Lecture Signal processing: The z – Transform include all of the following: The z – transform, the inverse z – transform, properties of the z – transform, system function of LTI systems, LTI systems characterized by linear constant – coefficient difference equations, connections between pole – zero locations and time – domain behavior, the one – sided z – transform.
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 The z – Transform The z – Transform The Inverse z – Transform Properties of the z – Transform System Function of LTI Systems LTI Systems Characterized by Linear Constant – Coefficient Difference Equations Connections between Pole – Zero Locations and Time – Domain Behavior The One – Sided z – Transform sites.google.com/site/ncpdhbkhn The z – Transform (1) x[n ] = ∞ ∑ x[k ]δ [n − k ] → y[ n ] = k =−∞ x[n ] = z , for all n z = Re( z ) + j Im( z ) n → y[ n ] = ∞ ∞ k =−∞ k =−∞ ∑ x[k ]h[n − k ] = ∑ h[k ]x[n − k ] ∞ ∑ h[k ]z k =−∞ n−k ∞ = ∑ h[k ] z − k z n , k =−∞ H (z) = ∞ ∑ h[k ]z for all n −k k =−∞ → y[ n ] = H ( z ) z n , x[n ] = ∑ ck zkn , k for all n → y[n] = ∑ ck H ( zk ) zkn , for all n for all n k sites.google.com/site/ncpdhbkhn The z – Transform (2) X ( z) = ∞ Im( z ) z = e jω z – plane ∑ x[n]z −n ω n =−∞ • ROC (region of convergence): the set of values of z for which X(z) converges • Zeros: the values of z for which X(z) = • Poles: the values of z for which X(z) is infinite sites.google.com/site/ncpdhbkhn Re( z ) Unit circle Im( z ) z – plane r sin ω z = re jω r ω Re( z ) r cos ω The z – Transform (3) Ex ∞ Given x1[n ] = {1 5}, x2 [ n] = {1 5} ↑ ↑ X ( z) = x[ n]z − n Determine their z – transforms? n =−∞ ∑ X ( z ) = x1[0]z + x1[1] z −1 + x1[2] z −2 + x1[3]z −3 + x1[4]z −4 = + z −1 + 3z −2 + z −3 + z −4 ROC: entire z – plane except z = X ( z ) = x2 [ −2]z − ( −2 ) + x2 [ −1]z − ( −1) + x2 [0]z + x2 [1]z −1 + x2 [2]z −2 = 1z + z + 3z + z −1 + 5z −2 = z + z + + z −1 + z −2 ROC: entire z – plane except z = & z = ∞ sites.google.com/site/ncpdhbkhn The z – Transform (4) Ex x1[n ] = δ [n], x2 [n] = δ [ n − k ], x3[n ] = δ [n + k ], k > Determine their z – transforms? X ( z) = ∞ ∑ x[n]z n =−∞ X ( z ) = + x1[ −1] z − ( −1) + x1[0]z + x1[1]z −1 + x1[2] z −2 + = + z − ( −1) + 1z + z −1 + z −2 + = ROC: entire z – plane sites.google.com/site/ncpdhbkhn −n The z – Transform (5) Ex 1, Find the z – transforms of the square – pulse sequence x[ n] = 0, X (z) = ∞ ∑ n =−∞ x[n ]z −n 0≤n≤ M otherwise M = ∑1z − n n =0 M +1 − A + A + A2 + A3 + + AN = , if A < 1− A Im − z −( M +1) → X (z) = − z −1 z −1 Re 1 ROC: |z| > sites.google.com/site/ncpdhbkhn ROC The z – Transform (6) Ex Find the z – transforms of the sequence x[n] = anu[n]? X (z) = ∞ ∑ n =−∞ ∞ x[n ]z − n = ∑ a z + A + A2 + A3 + = n =0 n −n ∞ = ∑ (az −1 )n n=0 , if A < 1− A → X (z) = az −1 < → z > a sites.google.com/site/ncpdhbkhn z = − az −1 z − a Zero: z = Pole: z = a ROC: |z| > a The z – Transform (7) Ex Find the z – transforms of the sequence x[n] = anu[n]? X ( z ) = z = − az −1 z − a Zero: z = 0; pole: z = a; ROC: |z| > a 0< a 1 a =1 … … … … … n n Im a n Im Re Im Re 0 ROC … ROC sites.google.com/site/ncpdhbkhn a Re ROC 10 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| = • 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 System Function of LTI Systems (4) H1(z) x[n ] y[n ] + y[ n ] x[n] H1(z)+H2(z) H2(z) y[ n ] x[ n ] H1(z) y[n ] x[n] H2(z) sites.google.com/site/ncpdhbkhn H1(z)H2(z) 29 The z – Transform The z – Transform The Inverse z – Transform Properties of the z – Transform System Function of LTI Systems LTI Systems Characterized by Linear Constant – Coefficient Difference Equations Connections between Pole – Zero Locations and Time – Domain Behavior The One – Sided z – Transform sites.google.com/site/ncpdhbkhn 30 LTI Systems Characterized by LCCDE (1) N M k =1 k =1 y[ n] = − ∑ ak y[n − k ] + ∑ bk x[ n − k ] N M k =1 k =1 → y[ n] + ∑ ak y[n − k ] = ∑ bk x[n − k ] y[ n] → Y ( z ) N −k y[ n − k ] → z Y ( z ) : ∑ ak y[n − k ] → ∑ ak z Y ( z ) k =1 k =1 −k N M −k −k x[n − k ] → z X ( z ) : ∑ bk x[n − k ] → ∑ bk z X ( z ) k =1 k =1 M N M → + ∑ ak z − k Y ( z ) = ∑ bk z − k X ( z ) k =1 k =1 sites.google.com/site/ncpdhbkhn 31 LTI Systems Characterized by LCCDE (2) N M −k y[n ] = −∑ ak y[ n − k ] + ∑ bk x[n − k ] → + ∑ ak z Y ( z) = ∑ bk z − k X ( z) k =1 k =1 k =1 k =1 N M M → Y ( z) = X (z) ∑b z k =1 N −k k + ∑ ak z − k = H ( z) k =1 M B( z) = ∑ bk z k =1 −k = b0 z −M M b1 M −1 bM z + z + + b0 b0 −M = b z ( z − z1 ) ( z − zM ) N A( z ) = + ∑ ak z − k = z − N ( z N + a1 z N −1 + + a N ) = z − N ( z − p1 ) ( z − pN ) k =1 M B( z ) z −M → H (z) = = b0 − N A( z ) z ∏ ( z − zk ) k =1 N ∏ ( z − pk ) k =1 sites.google.com/site/ncpdhbkhn M = b0 −1 ( − z z ∏ k ) k =1 N −1 ( − p z ) ∏ k k =1 32 Ex LTI Systems Characterized by LCCDE (3) Y ( z ) − z −1 + 3z −2 = Consider a system function H ( z ) = −1 −2 X ( z) − z + 3z Find its corresponding difference equation? (1 − z −1 + 3z −2 )Y ( z ) = (5 − z −1 + 3z −2 ) X ( z ) z −1 X ( z ) → x[n − 1] z −2 X ( z ) → x[n − 2] → y[n] − y[n − 1] + y[n − 2] = x[n] − x[n − 1] + x[n − 2] → y[n] = y[n − 1] − y[n − 2] + x[n] − x[n − 1] + x[n − 2] sites.google.com/site/ncpdhbkhn 33 LTI Systems Characterized by LCCDE (4) M H ( z) = −k b z ∑k k =1 N + ∑ ak z − k = M −N ∑ Ck z −k N Ak −1 − p z k =1 k +∑ k =1 k =1 → h[n] = M −N ∑ C δ [n − k ] + ∑ A ( p ) u[n] n k =1 Stable : N k k =1 ∞ M −N n=0 k =1 ∑ h[n] = ∑ k k N Ck + ∑ Ak k =1 ∞ ∑ n =0 n pk < ∞ → pk < for k = 1, , 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 sites.google.com/site/ncpdhbkhn 34 LTI Systems Characterized by LCCDE (5) M H (z) = ∑b z k =1 N −k k + ∑ ak z − k M −N M −N N Ak n = ∑ Ck z + ∑ → h [ n ] = C δ [ n − k ] + A ( p ) u[n ] ∑ ∑ k k k −1 k =1 k =1 − pk z k =1 k =1 −k N k =1 • If N > 0: an Infinite Impulse Response (IIR) system bn , ≤ n ≤ M If N = 0: h[n] = ∑ bkδ [n − k ] = k =0 0, otherwise 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 M • • • • • sites.google.com/site/ncpdhbkhn 35 The z – Transform The z – Transform The Inverse z – Transform Properties of the z – Transform System Function of LTI Systems LTI Systems Characterized by Linear Constant – Coefficient Difference Equations Connections between Pole – Zero Locations and Time – Domain Behavior The One – Sided z – Transform sites.google.com/site/ncpdhbkhn 36 Connections between Pole – Zero Locations and Time – Domain Behavior (1) M H (z) = ∑b z k =1 N −k k + ∑ ak z − k k =1 = M −N ∑ Ck z −k k =1 N Ak −1 k =1 − pk z +∑ ( M − N ) th order FIR N first-order system N th order system N The equation + ∑ ak z − k = has K1 real roots & 2K2 complex conjugate roots k =1 A A* b0 + b1 z −1 + = −1 * −1 − pz 1− p z + a1 z −1 + a2 z −2 H (z) = M −N ∑Cz k =1 k −k K2 Ak bk + bk z −1 +∑ +∑ −1 −1 − p z + a k z −2 k =1 k =1 + a k z k K1 sites.google.com/site/ncpdhbkhn 37 Connections between Pole – Zero Locations and Time – Domain Behavior (2) H (z) = b n , a , b real → h [ n ] = ba u[n ] −1 − az 1 Decaying alternating exponential … … n Im n Unit step Unit alternating step 1 Re … … … … Decaying exponential Growing alternating exponential n … … n Growing exponential … … n … … sites.google.com/site/ncpdhbkhn n 38 Connections between Pole – Zero Locations and Time – Domain Behavior (3) bk + bk 1z −1 z (b0 z + b1 ) H (z) = = + ak 1z −1 + ak z −2 z + a1z + a2 −b Zero : z1 = 0; z2 = , b0 − a1 ± a12 − 4a2 pole : p1,2 = h[n] = A r n cos(ω0n + θ )u[n] h[n] rn Im r 1 Re … … sites.google.com/site/ncpdhbkhn n 40 The z – Transform The z – Transform The Inverse z – Transform Properties of the z – Transform System Function of LTI Systems LTI Systems Characterized by Linear Constant – Coefficient Difference Equations Connections between Pole – Zero Locations and Time – Domain Behavior The One – Sided z – Transform sites.google.com/site/ncpdhbkhn 41 The One – Sided z – Transform X ( z) = ∞ ∑ x[n] z − n ( two-sided/bilateral z -transform) n =−∞ ∞ X ( z ) = Z {x[n]} = ∑ x[ n]z − n ( one-sided/unilateral z -transform) + + n =0 Z +{x[ n − 1]} = x[ −1] + x[0]z −1 + x[1]z −2 + x[ 2] z −3 + = x[−1] + z −1 ( x[0] + x[1]z −1 + x[2]z −2 + ) = x[ −1] + z −1 X + ( z ) Z +{x[ n − 1]} = x[ −2] + x[ −1]z −1 + z −2 X + ( z ) k Z {x[n − k ]} = z X ( z ) + ∑ x[− m]z m−k + −k + m =1 sites.google.com/site/ncpdhbkhn 42 ... outside the ROC sites.google.com/site/ncpdhbkhn 16 The z – Transform (14) sites.google.com/site/ncpdhbkhn 17 The z – Transform The z – Transform The Inverse z – Transform Properties of the z – Transform. .. ∑ b z = z −b n =−∞ ∞ z −1 n −n If az < → z > a → ∑ a z = z a n =0 −1 n≥0 n −n z z → X ( z) = + z b z a Zero: z = Pole: z = a, b ROC: a < |z| < b sites.google.com/site/ncpdhbkhn 12 The z – Transform. .. az −1 < → z > a sites.google.com/site/ncpdhbkhn z = − az −1 z − a Zero: z = Pole: z = a ROC: |z| > a The z – Transform (7) Ex Find the z – transforms of the sequence x[n] = anu[n]? X ( z ) = z