Walsh-Transform Analysis of Discrete Dyadic-Invariant Systems

This short paper shows how the sampled output of a dyadic-invariant linear system with a given sequency-domain transfer function, in response to a sampled input, can be determined by 1) a term-wise multiplication of the sampled transfer function and the discrete Walsh transform of the sampled input function, followed by an inverse Walsh transform, or 2) a discrete dyadic convolution of the sampled impulse response and the sampled input directly in the time domain. Functions in both time and sequency domains are represented by column matrices, and discrete Walsh transformation is effected simply by the multiplication with a Walsh matrix. An example is included to illustrate both procedures. The validity of the solutions is further verified by showing that the governing dyadic differential equation of the system is satisfied.