1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

364 CHAPTER 9 • Z-TRANSFORMS AND APPLICATIONS 2 Calculate the residue for f(z) = Y(z)zn- I = (~-'z) 1 zn-I at the pole d R.€s[f( ...

9.2 • SECOND-ORDER HOMOGENEOUS DIFFERENCE EQUATIONS 365 Solution (a) Take the ~transform of each term z^2 (Y(z) - 1 - 5z-^1 ) - ...

366 CHAPTER. 9 • Z-TR.ANSFOR.MS AND APPLICATIONS at the poles .. z^4 - 6z^3 + 12z^2 - lOz n - l Res[f(z), 1 + il = z-1+• !rm. ( ...

9.2 • SECOND-ORDER HOMOGENEOUS DIFFERENCE EQUATIONS 367 Calculate the residues for f(z) = Y(z)zn-l poles 2z3 n- 1 t th (z+i)(z-i ...

368 CHAPTER 9 • z-TRANSFORMS AND APPLICATIONS which can be written as ( y[n] = e• ..._)n + ( e-• k)" = e i.!!..!!. • +e-i.!!..!! ...

9.2 • SECOND-OR.DER. HOMOGENEOUS DIFFER.ENCE EQUATIONS 369 Solution (a) The homogeneous difference equation has the form (9-8) w ...

370 CHAPTER 9 • z-TRANSFORMS AND APPLICATIONS Solution (b) The formula H(z) = z2-~~z+~ = 1/((z - ~^1 }i)(z - ~-'j1)) is the tran ...

9.2 • SECOND-ORDER HOMOGENEOUS DIFFERENCE EQUATIONS 371 y[11] 8 6 4 2 .. '^10 •^15 20 · " -2 Figure 9.3 A typical solution to y[ ...

372 CHAPTER 9 • z-TRANSFORMS AND APPLICATIONS 4. Solve the homogeneous difference equations. (a) y[2 + n) - 8y[l + n] + 15y(n] = ...

9.3 • D IGITAL SIGNAL FILTERS 373 9.3 Digital Signal Filters 9.3.1 Introduction to Filtering In the field of signal processing, ...

374 CHAPTER 9 • z-TRANSFORMS AND APPLICATIONS 9.3.2 The Basic Filters The following three simplified basic filters serve as illu ...

A(6) 2 9.3 • DIGITAL SIGNAL F ILTERS 3 7 5 and x [l ] = J2 has the solution x[n] = 2cos(in). Thus cos(i(n + 2)) is a solution to ...

376 CHAPTER 9 • z-TRANSFORMS AND APPLICATIONS x[n] y[n) 2 0.9 A 1 - -- ' ' ,' v ' ' 10 0 4 0 60 n " ' / 1 '' _, -2 ~ Figure 9. ...

9.3 • DIGITAL SIGNAL F ILTERS 377 x[n) Figure 9.6 The input x(n] = cos(O.lOn) + 0.20sin(^23 " n) and output y [n). • EXAMPLE 9. ...

378 CHAPTER 9 • z-TRANSFORMS AND APPLICATIONS A(8) 2.5 2 1.5 0.5 Figure 9.1 The amplitude response A( 8) = I (^1) - i V1 .;:^1 1 ...

9.3 • DIGITAL SIGNAL FILTERS 379 x{n] 2 y[n] 4 Figure 9.8 The input x(n] = cos( in)+ sin(fn) + sin(2.60n) and output y[n). where ...

380 CHAPTER 9 • Z·TRANSFORMS AND APPLICATIONS We can factor X(z) and Y(z) out of the s ummations and write this in an equi- vale ...

9.3 • D IGITAL SIGNAL FILTERS 381 freq uency f s that is at least twice the highest input signal frequency to avoid frequency fo ...

382 CHAPTER. 9 • Z-TR.ANSFOR.MS AND APPLICATIONS The fundamental theorem of algebra implies that the numerator has Q roots (call ...

9.3 • DIGITAL SIGNAL F ILTERS 383 and ( z -pei") z = 1 + pz-^1 if 0 < p < 1 and <f> = 7r, and ( -z-z -P) = l -pz-,^1 ...