SS_09-3

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9.3 System Characterization with Laplace Transform x(t) y(t) = x(t)＊h(t)

h(t) X(s) H(s) Y(s) = X(s)H(s)

system function

transfer function

 Causality

- A causal system has an H(s) whose ROC is a right-half plane

h(t) is right-sided

- For a system with a rational H(s), causality is equivalent to its ROC being the right-half plane to the right of the rightmost pole - Anticausality

a system is anticausal if h(t) = 0, t > 0 an anticausal system has an H(s) whose ROC is a left-half plane, etc.

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 Stability

- A system is stable if and only if ROC of H(s) includes the j-axis

h(t) absolutely integrable, or Fourier transform converges

See Fig. 9.25, p. 696 of text

- A causal system with a rational H(s) is stable if and only if all poles of H(s) lie in the left-half of s-plane

ROC is to the right of the rightmost pole  Systems Characterized by Differential Equations

N k= 0  ak k =  bk dtk dt

k

N k= 0

dy(t)

k

M k= 0 dkx(t)

(  ak s ) Y(s) = (  bk s) X(s)

k= 0 M  k= 0 bk sk H(s) = X(s) = zeros N

M

k

Y(s)

 ask poles

k= 0  System Function Algebra - Parallel

h(t) = h1(t) + h2(t) H(s) = H1(s) + H2(s) - Cascade h(t) = h1(t)＊h2(t) H(s) = H1(s) H2(s) - Feedback x(t) + e(t)   + h1(t) H1(s)

z(t) h2(t) H2(s)

Y(s) = H1(s)[ X(s)  H2(s)Y(s)] H(s) = Y(s)

= H1(s)

X(s) 1+H1(s)H2(s)

poles 3

y(t)