Introduction to convolution

In addition to the operations performed on signals in the Signals chapter there are several more. The most important operation is linear filtering, which can beperformed by convolution . The reason that linear filtering is so important to signal processing is that it solves many problemsand that is relatively simple to describe mathematically. In this chapter we will be looking at convolution.

Convolution helps to determine the effect a system has on an input signal. It can be shown that a linear, time-invariant system is completely characterized by its impulse response.Using the sampling property of the delta function for for continuous time signals and the unit sample for discrete time signals we can decompose a signal into an infinitesum / integral of scaled and shifted impulses. By knowing how a system affects a single impulse, and by understanding the waya signal is comprised of scaled and summed impulses, it seems reasonable that it should be possible to scale and sum theimpulse responses of a system in order to determine what output signal will results from a particular input. This isprecisely what convolution does - convolution determines the system's output from knowledge of the inputand the system's impulse response .

Contents of this chapter

• Introduction (current module)
• Convolution - Discrete time
• Convolution - Continuous time
• Properties of convolution