CH6 - CHAPTER 6 Convolution Convolution is a mathematical...

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107 CHAPTER 6 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. Using the strategy of impulse decomposition, systems are described by a signal called the impulse response . Convolution is important because it relates the three signals of interest: the input signal, the output signal, and the impulse response. This chapter presents convolution from two different viewpoints, called the input side algorithm and the output side algorithm. Convolution provides the mathematical framework for DSP; there is nothing more important in this book. The Delta Function and Impulse Response The previous chapter describes how a signal can be decomposed into a group of components called impulses . An impulse is a signal composed of all zeros, except a single nonzero point. In effect, impulse decomposition provides a way to analyze signals one sample at a time. The previous chapter also presented the fundamental concept of DSP: the input signal is decomposed into simple additive components, each of these components is passed through a linear system, and the resulting output components are synthesized (added). The signal resulting from this divide-and-conquer procedure is identical to that obtained by directly passing the original signal through the system. While many different decompositions are possible, two form the backbone of signal processing: impulse decomposition and Fourier decomposition. When impulse decomposition is used, the procedure can be described by a mathematical operation called convolution . In this chapter (and most of the following ones) we will only be dealing with discrete signals. Convolution also applies to continuous signals, but the mathematics is more complicated. We will look at how continious signals are processed in Chapter 13. Figure 6-1 defines two important terms used in DSP. The first is the delta function , symbolized by the Greek letter delta, . The delta function is * [ n ] a normalized impulse, that is, sample number zero has a value of one, while
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The Scientist and Engineer's Guide to Digital Signal Processing 108 all other samples have a value of zero. For this reason, the delta function is frequently called the unit impulse . The second term defined in Fig. 6-1 is the impulse response . As the name suggests, the impulse response is the signal that exits a system when a delta function (unit impulse) is the input. If two systems are different in any way, they will have different impulse responses. Just as the input and output signals are often called and , the impulse response is usually given the x [ n ] y [ n ] symbol, . Of course, this can be changed if a more descriptive name is h [ n ] available, for instance, might be used to identify the impulse response of f [ n ] a filter .
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CH6 - CHAPTER 6 Convolution Convolution is a mathematical...

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