Issue No. 01 - January (1969 vol. 18)
A. Hausner , Harry Diamond Labs.
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<inf>j</inf>< t< t<inf>j+i</inf>, 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<inf>j</inf>< t< t<inf>j+i</inf>, but change at each transition time t<inf>j+i</inf>. With the definition (1) satisfies the differential system only if the output of each ft undergoes abrupt jumps at t<inf>j+i</inf>, From (1), the jumps s<inf>k,j</inf>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<inf>k</inf>. Let Now, since f<inf>k</inf>=g<inf>k</inf>, 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<inf>k+1,j</inf>with potentiometers), and one summer (to implement f<inf>o</inf>=g<inf>o</inf>+s<inf>o,j</inf>). The authors derive (6) in an alternate manner using the Laplace transform and slightly different notation.
A. Hausner, "R69-2 The Synthesis and Processing of Signals with Discontinuities in the Time Domain", IEEE Transactions on Computers, vol. 18, no. , pp. 83-84, January 1969, doi:10.1109/T-C.1969.222536