Closed-Form Impulse Responses of Linear Time-Invariant Systems: A Unifying Approach [Lecture Notes]

In many signal processing applications, filtering is accomplished through linear time-invariant (LTI) systems described by linear constant-coefficient differential and difference equations since they are conveniently implemented using either analog or digital hardware [1]. An LTI system can be completely characterized in the time domain by its impulse response or in the frequency domain by its frequency response, which is the Fourier transform of the system's impulse response. Equivalently, using the Laplace transform [or the z-transform in the case of discrete-time (DT) systems] as a generalization of the Fourier transform, any continuous-time (CT) or DT LTI system can be characterized by its transfer function (or system function) in the s-domain or the z-domain, respectively. In this article, we explain how to find impulse responses of LTI systems described by differential and difference equations directly in the time domain without resorting to any transform methods or recursive procedures.