Fast Computation of a Complex Quadratic Form

with t t t j ε ε ε Im Re + = , where } {Re t ε and } {Im t ε are independent real Gaussian white noise processes each with zero mean and variance 2 2 σ . These processes have been widely used to model the noise component in data models consisting of a deterministic signal and an additive noise. The pure harmonic signal, the damped harmonic signal, and the polynomial phase signal can be given as examples of deterministic signals. Let Λ be the N N × covariance matrix of T N e e ] , , [ 1 0 − K . Computation of the quadratic form y xH 1 − Λ , in which T N x x x ] , , [ 1 0 − = K and T N y y y ] , , [ 1 0 − = K are arbitrary 1 × N complex vectors, is an important issue in estimating the parameters of the above-mentioned signal-plus-noise data models. For example, the evaluation of the so-called Fisher information matrix for these data models necessitates computations of such quadratic forms. In this paper, we derive a fast algorithm for computing the quadratic form y xH 1 − Λ . The derivation is based on the Gohberg and Semencul representation (e.g., Pal (1993)) of the inverse covariance matrix 1 − Λ . A fast algorithm for the computation of the real quadratic form y xT 1 − Λ , where 1 − Λ is the inverse of an N N × covariance matrix of a real, Gaussian autoregressive process, has been derived in Ghogho and Swami (1999) by using a result from (Box and Jenkins, 1971). However, a similar