Đang tải... (xem toàn văn)
Chapter 3 introduce the z-transform. This chapter presents the following content: z-transform, properties of the ROC, inverse z-transform, properties of z-transform.
Digital Signal Processing, Fall 2006 Lecture 3: The z-transform Zheng-Hua Tan Department of Electronic Systems Aalborg University, Denmark zt@kom.aau.dk Digital Signal Processing, III, Zheng-Hua Tan, 2006 Course at a glance MM1 Discrete-time signals and systems MM2 Fourier-domain representation Sampling and reconstruction System System structures System analysis MM6 MM5 Filter MM4 z-transform MM3 DFT/FFT Filter structures MM9,MM10 MM7 Filter design MM8 Digital Signal Processing, III, Zheng-Hua Tan, 2006 Part I: z-transform z-transform Properties of the ROC Inverse z-transform Properties of z-transform Digital Signal Processing, III, Zheng-Hua Tan, 2006 Limitation of Fourier transform Fourier transform ∞ ∑ x[n]e − jωn X ( e jω ) = n = −∞ x[ n] = 2π π ∫−π X (e jω )e jωn dω Condition for the convergence of the infinite sum | X ( e jω ) | = | ∞ ∞ ∞ n = −∞ n = −∞ n = −∞ ∑ x[n]e − jωn | ≤ ∑ | x[n] ||e − jωn | ≤ ∑ | x[n] | < ∞ If x[n] is absolutely summable, its Fourier transform exists (sufficient condition) x[n] = a n u[n] | a |< : X (e jω ) = Example − ae − jω a = : X ( e jω ) = ∞ + ∑ πδ (ω + 2πk ) − e − jω k = −∞ | a |> : Digital Signal Processing, III, Zheng-Hua Tan, 2006 z-transform Fourier transform X ( e jω ) = ∞ ∑ x[n] e − jω n n = −∞ z-transform ∞ X ( z) = ∑ x[n] z−n Z x[n] ↔ X ( z ) n = −∞ The complex variable z in polar form z = re jω ∞ ∞ −∞ −∞ X ( z ) = X ( re jω ) = ∑ x[ n](re jω ) − n = ∑ ( x[n]r − n )e − jωn |z| = r = 1, X ( z ) = X (e jω ) Digital Signal Processing, III, Zheng-Hua Tan, 2006 z-plane z-transform is a function of a complex variable Ỉ using the complex z-plane Z-transform on unit circle Fourier transform Linear frequency axis in Fourier transform ỈUnit circle in z-transform (periodicity in freq of Fourier transform) Digital Signal Processing, III, Zheng-Hua Tan, 2006 Region of convergence – ROC Fourier transform does not converge for all ∞ sequences jω − j ωn X (e ) = ∑ x[n]e n = −∞ z-transform does not converge for all sequences or for all values of z X(z) = X ( re jω ) = ∞ ∑ x[n] z −n = n = −∞ ∞ ∑ ( x[n]r − n )e − jωn n = −∞ ROC – for any given seq., the set of values of z for which the z-transform converges ∞ ∑ | x[n]r − n |< ∞ n = −∞ ∞ ∑ | x[n] | | z | − n < ∞ ROC is ring! n = −∞ Digital Signal Processing, III, Zheng-Hua Tan, 2006 ROC Outer boundary is a circle (may extend to infinity) Inner boundary is a circle (may extend to include the origin) If ROC includes unit circle, Fourier transform converges Digital Signal Processing, III, Zheng-Hua Tan, 2006 Zeros and poles The most important and useful z-transforms – rational function: P( z ) Q( z ) P ( z ) and Q ( z ) are polynomials in z X ( z) = Zeros: values of z for which X(z)=0 Poles : values of z for which X(z) is infinite Close relation between poles and ROC Digital Signal Processing, III, Zheng-Hua Tan, 2006 Right-sided exponential sequence x[n] = a n u[n] ∞ ∑ a n u[n]z − n X ( z) = n = −∞ ∞ = ∑ (az −1 ) n n=0 ROC ∞ ∑ | az −1 |n < ∞ n=0 z-transform ∞ X ( z ) = ∑ ( az −1 ) n = n=0 Z u[n] ↔ 10 , − z −1 z = , − az −1 z − a | z |>| a | | z |> Digital Signal Processing, III, Zheng-Hua Tan, 2006 Left-sided exponential sequence x[n] = − a n u[− n − 1] ∞ X ( z ) = − ∑ a n u[− n − 1]z − n n = −∞ −1 ∞ n = −∞ n=0 = − ∑ a n z − n = − ∑ (a −1 z ) n ROC ∞ ∑ | a −1 z |n < ∞ n=0 z-transform X ( z) = 11 z = , − az −1 z − a | z | and | z |> | z |> | 12 Digital Signal Processing, III, Zheng-Hua Tan, 2006 Sum of two exponential sequence Another way to calculate: 1 x[n] = ( ) n u[n] + (− ) n u[n] Z 1 ( ) n u[n] ↔ , | z |> 2 − z −1 Z 1 , | z |> (− ) n u[n] ↔ 3 + z −1 Z 1 1 + ( ) n u[n] + (− ) n u[n] ↔ 1 − z −1 + z −1 13 | z |> Digital Signal Processing, III, Zheng-Hua Tan, 2006 Two-sided exponential sequence 1 x[n] = (− ) n u[n] − ( ) n u[−n − 1] Z 1 (− ) n u[n] ↔ , | z |> 3 + z −1 Z 1 , | z |< − ( ) n u[ − n − 1] ↔ 2 − z −1 1 1 , | z |> , | z |< X ( z) = + −1 −1 1+ z 1− z ) 12 = 1 ( z − )( z + ) 2z( z − 14 Digital Signal Processing, III, Zheng-Hua Tan, 2006 Finite-length sequence ⎧ an , ≤ n ≤ N −1 x[n] = ⎨ otherwise ⎩0, N −1 N −1 X ( z ) = ∑ a n z − n = ∑ (az −1 ) n n =0 = − (az ) zN − aN = N −1 −1 − az z z−a N −1 ROC n =0 −1 N ∑ | az −1 n | ), − az −1 | z |>| a | x[n] = ( ) n u[n] of course, | z |< ? x[n] = −( ) n u[−n − 1] Digital Signal Processing, III, Zheng-Hua Tan, 2006 11 Partial fraction expansion 23 For rational function, get the format of a sum of simpler terms, and then use the inspection method Digital Signal Processing, III, Zheng-Hua Tan, 2006 Second-order z-transform X ( z) = 1− X ( z) = 1 , | z |> −1 − 2 z + z 1 −1 (1 − z )(1 − z −1 ) A1 A2 + −1 (1 − z ) (1 − z −1 ) −1 A1 = (1 − z ) X ( z ) | z =1/ = −1 A2 = (1 − z −1 ) X ( z ) | z =1/ = 2 −1 X ( z) = + −1 −1 (1 − z ) (1 − z ) X ( z) = M X ( z) = ∑b z −k ∑a z −k k =0 N k =0 k k M b X ( z) = a0 ∑ (1 − c z k =1 N k ∑ (1 − d k =1 k −1 ) z −1 ) N Ak −1 − d k =1 kz X ( z) = ∑ Ak = X ( z )(1 − d k z −1 ) | z = d k 1 x[n] = 2( ) n u[n] − ( ) n u[n] 24 Digital Signal Processing, III, Zheng-Hua Tan, 2006 12 What about M>=N? X ( z) = + z −1 + z −2 , | z |> 1 − z −1 + z −2 2 X ( z) = A1 A2 + −1 (1 − z −1 ) (1 − z ) Found by long division − −1 −2 −1 z − z +1 z + 2z +1 2 z − − z −1 + X ( z ) = B0 + B0 = − z −1 − 1 A1 = (1 − z −1 ) X ( z ) | z =1/ = −9 A2 = (1 − z −1 ) X ( z ) | z =1 = X ( z) = − 25 + −1 (1 − z −1 ) (1 − z ) (1 + z −1 ) (1 − z −1 )(1 − z −1 ) x[n] = 2δ [n] − 9( ) n u[n] + 8u[n] Digital Signal Processing, III, Zheng-Hua Tan, 2006 Power series expansion By long division X ( z) = , | z |>| a | − az −1 + az −1 + a z −2 + − az −1 1 − az −1 az −1 az −1 − a z − a z − = + az −1 + a z − + − az −1 x[n] = a n u[n] 26 Digital Signal Processing, III, Zheng-Hua Tan, 2006 13 Finite-length sequence −1 z )(1 + z −1 )(1 − z −1 ) 1 X ( z ) = z − z − + z −1 2 X ( z ) = z (1 − 1 x[n] = δ [n + 2] − δ [n + 1] − δ [n] + δ [n − 1] 2 27 Digital Signal Processing, III, Zheng-Hua Tan, 2006 Part IV: Properties of z-transform 28 z-transform Properties of the ROC Inverse z-transform Properties of z-transform Digital Signal Processing, III, Zheng-Hua Tan, 2006 14 Linearity Z x1[n] ↔ X ( z ), ROC = Rx1 X ( z) = ∑ x[n] z −n n = −∞ Z x2 [n] ↔ X ( z ), ROC = Rx2 ∞ Linearity Z ax1[n] + bx2 [n] ↔ aX ( z ) + bX ( z ), ROC contains Rx1 ∩ Rx2 at least Z , | z |> 1 − z −1 Z z −1 u[n − 1] ↔ , | z |> 1 − z −1 Z − z −1 u[n] − u[n − 1] = δ [n] ↔ = 1, All z − z −1 u[n] ↔ 29 Digital Signal Processing, III, Zheng-Hua Tan, 2006 Time shifting Z x[n − n0 ] ↔ z − n0 X ( z ), X ( z) = ROC = Rx1 (except for or ∞ ) 30 Example ∞ ∑ x[n] z −n n = −∞ , | z |> z− z −1 X ( z) = = z −1 ( ) −1 −1 1− z 1− z 4 Z n ( ) u[n] ↔ − z −1 Z n −1 ( ) u[n − 1] ↔ z −1 ( ) − z −1 X ( z) = Digital Signal Processing, III, Zheng-Hua Tan, 2006 15 Multiplication by exponential sequence Z z0n x[n] ↔ X ( z / z ), ROC = |z 0|Rx ∞ ∑ x[n] z X ( z) = −n n = −∞ F e jω0 n x[n] ↔ X (e j (ω −ω0 ) ) Examples Z , | z |> 1 − z −1 Z 1 a nu[n] ↔ = , | z |> a −1 − ( z / a) − az −1 1 x[n] = cos(ω0 n)u[n] = (e jω ) n u[n] + (e − jω ) n u[n] 2 1 (1 − cos ω0 z −1 ) 2 X ( z) = + = , − e jω z −1 − e − jω z −1 − cos ω0 z −1 + z − u[n] ↔ 0 31 | z |> Digital Signal Processing, III, Zheng-Hua Tan, 2006 Differentiation of X(z) dX ( z ) nx[n] ↔ − z , ROC = Rx dz Z Example X ( z) = ∞ ∑ x[n] z −n n = −∞ X ( z ) = log(1 + az −1 ), | z |>| a | dX ( z ) − az − = dz + az −1 Z dX ( z ) az −1 nx[n] ↔ − z = , | z |>| a | dz + az −1 nx[n] = a (− a ) n −1 u[n − 1] x[n] = 32 a (− a) n −1 u[n − 1] n Digital Signal Processing, III, Zheng-Hua Tan, 2006 16 Conjugation of a complex sequence Z x [ n] ↔ X ( z ), ROC = Rx * 33 * * X ( z) = ∞ ∑ x[n] z −n n = −∞ Digital Signal Processing, III, Zheng-Hua Tan, 2006 Time reversal Z x[ − n] ↔ X (1 / z ), ROC = / Rx ∞ ∑ x[n] z −n n = −∞ Example x[n] = a − nu[−n] X ( z) = 34 X ( z) = , | z || a | − az −1 Digital Signal Processing, III, Zheng-Hua Tan, 2006 17 Convolution of sequences Z x1[n] * x2 [n] ↔ X ( z ) X ( z ), X ( z) = x[n] = ( ) n u[n] h[n] = u[n] y[n] ? 35 1 , | z |> −1 1− z H ( z) = , | z |> 1 − z −1 1 Y ( z) = , | z |> −1 −1 (1 − z )(1 − z ) 1 ) = − ( −1 (1 − z −1 ) (1 − z ) 1− 2 X ( z) = Example y[n] = −n n = −∞ ROC contains Rx1 ∩ Rx2 ∞ ∑ x[n] z 1 (u[n] − ( ) n+1 u[n]) 1− Digital Signal Processing, III, Zheng-Hua Tan, 2006 Initial-value theorem X ( z) = x[0] = lim X ( z ) z →∞ lim X ( z ) = lim z →∞ = z →∞ ∞ ∑ x[n] lim n = −∞ z →∞ ∞ ∑ x[n] z −n n = −∞ ∞ ∑ x[n] z −n n = −∞ z− n = x[0] 36 Digital Signal Processing, III, Zheng-Hua Tan, 2006 18 Properties of z-transform 37 Digital Signal Processing, III, Zheng-Hua Tan, 2006 Summary 38 z-transform Properties of the ROC Inverse z-transform Properties of z-transform Digital Signal Processing, III, Zheng-Hua Tan, 2006 19 Course at a glance MM1 Discrete-time signals and systems MM2 Fourier-domain representation Sampling and reconstruction System System structures System analysis MM6 MM5 Filter MM4 z-transform MM3 39 DFT/FFT Filter structures MM9,MM10 MM7 Filter design MM8 Digital Signal Processing, III, Zheng-Hua Tan, 2006 20 ... = −∞ z− n = x[0] 36 Digital Signal Processing, III, Zheng-Hua Tan, 2006 18 Properties of z-transform 37 Digital Signal Processing, III, Zheng-Hua Tan, 2006 Summary 38 z-transform Properties... the ROC Inverse z-transform Properties of z-transform Digital Signal Processing, III, Zheng-Hua Tan, 2006 Properties of the ROC 18 Digital Signal Processing, III, Zheng-Hua Tan, 2006 Properties... , N − Digital Signal Processing, III, Zheng-Hua Tan, 2006 Some common z-transform pairs 16 Digital Signal Processing, III, Zheng-Hua Tan, 2006 Part II: Properties of the ROC 17 z-transform