14.3 : 卷积：数学、图形和离散信号

In any LTI system, the convolution of two signals is denoted using a convolution operator, assuming all initial conditions are zero.

The convolution integral is divided into two parts: the zero-input or natural response and the zero-state or forced response of the system, with t₀ indicating the initial time.

The convolution integral can be simplified by presuming that both the input signal and impulse response are zero for negative time values.

Graphical convolution 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 a slight shift, multiplication, and integration of the resulting signals. The resulting convolution is depicted graphically.

In discrete-time convolution, the response is determined by applying an input to a discrete-time system and using the impulse response and convolution integral.

The convolution of the discrete input signal and impulse response forms the convolution sum for the system response.