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The use of the inverse Fourier transform (IFT), Fourier transform (FT), Hadamard transform and discrete Fourier transform in today's digital communications problems is well established. These transforms are a special case of an orthogonal digital transmultiplexer, a structure that is equivalent to the class of FIR orthonormal filter banks. We study the design of general multirate FIR transmitter and receiver filters that satisfy the orthogonality property such that optimum energy concentration is achieved in the channel frequency bands. A new algorithm for finding the global optimum FIR filter for the single carrier case is presented. We then consider the noisy version with both white and colored additive Gaussian noise and show that in some cases the optimum solutions converge to the so-called spectrally efficient filter.