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Introduction to Convolution

Module by: Anders Gjendemsjø

Summary: Introduction to the convolution chapter

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 be performed by convolution. The reason that linear filtering is so important to signal processing is that it solves many problems and 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 infinite sum / integral of scaled and shifted impulses. By knowing how a system affects a single impulse, and by understanding the way a signal is comprised of scaled and summed impulses, it seems reasonable that it should be possible to scale and sum the impulse responses of a system in order to determine what output signal will results from a particular input. This is precisely what convolution does - convolution determines the system's output from knowledge of the input and the system's impulse response.

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