Input-output gains of linear periodic discrete-time systems with application to multirate signal processing

Two input-output gains of linear periodically time-varying (LPTV) systems are defined, namely, G(l/sub 2/, l/sub 2/)-the worst-case l/sub 2/ norm of the output over all inputs of unit l/sub 2/ norm-and G(RMS, RMS)-the worst-case RMS value of the output over all inputs of unit RMS value. It is proved for LPTV systems that these two gains are equal, a known fact for LTI systems. In addition, the relationship between two recently introduced generalized frequency responses for LPTV systems is derived. Finally, M-channel maximally decimated filter banks are considered, where, except in the ideal case of perfect reconstruction, aliasing distortion, magnitude distortion, and phase distortion are present. It is shown how these are kept small if the filter bank is designed by a method that optimizes the gain G(l/sub 2/, l/sub 2/) of an error system.