5' Typical DiscreteTime Systems

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5. Typical Discrete-Time Systems 5.1. All-Pass
Systems (5.5) 5.2. Minimum-Phase Systems
(5.6) 5.3. Generalized Linear-Phase Systems (5.7)
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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 constant.
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 all-pass
system. Proof. Letting zejw, we obtain
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(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 zejw in (5.2), we obtain
(5.5)
Since H(ej?)1, (5.5) can be written as
(5.6)
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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, alt1.
Thus, grdH(ej?) must be positive. Now
consider a stable system with the system function
(5.9)
Evidently, this system is an all-pass system.
Moreover, if causal, this system must have a
positive group delay.
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5.2. Minimum-Phase Systems A minimum-phase
system is defined as a causal and stable system
with all its zeros inside the unit circle. A
minimum-phase system must have a causal and
stable inverse. 5.2.1. All-Pass and Minimum-Phase
Decomposition A causal, stable system
characterized by a rational system function can
be decomposed into the cascade of a causal
all-pass system and a minimum-phase system.
Suppose that a causal, stable system is
characterized by a rational system function H(z)
and H(z) has only one zero za outside the unit
circle. Then,
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(5.10)
Hmin(z)
Hap(z)
Note that Hap(z) is a causal all-pass system if
we choose zgt1/a as the ROC, and Hmin(z) is a
minimum-phase system. Thus, (5.10) shows that
system H(z) can be decomposed 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. Decompose the
following causal, stable systems into the cascade
of a causal all-pass system and a minimum-phase
system.
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5.2.2. Amplitude-Spectrum Restoration Let us
consider an application of the all-pass and
minimum-phase decomposition. 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).
x(n)
w(n)
y(n)
Distorting System Hd(z)
Restoring System Hr(z)
Figure 5.1. Amplitude-Spectrum Restoration.
Hd(z) can be expressed as Hd(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
select
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the causal, stable inverse of Hmin(z) as Hr(z),
i.e., Hr(z)1/Hmin(z) (5.12) with an
ROC of form zgtr. Then, Y(z)X(z)Hap(z).
(5.13) Letting zejw, we obtain Y(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(ejw) still exists. Example. A signal is
distorted by system
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Hd(z)(1-0.9ej0.6?z-1)(1-0.9e-j0.6?z-1)? (1-
1.25ej0.8?z-1)(1-1.25e-j0.8?z-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 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 systems, which have the same
amplitude response. Among these systems, the
minimum-phase system has the minimum group delay
and the minimum energy delay.
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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) and Hmin(z),
respectively. Then, the minimum energy-delay
property is formulated as
(5.20)
Especially, when n0, (5.20) becomes
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
response H(?)A(?)exp(-j??), (5.22)
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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(?). 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 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, some
other FIR systems and some IIR systems may also
belong to generalized linear-phase systems. In
addition, in next discussion, we assume that the
impulse response h(n) is real. 5.3.1. Type-I FIR
Generalized Linear-Phase Systems
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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), n0, 1,, N-1, (5.24) where
N is an 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
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(5.26)
Find the frequency response 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), n0, 1,, N-1, (5.27) where
N is an even number. When h(n) satisfies the
above condition, the frequency response of the
system can be expressed as
(5.28)
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(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 type-III FIR generalized
linear-phase system if its impulse response
satisfies the following antisymmetry
h(n)-h(N-1-n), n0, 1,, N-1, (5.30) where
N is an odd number. Letting n(N-1)/2 in
(5.30), we obtain h(N-1)/20.
(5.31)
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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 can be expressed as
(5.32)
(5.32) 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-2).
(5.33) Find the frequency response of the system.
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5.3.4. 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), n0, 1,, N-1,
(5.34) where N is an even number. If h(n)
satisfies the above condition, the 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.
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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 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-phase 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
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(1) A type-II FIR generalized linear-phase system
has zero -1 and cannot be used as a high-pass
filter. (2) A 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. Relation of Four Types of FIR
Generalized Linear-Phase Systems to FIR
Minimum-Phase Systems Suppose that H(z) is
the system function of a system belonging to four
types of FIR generalized linear-phase systems.
Then, H(z)Hmin(z)Hmax(z)Huc(z).
(5.37)
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Hmin(z) characterizes an FIR minimum-phase
system, and has all its zeros inside the unit
circle. Hmax(z) characterizes an FIR
maximum-phase system, and has all its zeros
outside the unit circle. The zeros of Hmax(z) are
the reciprocals of the zeros of Hmin(z)
correspondingly. Huc(z) has all its zeros on the
unit circle. Example. An FIR minimum-phase
system has system function Hmin(z)(1-0.9ej0.6?z
-1)(1-0.9e-j0.6?z-1) ?(1-0.8ej0.8?z-1)(1-0.8e-j
0.8?z-1). (5.38) Find an FIR maximum-phase
system such that the cascade of the two systems
belongs to four types of FIR generalized
linear-phase system.