1=\frac{\omega}{c_0}n$ is the phase constant and $r$ the reflection fresnel coefficient between medium $n_0$ and $n_1$.

I would like to tranform $H(\omega)$ into the time domian $H(t)$ using the inverse fourier transform. I know from simulations that the time domain response is complex, but for a detail look I need to know the analytic expression $H(t)$. Furthermore I tried to use tables of fourier transform pairs but could not achieve a reliable solution.

I would be very thankful for any advises to attack this problem. Its probably quite easy but somehow I am missing an important step.

EDIT:

I gave it try, could someone double check my approach?

$$h(t)=\mathscr{F}^{-1}\biggl(\mathrm{rect}\biggl(\frac{\omega-\omega_{c}}{\Delta\omega}\biggl) \biggl)*\mathscr{F}^{-1}\biggl(r-re^{j2\beta d}\biggl)*\mathscr{F}^{-1}\biggl(\frac{1}{1-r^2 e^{j2\beta d}}\biggl)$$

Using frequency shift and scaling relation $$(1)\quad\mathscr{F}^{-1}\biggl(\mathrm{rect}\biggl(\frac{\omega-\omega_{c}}{\Delta\omega}\biggl) \biggl)=\frac{|\Delta\omega|}{\sqrt{2\pi}}\mathrm{sinc}(\frac{\Delta\omega t}{2})e^{-j\omega_ct}$$

Using time domain shift $$(2)\quad\mathscr{F}^{-1}\biggl(r-re^{j2\beta d}\biggl)=\sqrt{2\pi}r\delta(t)-\sqrt{2\pi}r\delta(t-\tau n)$$

Using $\frac{1}{1-q}=\sum\nolimits_{k=0}^{\infty}q^k$ for $|q|<1$

$$(3)\quad\mathscr{F}^{-1}\biggl(\frac{1}{1-r^2 e^{j2\beta d}}\biggl)=\sum_{k=0}^{\infty}\sqrt{2\pi}r^{2k}\delta(t-k\tau n)$$

The Convolution of (1) and (2) results in sinc at same position at which the dirac distributions are placed. Is the phase term of the sincs affected by the time shift of the dirac distributions, or does it just cause a time shift of the sinc?

• What I know, that the rect function is equal to scanning with si-pulse in time domain, which results to a si pulse in the time domain with a complex phase $si(\Delta\omega t) e^{j\omega_c t}$ factor caused by the shift $\omega_c$. The transfer function should add another phase shift though – Pavel Jul 9 '18 at 18:34