This paper gives the finite-length analysis of a spectrum estimation method for the case that the samples are obtained at a rate lower than the Nyquist rate. The method is referred to as the averaged correlogram for undersampled data. It is based on partitioning the spectrum into a number of segments and estimating the average power within each spectral segment. This method is able to estimate the power spectrum density of a signal from undersampled data without essentially requiring the signal to be sparse. We derive the bias and the variance of the spectrum estimator, and show that there is a tradeoff between the accuracy of the estimation, the frequency resolution, and the complexity of the estimator. A closed-form approximation of the estimation variance is derived, which clearly shows how the variance is related to different parameters. The asymptotic behavior of the estimator is also investigated, and it is proved that in the case of a white Gaussian process, this spectrum estimator is consistent. Moreover, the estimation made for different spectral segments becomes uncorrelated as the signal length tends to infinity. Finally, numerical examples and simulation results are provided, which approve the theoretical conclusions.
- Spectral analysis