Discrete scale vectors and decomposition of time-frequency kernels

Previous work has shown that time-frequency distributions (TFDs) belonging to Cohen's class can be represented as a sum of weighted spectrograms(1). This representation offers the means of reducing the computational complexity of TFDs. The windows in the spectrogram representation may either be the eigenfunctions obtained from an eigendecomposition of the kernel or any complete set of orthonormal basis functions. The efficiency of the computation can further be increased by using a set of scaled and shifted functions like wavelets. In this paper, the concept of scaling is considered in discrete-time domain. The scale operator in the frequency domain is formulated and the vectors which correspond to the solutions of this eigenvalue problem in discrete-time are derived. These new eigenvectors are very similar in structure to the eigenvectors obtained from eigensystem decomposition of reduced interference distribution (RID) kernels(2). The relationship between these two sets of window functions is illustrated and a new efficient way of decomposing time-frequency kernels is introduced. The results are compared to the previous decomposition methods. Finally, some possible applications of these discrete scale functions in obtaining new time-frequency distributions are discussed.