Threshold performance of MUSIC when using the forward-backward data matrix

The multiple signal classification (MUSIC) frequency estimator is a suboptimal method for estimating the frequencies of multiple sinusoids buried in white noise and has a threshold of around 14 dB for the well-known two-sinusoid example (equal amplitudes f₁=0.52, f₂=0.5, φ₁=π/4, and φ₂=0). We point out that this threshold value is the result of estimating the autocorrelation estimates using the forward data matrix alone. Instead, if the autocorrelation estimates are obtained from the forward-backward data matrix, the threshold is lowered to 4 dB (lower than Kumaresan-Tufts method\´s 7 dB and within 1 dB of maximum-likelihood estimator). We offer an explanation of why the threshold is lowered by examining the noiseless autocorrelation matrix based on the forward and forward-backward data matrices. Also, it is well known that the Crame´r-Rao lower bound (CRLB) is also a function of the relative phases. We point out that when φ₁=π/2, the estimates obtained using MUSIC become increasingly biased and cause the variance to fall below CRLB at 23 dB for the "forward-backward" root MUSIC and at 25 dB for "forward-only" root MUSIC. The use of the forward-backward data matrix in spectral estimation is not novel, but to our knowledge, the improvement in threshold for φ₁=π/4 has not been reported, nor the comparative performance as φ₁ varies.