Representation of Linear Time‐Invariant Systems by Impulse Response and Convolution Operation

Summary

This chapter defines a unique function, called the impulse response, which represents linear time‐invariant (LTI) systems. Impulse response is defined as the output of an LTI system, when the input is a unit impulse function. We describe an essential operation, called convolution, which relates the input‐output pair of an LTI system through the impulse response. As an alternative to convolution, we also define correlation and autocorrelation operations, which are widely used in machine learning applications. Stability, memory, and invertibility properties of LTI systems are revisited, using their mathematical representation with the impulse response.