In satellite gravimetry, the recovery of lumped coefficients using the torus approach is a two-dimensional Fourier series problem on a torus. In this approach, the sampling on the torus is quasi-regular, which prevents the direct use of the Fast Fourier Transform (FFT), as FFT requires uniform sampling. At present, this is addressed by interpolating the data onto a regular grid on the torus, but this approach is suboptimal because interpolation introduces errors. Since the domain of the torus is the topological product of two circles, we begin our investigation with a circle. Several techniques are available for spectral analysis of non-uniformly sampled data, with the Lomb-Scargle periodogram being the most widely used. However, this method is generally applied to time-series observations, where the sampling domain is a line, and therefore requires careful modifications before being used in the circular domain. The objective of this study is to understand the spectral space of data sampled non-uniformly on a circle. We investigate the challenges posed by non-uniform sampling in spectrum estimation and discuss the fundamental differences in spectral analysis on a circle and a line. We outline the necessary modifications to adapt the Lomb-Scargle periodogram for the circular domain. We then estimate the spectrum of the simulated observations using both the Fourier series and the generalized Lomb-Scargle periodogram. We observe several spurious peaks in the estimated spectrum. We demonstrate that combining the false-alarm probability with the sampling spectrum enables the identification of spurious peaks and improves the spectral estimate for non-uniformly sampled data on a circle. This approach can further be used in the torus approach of gravity field recovery.