UploadFile_6417

System Representation in the z-Domain 125
we have
H(z) =1.2346 +
Hence from Table 4.1
−0.6173 + j0.9979 −0.6173 − j0.9979
+ , |z| > 0.9
1 − 0.9e −jπ/3 z−1 1 − 0.9e jπ/3 z −1
h(n) =1.2346δ(n)+[(−0.6173 + j0.9979)0.9 n e −jπn/3
+(−0.6173 − j0.9979)0.9 n e jπn/3 ]u(n)
=1.2346δ(n)+0.9 n [−1.2346 cos(πn/3) + 1.9958 sin(πn/3)]u(n)
=0.9 n [−1.2346 cos(πn/3) + 1.9958 sin(πn/3)]u(n − 1)
The last step results from the fact that h(0) =0.
□
4.4.4 RELATIONSHIPS BETWEEN SYSTEM REPRESENTATIONS
In this and the previous two chapters, we developed several system representations.
Figure 4.9 depicts the relationships among these representations
in a graphical form.
Express H(z) in z –1 ,
cross multiply, and
take inverse
Take
z-transform,
solve for Y /X
H(z)
Take
z-transform
Take inverse
z-transform
Diff Equation
h(n)
Substitute
z = e jω
Take inverse
DTFT
Take DTFT,
solve for Y/X
H(e jω )
Take Fourier
transform
FIGURE 4.9
System representations in pictorial form
4.4.5 STABILITY AND CAUSALITY
ForLTI systems, the BIBO stability is equivalent to ∑ ∞
−∞
|h(k)| < ∞.
From the existence of the discrete-time Fourier transform, this stability
implies that H(e jω ) exists, which further implies that the unit circle |z| =
1must be in the ROC of H(z). This result is called the z-domain stability
theorem; therefore the dashed paths in Figure 4.9 exist only if the system
is stable.
Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.