2. Time-Domain Representations of LTI Systems
2. Time-Domain Representations of LTI Systems
2. Time-Domain Representations of LTI Systems
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<strong>2.</strong>10 SolvingDifferential and Difference EquationsExample <strong>2.</strong>21: First-Order Recursive System : Complete SolutionFind the complete solution for the first-order recursive systemdescribed by the difference equation 1y[ n] y[n 1]x[ n](<strong>2.</strong>46)4if the input is x[n] = (1/2) n u[n] and the initial condition is y[1] = 8.1. Homogeneous sol. : <strong>2.</strong> Particular solution :3. Complete solution : (<strong>2.</strong>47)4. Coefficient c 1 determined by I.C. :I.C. : y 0 x 0 1 4 y 1We substitute y[0] = 3into Eq. (<strong>2.</strong>47), yielding5. Final solution :for n 0<strong>Time</strong>-<strong>Domain</strong> <strong>Representations</strong> <strong>of</strong><strong>LTI</strong> <strong>Systems</strong>69
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