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barcode generator for ssrs H (z) = A k in Software
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What distinguishes the ztransform of a causal signal from one that is not is the fact that X(z) does not contain any positive powers of z. Consequently, if we let lzl t m, X(z) + x(O), which is a statement of the initial value theorem. It follows, therefore, that if .r(n) is causal, this limit must be finite. For noncausal signals, on the other hand, this limit will tend to infinity, because the ztransform will contain positive powers of z. For example, the sequence x(n) = u ( n I) has a ztransform lim X(z) = m
:ICU
Thus, a ztransform may be the system function of a causal system only if
Of the transforms listed, only ( a ) and ( c )have a finite limit as Izl + rn and, therefore, are the only ones that could be the ztransform of a causal signal. The result of a particular computeraided filter design is the following causal secondorder filter: H (z) = 1  2zI
+ 22' + zp2 +1.33~~
Show that this filter is unstable, and find a causal and stable filter that has the same magnitude response as H(z). This filter is clearly not stable, because the coefficient for z' in the denominator, which is the product of the roots of H(z), is greater than 1. Specifically, if the poles of H(z) are crl and crz, then a l . a = 1.33, and this implies that 2 at least one of the roots is outside the unit circle. Because the discriminant of the polynomial is negative, the roots are complex with crl = reJ%nd cr2 = reIH where , = and % = c o s p ' ( l / m ) . Recall that if we form a new system function given by Hf(z)= H(z)G,,(z), where Gap(z)is an allpass filter of the form I ~ ' ( e j " ) l= I H (ejW)l.Therefore, if
Gap(z)= I  2:I 1 .33zr2 1.33  22I + z r 2
the effect of Gap(:) is to replace the pair of complex poles in H(z) that are outside the unit circle with a complex pole pair inside the unit circle at the reciprocal locations while preserving the magnitude response. Thus. a stable filter that has a frequency response with the same magnitude as H(eJW) the following: is H '(z) = 1 2zI + z 1.33  221 + zp2
The system function of a discretetime linear shiftinvariant system is H (z). Assume that H ( z ) is a rational function of z and that H ( z ) is causal and stable. Determine which of the following systems are stable and which are causal: (a) G ( z ) = H(z)H*(z*) d (b) G ( z ) = H1(z) where H1(z) = [ H ( z ) ] CHAP. 51
TRANSFORM ANALYSIS OF SYSTEMS
(c) G(z) = ~ ( z  I ) (d) G ( z ) = H(Z) With H(z) a rational function of z , if h(n) is stable and causal, the poles of H(z) (if any) are inside the unit circle, and the region of convergence is the extenor of a circle and includes the unit circle. ( a ) If H(z) is the ztransform of h(n), then H*(z*)is the ztransfonn of h*(n), and the region of convergence is the same as that for H(z). Because the region of convergence of G(:) = H(z)H*(i*)includes the regions of convergence of H(z) and H*(z*),the region of convergence of G(z) will be the exterior of a circle and include the unit circle. Therefore, g(n) is stable and causal. ( h ) Recall that if H(z) is the ztransform of h(n), Therefore, delaying the sequence nh(n) by I yields the following ztransform pair:

