# Causal Periodic Pulse Response

## Abstract

The causal periodic pulse response is of great importance especially for digital circuits. For example one of the most basic signals on a chip (if not every chip) is the clock, which is nothing other than a periodic pulse! We can find the causal periodic response, utilizing the transfer function via at least two methods. The first one is, since we know the causal sine/cosine response (from last chapter), and using the Fourier series we can build the causal periodic response one harmonic at a time! The second method is using the Laplace transform of periodic functions. We have an expression for the transform of a periodic pulse of width *τ* and period *T* as \(X(s)= \frac {1 - e^{-\tau s}}{s(1 - e^{-Ts})}\). If we take this, multiply by the system transfer function then find the inverse transform (analytically, numerically, or using a fit) we are then able to find the sought solution in the time domain. We illustrate both methods in the chapter on a few *RLC* circuits. We also examine the impact of multiplying the Laplace transform of the causal period pulse times the transfer function. Of course we can also find the causal period pulse simply by iterating the shifted pulse response.