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TRANSFORM ANALYSIS O F SYSTEMS
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[CHAP. 5
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Thus,$ipping one or more zeros of the system function about the unit circle does not change the magnitude of the frequency response.
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EXAMPLE 5.4.1 For a filter that has a system function
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the magnitude of the frequency response will not be changed if it is cascaded with the allpass filter
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This allpass filter flips the zero at z = 0.2 in H(z) to its reciprocal location, z = 5, and the new filter has a system function
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5.5 MINIMUM PHASE SYSTEMS
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A stable and causal linear shift-invariant system with a rational system function of the form given in Eq. (5.2) has all of its poles inside the unit circle, lakI < 1. The zeros, however, may lie anywhere in the z-plane. In some applications, it is necessary to constrain a system so that its inverse, G(z) = l/H(z), is also stable and causal. I This requires that the zeros of H (z) lie inside the unit circle, JBk < I. A stable and causal filter that has a stable and causal inverse is said to have minimum phase. Equivalently, we have the following definition:
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Definition: A rational system function with all of its poles and zeros inside the unit circle is said to be have minimum phase.
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A minimum phase system is uniquely defined by the magnitude of its Fourier transform, IH(eJm)(.The procedure to find H (z) from 1 H (ejm)\is as follows. Given \ H (ejW)l,we find 1 H (ej")12, which is a function of cos(ko). Then, by replacing cos(kw) with C(zk zpk), we have
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Finally, the minimum phase system is then formed from the poles and zeros of G ( z )that are inside the unit circle.
EXAMPLE 5.5.1
Let H(z) be a minimum phase system with a Fourier transform magnitude
I ~(e'")l*= $ - f cosw
Expressing cos w in terms of complex exponentials,
and replacing
with z and e-1" with z-', we have
Thus, the minimum phase system is
CHAP. 51
TRANSFORM ANALYSIS OF SYSTEMS
A stable and causal system may always be factored into a product of a minimum phase system with an allpass system: (5.17) H ( z ) = Hmin(z).Hap(z)
The procedure for performing this factorization is as follows. First, all of the zeros of H ( z ) that are outside the unit circle are reflected inside the unit circle to their conjugate reciprocal location. The resulting system function is minimum phase, Hmin(z). Then, the allpass filter is selected so that it reflects the appropriate set of zeros of Hmin(z) back outside the unit circle. EXAMPLE 5.5.2 For the system function
the minimum phase factor is
Then, to reflect the zero at z = 0.5 back outside the unit circle to z = 2, we use the allpass factor
Two properties of minimum phase systems are as follows. First, of all systems that have the same Fourier transform magnitude, the minimum phase system has the minimum group delay. This follows from the factorbe ization given in Eq. (5.17). Specifically, let Hmin(z) a minimum phase system, and let H (z) be another system with the same magnitude. The group delay for H ( z ) may be written as
where rap(u) the group delay of a stable and causal allpass system. Because tap(u) 0, the group delay of H ( z ) is > will be larger than the group delay of the minimum phase system Hmi,,(z). Furthermore, because the phase is the negative of the integral of the group delay, the minimum phase system is also said to have the minimum phase-lag. The second property of minimum phase systems is that they have the minimum energy delay. Specifically, if hmin(n) the unit sample response of a minimum phase system, and h ( n ) is the unit sample response of another is causal system that has the same magnitude response,
for any n 3 0.
FEEDBACK SYSTEMS
Feedback systems are used in many applications such as stabilization of unstable systems, compensation of nonideal elements, tracking systems, and inverse system design. The general configuration of a discrete-time feedback system is shown in Fig. 5-6. The system N ( z ) is referred to as the system function of the forward path, and G ( z )is referred to as the system function of thefeedbuck path. The system function relating the input x ( n ) to the output y(n) is called the closed-loop system function and is denoted by Q(z). Because
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