On the generalised Burg technique

A new maximum-likelihood algorithm for Toeplitz covariance matrix estimation is introduced. The new algorithm requires substantially less computational cost than the generalised procedure proposed by Burg et al. It solves for the reflection coefficients rather than the covariance lags, and it calculates these coefficients successively starting from filter length one: the kth reflection coefficient akk is computed by maximising the associated likelihood function and assuming that the reflection coefficients ann, n = 1, ¿¿, k ¿¿ 1, have been previously calculated. The algorithm does not assume that the noise data are expressed by an autoregressive (AR) filter. This assumption is incorrect since noise is always present in real signals and usually has serious consequences at low signal/noise ratios where the approximation due to the assumption is significant and the true spectrum is required for reliable detection. The optimisation associated with the kth-order recursion specifying the kth reflection coefficient akk, produces a cubic equation whose root with absolute value less than one is taken equal to akk; if more than one root satisfies this condition, the one with the higher likelihood is chosen. The rest of the kth-order filter coefficients are computed using the Levinson recursion. The stability of the algorithm is guaranteed as the reflection coefficients are constrained to have absolute values less than one during the recursion. As is verified by simulation results, the spectrum of the new algorithm exhibits no line splitting and is not affected by the initial phase.