  The nonlinear Schr\"odinger equation (NSE) is well-known to model an ideal
fiber-optic communication channel. Even though the NSE is a nonlinear evolution
equation, it can be solved analytically using a nonlinear Fourier transform
(NFT). Recently, there has been much interest in transceiver concepts that
utilize this NFT and its inverse to (de-)modulate data. Fast algorithms for the
(inverse) NFT are a key requirement for the simulation and real-time
implementation of fiber-optic communication systems based on NFTs. While much
progress has already been made for accelerating the forward NFT, less is known
on its inverse. The nonlinear Fourier spectrum comprises a continuous and a
discrete part in general, but so far only fast inverse NFTs for signals whose
continuous spectrum is null have been reported in the literature. In this
paper, we investigate the complementary case and propose the first fast inverse
NFT for signals whose discrete spectrum is empty. This is the case required by
transmitters in the recently proposed nonlinear inverse synthesis scheme, but
the problem also occurs in different application areas such as fiber Bragg
grating design. Our algorithms require only $\mathcal{O}(D\log^{2}D)$ floating
point operations to generate $D$ samples of the desired signal, which is almost
an order of magnitude faster than the current state of the art,
$\mathcal{O}(D^{2})$. This paper also quantifies, apparently for the first
time, the impact that truncating a signal in the time-domain has on the NFT.
