Impulse Response of LTI Systems
It was shown in Chap. 5 that the impulse response of a LTI system is the inverse Fourier transform of the frequency response. The immediately apparent difficulty in the calculation of h(t) is that the function H(ω) is a complex function of ω in the general case. The integral cannot generally be evaluated simply by the methods of integration of real functions. The integration is then performed in the complex plane by integration over a closed contour. The principles of the analysis and integration of a complex function are presented in Appendix A1. We use as examples the calculations of the impulse responses of first- and second-order systems. It appears the important result that the causality of a stable physical system is implied by the position of the poles of the transfer function in the half complex plane with negative real parts. The residue method is generally used to calculate the integral.