The Optimum Approximation of a Matrix Filter Bank for Signals in the Frequency Domain with No Sharp Peak of a Limited Energy

With respect to a set of input matrix-signals F(ω) satisfying that a weighted absolute squared sum of the aliasing components in the frequency domain is smaller than a given positive constant and a set of output matrix-signals Y(ω) of a matrix-filterbank Y(ω)=α[F(ω)] that has a given analysis-matrix H(ω) with column array of sub-band matrix-filters H_1(ω), H_2(ω),...,H_M-1(ω) in it, we present the optimum approximation that minimizes all the worst-case measures of matrix-error-signals E(ω)=F(ω)-Y(ω) in the frequency domain Ω={ω} at the same time. We use the sample values of the output of the analysis-matrix H(ω) at the uniform sample points in the time-domain. The optimum approximation presented in this paper succeeds to the simultaneous minimization of all the upper-limit measures sup_F(ω)eΘβ{E(ω)}, where Θand β are the given set of input matrix-signals and an arbitrary operator of the error-matrix E(ω) in the frequency domain. It is assumed the weight approximates the finite amplitude of the low-pass characteristic of the on-line or the wireless communication environment and, by this restriction, we can escape unwanted disturbance-noise in the frequency domain having a high peak with a finite energy.