Quantum Signal Processing (QSP) is an algorithmic process by which one represents a function $f$ on the unit interval as the upper left entry of a product of $SU(2)$ matrices parametrized by the input variable $x ∈ [0,1]$ and some “phase factors” $\{ψ_k\}_{k ≥ 0}$ depending on $f$. We show that, after a change of variables, QSP is actually the $SU(2)$-valued nonlinear Fourier transform, and the phase factors correspond to the nonlinear Fourier coefficients. By exploiting a nonlinear Plancherel identity and using some basic spectral theory, we extend QSP to represent any function $f$ satisfying a mild $\log$ integrability condition.