Discrete step-sifting theorems for signal and system analyses

Abstract

Generalized step sifting theorems (GSSTs) that can be used to sift unfolded and folded step functions through the summation sign are presented. The theorems are shown to result in an unsegmented answer that contains step function multipliers that turn the terms on or off at the proper times. The simplified step sifting theorem for unfolded functions (SSST-UF), together with the step sifting theorem for convolution (SST-C) and the identity delta /sub 1/(-n)=1- delta /sub 1/(n-1), can be used to solve all piecewise convolution problems easily without the need for sketches. The GSST-UF is easiest to remember and can be used for folded functions by using the above identity. The SSST-UF proves to be the most useful (applicable about 90% of the time). These theorems can greatly reduce the labor involved in signal and system analysis and lead to more meaningful insight and solutions.<>