Simulating a Gaussian stochastic gravitational wave background signal in pulsar timing arrays

We revisit the theoretical modeling and simulation of a Gaussian stochastic gravitational wave background (SGWB) signal in a pulsar timing array (PTA). We show that the correlation between Fourier components of pulsar timing residuals can be expressed using transfer functions; that are indicative of characteristic temporal correlations in a SGWB signal observed in a finite time window. These transfer functions, when convolved with the SGWB power spectrum and spatial correlation (Hellings \& Downs curve), describe the variances and correlations of the pulsar timing residuals' Fourier coefficients. The convolutions are the exact frequency- and Fourier-domain representations of the time-domain covariance function. We derive explicit forms for the transfer functions for unpolarized and circularly polarized SGWB signals. We validate our results by comparing Gaussian theoretical expectation values with standard simulations based on point sources and our own covariance-matrix-based approach. The unified frequency- and Fourier-domain formalism provides a robust foundation for future PTA precision analyses and highlights the importance of temporal correlations in interpreting GW signals.