5.1 INTRODUCTION
Given a linear shift-invariant system with a unit sample response
h ( n ) ,
the input and output are related by a convolution sumAs discussed in Chap.
2,
this relationship implies that~ ( e j " )
=x ( e j W ) ~ ( e j w )
whereH ( e j w ) ,
the frequency response of the system, is the discrete-time Fourier transform ofh ( n ) .
This relationship betweenx ( n )
andy ( n )
may also be expressed in the z-transform domain asY ( z )
= X( z ) H ( z )
where H(z), the z-transform of
h ( n ) ,
is thesystem function
of theLSI
system. The system function is very useful in the description and analysis ofLSI
systems. In this chapter, we look at the characterization of a linear shift-invariant system in terms of its system function and discuss special types ofLSI
systems such as linear phase systems, allpass systems, minimum phase systems, and feedback networks.5.2 SYSTEM FUNCTION
The frequency response of
a
linear shift-invariant system is the discrete-time Fourier transform of the unit sample response, and thesystem function
is the z-transform of the unit sample response:The frequency response may be derived from the system function by evaluating H(z) around the unit circle:
If the z-transform of the input to a linear shift-invariant system with a system function H ( z ) is X(z), the z-transform of the output
is
Y ( z )
=H
(z)X (z)For linear shift-invariant systems that are described by a linear constant coefficient difference equation,
the system function is a rational function of z:
184 TRANSFORM ANALYSIS O F SYSTEMS [CHAP. 5 Therefore, the system function is defined, to within a scale factor, by the location of its poles, ( ~ k , and zeros,
Bk.
Note that each term in the numerator
contributes a zero to the system function at z = #ln and a pole to the system function at z =
0.
Similarly, each term in the denominator contributes a pole at z = crk and a zero at z =0.
Therefore, including the poles and zeros that may lie at z =0
or z = oo, the number of zeros in H (z) is equal to the number of poles.If the unit sample response is real-valued, H (z) is a conjugate symmetric function of z, H (z) = H*(z*)
and the complex poles and zeros occur in conjugate symmetric pairs (i.e., if there is a complex pole (zero) at z = zo, there is also a complex pole (zero) at z = z;).
5.2.1
Stability and Causality
Stability and causality impose some constraints on the system function of a linear shift-invariant system.
Stability
The unit sample response of a stable system must be absolutely summable:
Note that because this is equivalent to the condition that
for lzl = 1, the region of convergence of the system function must include the unit circle if the system is stable.
Causality
Because the unit sample response of a causal system is right-sided, h(n) =
0
for n <0,
the region of convergence of H(z) will be the exterior of a circle, Izl > a. Because no poles may lie within the region of convergence, all of the poles of H(z) must lie on or inside the circle lzl 5 a.Causality imposes some tight constraints on a linear shift-invariant system. The first of these is the Paley- Wiener theorem.
Paley-Wiener Theorem:
If h(n) has finite energy and h(n) =0
for n <0,
One of the consequences of this theorem is that the frequency response of a stable and causal system cannot be zero over any finite band of frequencies. Therefore, any stable ideal frequency selective filter will be noncausal.
Causality also places restrictions on the real and imaginary parts of the frequency response. For example, if h(n) is real, h(n) may be decomposed into its even and odd parts as follows:
CHAP. 51 TRANSFORM ANALYSIS OF SYSTEMS H ( e J W ) is uniquely defined by its real part. This implies a relationship between the real and imaginary parts of H (el"), which is given by
A
realizable system is one that is both stable and causal.A
realizable system will have a system function with a region of convergence of the form lzl > a where 0 5 a < 1. Therefore, any poles of H ( z ) must lie inside the unit circle. For example, the first-order systemwill be realizable (stable and causal) if and only if
For the second-order system,
These constraints define a stability triangle in the coefficient plane as shown in Fig.
5-
1. Thus, a causal second- order system will be stable if and only if the coefficients a ( ] ) and a ( 2 ) lie inside this triangle. This result is of special interest, because second-order systems are the basic building blocks for higher-order systems. If the coefficients lie in the shaded region above the parabola defined by the equationthe roots are complex; otherwise they are real.
TRANSFORM ANALYSIS O F SYSTEMS [CHAP. 5
Fig. 5-1. The stability triangle, which is defined by the lines la(2)I < 1 and l a ( l ) ( < 1 +a(2). The shaded region above theparabolaa2(1)- 4a(2) = 0 contains the values of a ( l ) and a(2) that correspond to complex roots.
5.2.2 Inverse Systems
For a linear shift-invariant system with a system function H ( z ) , the inverse system is defined to be the system that has a system function G ( z ) such that
H ( z ) . G ( z ) = 1
In other words, the cascade of H ( z ) with G ( z ) produces the identity system. In terms of H ( z ) , the inverse is simply
G ( z ) = - 1 H ( z )
For example, if H ( z ) is a rational function of z as given in
Eq. (5.2).
the inverse system isThus, the poles of H ( z ) become the zeros of G ( z ) , and the zeros of H ( z ) become the poles of G ( z ) . The region of convergence that
is
associated with the inverse system is determined by the requirement that H ( z ) and G ( z ) have overlapping regions of convergence.'EXAMPLE 5.2.1 If
the inverse system is
There are two possible regions of convergence for ,q(n). The ti rst is IzJ >
f
, and the second is lzI i $. Because lzl < f does not overlap the region of convergence for H(z), the only possibility for the inverse system is l z l >{.
In this case, the unit sample response is~ ( n ) = (1)" u(n) - 0 . 8 ( ~ ) " - ' u ( n - I )
'If this were not the case, H ( r ) G ( z ) would not be the identity system. because the region of convergence would be empty.
CHAP. 51 TRANSFORM ANALYSIS OF SYSTEMS which is stable and causal. However, suppose that
0.5 - Z - '
H (z) = Izl > 0.8 I
-
0.82-'In this case, the inverse system is
where the region of convergence may be either (zI > 2 or IzI c 2. Because both regions of convergence overlap the region of convergence of H ( z ) , both are valid inverse systems. The first, which has a region of convergence li( > 2, has a unit sample response
~ ( n ) = 2(2)"u(n)
-
1,6(2)"-'u(n--
I)and is causal but unstable. The second, with a region of convergence lzl < 2, has a unit sample response
and is stable but noncausal.