  In this article, we study the properties of the nonlinear Fourier spectrum in
order to gain better control of the temporal support of the signals synthesized
using the inverse nonlinear Fourier transform (NFT). In particular, we provide
necessary and sufficient conditions satisfied by the nonlinear Fourier spectrum
such that the generated signal has a prescribed support. In our exposition, we
assume that the support is a simply connected domain that is either a bounded
interval or the half-line, which amounts to studying the class of signals which
are either time-limited or one-sided, respectively. Further, it is shown that
the analyticity properties of the scattering coefficients of the aforementioned
classes of signals can be exploited to improve the numerical conditioning of
the differential approach of inverse scattering. Here, we also revisit the
integral approach of inverse scattering and provide the correct derivation of
the so called T\"oplitz inner-bordering algorithm. Finally, we conduct
extensive numerical tests in order to verify the analytical results presented
in the article. These tests also provide us an opportunity to compare the
performance of the two aforementioned numerical approaches in terms of accuracy
and complexity of computations.
