ABSTRACT
The impulse or Dirac delta function and the eternal exponential are two mathematical objects which are about as dissimilar a pair as one can conceive, yet they are siblings and play a complementary role in the analysis of systems. The linear time invariant (LTI) operator eigenfunction ϵst is continuous and nonzero -∞ < t < + ∞. The impulse function δ(t) is zero everywhere except at the origin, where it has an infinite discontinuity. The system function is found by conventional phasor analysis, which in turn permits the calculation of the impulse response by inversion of the Laplace transform. The time domain response to an arbitrary excitation can then be found by convolution. Particularly in the case of rational system functions, where the only singularities are poles, it is useful in implementing the above analysis procedure to use contour integration in the complex plane and the method of residues to evaluate the inverse Laplace transform.
