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23 PROPERTIES OF CONTINUOUS-TIME LTI SYSTEMS .
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Systems with or without Memory:
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Since the output y(t) of a memoryless system depends on only the present input x(t), then, if the system is also linear and time-invariant, this relationship can only be of the form
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(2.14) Y ( [ )= Kx(t) where K is a (gain) constant. Thus, the corresponding impulse response h(f) is simply h ( t ) = K6(t) Therefore, if h(tJ
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(2.15)
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0, the continuous-time LTI system has memory.
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B. Causality:
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As discussed in Sec. 1.5D, a causal system does not respond to an input event until that event actually occurs. Therefore, for a causal continuous-time LTI system, we have
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Applying the causality condition (2.16) to Eq. (2.101, the output of a causal continuous-time
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CHAP. 21
LINEAR TIME-INVARIANT SYSTEMS
LTI system is expressed as
Alternatively, applying the causality condition ( 2.16) to Eq. (2.61, we have y(t) =
lt x ( r ) h ( t - T )d r w
(2.18)
Equation (2.18) shows that the only values of the input x ( t ) used to evaluate the output y( t ) are those for r 5 t. Based on the causality condition (2.161, any signal x(t) is called causal if and is called anticausal if x(t) = 0 t>O Then, from Eqs. (2.17), (2. I8), and (2. Iga), when the input x ( t ) is causal, the output y(t ) of a causal continuous-time LTI system is given by
C. Stability: The BIBO (bounded-input/bounded-output)stability of an LTI system (Sec. 1.5H) is readily ascertained from its impulse response. It can be shown (Prob. 2.13) that a continuous-time LTI system is BIBO stable if its impulse response is absolutely integrable, that is,
2.4 EIGENFUNCTIONS OF CONTINUOUS-TIME LTI SYSTEMS In Chap. 1 (Prob. 1.44) we saw that the eigenfunctions of continuous-time LTI systems represented by T are the complex exponentials eS',with s a complex variable. That is, where h is the eigenvalue of T associated with e"'. Setting x(t) = es' in Eq. (2.10), we have
where Thus, the eigenvalue of a continuous-time LTI system associated with the eigenfunction es' is given by H ( s ) which is a complex constant whose value is determined by the value of s via Eq. (2.24). Note from Eq. (2.23) that y(0) = H ( s ) (see Prob. 1.44). The above results underlie the definitions of the Laplace transform and Fourier transform which will be discussed in Chaps. 3 and 5.
LINEAR TIME-INVARIANT SYSTEMS
[CHAP. 2
SYSTEMS DESCRIBED BY DIFFERENTIAL EQUATIONS Linear Constant-Coefficient Differential Equations:
A general Nth-order linear constant-coefficient differential equation is given by
where coefficients a , and b, are real constants. The order N refers to the highest derivative of y ( 0 in Eq. (2.25).Such differential equations play a central role in describing the input-output relationships of a wide variety of electrical, mechanical, chemical, and biological systems. For instance, in the RC circuit considered in Prob. 1.32, the input x ( 0 = il,(O and the output y ( l ) = i-,.(t)are related by a first-order constant-coefficient differential equation [Eq. ( l . 1 0 5 ) ]
The general solution of Eq. (2.25) for a particular input x ( t ) is given by
where y , ( t ) is a particular solution satisfying Eq. (2.25) and y h ( t ) is a homogeneous solution (or complementary solution) satisfying the homogeneous differential equation
The exact form of y h ( t )is determined by N auxiliary conditions. Note that Eq. ( 2 . 2 5 )does not completely specify the output y ( t ) in terms of the input x ( t ) unless auxiliary conditions are specified. In general, a set of auxiliary conditions are the values of
at some point in time.
B. Linearity:
The system specified by Eq. (2.25) will be linear only if all of the auxiliary conditions are zero (see Prob. 2.21). If the auxiliary conditions are not zero, then the response y ( t ) of a system can be expressed as
where yzi(O,called the zero-input response, is the response to the auxiliary conditions, and yz,(t), called the zero-state response, is the response of a linear system with zero auxiliary conditions. This is illustrated in Fig. 2-2. 0 Note that y,,(t) # y h ( t ) and y,,(t) 2 y,(t) and that in general yZi( contains y h ( t ) and y,,( t ) contains both y h ( t and y,( t (see Prob. 2.20).
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