System Representations for the Zakai Class With Applications

The convergence behavior of a convolution representation of stable linear time-invariant (LTI) systems operating on the Zakai class of bandlimited signals is analyzed. It is shown that there are signals in the Zakai class for which the convolution integral diverges if the system is the Hilbert transform or the ideal low-pass filter with bandwidth less than or equal to the signal bandwidth. Moreover, using a previously obtained result of Habib, it is proved that the class of stable LTI systems that map the Zakai class into itself does not include the Hilbert transform and the ideal low-pass filter with bandwidth less than or equal to the signal bandwidth. Finally, it is shown that the concept of the analytical signal, which is used in communications, is problematic for the signal space Z_π, because the operator for its computation is unbounded and discontinuous.