Optimized singular value decomposition with applications to signal extrapolation and detection of sinusoid number

The singular value decomposition (SVD) is an excellent tool to attain reliable and stable least squares type of parameter estimate in ill-conditioned cases, by discarding small singular values adequately. However, large singular values and small singular values do not always separate neatly. In the present paper, we introduce multiple regularization parameters to modify the Moore-Penrose type of pseudo-inverse matrix, to stabilize the ill-posed least squares problems. The optimal values of the regularization parameters can be determined so as to minimize an estimated mean squares error (EMSE) criterion calculated by using only accessible signal data. Thus, the proposed scheme can give a threshold condition whether the singular value should be adopted or discarded. The relationship with the optimal truncation of the singular values are also investigated analytically. Proposed method and its properties are discussed through the applications to the optimal extrapolation of band-limited signals and the detection of the number of sinusoid in white noise.