R69-2 The Synthesis and Processing of Signals with Discontinuities in the Time Domain

Abstract

The authors consider generating (with integrators) a discontinuous function f(t) for use on an analog computer. It is assumed that f(t) is continuous on each of m subintervals t j j+i , j=0, * , m-1, and that within each subinterval, f can be approximated by a polynomial of degree n: In (1), the ai, are constant for t t j j+i , but change at each transition time t j+i . With the definition (1) satisfies the differential system only if the output of each ft undergoes abrupt jumps at t j+i , From (1), the jumps s k,j are found to be These jumps cannot be represented directly in (3); integrator outputs must be continuous. The way around this difficulty is to transform the integrator outputs to continuous variables g k . Let Now, since f k =g k , we have by direct substitution This gives rise to a circuit with n integrators, m-1 double-pole single-throw switches (to switch the positive and negative reference needed for obtaining S k+1,j with potentiometers), and one summer (to implement f o =g o +s o,j ). The authors derive (6) in an alternate manner using the Laplace transform and slightly different notation.