ECE 38-9 Solutio of Liear Costat-Coefficiet Differece Equatios Z. Aliyazicioglu Electrical ad Computer Egieerig Departmet Cal Poly Pomoa Solutio of Liear Costat-Coefficiet Differece Equatios Example: Determie the respose y ( ), of the system described by the secod-order differece equatio y ( ).7 y ( ). y ( ) x ( ) x ( ) to the iput x( ) 4 u( ) The homogeous solutio is y ( ).7 y ( ). y ( ) λ λ λ.7. ( ) λ λ λ.7. λ ad.5 λ. y ( ) c.5 c. h ECE 38-9
Solutio of Liear Costat-Coefficiet Differece Equatios Particular Solutio: y ( ) K4 u( ) p K u K u K u u u 4 ( ).7 4 ( ). 4 ( ) ()4 ( ) 4 ( ) K4.7K4.K4 (4) 4 6K.8K.K 3 y ( ).33(4) u( ) p The total solutio 3 K.33 3.3 y ( ) c.5 c..33(4) u ( ) ECE 38-9 3 Solutio of Liear Costat-Coefficiet Differece Equatios To fid c ad c For : From differece equatio, y().7 y( ). y( ) x() x( ) y() From the total solutio, y() c c.33, For : From differece equatio, y().7 y( ). y( ) x() x( ) y().4 8 9.4 From differece equatio, y().5c.c 9.3 Therefore, c c.33 9.4.5c.c 9.3 c.87 c.466 Total Solutio y ( ).466(.5).87(.).33(4) u ( ) ECE 38-9 4
The Impulse Respose of a LTI recursive system I geeral case If the iput y ( ) h( ) x( ), zs x ( ) δ ( ), the we obtai y ( ) h( ) zs The impulse respose ca be obtaied from the liear costatcoefficiet differece equatio. That is the solutio of homogeeous equatio ad particular solutio to the excitatio fuctio. I the case where the excitatio fuctio is a impulse fuctio. The particular solutio is zero y ( p ), sice x( ) for >. ECE 38-9 5 The Impulse Respose of a LTI recursive system Example: Fid the impulse respose form the followig equatio, y ( ) 3 y ( ) 4 y ( ) x ( ) x ( ) The homogeous solutio is y ( ) C ( ) C 4 h The particular solutio is zero whe To fid C ad C, we evaluate differece equatio ad homogeous solutio for ad.( y( ), y( ), sice the system must be relaxed) y() y() 3 y() 5 The impulse respose is y() C C y() C 4C x( ) δ ( ) C 6 5 6 h ( ) ( ) 4 u ( ) 5 5 C 5 ECE 38-9 6
Implemetatio of Discrete-Time Systems A system ca be described by a liear costat-coefficiet differece equatio. Let s cosider the first order system y ( ) ay ( ) bx ( ) bx ( ) The system ca be described by two systems i cascade. The first is a orecursive system described by the equatio v ( ) bx ( ) bx ( ) The secod part is recursive system x() y ( ) ay ( ) v ( ) b z - z -a - b v() y() This is called a direct form I structure. ECE 38-9 7 Implemetatio of Discrete-Time Systems Usig covolutio properties, we ca iterchage the order of the recursive ad orecursive system x() w() b y() z - z - -a b w(-) w(-) w ( ) aw ( ) x ( ) y ( ) bw ( ) bw ( ) x() w() z - b -a b w(-) y() Usig oly oe delay. It is more efficiet i terms of memory requiremet. It is called the direct form II structure ECE 38-9 8
Implemetatio of Discrete-Time Systems I geeral form, The differece equatio is give by y( ) a y( ) b x( ) The orecursive system is The recursive system is v ( ) bx ( ) y ( ) ay ( ) v ( ) Direct form II structure, the recursive system is w ( ) aw ( ) x ( ) The orecursive system is y ( ) bw ( ) ECE 38-9 9 Implemetatio of Discrete-Time Systems x() b v() z - z - b -a z - z - -a b y() x() w() z - b -a b z - -a b y() z - z - -a b M z - -a - b M z - -a w(-) Step- Step-3 ECE 38-9
Implemetatio of Discrete-Time Systems A special case of geeral case a,,,..., y ( ) bx ( ) which is a orecursive LTI system. Such a system coefficiets b b M h ( ) otherwise The secod part, set M to obtai the geeral case differece equatio y ( ) ay ( ) bx ( ) ECE 38-9 Crosscorrelatio ad Autocorrelatio Sequece The crosscorelatio of x() ad y() is s sequece defied as Or r ( l) x( ) y( l), l, ±, ±,... xy r () l x( l) y( ), l, ±, ±,... xy The reverse crosscorrelatio is Or r ( l) y( ) x( l), l, ±, ±,... yx r ( l) y( l) x( ), l, ±, ±,... yx rxy () l is r () l r ( l) xy y x ECE 38-9
Crosscorrelatio ad Autocorrelatio Sequece I special case, we have autocorrelatio, which is defied as So, that r ( ) ( ) ( ) xx l xx l rxx () l x ( ) x ( ) Or Example: x ( ) {...,,,,3,7,,, 3,,,...} y ( ) {...,,,,,,4,,,5,,,...} xy xy r () x( ) y( ) 6 4 4 6 7 r () x( ) y( ) 3 4 8 3 3 rxy ( l ) {, 9,9,36, 4,33,,7,3, 8,6, 7,5, 3} ECE 38-9 3 Problem 43.a Problem Solutios Determie the direct form II realizatio for the followig LTI system y ( ) y ( ) 4 y ( 3) x ( ) 3 x ( 5) 3 y ( ) y ( ) y ( 3) x ( ) x ( 5) x() / z - y() -/ z - z - z - z - 3/ ECE 38-9 4
Problem 44.a Problem Solutios Compute the first samples of its impulse respose x() y() / z - x( ) δ ( ) x ( ) {,,,... } y ( ) y ( ) x ( ) x ( ) y() x() 3 y() y() x() x() 3 y() y() x() x() 4 3 y(3) y() x(3) x() 8 3 3 3 3 3 3 y ( ),,,,,,,... 4 8 `6 3 64 ECE 38-9 5 Problem Solutios Problem 44.b Fid the iput output relatio y ( ) y ( ) x ( ) x ( ) Problem 44.c x ( ) {,,,... } The iput y ( ) y ( ) x ( ) x ( ) y() x() 5 y() y() x() x() 3 y() y() x() x() 4 9 y(3) y() x(3) x() 8 ECE 38-9 6
Problem 44.d Problem Solutios Use covolutio y ( ) u ( * h ( ) uh ( ) ( ) h ( ) y() h() 5 y() h() h() 3 y() h() h() h() 4 9 y(3) h() h() h() h(3) 8 ECE 38-9 7 Problem 54 Problem Solutios Fid the y() for the followig equatio y ( ) 4 y ( ) 4 y ( ) x ( ) x ( ) x( ) ( ) u( ) The characteristic equatio λ 4λ 4 λ, λ The homogeous solutio is y ( ) c c The particular solutio is h y ( ) ( ) u( ) p u u u u u ( ) ( ) 4 ( ) ( ) 4 ( ) ( ) ( ) ( ) ( ) ( ) For 4 4 9 ECE 38-9 8
Problem Solutios The total solutio is y ( ) yh( ) yp( ) c c ( ) ( ) 9 u Usig the iitial coditio, y( ) y( ) we ca obtai from differece equatio at y() 4 y( ) 4 y( ) x() x( ) y() y() 4 y() 4 y( ) x() x() y() From the total solutio y() c y() c c 9 9 7 9 c c 3 The total solutio y() c c 9 7 y ( ) ( ) u ( ) 9 3 9 ECE 38-9 9 Problem Solutios Problem 55 Fid the impulse respose h() for the followig equatio y ( ) 4 y ( ) 4 y ( ) x ( ) x ( ) The homogeous solutio is So system respose To fid the costat y ( ) c c h y() 4 y( ) 4 y( ) δ () δ ( ) y() h ( ) c c u ( ) y() 4 y() 4 y( ) δ () δ () y() 3 From the system respose h() y() c y() c c 3 c h ( ) u ( ) ECE 38-9