CT203 Reference Material-2 Cross Correlation

# CT203 Reference Material-2 Cross Correlation - PPye ub S m...

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PPye ub S m e s for the realizationo-v- -- FIR%= (c) "purely recursive system" 2.5.2 Recursive and NonrecursJve ~tzations of FIR Systems I;.. - % H . . We have already made the distinction between FIR and IIR systems, based on whether the impulse-response h(n) of the system has a linite duration, or an infi- vite duration. We have also made the distinction between recursive and nonrecur- sive systems. Basically, a causal recursive system is described by an input-output &quation of the fern " 1 h &13'* ' -y(n) = F[y(n -I), . . . . y(n - N),x(n), . . . .x(n - M)] (2.5.17) Qnd for a linear time-invariant system specifically, by the difference equation

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: - * . - - Sec. 2.5 implementation oPDisc&te-The Sgs&m\$d .:. I On the other hand, causal nonreeurs the output and hence are described by y(n) = F[x(n), x(n - I), . . . , and for linear time-invariant systems [email protected] (2.5.18) with ak = 0 for k = 1, 2, . . . , N. In the case of FIR systems, we have already ob to realize such systems nonrecursively. In fact; ~\$8. (2.5.18), we have a system with an input-output e M ?>+ y ( n ) = C b k x ( n - k , ,\$ k=O This is a nonrecursive and FIR system. As in response of the system is simply equal to the co system can be realized nonrecursively. On the o also be realized recursively. Although the general later, we shall give a simple example to illustrate Suppose that we have an FIR system of the , 2'. for computing the moving average of a & g d impulse response . w ; P ; z ~ -Figure 2.36 illustrates the;&?* Now, suppose that we e q q - i
Discrete-Tlrne Signals and Systems Chap. 2 Now, (2.5.22) represents a recursive realization of the FIR system. The structure of this recursive realization of the moving average system is illustrated in Fig. 2.37. In summary, we can think of the terms FIR and IIR as general characteristics that distinguish a type of linear time-invariant system, and of the terms recursive and nonrecursive as descriptions of the structures for realizing or implementing the system. Figure 237 Recursive realization of an FIR moving average system. 2.6 CORRELATION OF DISCRETE-TIME SIGNALS A mathematical operation that closely resembles convolution is correlation. Just as in the case of convolution, two signal sequences are involved in correlation. ---- In contrast to convolution, however, our objective in computing the correlation - between the two signals is to measure the degree to which the two signals are similar and thus to extract some information that depends to a large extent on the application. Correlation of signals is often encountered in radar, sonar, digital communications, geology, and other areas in science and engineering. To be specific, let us suppose that we have two signal sequences x(ri) and y(n) that we wish to compare. In radar and active sonar applications, x(n) can represent the sampled version of the transmitted signal and y (n) can represent the sampled version of the received signal at the output of the analog-to-digital (AID) .

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