Generalized Walsh functions: Theory and applications in digital systems

By defining a generalized dyadic multiplication and a generalized Gray code, Walsh functions are generalized to include nonintegral orders. A dyadic differentiation is defined which is compatible with Walsh functions and generalized Walsh transformations. Walsh transform method is used to determine the response of discrete dyadic-invariant systems, and a fast dyadic convolution scheme is devised which requires fewer operations than that required for direct computation when the input signal and the system response are given in the time domain. An approach for the matrix representation of nonlinear elements and for solving nonlinear differential equations is formulated and found to yield steady-state solutions in a straightforward manner. The effect of channel noise in a binary transmission system is analyzed. Systematic procedures for both the bipolar encoder and the natural encoder cases are developed.