Connection between DCT and discrete-time fractional Brownian motion

摘要

In this paper, we have shown that the orthogonal matrix Q that diagonalizes the auto-covariance matrix of 1st order discrete-time fBm (dfBm) process is close to the columns of a paticular DCT matrix. This implies that DCT basis acts as approximate discrete Karhunen-Love trans- form (DKLT) for these processes and hence, may act as sparsifying basis. We illus- trated this via compressive sensing (CS) based reconstruction of nancial time series modeled as fBm signals. We compared results over DFT basis, DWT basis, and DST basis by averaging over 20 random sensing matrices. The best SNR is obtained with the above DCT basis for di erent sub-sampling factors of 2,4, and 8 considered. This result can be of great signi cance in applications where one is looking for an appropriate basis to work with dfBm processes.