Maximum-Likelihood Autoregressive Estimation on Incomplete Spectra

Frequency-Selective Noise-Compensated Autoregressive Estimation

“…The present author recently provided his own particular derivation [18]. The suitability of this criterion for autoregressive estimation on incomplete spectra has been demonstrated anew in [19].…”

“…A known fact is that the asymptotic Cramér-Rao bound is inversely proportional to the number of samples Supported on this well-known fact and on the frequency-selective nature of the maximum-likelihood solution (20), we present here an intuitive approximation of that bound: since the spectral weight (19) sort of informs on how valuable (from 0 to 1) each spectral sample is, we suggest replacing the number of samples in the noise-free Cramér-Rao bound by the effective number of available samples under the presence of noise; in consequence, this number is obtained as follows: (29) The proposed estimation bound for the autoregressive coefficients in noise finally results in (30) Although the proposed bound may not correspond to the correct formula of the lower bound of the estimation variance, a supportive fact to this proposal is that the Fisher information matrix (the inverse of the Cramér-Rao bound) is built with the logarithm of the likelihood, which corresponds indeed to the core of the proposed estimator (16). Such a parallel further suggests that the proposed bound should be close in form to the actual asymptotic Cramér-Rao bound.…”

“…• The real-valued nonnegative factor (19), built with the square of the Wiener filter, will be referred to here as "spectral weight"; this factor sets the relevance of the spectral error at each frequency, in a way that samples at frequencies with higher local SNR are more relevant than those with lower SNR. These two characteristics reveal more qualitative differences with the MSE solution (12): firstly, the difference between the data and the model is not present there, and, secondly, the Wiener filter factor (13a) in the MSE equation cannot be considered a spectral weight but a filter.…”

“…In this paper, we extend that framework to the problem of autoregressive analysis in noise, presenting a novel estimation method thereto. Given that the ML criterion is suitable for scenarios with incomplete samples [19], its adoption here appears sensate because noise somewhat "hides" the spectral samples of the actual AR process. The main assumptions (alike with the methods included in this review) are that the second-order statistics of the noise, 1 and the AR order are known.…”

“…In the past years, we explored the maximum-likelihood (ML) criterion on the frequency domain for noiseless autoregressive analysis [18], [19]. In this paper, we extend that framework to the problem of autoregressive analysis in noise, presenting a novel estimation method thereto.…”