2. Typical Discrete-Time Systems All-Pass Systems (5.5) 2.2. Minimum-Phase Systems (5.6) 2.3. Generalized Linear-Phase Systems (5.

Transcription

1 . Typical Discrete-Time Systems.1. All-Pass Systems (5.5).. Minimum-Phase Systems (5.6).3. Generalized Linear-Phase Systems (5.7)

2 .1. All-Pass Systems An all-pass system is defined as a system which has a constant amplitude response. That is, H() =A, (.1) where H() is the frequency response of the system, and A is a constant. Now consider a typical all-pass system. Assume that a stable system has the system function 1 z a (z). (.) 1 az H 1 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 all-pass system. Proof. Letting z=e j in (.), we obtain

3 H(e j From (.3), we obtain ) j e a j 1 ae e j Thus, this system is an all-pass system. (1 ae 1 ae j ) j. (.3) H(e j ) =1. (.4) 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=e j in (.), we obtain H(e Since H(e j ) =1, (.5) can be written as j j e a ). (.5) j 1 ae exp[ j j e a j H(e )]. (.6) j 1 ae

4 Differentiating both the sides of (.6) with respect to, we obtain grd [H(e Substituting (.6) into (.7), we obtain j j (1 a )e )]. (.7) j j (1 ae ) exp[ jh(e )] j 1 a grd [H(e )]. j (.8) 1 ae Since the system is causal and stable, a <1. Thus, grd[h(e j )] must be positive. Now consider a stable system with the system function M 1 z a m H(z). (.9) 1 1 a z m1 Evidently, this system is an all-pass system. Moreover, if causal, this system must have a positive group delay. m

5 .. Minimum-Phase Systems..1. Definition of Minimum-Phase Systems A system is a minimum-phase system if it has a rational system function, is causal and stable, and has a causal, stable inverse. Besides a causal, stable inverse, a minimum-phase system may have other inverses. If a minimum-phase system has system function H(z), then H(z) has the following properties: (1) All the poles of H(z) are inside the unit circle centered about the origin. () All the zeros of H(z) are inside the unit circle centered about the origin. (3) The denominator and the numerator of H(z) have equal orders of z.

6 ... All-Pass and Minimum-Phase Decomposition Let us assume that a system is not a minimum-phase system only because its system function has zeros outside z =1. Then, the system can be decomposed into the cascade of a causal all-pass system with a unit amplitude response and a minimum-phase system. Suppose that H(z) is the system function of the above system, and it has only one zero z=a outside z =1. Then, H(z) (1 az 1 1 az 1 1 (a ) )H 1 z 0 1 (z) [1 (a 1 ) z 1 ] H 0 (z) 1 z a 1 1 (a ) 1 z 1 ( a)[1 (a 1 ) z 1 ] H 0 (z). (.10) H ap (z) H min (z)

7 H ap (z) is a causal all-pass system with a unit amplitude response if z >1/ a is chosen as the ROC. H min (z) is a minimum-phase system. (.10) shows that system H(z) can be decomposed into the cascade of H ap (z), a causal all-pass system with a unit amplitude response, and H min (z), a minimum-phase system. The above argument can be generalized. Example. Assume that the following systems are causal and stable. Decompose each of them into the cascade of a causal all-pass system with a unit amplitude response and a minimum-phase system z (1) H(z) z 3 1 e () H(z). j/ 4 1 z z e j/ 4 z 1.

8 ..3. Amplitude-Spectrum Restoration An application of the all-pass and minimum-phase decomposition is given next. Assume that a signal is distorted. The distorting system is not a minimum-phase system only because its system function has zeros outside z =1. We hope to restore the amplitude spectrum of the signal by a causal, stable system (figure.1). x(n) Distorting System H d (z) w(n) Restoring System H r (z) y(n) Figure.1. Amplitude-Spectrum Restoration. H d (z) can be expressed as H d (z)=h ap (z)h min (z). (.11) Here H ap (z) describes a causal all-pass system with a unit amplitude response, and H min (z) describes a minimum-phase system. We select

9 the causal, stable inverse of H min (z) as H r (z), i.e., H r (z)=1/h min (z) (.1) with an ROC of form z >r. Then, Y(z)=X(z)H ap (z). (.13) Letting z=e j in (.13), we obtain Y(e jw )=X(e jw )H ap (e jw ). (.14) That is, Y(e jw ) = X(e jw ), (.15) Y(e jw )=X(e jw )+H ap (e jw ). (.16) Thus, X(e jw ) is restored although a phase error H ap (e jw ) still exists. Example. A signal is distorted by system H d (z)=(10.9e j0.6 z 1 )(10.9e j0.6 z 1 )

10 (11.5e j0.8 z 1 )(11.5e j0.8 z 1 ). (.17) Find a causal, stable system to restore the amplitude spectrum of the signal...4. Properties of Minimum-Phase Systems Let us assume that a system is not a minimum-phase system only because its system function has zeros outside z =1. Then, the system function of the system can be written as H(z)=H ap (z)h min (z), (.18) where H ap (z) and H min (z) characterize a causal all-pass system with a unit amplitude response and a minimum-phase system, respectively. If H min (z) is fixed and H ap (z) is given different choices, we will obtain a class of systems, which have the same amplitude response. Among these systems, the minimum-phase system has the minimum group delay and the minimum energy delay.

11 The minimum group-delay property is formulated as grd[h(e jw )]grd[h min (e jw )]. (.19) Let h(n) and h min (n) be the impulse responses corresponding to H(z) and H min (z), respectively. Then, the minimum energy-delay property is formulated as n m0 h(m) n h (m). (.0) m0 Especially, when n=0, (.0) becomes.3. Generalized Linear-Phase Systems min h(0) h min (0). (.1) A system is referred to as a linear-phase system if it has frequency response H()=A()exp(j), (.)

12 where A() is a nonnegative real function, and is a real constant. A() is essentially the amplitude of H(). is essentially the group delay of H(). A system is referred to as a generalized linear-phase system if its frequency response has the form H()=A()exp(j+j), (.3) where A() is a real function (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, other FIR systems and some IIR systems may also belong to generalized linearphase systems. We assume that the impulse response h(n) is a real sequence over 0nN1 in the next discussion.

13 .3.1. Type-I FIR Generalized Linear-Phase Systems A system is a type-i FIR generalized linear-phase system if h(n) is symmetric, i.e., and N is an odd number. h(n)=h(n1n), n=0, 1,, N1, (.4) If h(n) satisfies the above conditions, the frequency response of the system can be expressed as H( ) h N 1 exp j (N n0 3)/ N 1. h(n) cos n N 1 (.5) (.5) shows that H() has the form in (.3), and thus the system is a generalized linear-phase system.

14 .3.. Type-II FIR Generalized Linear-Phase Systems A system is a type-ii FIR generalized linear-phase system if h(n) is symmetric, i.e., and N is an even number. h(n)=h(n1n), n=0, 1,, N1, (.6) If h(n) satisfies the above conditions, the frequency response of the system can be expressed as H( ) N/ 1 n0 h(n) cos n N 1 exp j Thus, the system is a generalized linear-phase system. N Type-III FIR Generalized Linear-Phase Systems (.7) A system is a type-iii FIR generalized linear-phase system if h(n)

15 is antisymmetric, i.e., and N is an odd number. h(n)=h(n1n), n=0, 1,, N1, (.8) Letting n=(n1)/ in (.8), we obtain h[(n1)/]=0. (.9) If h(n) satisfies the above conditions, the frequency response of the system can be expressed as H( ) (N n0 3)/ exp h(n) sin n j N 1 j. N 1 Thus, the system is a generalized linear-phase system. (.30)

16 .3.4. Type-IV FIR Generalized Linear-Phase Systems A system is a type-iv FIR generalized linear-phase system if h(n) is antisymmetric, i.e., and N is an even number. h(n)=h(n1n), n=0, 1,, N1, (.31) If h(n) satisfies the above conditions, the frequency response of the system can be expressed as H( ) N / 1 n0 exp h(n) sin n j N 1 j. N 1 (.3) (.3) shows that H() has the form in (.3), and thus the system is a generalized linear-phase system.

17 .3.5. Zeros for Four Types of FIR Generalized Linear-Phase Systems If a system belongs to the four types of FIR generalized linearphase systems, the zeros of the system can be divided into groups with form a, a, 1/a, and 1/a. There are four cases about the zeros, a, a, 1/a, and 1/a : (1) If a=1, the zeros are one zero, a. () If a1 but Im(a)=0, the zeros are two zeros, a and 1/a. (3) If a1 but a =1, the zeros are two zeros, a and a. (4) Otherwise, the zeros are four zeros, a, a, 1/a, and 1/a. For an FIR system, if the zeros of the system can be divided into groups with form a, a, 1/a, and 1/a, then the system belongs to the four types of FIR generalized linear-phase systems.

18 Example. A system has system function H(z)=(10.9e j0.6 z 1 )(10.9e j0.6 z 1 ) (10.8e j0.8 z 1 )(10.8e j0.8 z 1 ). (.33) Find a system such that their cascade belongs to the four types of FIR generalized linear-phase systems. The four types of FIR generalized linear-phase systems also have some other properties, as we will see in the next example. Example. Prove the following statements: (1) Any type-ii FIR generalized linear-phase system has zero 1 and cannot be used as a high-pass filter. () Any type-iii FIR generalized linear-phase system has zero 1 and cannot be used as a low-pass filter. (3) Any type-iii FIR generalized linear-phase system has zero 1 and

19 cannot be used as a high-pass filter. (4) Any type-iv FIR generalized linear-phase system has zero 1 and cannot be used as a low-pass filter.