A constrained weighted least squares approach for time-frequency distribution kernel design

In most applications of time-frequency (t-f) distributions, the t-f kernel is of finite extent and applied to discrete time signals. This paper introduces a matrix-based approach for t-f distribution kernel design. In this new approach, the optimum kernel is obtained as the solution of a linearly constrained weighted least squares minimization problem in which the kernel is vectorial and the constraints form a linear subspace. Similar to FIR temporal and spatial constrained least squares (LS) design methods, the passband, stopband, and transition band of an ideal kernel are first specified. The optimum kernel that best approximates the ideal kernel in the LS error sense, and simultaneously satisfies the multiple linear constraints, is then obtained using closed-form expressions. This proposed design method embodies a well-structured procedure for obtaining fixed and data-dependent kernels that are difficult to obtain using other design approaches.