  Given the wide range of applications of time-domain spectroscopy, and
particularly THz time-domain spectroscopy, the modelling of a probe pulse
propagating through a multilayered structure is often required. Due to the fact
that the multilayers are usually grown on a substrate much thicker than the
other layers, the transmission of a probe pulse includes a series of echo
pulses caused by the multiple reflections at the substrate interfaces.
Experiments often measure the time profile and construct the transmitted
spectrum only from the first transmitted pulse. Due to the fact that typical
substrates lead to times of crossing comparable to the spectral bandwidth, the
first transmitted pulse's spectrum and the full transmitted spectrum can be
importantly different. It is therefore important to theoretically model the
transmission without the echo, to be able to directly compare with experimental
results. Here we propose a method to elegantly and easily theoretically remove
the echo from the transmission spectrum, without Fourier transforming the
signal twice. The spectrum of the transmitted pulse without echo will be
produced analytically without additional numerical steps, reducing
computational time and program complexity when dealing with larger scale
computing.
