Signal processing - The z – Transform

1. The z – Transform 2. The Inverse z – Transform 3. Properties of the z – Transform 4. System Function of LTI Systems 5. LTI Systems Characterized by Linear Constant – Coefficient Difference Equations 6. Connections between Pole – Zero Locations and Time – Domain Behavior 7. The One – Sided z – Transform

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Nguyễn Công Phương SIGNAL PROCESSING The z – Transform Contents I. Introduction II. Discrete – Time Signals and Systems III.The z – Transform IV. Fourier Representation of Signals V. Transform Analysis of LTI Systems VI. Sampling of Continuous – Time Signals VII.The Discrete Fourier Transform VIII.Structures for Discrete – Time Systems IX. Design of FIR Filters X. Design of IIR Filters XI. Random Signal Processing sites.google.com/site/ncpdhbkhn 2 The z – Transform 1. The z – Transform 2. The Inverse z – Transform 3. Properties of the z – Transform 4. System Function of LTI Systems 5. LTI Systems Characterized by Linear Constant – Coefficient Difference Equations 6. Connections between Pole – Zero Locations and Time – Domain Behavior 7. The One – Sided z – Transform sites.google.com/site/ncpdhbkhn 3 sites.google.com/site/ncpdhbkhn 4 The z – Transform (1) [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] k k k x n x k n k y n x k h n k h k x n kδ ∞ ∞ ∞ =−∞ =−∞ =−∞ = − → = − = −∑ ∑ ∑ for all[ ] , Re( ) Im( ) nx n z n z z j z = = + for all[ ] [ ] [ ] ,n k k n k k y n h k z h k z z n ∞ ∞ − − =−∞ =−∞   → = =     ∑ ∑ ( ) [ ] k k H z h k z ∞ − =−∞ = ∑ for all[ ] ( ) ,ny n H z z n→ = for all for all[ ] , [ ] ( ) ,n nk k k k k k k x n c z n y n c H z z n= → =∑ ∑ The z – Transform (2) • ROC (region of convergence): the set of values of z for which X(z) converges • Zeros: the values of z for which X(z) = 0 • Poles: the values of z for which X(z) is infinite sites.google.com/site/ncpdhbkhn 5 ( ) [ ] n n X z x n z ∞ − =−∞ = ∑ 0 ω jz e ω= 1 Im( )z Re( )z Unit circle z – plane 0 ω jz re ω= cosr ω Im( )z Re( )z z – plane sinr ω r sites.google.com/site/ncpdhbkhn 6 The z – Transform (3) Ex. 1 1 21 2 3 4 5 1 2 3 4 5[ ] { }, [ ] { }.x n x n↑ ↑= =Given Determine their z – transforms? ( ) [ ] n n X z x n z ∞ − =−∞ = ∑ 0 1 2 3 4 1 1 1 1 1 10 1 2 3 4( ) [ ] [ ] [ ] [ ] [ ]X z x z x z x z x z x z − − − − = + + + + 1 2 3 41 2 3 4 5z z z z− − − −= + + + + ROC: entire z – plane except z = 0 2 1 0 1 2 2 2 2 2 2 22 1 0 1 2 ( ) ( )( ) [ ] [ ] [ ] [ ] [ ]X z x z x z x z x z x z− − − − − −= − + − + + + 2 0 1 21 2 3 4 5z z z z z− −= + + + + 2 1 22 3 4 5z z z z− −= + + + + ROC: entire z – plane except z = 0 & z = ∞ sites.google.com/site/ncpdhbkhn 7 The z – Transform (4) Ex. 2 1 2 3 0[ ] [ ], [ ] [ ], [ ] [ ],x n n x n n k x n n k kδ δ δ= = − = + > Determine their z – transforms? ( ) [ ] n n X z x n z ∞ − =−∞ = ∑ 1 0 1 2 1 1 1 1 11 0 1 2 ( )( ) ... [ ] [ ] [ ] [ ] ...X z x z x z x z x z− − − −= + − + + + + 1 0 1 20 1 0 0 1( )... ...z z z z− − − −= + + + + + = ROC: entire z – plane sites.google.com/site/ncpdhbkhn 8 The z – Transform (5) Ex. 3 1 0 0 otherwise , [ ] , n M x n ≤ ≤ =   Find the z – transforms of the square – pulse sequence ( ) [ ] n n X z x n z ∞ − =−∞ = ∑ ROC: |z| > 1 0 1 M n n z− = =∑ 1 2 3 11 if 1 1 ... , M N AA A A A A A + − + + + + + = < − 1 1 1 1 ( ) ( ) MzX z z − + − − → = − 1 1 1z z− 0 1 Im Re ROC sites.google.com/site/ncpdhbkhn 9 The z – Transform (6) Ex. 4 Find the z – transforms of the sequence x[n] = anu[n]? ( ) [ ] n n X z x n z ∞ − =−∞ = ∑ 1 0 0 ( ) n n n n n a z az ∞ ∞ − − = = = =∑ ∑ 2 3 11 if 1 1 ... ,A A A A A + + + + = < − 1 1 1 ( ) zX z az z a− → = = − − Zero: z = 0 Pole: z = a 1 1az z a− ROC: |z| > a sites.google.com/site/ncpdhbkhn 10 The z – Transform (7) Ex. 4 Find the z – transforms of the sequence x[n] = anu[n]? 1 1 1 ( ) zX z az z a− = = − − Zero: z = 0; pole: z = a; ROC: |z| > a 0 1 n 0 1a< < 0 1 n 1a = 0 1 n 1a > 0 1 Im Re ROC a 0 1 Im Re ROC 0 1 Im Re ROC a sites.google.com/site/ncpdhbkhn 11 The z – Transform (8) Ex. 5 0 0 1 0 , [ ] [ ] , n n n x n a u n a n ≥ = − − − =  − < Find the z – transforms of the sequence ( ) [ ] n n X z x n z ∞ − =−∞ = ∑ 1 1 1 ( ) n n n n n a z az − − − − =−∞ =−∞ = − = −∑ ∑ 2 3 11 if 1 1 ... ,A A A A A + + + + = < − 1 1 1 1 ( ) zX z a z a z z a − − → = − = − − Zero: z = 0 Pole: z = a 1 1a z z a− < → < ROC: |z| < a 1 1 2 21( ...)a z a z a z− − −= − + + + [ ] [ ] ( )n z x n a u n X z z a = → = − Zero: z = 0; pole: z = a; ROC: |z| > a 0 n sites.google.com/site/ncpdhbkhn 12 The z – Transform (9) Ex. 6 0 0 , [ ] , n n a n x n b n  ≥ =  − < Find the z – transforms of the sequence ( ) [ ] n n X z x n z ∞ − =−∞ = ∑ 1 0 n n n n n n b z a z − ∞ − − =−∞ = = − +∑ ∑ 1 0 If 1 n n n z az z a a z z a ∞ − − = → = − ∑ ( ) z zX z z b z a → = + − − 1 1If 1 n n n zb z z b b z z b − − − =−∞ < → < → − = − ∑ Zero: z = 0 Pole: z = a, b ROC: a < |z| < b sites.google.com/site/ncpdhbkhn 13 The z – Transform (10) 0 n 0 Im Re ROC a 0 n 0 Im Re ROC b b a Im Re ROC b0 a 0 n ROC: |z| > a ROC: |z| < a ROC: a < |z| < b sites.google.com/site/ncpdhbkhn 14 The z – Transform (11) Ex. 7 0 00 0 2[ ] (cos ) [ ], , nx n r n u n rω ω pi= > ≤ ≤Find the z – transforms of the sequence 0 0 (cos ) n n n r n zω ∞ − = =∑( ) [ ] n n X z x n z ∞ − =−∞ = ∑ 1 1 2 2 cos sin cos j j j e j e eθ θ θθ θ θ −= + → = + 0 01 1 0 0 1 1 2 2 ( ) ( ) ( ) j jn n n n X z re z re zω ω ∞ ∞ −− − = = → = +∑ ∑ 2 2 1cos sin cos sinje jθ θ θ θ θ= + = + = 0 01 1 11 1 1j jre z re z rz z rω ω−− − −& 0 01 1 1 1 1 1 ROC 2 1 2 1 ( ) , :j jX z z r re z re zω ω−− − → = + > − − sites.google.com/site/ncpdhbkhn 15 The z – Transform (12) Ex. 7 0 00 0 2[ ] (cos ) [ ], , nx n r n u n rω ω pi= > ≤ ≤Find the z – transforms of the sequence 0 0 0 0 0 0 1 1 1 1 1 1 2 2 0 1 1 2 2 1 1 2 1 2 ( ) ( ) ( ) ( )( ) [ ( cos ) ] j j j j j j re z re z rz e e re z re z r z r z ω ω ω ω ω ω ω − −− − − −− − − − − + − − + = = − − − + 0 01 1 1 1 1 1 2 1 2 1 ( ) j jX z re z re zω ω−− − = + − − 2cos sin cosj j je j e eθ θ θθ θ θ−= + → + = 1 0 1 2 2 0 1 1 2 (cos ) ( ) ( cos ) r zX z r z r z ω ω − − − − → = − + 0 0 0( cos ) ( )( ) j j z z r z re z reω ω ω − − = − − Zero: z1 = 0; z2 = rcosω0 ROC: |z| > r 0 0 1 2Pole : ; j jp re p reω ω−= = 0 r Im Re ROC 1z 2z 1p 2p The z – Transform (13) • ROC – There is no pole inside a ROC – The ROC is a connected region – For finite duration sequences, the ROC is the entire z – plane, sometimes except for z = 0 and z = ∞ • The z – transform –We need both X(z) and its ROC – X(z) is not defined outside the ROC sites.google.com/site/ncpdhbkhn 16 sites.google.com/site/ncpdhbkhn 17 The z – Transform (14) The z – Transform 1. The z – Transform 2. The Inverse z – Transform 3. Properties of the z – Transform 4. System Function of LTI Systems 5. LTI Systems Characterized by Linear Constant – Coefficient Difference Equations 6. Connections between Pole – Zero Locations and Time – Domain Behavior 7. The One – Sided z – Transform sites.google.com/site/ncpdhbkhn 18 sites.google.com/site/ncpdhbkhn 19 The Inverse z – Transform (1) 11 2 [ ] ( ) n C x n X z z dzjpi − = ∫ 1 2 1 0 1 2 1 1 2 1 21 ( )... ( ) ... N N N N b b z b z b zX z a z a z a z − − − − − − − − + + + = + + + sites.google.com/site/ncpdhbkhn 20 The Inverse z – Transform (2) Ex. 1 1 1 1 1 1 1 0 2 ( ) ( )( . ) zX z z z − − − + = − − 1 1 2 1 1 1 1 1 1 1 0 2 1 1 0 2( )( . ) . z K K z z z z − − − − − + = + − − − − 1 1 1 1 21 1 0 2 1( . ) ( )z K z K z − − −→ + = − + − 1 21 1 1 1 0 2 1 1 1( . ) ( )z K K= → + = − × + − 1 2 5.K→ = 2 1 5.K→ = −1 20 2 1 5 1 0 2 5 1 5. ( . ) ( )z K K= → + = − × + − 1 1 2 5 1 5 1 1 0 2 . . ( ) . X z z z− − → = − − − sites.google.com/site/ncpdhbkhn 21 The Inverse z – Transform (3) Ex. 1 1 1 1 [ ] , ROC : n a u n z a az− → > − If 1z > 1 1 2 5 2 5 1 1 5 1 5 0 2 1 0 2 . . [ ] . . ( . ) [ ] . n u n z u n z − −  → −→  − → −  − 1 1 1 1 1 1 0 2 ( ) ( )( . ) zX z z z − − − + = − − 1 1 2 5 1 5 1 1 0 2 . . .z z− − = − − − 2 5 1 5 0 2[ ] . [ ] . ( . ) [ ]nx n u n u n→ = − 1 11 1 [ ] , ROC : n a u n z a az− − − − → < − If 0 2.z < 1 1 2 5 2 5 1 1 1 5 1 5 0 2 1 1 0 2 . . [ ] . . ( . ) [ ] . n u n z u n z − −  → − − − −→  − → − −  − 2 5 1 1 5 0 2 1[ ] . [ ] . ( . ) [ ]nx n u n u n→ = − − − + − − If 0 2 1. z< < 1 1 2 5 2 5 1 1 1 5 1 5 0 2 1 0 2 . . [ ] . . ( . ) [ ] . n u n z u n z − −  → − − − −→  − → −  − 2 5 1 1 5 0 2[ ] . [ ] . ( . ) [ ]nx n u n u n→ = − − − − sites.google.com/site/ncpdhbkhn 22 The Inverse z – Transform (4) Ex. 2 1 1 2 1 1 2 5 ( ) . zX z z z − − − + = − + 1 2 1 25 1 21 2 5 0 0 5 1 5 1 58 . , . . . . jz z p j e− − ±− + = → = ± = 1 1 2 1 2 1 1 1 2 1 1 2 5 1 1. z K K z z p z p z − − − − − + = + − + − − 1 1 1 1 2 2 11 1 1( ) ( )z K p z K p z − − −→ + = − + − 1 1 1 2 1 21 1 1 1 1/ ( / ) ( )z p p K p p K= → + = − + − 0 93 1 0 5 0 67 0 83 .. . . jK j e−→ = − = 0 93 2 0 5 0 67 0 83 .. . . jK j e→ = + =2 2 1 2 1 21 1 1 1 1/ ( ) ( / )z p p K K p p= → + = − + − 0 93 1 25 0 93 1 250 83 1 58 0 83 1 58. . . .[ ] . ( . ) [ ] . ( . ) [ ]j j n j j nx n e e u n e e u n− −→ = + 1 25 0 93 1 25 0 930 83 1 58 ( . . ) ( . . ). ( . ) ( )n j n j ne e− − −= + 1 25 0 93 1 25 0 93 2 1 25 0 93( . . ) ( . . ) cos( . . )j n j ne e n− − −+ = − 1 67 1 58 1 25 0 93 1 67 1 58 1 25 53 13o[ ] . ( . ) cos( . . ) [ ] . ( . ) cos( . . ) [ ]n nx n n u n n u n→ = − = − The z – Transform 1. The z – Transform 2. The Inverse z – Transform 3. Properties of the z – Transform 4. System Function of LTI Systems 5. LTI Systems Characterized by Linear Constant – Coefficient Difference Equations 6. Connections between Pole – Zero Locations and Time – Domain Behavior 7. The One – Sided z – Transform sites.google.com/site/ncpdhbkhn 23 sites.google.com/site/ncpdhbkhn 24 Properties of the z – Transform The z – Transform 1. The z – Transform 2. The Inverse z – Transform 3. Properties of the z – Transform 4. System Function of LTI Systems 5. LTI Systems Characterized by Linear Constant – Coefficient Difference Equations 6. Connections between Pole – Zero Locations and Time – Domain Behavior sites.google.com/site/ncpdhbkhn 25 sites.google.com/site/ncpdhbkhn 26 System Function of LTI Systems (1) [ ] [ ]* [ ] [ ] [ ] k y n x n h n x k h n k ∞ =−∞ = = −∑ z – transform [ ]x n  [ ]y n( ) ( ) ( )Y z X z H z= z – transform [ ]h n Inverse z – transform ( )X z ( )H z x n h n X z H z→[ ]* [ ] ( ) ( ) sites.google.com/site/ncpdhbkhn 27 System Function of LTI Systems (2)Ex. Given a system with impulse response h[n] = anu[n], |a| < 1, and an input x[n] = u[n]. Find the output? [ ] [ ]* [ ]y n h n x n= ( ) ( ) ( )Y z H z X z→ = 1 0 1 1 ( ) ,n n n H z a z z a az ∞ − − = = = > − ∑ 1 0 1 1 1 ( ) ,n n X z z z z ∞ − − = = = > − ∑ 1 1 1 1 1 1 1 ( ) , max{ , } ( )( ) Y z z a az z− − = > = − − 1 1 1 1 1 1 1 1 , a z a z az− −   = − >  − − −  1 11 1 1 1 [ ] ( [ ] [ ]) [ ] n n ay n u n a u n u n a a + + − = − = − − System Function of LTI Systems (3) • A system function H(z) with the ROC that is the exterior of a circle, extending to infinity, is a necessary condition for a discrete – time LTI system to be causal, but not a sufficient one • An LTI system is stable if and only if the ROC of the system function H(z) includes the unit circle |z| = 1 • An LTI system with rational H(z) is both causal and stable if and only if all poles of H(z) are inside the unit circle and its ROC is on the exterior fo a circle, extending to infinity sites.google.com/site/ncpdhbkhn 28 sites.google.com/site/ncpdhbkhn 29 System Function of LTI Systems (4) H1(z) [ ]x n y n[ ] + H2(z) H1(z)+H2(z) [ ]x n [ ]y n H1(z) [ ]x n [ ]y n H2(z) H1(z)H2(z) [ ]x n [ ]y n The z – Transform 1. The z – Transform 2. The Inverse z – Transform 3. Properties of the z – Transform 4. System Function of LTI Systems 5. LTI Systems Characterized by Linear Constant – Coefficient Difference Equations 6. Connections between Pole – Zero Locations and Time – Domain Behavior 7. The One – Sided z – Transform sites.google.com/site/ncpdhbkhn 30 sites.google.com/site/ncpdhbkhn 31 LTI Systems Characterized by LCCDE (1) 1 1 [ ] [ ] [ ] N M k k k k y n a y n k b x n k = = = − − + −∑ ∑ 1 1 [ ] [ ] [ ] N M k k k k y n a y n k b x n k = = → + − = −∑ ∑ [ ] ( )y n Y z→ 1 1 [ ] ( ) : [ ] ( ) N N k k k k k k y n k z Y z a y n k a z Y z− − = =   − → − →     ∑ ∑ 1 1 [ ] ( ) : [ ] ( ) M M k k k k k k x n k z X z b x n k b z X z− − = =   − → − →     ∑ ∑ 1 1 1 ( ) ( ) N M k k k k k k a z Y z b z X z− − = =     → + =        ∑ ∑ sites.google.com/site/ncpdhbkhn 32 LTI Systems Characterized by LCCDE (2) 1 1 [ ] [ ] [ ] N M k k k k y n a y n k b x n k = = = − − + −∑ ∑ 1 1 1 ( ) ( ) N M k k k k k k a z Y z b z X z− − = =     → + =        ∑ ∑ 1 1 1 ( ) ( ) ( ) M k k k N k k k b z Y z H z X z a z − = − = → = = + ∑ ∑ 11 0 0 1 1 0 0 ( ) ... ( )...( ) M k M M M MM k M k b bB z b z b z z z b z z z z z b b − − − − =   = = + + + = − −    ∑ 1 1 1 1 1( ) ( ... ) ( )...( ) N k N N N N k N N k A z a z z z a z a z z p z p− − − − = = + = + + + = − −∑ 1 1 1 0 0 1 1 1 1 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) M M M k k k k N NN k k k k z z z z B z zH z b b A z z z p p z − − = = − − = = − − → = = = − − ∏ ∏ ∏ ∏ sites.google.com/site/ncpdhbkhn 33 LTI Systems Characterized by LCCDE (3) 1 2 1 21 4 3 5 4 3( ) ( ) ( ) ( )z z Y z z z X z− − − −− + = − + Ex. 1 2 1 2 5 4 3 1 4 3 ( ) ( ) . ( ) Y z z zH z X z z z − − − − − + = = − + Consider a system function Find its corresponding difference equation? 1 2 1 2 ( ) [ ] ( ) [ ] z X z x n z X z x n − − → − → − 4 1 3 2 5 4 1 3 2[ ] [ ] [ ] [ ] [ ] [ ]y n y n y n x n x n x n→ − − + − = − − + − 4 1 3 2 5 4 1 3 2[ ] [ ] [ ] [ ] [ ] [ ]y n y n y n x n x n x n→ = − − − + − − + − sites.google.com/site/ncpdhbkhn 34 LTI Systems Characterized by LCCDE (4) 1 1 1 ( ) M k k k N k k k b z H z a z − = − = = + ∑ ∑ 1 1 11 M N N k k k k k k AC z p z − − − = = = + − ∑ ∑ 1 1 [ ] [ ] ( ) [ ] M N N n k k k k k h n C n k A p u nδ − = = → = − +∑ ∑ 0 1 1 0 Stable : [ ] M N N n k k k n k k n h n C A p ∞ − ∞ = = = = = + < ∞∑ ∑ ∑ ∑ 1 for 1,...,kp k N→ < = A causal LTI with a rational system function is stable if and only if all poles of H(z) are inside the unit circle in the z – plane. The zeros can be anywhere. LTI Systems Characterized by LCCDE (5) • If N > 0: an Infinite Impulse Response (IIR) system • If N = 0:  a Finite Impulse Response (FIR) system • If N ≥ 1: a recursive system • If N = 0: a nonrecursive system • If ak = 0, for k = 1, , N: all – zero system • If bk = 0, for k = 1, , M: all – pole system sites.google.com/site/ncpdhbkhn 35 1 1 1 1 1 1 1 11 ( ) [ ] [ ] ( ) [ ] M k k M N N M N N k nk k k k k kN k k k k kk k k b z AH z C z h n C n k A p u n p z a z δ − − − −= − − = = = = = = = + → = − + − + ∑ ∑ ∑ ∑ ∑ ∑ 0 0 0 , [ ] [ ] , otherwise M n k k b n M h n b n kδ = ≤ ≤ = − =   ∑ The z – Transform 1. The z – Transform 2. The Inverse z – Transform 3. Properties of the z – Transform 4. System Function of LTI Systems 5. LTI Systems Characterized by Linear Constant – Coefficient Difference Equations 6. Connections between Pole – Zero Locations and Time – Domain Behavior 7. The One – Sided z – Transform sites.google.com/site/ncpdhbkhn 36 sites.google.com/site/ncpdhbkhn 37 Connections between Pole – Zero Locations and Time – Domain Behavior (1) th th 1 1 1 1 first-order systemorder FIR1 order system 11 ( ) ( ) M k k M N N kk k kN k k k k k NM Nk N b z AH z C z p z a z − − −= − − = = −= = = + − + ∑ ∑ ∑ ∑    1 1 0 N k k k a z− = + =∑The equation has K1 real roots & 2K2 complex conjugate roots 1 0 1 1 1 1 2 1 21 1 1 * * b b zA A pz p z a z a z − − − − − + + = − − + + 1 2 1 0 1 1 1 2 1 1 1 1 21 1 ( ) K KM N k k k k k k k kk k k A b b zH z C z p z a z a z − − − − − − = = = + = + + − + + ∑ ∑ ∑ sites.google.com/site/ncpdhbkhn 38 Connections between Pole – Zero Locations and Time – Domain Behavior (2) 11 ( ) , , real [ ] [ ]n bH z a b h n ba u n az− = → = − 0 1 Im Re 0 1 n Decaying alternating exponential 0 1 n Unit alternating step 0 1 n Growing alternating exponential 0 1 n Decaying exponential 0 1 n Growing exponential 0 1 n Unit step sites.google.com/site/ncpdhbkhn 39 Connections between Pole – Zero Locations and Time – Domain Behavior (3) 1 0 1 0 1 1 2 2 1 2 1 21 ( ) ( ) k k k k b b z z b z bH z a z a z z a z a − − − + + = = + + + + 2 1 1 21 1 2 1 2 0 4 0 2, Zero : ; , pole : a a ab z z p b − ± − − = = = 02[ ] cos( ) [ ] nh n A r n u nω θ= + 0 n n r [ ]h n 0 1 Im Rer1r < sites.google.com/site/ncpdhbkhn 40 Connections between Pole – Zero Locations and Time – Domain Behavior (4) 0 n n r [ ]h n 0 1 Im Re1r > 0 1 Im Re1r = 0 n 1r =[ ]h n The z – Transform 1. The z – Transform 2. The Inverse z – Transform 3. Properties of the z – Transform 4. System Function of LTI Systems 5. LTI Systems Characterized by Linear Constant – Coefficient Difference Equations 6. Connections between Pole – Zero Locations and Time – Domain Behavior 7. The One – Sided z – Transform sites.google.com/site/ncpdhbkhn 41 sites.google.com/site/ncpdhbkhn 42 The One – Sided z – Transform two-sided/bilateral -transform( ) [ ] ( )n n X z x n z z ∞ − =−∞ = ∑ 0 one-sided/unilateral -transform( ) { [ ]} [ ] ( )n n X z Z x n x n z z ∞ + + − = = =∑ 1 2 31 1 0 1 2{ [ ]} [ ] [ ] [ ] [ ] ...Z x n x x z x z x z+ − − −− = − + + + + 1 1 21 0 1 2[ ] ( [ ] [ ] [ ] ...)x z x x z x z− − −= − + + + + 11[ ] ( )x z X z− += − + 1 21 2 1{ [ ]} [ ] [ ] ( )Z x n x x z z X z+ − − +− = − + − + 1 { [ ]} ( ) [ ] k k m k m Z x n k z X z x m z+ − + − = − = + −∑

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