Structured total least squares methods in signal processing

In many signal processing applications, one has to solve an overdetermined system of linear equations Ax ≈ b, while minimizing the errors on A and b. The Total Least Squares (TLS) method calculates corrections ΔA and Δb such that (A + ΔA)x = b + Δb and ||[ΔA Δb]||
F
is minimal. The resulting parameter vector x is ä Maximum Likelihood (ML) estimate when the noise on the different entries of [A b] is i.i.d. Gaussian noise with zero mean and equal variance. In many applications, these last conditions do not hold because of the structure present in [ΔA Δb]. Under those circumstances, the TLS will not yield a ML estimate of the parameter vector x since the SVD (which is the standard way to obtain the TLS solution) is not structure preserving. Therefore, several structured Total Least Squares methods have been developed in recent years: Constrained Total Least Squares (CTLS) method [1][2], the Structured Total Least Squares (STLS) method [3] and the Structured Total Least Norm (STLN) method [8] [7]. As opposed to the ordinary TLS these methods yield a ML estimate of the parameter vector x, by imposing the structure of [A b] to [ΔA Δb].