A systematic approach to parametrically analyze compressible time-periodic flows is proposed. Instead of time-stepping simulations, a Fourier–Galerkin strategy is adopted, which consists of the projection of the periodic solution onto a truncated Fourier series. Starting from the compressible Navier–Stokes equations, a truncated nonlinear problem coupling the 2N+1 fields, representing the Fourier discretization, is derived. This nonlinear problem is solved by Newton iteration and a set of efficient algorithms for the resolution of the arising inner linear systems is proposed. Compared to alternative methods, the formulation is free of any numerical constraint affecting performance, e.g. a CFL-like condition. It is also free of aliasing effects arising in other spectral techniques. The efficiency of the method is illustrated for two configurations where sound is radiated due to a flow instability, namely the flow around a circular cylinder and the flow through two successive apertures, i.e. the hole-tone configuration.