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CH7 - CHAPTER 7 Properties of Convolution A linear system's...

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123 CHAPTER 7 EQUATION 7-1 The delta function is the identity for convolution. Any signal convolved with a delta function is left unchanged. x [ n ] ( * [ n ] x [ n ] Properties of Convolution A linear system's characteristics are completely specified by the system's impulse response, as governed by the mathematics of convolution. This is the basis of many signal processing techniques. For example: Digital filters are created by designing an appropriate impulse response. Enemy aircraft are detected with radar by analyzing a measured impulse response. Echo suppression in long distance telephone calls is accomplished by creating an impulse response that counteracts the impulse response of the reverberation. The list goes on and on. This chapter expands on the properties and usage of convolution in several areas. First, several common impulse responses are discussed. Second, methods are presented for dealing with cascade and parallel combinations of linear systems. Third, the technique of correlation is introduced. Fourth, a nasty problem with convolution is examined, the computation time can be unacceptably long using conventional algorithms and computers . Common Impulse Responses Delta Function The simplest impulse response is nothing more that a delta function, as shown in Fig. 7-1a. That is, an impulse on the input produces an identical impulse on the output. This means that all signals are passed through the system without change . Convolving any signal with a delta function results in exactly the same signal. Mathematically, this is written: This property makes the delta function the identity for convolution. This is analogous to zero being the identity for addition , and one being the ( a % 0 a ) identity for multiplication . At first glance, this type of system ( a ×1 a )
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The Scientist and Engineer's Guide to Digital Signal Processing 124 EQUATION 7-2 A system that amplifies or attenuates has a scaled delta function for an impulse response. In this equation, k determines the amplification or attenuation. x [ n ] ( k * [ n ] kx [ n ] EQUATION 7-3 A relative shift between the input and output signals corresponds to an impulse response that is a shifted delta function. The variable, s, determines the amount of shift in this equation. x [ n ] ( * [ n % s ] x [ n % s ] may seem trivial and uninteresting. Not so! Such systems are the ideal for data storage, communication and measurement. Much of DSP is concerned with passing information through systems without change or degradation. Figure 7-1b shows a slight modification to the delta function impulse response. If the delta function is made larger or smaller in amplitude, the resulting system is an amplifier or attenuator , respectively. In equation form, amplification results if k is greater than one , and attenuation results if k is less than one : The impulse response in Fig. 7-1c is a delta function with a shift . This results in a system that introduces an identical shift between the input and output signals. This could be described as a signal delay , or a signal advance , depending on the direction of the shift.
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