Stability and Causality

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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:

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Note that because this is equivalent to the condition that

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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 PaleyWiener theorem.

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Paley-Wiener Theorem: If h(n) has finite energy and h(n) = 0 for n < 0,

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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:

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CHAP. 51

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TRANSFORM ANALYSIS OF SYSTEMS

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h,(n) = i [ h ( n )- h(-n)]

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If h ( n ) is causal, it is uniquely defined by its even part:

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h ( n ) = 2h,(n)u(n) - h,(n)6(n)

If h ( n ) is absolutely summable, the DTFT of h ( n ) exists, and H(eja1) may be written in terms of its real and imaginary parts as follows:

H (eiW) = HR(ejCU)j H I (eiCu)

Therefore, because HR(ejw) the DTFT of the even part of h(n),it follows that if h ( n ) is real, stable, and causal, is H ( e J Wis uniquely defined by its real part. This implies a relationship between the real and imaginary parts of ) H (el"), which is given by

HI (ei") = - 2n

S" -"

HR(eie)cot

This integral is called a discrete Hilberr transform. Specifically, Hl(ejw)is the discrete Hilbert transform of HR(eJ").

Realizable Systems

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 system

will be realizable (stable and causal) if and only if

For the second-order system,

H(z)= H ( z )has two zeros at the origin and poles at

+ a(l)z-I + a ( 2 ) z Z

b(O)

These roots satisfy the following two equations:

a ( ] )= -(a, a2) a(2) = a , .a;

From these equations, it follows that the roots of H ( z ) will be inside the unit circle if and only if (see Prob. 8.29)

law1 < 1 la(l>l < 1 + a @ )

These constraints define a stability triangle in the coefficient plane as shown in Fig. 5- 1. Thus, a causal secondorder 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 equation

the 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 la(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 1 G(z) = H(z)