On the Correlation Concentration of Discrete Prolate Spheroidal Sequences

For theessential set of discrete prolate spheroidal sequences (DPSS) corresponding to a given discrete-time-bandwidth product, we show that the auto and cross-correlations between the member sequences of the set also exhibit high concentration over a correlation length twice the concentration length of the member sequences. We start by deriving one upper bound and two lower bounds for the energy of the correlation sequence of index-limited DPSSs from which we derive similar bounds for the energy of DPSS correlation sequences. Then, we obtain an upper bound for the tail energy of DPSS correlation sequences. Finally, we obtain a lower bound for the concentration ratio of DPSS correlation sequences. All derived bounds require knowledge of the following parameters for each DPSS sequence in the correlation pair: corresponding eigen-value, Fourier transform evaluated at two special frequency points, derivative of the Fourier transform evaluated at the same two points. Most of the aforementioned parameters can be expressed or bounded by functions in the global parameters: sequence length and discrete-time-bandwidth.