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Four somewhat different methods are presented. Method I is an interpolative method, suitable for many all-digital simulations. Method II is an extrapolative method, suitable for hybrid analog-digital simulations and cases involving closed-loop systems with nonlinearities. Method I calculates values of the output vector y and state vector q at t = nT, using values of the input vector, x, at times t = nT and (n-1) T. Method II is quite similar, except that the calculations use values of the input at t=(n-1)T and (n-2)T, so that these samples may be received from an external signal source (e.g., an A/D converter) and used to calculate output samples at t = nT. Both methods use straight-line approximations for the input signals between sample instants. Method I uses closed linear interpolation analogous to a nonrealizable first-order hold. Method II uses open linear extrapolation analogous to a realizable first-order hold. Methods III and IV are preferable for simulating systems with step (discontinuous) inputs.