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 2017/4/17 Zhongguo Liu_Biomedical Engineering_Shandong Univ.

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 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. 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. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. 3.0 Introduction Motivation of z-transform: The Fourier transform does not converge for all sequences and it is useful to have a generalization of the Fourier transform. In analytical problems the z-Transform notation is more convenient than the Fourier transform notation. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. 3.1 z-Transform z-Transform: two-sided, bilateral z-transform one-sided, unilateral z-transform If , z-transform is Fourier transform. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Relationship between z-transform and Fourier transform Express the complex variable z in polar form as The Fourier transform of the product of and the exponential sequence Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. Complex z plane Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

periodic sampling Review T：sample period; fs=1/T:sample rate

Complex frequency domain： Review Relation between Laplace Transform and Z-transform Continuous Time domain： Laplace transform Complex frequency domain：

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： Laplace transform continuous time signal z-transform discrete-time signal so： 

DTFT : Discrete Time Fourier Transform S plane Z plane - Go

plane

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 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

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. 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) Neither of them is absolutely summable, neither of them multiplied by r-n (-∞<n<∞) would be absolutely summable for any value of r. Thus, neither them has a z-transform that converges absolutely. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

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. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. 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 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

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: Solution: ROC: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. : zeros  : poles Gray region: ROC Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Ex. 3.2 Left-sided exponential sequence Determine the z-transform, including the ROC, pole-zero-plot, for sequence: Solution: ROC: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Ex. 3.3 Sum of two exponential sequences Determine the z-transform, including the ROC, pole-zero-plot, for sequence: Solution: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Example 3.3: Sum of two exponential sequences ROC: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Example 3.4: Sum of two exponential Solution: Zhongguo Liu_Biomedical Engineering_Shandong Univ. ROC: 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Example 3.5: Two-sided exponential sequence Solution: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. ROC, pole-zero-plot Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Finite-length sequence Example : Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Example 3.6: Finite-length sequence Determine the z-transform, including the ROC, pole-zero-plot, for sequence: Solution: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. N=16, a is real pole-zero-plot Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. z-transform pairs Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. z-transform pairs Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. z-transform pairs Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. z-transform pairs Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. z-transform pairs Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

3.2 Properties of the ROC for the z-transform Property 1: The ROC is a ring or disk in the z-plane centered at the origin. For a given x[n], ROC is dependent only on .

3.2 Properties of the ROC for the z-transform Property 2: The Fourier transform of converges absolutely if the ROC of the z-transform of includes the unit circle. The z-transform reduces to the Fourier transform when ie. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

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. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

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 possible or Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

3.2 Properties of the ROC for the z-transform Property 5: If is a right-sided sequence, i.e., a sequence that is zero for , the ROC extends outward from the outermost finite pole in to (may including) Proof: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

3.2 Properties of the ROC for the z-transform Property 6: If is a left-sided sequence, i.e., a sequence that is zero for , the ROC extends inward from the innermost nonzero pole in to . Proof: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

3.2 Properties of the ROC for the z-transform 2017年4月17日3时34分 3.2 Properties of the ROC for the z-transform Property 5: If is a right-sided sequence, i.e., a sequence that is zero for , the ROC extends outward from the outermost finite pole in to (possibly including) Proof:

Zhongguo Liu_Biomedical Engineering_Shandong Univ. Property 5: right-sided sequence for the z-transform: For other terms: ROC Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

3.2 Properties of the ROC for the z-transform Property 6: If is a left-sided sequence, i.e., a sequence that is zero for , the ROC extends inward from the innermost nonzero pole in to Proof: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. Property 6: left-sided sequence for the z-transform: For other terms: ROC Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

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 not containing any poles. 2017/4/17

3.2 Properties of the ROC for the z-transform Property 8: ROC must be a connected region. for finite-duration sequence possible ROC: for right-sided sequence possible ROC: for left-sided sequence possible ROC: for two-sided sequence Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Example: Different possibilities of the ROC define different sequences A system with three poles Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. Different possibilities of the ROC. (c) ROC to a left-handed sequence (b) ROC to a right-sided sequence Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. (e) ROC to another two-sided sequence Unit-circle included (d) ROC to a two-sided sequence. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

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 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Ex. 3.7 Stability, Causality, and the ROC Consider a LTI system with impulse response h[n]. The z-transform of h[n] i.e. the system function H (z) has the pole-zero plot shown in Figure. Determine the ROC, if the system is: (1) stable system: (ROC include unit-circle) (2) causal system: (right sided sequence) Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Ex. 3.7 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. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Ex. 3.7 Stability, Causality, and the ROC (2) causal system: (right sided sequence) ROC: ,the impulse response is right-sided. system is causal but unstable. A system is causal and stable if all the poles are inside the unit circle. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Ex. 3.7 Stability, Causality, and the ROC ROC: , the impulse response is left-sided, system is non-causal, unstable since the ROC does not include unit circle. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

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。 Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

3.3 The Inverse Z-Transform Less formal ways are sufficient and preferable in finding the inverse z-transform. : Inspection method Partial fraction expansion Power series expansion Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

3.3 The inverse z-Transform 3.3.1 Inspection Method Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

3.3 The inverse z-Transform 3.3.1 Inspection Method Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

3.3 The inverse z-Transform 3.3.2 Partial Fraction Expansion Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Example 3.8 Second-Order z-Transform Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Example 3.8 Second-Order z-Transform Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Inverse Z-Transform by Partial Fraction Expansion Br is obtained by long division Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Inverse Z-Transform by Partial Fraction Expansion if M>N, and has a pole of order s at d=di Br is obtained by long division Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Example 3.9: Inverse by Partial Fractions Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

LTI system Stability, Causality, and ROC Review For a LTI system with impulse response h[n], if it is causal, what do we know about h[n]? Is h[n] one-sided or two-sided sequence? Left-sided or right-sided? Then what do we know about the ROC of the system function H (z)? If the poles of H (z) are all in the unit circle, is the system stable? Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

LTI system Stability, Causality, and ROC For H (z) with the poles as shown in figure , Review can we uniquely determine h[n] ? Unit-circle included If ROC of H(z) is as shown in figure, can we uniquely determine h[n] ? is the system stable ? Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

LTI system Stability, Causality, and ROC For H (z) with the poles as shown in figure , Review If the system is causal (h[n]=0,for n<0,right-sided ), What’s the ROC like? If ROC is as shown in figure, is h[n] one-sided or two-sided? Is the system causal or stable? Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

3.3 The Inverse Z-Transform Review Inspection method Partial fraction expansion Power series expansion Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Partial Fraction Expansion Review Partial Fraction Expansion if M>N, and has a pole of order s at d=di Br is obtained by long division Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

3.3 The inverse z-Transform 3.3.3 Power Series Expansion Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Example 3.10: Finite-Length Sequence Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Ex. 3.11: Inverse Transform by power series expansion Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Example 3.12: Power Series Expansion by Long Division Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Example 3.13: Power Series Expansion for a Left-sided Sequence Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

3.4 z-Transform Properties 3.4.1 Linearity Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. Example of Linearity Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. 3.4.2 Time Shifting is an integer is positive, is shifted right is negative, is shifted left Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. Time Shifting: Proof Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Example 3.14: Shifted Exponential Sequence Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

3.4.3 Multiplication by an Exponential sequence Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Example 3.15: Exponential Multiplication Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

3.4.4 Differentiation of X(z) Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Example 3.16: Inverse of Non-Rational z-Transform Look Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Example 3.17: Second-Order Pole Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

3.4.5 Conjugation of a complex Sequence Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. 3.4. 6 Time Reversal Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Example 3.18: Time-Reverse Exponential Sequence Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

3.4. 7 Convolution of Sequences Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Ex. 3.19: Evaluating a Convolution Using the z-transform Solution: Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Example 3.19: Evaluating a Convolution Using the z-transform Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. 3.4. 8 Initial Value Theorem Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Region of convergence (ROC) does not converge uniformly to the discontinuous function . -∞<n<∞ Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Zhongguo Liu_Biomedical Engineering_Shandong Univ. Example There’s no Z-Transform for For Zhongguo Liu_Biomedical Engineering_Shandong Univ. 2017/4/17

Chapter 3 HW 3.3, 3.4, 3.9, 3.16 3.2 , 3.8, 3.11, 3.20 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Zhongguo Liu_Biomedical Engineering_Shandong Univ. 102 2017/4/17 2017/4/17 返 回 上一页 下一页