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Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fourth Edition 6 11 Copyright © Orchard Publications Graphical Evaluation of the Convolution Integral Figure 6.13. Convolution of at for Example 6.4 Figure 6.14. Convolution for interval of Example 6.4 Using the convolution integral, we find that the area for the interval is (6.25) Thus, for , the area decreases in accordance with . Evaluating (6.25) at , we find that . For , the product is zero since there is no overlap between these two signals. The convolution of these signals for , is shown in Figure 6.15. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.1 0.2 0.3 0.4 0.5 tt 2 2 u τ () *h τ t1 = 1 0 A 1 t 1 1 0 1 2 A t ut τ 1 , t2 << h τ h τ τ , = τ τ 1t2 τ h τ ()τ d τ h τ d 1 1 () τ 1 + τ d 1 τ τ 2 2 ---- 1 === 1 1 2 -- t 1 t 2 2t 1 + 2 ----------------------- + t 2 2 --- 2 t 2 + = = t 2 2 2 + = u τ τ 0 = > τ h τ 0t2 ≤≤

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Chapter 6 The Impulse Response and Convolution 6 12 Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fourth Edition Copyright © Orchard Publications Figure 6.15. Convolution for of the signals of Example 6.4 The plot of Figure 6.15 was obtained with the MATLAB script below. t1=0:0.01:1; x=t1 t1.^2./2; axis([0 1 0 0.5]);. .. t2=1:0.01:2; y=t2.^2./2 2.*t2+2; axis([1 2 0 0.5]); plot(t1,x,t2,y); grid Example 6.5 The signals and are as shown in Figure 6.16. Compute using the graphical evaluation method. Figure 6.16. Signals for Example 6.5 Solution: Following the same procedure as in the previous example, we form by first constructing the image of . This is shown as in Figure 6.17. Figure 6.17. Construction of for Example 6.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.1 0.2 0.3 0.4 0.5 tt 2 2 t 2 2 2t 2 + 0 τ 2 ≤≤ ht () ut *u t 1 1 1 t t 0 0 e t = u 0 t u 0 t1 = τ u τ u τ 1 1 0 u τ τ u τ
Signals and Systems with MATLAB ® Computing and Simulink ® Modeling, Fourth Edition 6 13 Copyright © Orchard Publications Graphical Evaluation of the Convolution Integral Next, we form by shifting to the right by some value as shown in Figure 6.18.

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