1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

344 CHAPTER 9 • Z-TRANSFORMS AND APPLICATIONS The following two theorems about z-tra.nsforms are useful in finding the solution ...

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 9.1 • THE Z..TRANSFORM 345 Solution Let both Xn =land Yn = 1 be the unit step seque ...

346 CHAPTER 9 • Z-TRANSFORMS AND APPLICATIONS Applying the time delay properties to the z..transforms of ea.ch term in {9-4), we ...

9.1 • THE Z- TRANSFORM 347 Therefore, the sequence x[nJ = 3-^1 (X(z)J is x [nJ = 3-^1.. [ z2 l (z-e't) (z-e-:") = Res[X(z)z"-^1 ...

348 CHAPTER 9 • z -TRANSFORMS AND APPLICATIONS Difference Equation Model Exponential growth or decay y[n+ l] = (1 + r)y[n] Newto ...

9.1 • THE Z-TRANSFORM 349 (iii) Use partial fractions to expand Y (z ) in a sum of terms and look up the inverse z-transform(s), ...

3 50 CHAPTER 9 • Z-TRANSFORMS AND APPLICATIONS Solution (a) Take the z-transform of both sides z z(Y(z) - 2) - 2Y(z) = - 3 . z - ...

9.1 • THE Z-T RANSFO.RM 351 Solution (a) Take the z..transform of both sides z z(Y(z ) - 1) - 2Y(z) = (z _ l ) 2. Solve for Y(z) ...

352 CHAPTER 9 • z -TRANSFORMS AND APPLICAT IONS ( c) Use residues to find the solution. ( d) Use convolution to find the solutio ...

9.1 • THE Z-TRANSFORM 353 Find the inverse z-transform of each term yn ( l =yo3-l[ --z I +--3 ab _^11 --^1 l ---3 b - 1! --^1 I ...

354 CHAPTER 9 • Z.TRANSFORMS ANO APPLICATIONS The particular solution is calculated with t he formula Yp [n) = 3 -^1 (X ( z )H ( ...

9.1 • THE Z-TRANSFOR.M 355 y ( 11 ] (^5).. ....... 4 3 2 Figure 9.1 The solution to y[n + 1] = ~y[n] + 1 with Yo = 0. -------.-E ...

356 CHAPTER 9 • Z-TRANSFORMS AND APPLI CATIONS (c) Time forward 1 tap: 3[xn+1) = z(X(z) - xo). (d) Differentiation: 3[nx.,J = -z ...

9.1 • THE Z- TRANSFORM 357 (a) Use the z-transform and Thbles 9.1-9.3 to find the solution. (b) Use residues to find the solutio ...

358 CHAPTER. 9 • Z-TR.ANSFOR.MS AND APPLICATI ONS 9.2 Second-Order Homogeneous Difference Equations Before proceeding with the z ...

9.2 • SECOND-ORDER HOMOGENEOUS D IFFERENCE EQUATIONS 359 9.2.1 Remark About Stability Stability depends on the location of the r ...

360 CHAPTER 9 • z-TRANSFORMS AND APPLICATIONS y[n) = c 1 (2 + i)" + c2(2 - i)". Use the initial conditions and form the linear s ...

9.2 • SECOND-ORDER HOMOGENEOUS DIFFERENCE EQUATIONS 361 This can be rearranged as Y(z)(l + L:=l apz-P) = X (z) L~=O bqz-q a nd t ...

362 CHAPTER 9 • Z·TRANSFORMS AND APPLI CATIONS Step (ii) Solve equation (9-20) for Y(z). Step (iii) Use partial fractions to exp ...

9.2 • SECOND-ORDER HOMOGENEOUS DIFFERENCE EQUATIONS 363 Thus the solution is y[n] = Res[f(z), 1] + Res[f(z), 3) = -1+2. 3», whic ...