Convolution: Math, Graphics, and Discrete Signals

In any LTI (Linear Time-Invariant) system, the convolution of two signals is denoted using a convolution operator, assuming all initial conditions are zero. The convolution integral can be divided into two parts: the zero-input or natural response and the zero-state or forced response, with t₀ indicating the initial time.

To simplify the convolution integral, it is assumed that both the input signal and impulse response are zero for negative time values. The graphical convolution process involves four steps: folding, shifting, multiplication, and integration.

Consider an RC circuit with a specified input pulse signal and output response. Initially, folding is performed by creating a mirror image of the input signal along the y-axis. This is followed by shifting, where the folded signal is slid along the time axis. Next, the multiplication of the folded and shifted signals is done point-by-point. Finally, the integration of the resulting signal over time provides the convolution result. This process can be depicted graphically.

In discrete-time convolution, the system's response is determined by applying an input to a discrete-time system and using the impulse response and convolution sum. The convolution of the discrete input signal x[n] and the impulse response h[n] forms the convolution sum for the system response:

This sum computes the output signal y[n] at each discrete time step n. Understanding both continuous and discrete convolution is essential for analyzing LTI systems and predicting their behavior in response to various inputs.