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Lti system

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CT convolution
DT convolution

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Lti system

1. 1. LINEAR TIME-INVARIANT SYSTEM 1) RESPONSE OF A CONTINOUS-TIME LTI SYSTEM 2) CONVOLUTION CT 3) RESPONSE OF DISCRETE-TIME LTI SYSTEM 4) CONVOLUTION DT
2. 2. Convolution Convolution is the most important and fundamental concept in signal processing and analysis. By using convolution, we can construct the output of system for any arbitrary input signal, if we know the impulse response of system. How is it possible that knowing only impulse response of system can determine the output for any given input signal? We will find out the meaning of convolution.
3. 3. INTRODUCTION CONVOLUTION Convolution is a mathematical way of combining two signals to form a third signal. Convolution is a formal mathematical operation, just as multiplication, addition, and integration. Addition takes two numbers and produces a third number, while convolution takes two signals and produces a third signal.
4. 4. First, the input signal can be decomposed into a set of impulses, each of which can be viewed as a scaled and shifted delta function. Second, the output resulting from each impulse is a scaled and shifted version of the impulse response. Third, the overall output signal can be found by adding these scaled and shifted impulse responses.
5. 5. Definition The mathematical definition of convolution in discrete time domain is (We will discuss in discrete time domain only.) where x[n] is input signal, h[n] is impulse response, and y[n] is output. * denotes convolution. Notice that we multiply the terms of x[k] by the terms of a time-shifted h[n] and add them up.
6. 6. Applications In digital signal processing and image processing applications, the entire input function is often available for computing every sample of the output function. Convolution amplifies or attenuates each frequency component of the input independently of the other components. In digital image processing, convolution filtering plays an important role in many important algorithms in edge detection and related processes.
7. 7. Procedure for evaluating convolution 1) Folding (flip)to obtain x2(-k) 2) Shifting to obtain x2(n-k) 3) Multiplication to obtain the product sequence x1(k).x2(n-k)
8. 8. Procedure for evaluating convolution 1) Folding (flip)to obtain x2(-k) 2) Shifting to obtain x2(n-k) 3) Multiplication to obtain the product sequence x1(k).x2(n-k) 4) Summation to obtain y(k)
9. 9. EXAMPLE 1 • Determine the convolution, x3p[n] of the circular sequences x1p[n] and x2p [n] of length N=3 as shown below.
10. 10. EXAMPLE 2: Given are two periodic sequence, x1[n] = { …..,3,1,2,3,1,2,….} and x2[n] = {…,1,-1,1,1,-1,1,…..} Find the convolution y[n] for x1[n] and x2[n].