819信號系統(tǒng)與信號處理教學課件discrete-time systems

1、5. Typical Discrete-Time Systems5.1. All-Pass Systems (5.5)5.2. Minimum-Phase Systems (5.6)5.3. Generalized Linear-Phase Systems (5.7)5.1. All-Pass Systems An all-pass system is defined as a system which has a constant amplitude response. Let H() be the frequency response of the system and A be a co

2、nstant. Then, |H()|=A. (5.1) Now consider a typical all-pass system. Assume that a stable system has the system function(5.2)Note that the zero and the pole of H(z) are conjugate reciprocal (that is, they have reciprocal amplitudes and the same phase). Then, it can be shown that this system is an al

3、l-pass system. Proof. Letting z=ejw, we obtain(5.3)From (5.3), we obtain |H(ej)|=1. (5.4)Thus, this system is an all-pass system. Now assume that the above all-pass system is causal. Then, it can be shown that this system must have a positive group delay. Proof. Letting z=ejw in (5.2), we obtain(5.5

4、)Since |H(ej)|=1, (5.5) can be written as(5.6)Differentiating both the sides of (5.6) with respect to , we obtain(5.7)Substituting (5.6) into (5.7), we obtain(5.8)Since the system is causal and stable, |a|1/a| as the ROC, and Hmin(z) is a minimum-phase system. Thus, (5.10) shows that system H(z) can

5、 be posed into the cascade of a causal all-pass system Hap(z) and a minimum-phase system Hmin(z). The above argument can be generalized. Example. pose the following causal, stable systems into the cascade of a causal all-pass system and a minimum-phase system.5.2.2. Amplitude-Spectrum Restoration Le

6、t us consider an application of the all-pass and minimum-phase position. Assume that a signal is distorted. The distorting system is causal and stable, and is characterized by a rational system function. We hope to restore the amplitude spectrum of the signal by a causal, stable system (figure 5.1).

7、x(n)w(n)Distorting System Hd(z)Restoring System Hr(z)y(n)Figure 5.1. Amplitude-Spectrum Restoration. Hd(z) can be expressed asHd(z)=Hap(z)Hmin(z). (5.11)Here, Hap(z) describes a causal all-pass system with a unit amplitude response, and Hmin(z) describes a minimum-phase system. We selectthe causal,

8、stable inverse of Hmin(z) as Hr(z), i.e., Hr(z)=1/Hmin(z) (5.12)with an ROC of form |z|r. Then, Y(z)=X(z)Hap(z). (5.13)Letting z=ejw, we obtainY(ejw)=X(ejw)Hap(ejw). (5.14)That is, |Y(ejw)|=|X(ejw)|, (5.15) Y(ejw)=X(ejw)+Hap(ejw). (5.16)As we can see, |X(ejw)| is restored. However, a phase error Hap

9、(ejw) still exists. Example. A signal is distorted by system Hd(z)=(1-0.9ej0.6z-1)(1-0.9e-j0.6z-1)(1-1.25ej0.8z-1)(1-1.25e-j0.8z-1). (5.17)Find a causal, stable system to restore the amplitude spectrum of the signal.5.2.3. Properties of Minimum-Phase Systems Assume that a causal, stable system has a

10、 rational system function H(z). Then,H(z)=Hap(z)Hmin(z), (5.18)where Hap(z) and Hmin(z) characterize a causal all-pass system with a unit amplitude response and a minimum-phase system, respectively. If Hmin(z) is fixed and Hap(z) is given different choices, we will obtain a class of causal, stable s

11、ystems, which have the same amplitude response. Among these systems, the minimum-phase system has the minimum group delay and the minimum energy delay. The minimum group-delay property is formulated as grdH(ejw)grdHmin(ejw). (5.19)Let h(n) and hmin(n) be the impulse responses corresponding to H(z) a

12、nd Hmin(z), respectively. Then, the minimum energy-delay property is formulated as(5.20)Especially, when n=0, (5.20) es |h(0)|hmin(0)|. (5.21)5.3. Generalized Linear-Phase Systems A system is referred to as a linear-phase system if it has frequency responseH()=A()exp(-j), (5.22)where A() is a nonneg

13、ative real function, and is a real constant. A() is essentially the amplitude of H(). is essentially the group delay of H(). It can be an integer or not. A system is referred to as a generalized linear-phase system if its frequency response has the form H()=A()exp(-j+j), (5.23)where A() is a real fu

14、nction (it does not have to be nonnegative), and and are two real constants. Next we will introduce four types of FIR generalized linear-phase systems. Note that besides the four types of FIR systems, some other FIR systems and some IIR systems may also belong to generalized linear-phase systems. In

15、 addition, in next discussion, we assume that the impulse response h(n) is real.5.3.1. Type-I FIR Generalized Linear-Phase Systems A system is called a type-I FIR generalized linear-phase system if its impulse response satisfies the following symmetry: h(n)=h(N-1-n), n=0, 1, N-1, (5.24)where N is an

16、 odd number. If h(n) satisfies the above condition, the frequency response of the system can be expressed as(5.25)(5.25) shows that H() has the form in (5.23) and thus the system is a generalized linear-phase system. Example. Assume that a system has the impulse response(5.26)Find the frequency resp

17、onse of the system.5.3.2. Type-II FIR Generalized Linear-Phase Systems A system is called a type-II FIR generalized linear-phase system if its impulse response satisfies the following symmetry: h(n)=h(N-1-n), n=0, 1, N-1, (5.27)where N is an even number. When h(n) satisfies the above condition, the

18、frequency response of the system can be expressed as(5.28)(5.28) shows that the system is a generalized linear-phase system. Example. Assume that a system has the impulse response(5.29)Find the frequency response of the system.5.3.3. Type-III FIR Generalized Linear-Phase Systems A system is called a

19、 type-III FIR generalized linear-phase system if its impulse response satisfies the following antisymmetry: h(n)=-h(N-1-n), n=0, 1, N-1, (5.30)where N is an odd number. Letting n=(N-1)/2 in (5.30), we obtain h(N-1)/2=0. (5.31)(5.32)(5.32) shows that H() has the form in (5.23), and thus the system is

20、 a generalized linear-phase system. Example. Assume that a system has the impulse response h(n)=(n)-(n-2). (5.33)Find the frequency response of the system.This is a property of type-III FIR generalized linear-phase systems. If h(n) satisfies the above condition, the frequency response of the system

21、can be expressed as5.3.4. Type-IV FIR Generalized Linear-Phase Systems A system is called a type-IV FIR generalized linear-phase system if its impulse response satisfies the following antisymmetry: h(n)=-h(N-1-n), n=0, 1, N-1, (5.34)where N is an even number. If h(n) satisfies the above condition, t

22、he frequency response of the system can be expressed as(5.35)(5.35) shows that H() has the form in (5.23), and thus the system is a generalized linear-phase system. Example. Assume that a system has the impulse response h(n)=(n)-(n-1). (5.36)Find the frequency response of the system.5.3.5. Locations

23、 of Zeros for Four Types of FIR Generalized Linear-Phase Systems Suppose that a system belongs to four types of FIR generalized linear-phase systems. If a is a zero of the system, then a*, 1/a and 1/a* are also the zeros of the system. An FIR system belongs to four types of FIR generalized linear-ph

24、ase systems if its zeros have the above property. Let us introduce more results about four types of FIR generalized linear-phase systems by an example. Example. Prove the following statements:(1) A type-II FIR generalized linear-phase system has zero -1 and cannot be used as a high-pass filter.(2) A

25、 type-III FIR generalized linear-phase system has zero 1 and cannot be used as a low-pass filter.(3) A type-III FIR generalized linear-phase system has zero -1 and cannot be used as a high-pass filter.(4) A type-IV FIR generalized linear-phase system has zero 1 and cannot be used as a low-pass filter.5.3.6. Relati