Mask operations in discrete fractional Fourier transform domains with nearly white real valued wide sense stationary output signals

• There is only one degree of freedom for designing the mask coefficients by using the discrete fractional Fourier transform.
• There are N degrees of freedom for designing the mask coefficients by using the discrete Fourier transform.
• Conditions for obtaining real valued processed signals imply conditions for obtaining WSS processed signals.
• Design of the mask coefficients are formulated as an optimization problem with an L1L1 norm nonconvex function.
• A vector space approach is employed for deriving the analytical solution.

For linear time invariant transforms, the multiplications in these transformed domains are referred to as filtering. On the other hand, the multiplications in linear time varying transformed domains are referred to as mask operations. Discrete fractional Fourier transforms (DFrFTs) are linear time varying transforms which map signals from the time domain to the rotated time frequency domains. In this paper, effects of the rotational angles of the DFrFTs on the output signals after applying the mask operations are studied. It is proved in this paper that if the rotational angles of the DFrFTs are not integer multiples of π  , as well as they are not odd integer multiples of π2 when the signal lengths are odd, then there is only one degree of freedom for designing the mask coefficients. Otherwise, there are N   degrees of freedom for designing the mask coefficients. Moreover, it is proved in this paper that satisfying the conditions for obtaining real valued output signals will automatically satisfy the conditions for obtaining wide sense stationary (WSS) output signals. Based on this result, designs of the mask coefficients are formulated as optimization problems with L1L1 norm nonconvex objective functions only subject to the conditions for obtaining real valued output signals. These constrained optimization problems are further reformulated to unconstrained optimization problems by a vector space approach. Finally, when there is only one degree of freedom for designing the mask coefficients, the globally optimal solutions of the unconstrained optimization problems are derived analytically. Computer numerical simulation results are presented for illustrations.