This work presents a Fourier analysis framework for the non-interactive source simulation (NISS) problem. Two distributed agents observe a pair of sequences $X^d$ and $Y^d$ drawn according to a joint distribution $P_{X^dY^d}$. The agents aim to generate outputs $U=f_d(X^d)$ and $V=g_d(Y^d)$ with a joint distribution sufficiently close in total variation to a target distribution $Q_{UV}$. Existing works have shown that the NISS problem with finite-alphabet outputs is decidable. For the binary-output NISS, an upper-bound to the input complexity was derived which is $O(\exp\operatorname{poly}(\frac{1}{\epsilon}))$. In this work, the input complexity and algorithm design are addressed in several classes of NISS scenarios. For binary-output NISS scenarios with doubly-symmetric binary inputs, it is shown that the input complexity is $\Theta(\log{\frac{1}{\epsilon}})$, thus providing a super-exponential improvement in input complexity. An explicit characterization of the simulating pair of functions is provided. For general finite-input scenarios, a constructive algorithm is introduced that explicitly finds the simulating functions $(f_d(X^d),g_d(Y^d))$. The approach relies on a novel Fourier analysis framework. Various numerical simulations of NISS scenarios with IID inputs are provided. Furthermore, to illustrate the general applicability of the Fourier framework, several examples with non-IID inputs, including entanglement-assisted NISS and NISS with Markovian inputs are provided.