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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 tj< t< tj+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 tj< t< tj+i, but change at each transition time tj+i. With the definition (1) satisfies the differential system only if the output of each ft undergoes abrupt jumps at tj+i, From (1), the jumps sk,jare 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 gk. Let Now, since fk=gk, 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 Sk+1,jwith potentiometers), and one summer (to implement fo=go+so,j). 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. 1, pp. 83-84, Jan. 1969, doi:10.1109/T-C.1969.222536
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