Apporiate Fourier Coefficients for a Frequency Domain version to reduce Pulsar Timing Array datasets and detect a Gravitational Wave Background.

Pulsar timing array (PTA) datasets are complicated and require many computational resources to analyze for gravitational wave (GW). We present a method that can lower this complexity by reducing pulsar's timing residuals to their red noise Fourier coefficients component and using them in a frequency domain version of the optimal statistic (FDOS). This version of the optimal statistic is remarkably simple to calculate as it only requires Fourier coefficients and power spectral densities for each pulsar. We explored both single-pulsar Gibbs sampling (SPGS) and a red noise covariance matrix decomposition method for generating Fourier coefficients and tested their viability in this FDOS at lower and higher frequency bins. At lower frequencies both methods work well when the GW is above the white noise, but at frequencies where the signal is close to or below the white noise, we see a large bias appear in the FDOS for SPGS Fourier coefficients. In this presentation we show our results on realistic simulated datasets, discuss where this bias comes from.