Discrete-Time Signal processing Chapter 3 the Z-transform Zhongguo Liu Biomedical Engineering School of Control Science and Engineering, Shandong University 山东省精品课程《生物医学信号处理(双语)》 http://course.sdu.edu.cn/bdsp.html

Chapter 3 The z-Transform 2018年12月7日5时57分 Chapter 3 The z-Transform 3.0 Introduction 3.1 z-Transform 3.2 Properties of the Region of Convergence for the z-transform 3.3 The inverse z-Transform 3.4 z-Transform Properties 3.5 z-Transform and LTI Systems 3.6 the Unilateral z-Transform

3.0 Introduction Fourier transform plays a key role in analyzing and representing discrete-time signals and systems, but does not converge for all signals. Continuous systems: Laplace transform is a generalization of the Fourier transform. Discrete systems : z-transform, generalization of DTFT, converges for a broader class of signals.

3.0 Introduction Motivation of z-transform: The Fourier transform does not converge for all sequences; It is useful to have a generalization of the Fourier transform→z-transform. In analytical problems the z-Transform notation is more convenient than the Fourier transform notation.

3.1 z-Transform Fourier transform z-transform If , z-Transform: two-sided, bilateral z-transform one-sided, unilateral z-transform Go

Relationship between z-transform and Fourier transform Express the z in polar form as The Fourier transform of the product of and (the exponential sequence ).

Region Of Convergence (ROC) Complex z plane Region Of Convergence (ROC)

periodic sampling T: sampling period (单位s); fs=1/T: sampling rate(Hz); Review periodic sampling T: sampling period (单位s); fs=1/T: sampling rate(Hz); Ωs=2π/T: sampling rate 模拟频率 模拟角频率 (rad/s) t n

Complex frequency domain： Review Relation between Laplace Transform and Z-transform Laplace transform Continuous Time domain： Complex frequency domain： Region Of Convergence (ROC)

Laplace Transform and Fourier transform Since So Fourier Transform frequency domain ： Fourier Transform is the Laplace transform when s have the value only in imaginary axis, s=jΩ

z-transform of discrete-time signal For sampling signal， the Laplace transform 令 z-transform of discrete-time signal

 let： so： 数字频率(单位rad)和模拟频率(rad/s)的关系 Laplace transform continuous time signal z-transform discrete-time signal so：  数字频率(单位rad)和模拟频率(rad/s)的关系

DTFT : Discrete Time Fourier Transform 2018年12月7日5时57分 DTFT : Discrete Time Fourier Transform S plane Z plane - Go

Region of convergence (ROC) For any given sequence, the set of values of z for which the z-transform converges is called the Region Of Convergence (ROC). Absolute Summability the ROC consists of all values of z such that the inequality in the above holds

Region of convergence (ROC) Convergence of the z-transform for a given sequence depends only on . if some value of z, say, z =z1, is in the ROC, z1 then all values of z on the circle defined by |z|=|z1| will also be in the ROC. 设|z|=|z1|=1在收敛域内 一致收敛uniform convergence if ROC includes unit circle, then Fourier transform and all its derivatives with respect to w must be continuous functions of w.

Region of convergence (ROC) The Fourier transforms are not continuous, infinitely differentiable functions, so they cannot result from evaluating a z-transform on the unit circle. it is not strictly correct to think of the Fourier transform as being the z-transform evaluated on the unit circle.

Zero and pole The z-transform is most useful when the infinite sum can be expressed in closed form, usually a ratio of polynomials in z (or z-1). Zero: The value of z for which Pole: The value of z for which

Example 3.1: Right-sided exponential sequence Determine the z-transform, including the ROC in z-plane and a sketch of the pole-zero-plot, for sequence: no DTFT DTFT Solution:   ROC:

: zeros  : poles Gray region: ROC

Ex. 3.2 Left-sided exponential sequence Determine the z-transform, including the ROC, pole-zero-plot, for sequence: Solution: ROC:

比较: : zeros  : poles Gray region: ROC

Ex. 3.3 Sum of two exponential sequences Determine the z-transform, including the ROC, pole-zero-plot, for sequence: Solution:

Example 3.3: Sum of two exponential sequences ROC:

Example 3.4: Sum of two exponential another solution for Ex.3.3 Solution: ROC:

Example 3.5: Two-sided exponential sequence Solution:

ROC, pole-zero-plot

Finite-length sequence Example : or entire z-plane, except z=0

Example 3.6: Finite-length sequence Determine the z-transform, the ROC, pole-zero-plot, for sequence: (N=16, 0<a<1) Solution: z-transform or , including z=a

with N=16, a is real, and 0<a<1 pole-zero-plot 零极点图 N=16

z-transform pairs 1. 2. 3. 4.

z-transform pairs 5. 6. 7. 8.

z-transform pairs 9. 10.

z-transform pairs 11. 12.

z-transform pairs 13.

3.2 Properties of the ROC for z-transform 2018年12月7日5时57分 3.2 Properties of the ROC for z-transform Property 1: The ROC will either be of the form or, in general the annulus, i.e.,

3.2 Properties of the ROC for z-transform 2018年12月7日5时57分 3.2 Properties of the ROC for z-transform Property 1: The ROC : For a given x[n], ROC is dependent only on . (not on the angle of z). Two-Sided Sequence:

3.2 Properties of the ROC for the z-transform Property 2: The Fourier transform of converges absolutely if and only if the ROC of the z-transform of includes the unit circle. The z-transform reduces to the Fourier transform when ie.

3.2 Properties of the ROC for the z-transform Property 3: The ROC cannot contain any poles. is infinite at a pole and therefore does not converge.

3.2 Properties of the ROC for the z-transform Property 4: If is a finite-duration sequence, i.e., a sequence that is zero except in a finite interval : then the ROC is the entire z-plane, except possibly or

3.2 Properties of the ROC for the z-transform Property 5: If is a right-sided sequence, i.e., for , then the ROC extends outward from the outermost finite pole in to (may including) . Proof: for ROC must let i.e. ROC proved.

3.2 Properties of the ROC for the z-transform Property 6: If is a left-sided sequence, i.e., for , then the ROC extends inward from the innermost nonzero pole in to . Proof: for ROC must let i.e. ROC proved.

3.2 Properties of the ROC for the z-transform Property 7: A two-sided sequence is an infinite-duration sequence that is neither right-sided nor left-sided. If is a two-sided sequence, the ROC will consist of a ring in the z-plane, bounded on the interior and exterior by a pole and consistent with Property 3, not containing any poles.

3.2 Properties of ROC for z-transform Property 8: ROC must be a connected region. for finite-duration sequence possibly ROC: for right-sided sequence possibly ROC: for left-sided sequence possibly ROC: for two-sided sequence If no overlap between ROCs of right- and left-sided parts; i.e., rL<rR, no z-transform exists.

Example 3.7: Non-Overlapping ROCs Solution: Since there is no overlap between x[n] has no z-transform (nor Fourier transform).

Example: Different possibilities of the ROC define different sequences A system with three poles

Different possibilities of the ROC. (c) ROC to a left-handed sequence (b) ROC to a right-sided sequence

(e) ROC to another Unit-circle included Unit-circle (d) ROC to a two-sided sequence two-sided sequence not stable stable

LTI system Stability, Causality, and ROC A z-transform does not uniquely determine a sequence without specifying the ROC It’s convenient to specify the ROC implicitly through time-domain property of a sequence Consider a LTI system with impulse response h[n]. The z-transform of h[n] is called the system function H (z) of the LTI system. stable system(h[n] is absolutely summable and therefore has a Fourier transform): ROC include unit-circle. causal system: h[n]=0,for n<0, right sided

Ex. 3.8 Stability, Causality, and the ROC A LTI system with impulse response h[n]. The z-transform of h[n] i.e. system function H(z) has pole-zero plot shown in Figure 3.9. Determine the Stability, Causality,ROC, for: (1) stable system: (ROC include unit-circle) (2) causal system: (right sided sequence) (3) non-causal, unstable system Figure 3.9

Ex. 3.8 Stability, Causality, and the ROC Solution: (1) stable system (ROC include unit-circle), ROC: , the impulse response is two-sided, system is non-causal. stable.

Ex. 3.8 Stability, Causality, and the ROC (2) causal system: (right sided sequence) ROC: ,the impulse response is right-sided. system is causal but unstable.

Ex. 3.8 Stability, Causality, and the ROC (2) causal system: (right sided sequence) ROC: ,the impulse response is right-sided. system is causal but unstable. Unit-circle A system is causal and stable if all the poles are inside the unit circle, and the ROC extends outward from the outermost pole to z=∞.

Ex. 3.8 Stability, Causality, and the ROC (3) ROC: , the impulse response is left-sided, system is non-causal, unstable since the ROC does not include unit circle.

3.3 The Inverse Z-Transform Formal inverse z-transform is based on a Cauchy integral theorem. （留数residue法） Zi是X(z)zn-1在围线C内的极点。 围线c：X(z)的环状收敛域内环绕原点的一条逆时针的闭合单围线。 Zi

3.3 The Inverse Z-Transform Less formal ways are sufficient and preferable in finding the inverse z-transform : Inspection method (观察法) Partial fraction expansion (部分分式展开法) Power series expansion (幂级数展开法)

3.3 The inverse z-Transform 3.3.1 Inspection Method(观察法)

3.3 The inverse z-Transform 3.3.1 Inspection Method

3.3 The inverse z-Transform 3.3.2 Partial Fraction Expansion部分分式展开法

Ex. 3.9 find inverse z-Transform of 2nd-Order z-Transform Solution:

Example 3.9 Second-Order z-Transform

Inverse Z-Transform by Partial Fraction Expansion Br is obtained by long division

Inverse Z-Transform by Partial Fraction Expansion s重极点di if M≥N, and has a pole of order s at d=di Br is obtained by long division;

Example 3.10: Inverse by Partial Fractions

3.3 The inverse z-Transform 3.3.3 Power Series Expansion幂级数展开法 If the z-transform is given as a power series: Laurent series where x[n] are the coefficients of z-n we can determine any particular value of the sequence x[n] by finding the coefficient of the appropriate power of z-1.

Example 3.11: find Finite-Length Sequence x[n] of X(z): Solution:

Ex. 3.12: find Inverse Transform by power series expansion Solution: Taylor series:

Example 3.13: Power Series Expansion by Long Division Solution:

Example 3.13(第二版): Power Series Expansion for a Left-sided Sequence Solution:

3.4 z-Transform Properties 3.4.1 Linearity

Example of Linearity Find X(z) of Solution:

3.4.2 Time Shifting is an integer is positive, is shifted right is negative, is shifted left

Time Shifting: Proof

Example 3.14: Shifted Exponential Sequence , Determine x[n]. Solution 2: Solution 1:

3.4.3 Multiplication by an Exponential sequence that is the frequency shifting of Fourier Transform

Ex. 3.15 Exponential Multiplication Use , determine Z-transform of Solution :

Example 3.15: Exponential Multiplication

3.4.4 Differentiation of X(z) Proof:

Example 3.16: Inverse of Non-Rational z-Transform Check Solution :

Example 3.17: Second-Order Pole 2018年12月7日5时57分 Example 3.17: Second-Order Pole determine X(z) of Solution :

3.4.5 Conjugation of a complex Sequence Proof:

3.4. 6 Time Reversal

Example 3.18: Find X(z) of Time-Reverse Exponential Sequence Solution:

3.4. 7 Convolution of Sequences Proof:

Ex. 3.19: Convolution of Finite-Length Sequences Given Solution:

3.4. 8(第二版) Initial Value Theorem Proof:

3.4. 8 Summary of Some z-Transform Properties TABLE 3.2

3.5 z-transforms and LTI systems 2018年12月7日5时57分 3.5 z-transforms and LTI systems z-transform can be used in the representation and analysis of LTI systems. h[n] x[n] y[n] X(z) Y(z) H(z) System Function: y[n] = h[n] *x[n] Y(z) = H(z) X(z)

3.5 z-transforms and LTI systems analysis of LTI system in difference equation: h[n] x[n] initial rest conditions → causal LTI system for LTI System:

Ex.3.20 Convolution of Infinite-Length Sequences Solution:

Example 3.20: Convolution of Infinite-Length Sequences Using the z-transform

3.5 z-transforms and LTI systems h[n] x[n] y[n] Z z-plane  unit circle for causal LTI System, and if stable all poles are in the unit circle.

Ex. 3.21 1st-order System for causal LTI System, Solution: 1). Iterate the difference equation, not easy to get a closed form solution; 2). convolution techniques, y[n] = h[n] *x[n] 3).

3.6 the Unilateral Z-transform two-sided, bilateral z-transform one-sided, unilateral z-transform ROC: If x[n]=0 for n< 0, the unilateral and bilateral z-transforms are identical, if x[n] is not zero for all n< 0, they will be different..

Ex. 3.22 Unilateral Transform of an Impulse 2018年12月7日5时57分 Ex. 3.22 Unilateral Transform of an Impulse compare bilateral and Unilateral z-transform of Solution: bilateral z-transform: Unilateral z-transform identical to different

3.6 the Unilateral Z-transform h[n] x[n] usually initial rest conditions. in noninitial rest conditions, the linearity and time-shifting properties of the unilateral z-transform are particularly useful tools. the linearity prop­erty is identical to that of the bilateral z-transform, time-shifting property is different: for

3.6 the Unilateral Z-transform By a similar analysis, Proof:

Ex. 3.23 Effect of Nonzero Initial Conditions 2018年12月7日5时57分 Ex. 3.23 Effect of Nonzero Initial Conditions initial condition at n= -1: y[-1], find y[n]= ? 2) if x[n]=Au[n]. Solution: Apply the Unilateral z-transform, using the linearity property and the time-shift property: if initial rest condition: system function: LTI system

Ex. 3.23 Effect of Nonzero Initial Conditions 2018年12月7日5时57分 Ex. 3.23 Effect of Nonzero Initial Conditions initial condition at n=-1: y[-1], find y[n]= ? 2) if x[n]=Au[n]. Solution: T{‧} 齐次性 not linear, ∵ 0x[n]=0, T{0x[n]}≠0y[n]=0.

Ex. 3.23 Effect of Nonzero Initial Conditions 2018年12月7日5时57分 Ex. 3.23 Effect of Nonzero Initial Conditions find y[n]= ? 2) if x[n]=Au[n]. Solution:

Ex. 3.23 Effect of Nonzero Initial Conditions 2018年12月7日5时57分 Ex. 3.23 Effect of Nonzero Initial Conditions 2) if x[n]=Au[n]. find y[n]= ? Solution: ZIR ZICR not linear, not time-shifting zero input response (ZIR) The zero initial conditions response (ZICR)

Zhongguo Liu_Biomedical Engineering_Shandong Univ. Chapter 3 HW 3.3, 3.4, 3.7, 3.16 3.2 , 3.8, 3.11, 3.20 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 122 2018/12/7 返 回 上一页 下一页