Two-dimensional spectral estimation

TL;DR: Effective methods for generating two-dimensional quarter-plane causal autoregressive (AR) and Autoregressive moving average (ARMA) spectral estimation models are developed to provide super resolution capabilities when compared to other more classical methods such as the Fourier transform.

Abstract: In this paper, effective methods for generating two-dimensional quarter-plane causal autoregressive (AR) and autoregressive moving average (ARMA) spectral estimation models are developed. These procedures are found to provide super resolution capabilities when compared to other more classical methods such as the Fourier transform. The ARMA method involves manipulation of the model equation \sum\min{k = 0}\max{p_{1}} \sum\min{k = 0}\max{p_{2}} a_{km}x(n_{1} - k, n_{2} - m) = \sum\min{k = 0}\max{q_{1}} \sum\min{k = 0}\max{q_{2}} b_{km}\epsilon(n_{1} - k, n_{2} - m) and utilizes the given finite set of observations x(n_{1}, n_{2}) for 1 \leq n_{1} \leq N_{1},1 \leq n_{2} \leq N_{2} . In the above relationship, the random excitation {\epsilon(n_{1}, n_{2})} is taken to be white. This ARMA model's autoregressive a km coefficients are selected to minimize a weighted least-squares criterion composed of error elements while the moving average b km coefficients are obtained using an alternative approach. The spectral estimation performance of the AR and ARMA methods will be empirically demonstrated by considering the problem of resolving two sinusoids embedded in noise.