Bilinear equation method for unbiased identification of linear FIR systems in the presence of input and output noises

The presence of contaminating noises in both the input and output of an FIR system usually results in a biased least squares (LS) parameter estimate. The total least squares (TLS) methods are known to be efficient in achieving unbiased estimation, if the ratio of the input noise variance to the output noise variance (NNR) is known. However, when the NNR is unknown, a simple analysis shows that the classical LS and TLS estimation methods usually have such insufficient degree of freedom that they can achieve the unbiased solution. In this paper, it is shown by analyzing the algebraic structure of the correlation matrix that the unbiased estimate of FIR parameters can be obtained by solving a special bilinear equation. Then we develop a bilinear equation method (BEM) for solving the bilinear equation associated with the unbiased solution of the FIR system or filtering under the unknown NNR. Unlike the available unbiased estimators, the main advantage is that the proposed method exploits much sufficiently the special structure of the correlation matrix and obtains much accurate estimation for FIR filtering in the presence of input and output noises. Two simulation examples are presented that show the good performance of the proposed method, including its superiority over the classical LS and TLS approaches, and the instrumental variable methods.