9.5 One-Sided Z-Transform Inverse 561
0 1 2 3 4 5 6 7 8 910
0
0.2
0.4
0.6
0.8
1
1.2
t, nTs
y(
t),
y[
n]
FIGURE 9.12
Solution of the differential equationd^2 vc(t)/dt^2 +dvc(t)/dt+vc(t)=vs(t)(solid line). Solution of the difference
equation approximating the differential equation forTs=0.1(dotted line). Exact and approximate solutions are
very close.
differential equation is well approximated by the solution of the difference equation obtained by
approximating the first and second derivatives.
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% Example 9.20
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syms s
vc = ilaplace(1/(s ˆ 3 + s ˆ 2 + s)); % exact solution
ezplot(vc,[0,10]);grid; hold on % plotting of exact solution
Ts = 0.1; % sampling period
a1 = 1/Tsˆ2 + 1/Ts + 1;a2 = -2/Ts ˆ 2 - 1/Ts;a3 = 1/Ts ˆ 2; % coefficients
a = [1 a2/a1 a3/a1];b = 1;
t = 0:Ts:10; N = length(t);
vs=ones(1,N); % input
vca = filter(b,a,vs);vca = vca/vca(N); % solution
n
9.5.5 Inverse of Two-Sided Z-Transforms
When finding the inverse of a two-sided Z-transform, or a noncausal discrete-time signal, it is impor-
tant to relate the poles to the causal and the anti-causal components. The region of convergence plays
