A fast multiple decomposition of a discrete-time signal on bases in vector spaces by the polynomial time-frequency transformation

It is shown that the redundant decomposition of a discrete-time signal by the block polynomial time-frequency transform (PTFT) can be implemented in a very efficient way. First, redundancy of decomposition of a discrete-time signal by a block transform defined by a special singular transformation matrix is discussed and its relation with an oversampled, power and allpass complementary, KN channel filter bank is illustrated. In the considered block transform the singular matrix can be partitioned into K subsets of unitary systems of vectors. Based on the parallels which exist between unitary transforms and filter banks, namely the parallel that any block unitary transform can be shown as a perfect reconstruction filter bank, allow us to relate the considered block transform with an oversampled KN channel filter bank which can be partitioned into K maximally decimated, power and allpass complementary, filter banks. It results in the fact that computation of frequency domain representation of a block of signal of length N, computed at M>N not necessarily uniformly spaced frequencies, can require less computation and can be more efficient than computation of the frequency domain representation which uses fast M-point FFT. It is shown that the fast decomposition of discrete time signal onto bases in vector spaces by the polynomial time-frequency transform is possible in a very similar way.