Convolution

CONVOLUTION

Convolution is a mathematical way of combining two signals to form a third signal. It is the single most important technique in Digital Signal Processing. Convolution is importantbecause it relates the three signals of interest: the input signal, the output signal, and the impulse response.

Input Side Algorithm: Figure 6-5 shows a simple convolution problem: a 9 point inputsignal, , x[n] is passed through a system with a 4 point impulse response, h[n] , resulting in a 9+4-1=12 point output signal, y[n]. In mathematical terms x[n], is h[n] convolved with to produce y [n].This first viewpoint of convolution is based on the fundamental concept of DSP: decompose the input, pass the components through the system, and synthesize the output. In this example, each of thenine samples in the input signal will contribute a scaled and shifted version of the impulse response to the output signal. These nine signals are shown in Fig. 6-6. Adding these nine signals producesthe output signal y[n].


Fig 6-5 Input Side Algorithm
[pic]
Example convolution problem. A nine point input signal, convolved with a four point impulse response,results in a twelve point output signal. Each point in the input signal contributes a scaled and shifted impulse response to the output signal. These nine scaled and shifted impulse responses are shown inFig. 6-6.

Fig 6-6 Input Side Algorithm
[pic]
Output signal components for the convolution in Fig. 6-5. In these signals, each point that results from a scaled and shiftedimpulse response is represented by a square marker. The remaining data points, represented by diamonds, are zeros that have been added as place holders.


The Output Side Algorithm: The firstviewpoint of convolution analyzes how each sample in the input
signal affects many samples in the output signal. In this second viewpoint, we reverse this by looking at individual samples in the output...