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Inverse system
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Fig. 1-43
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Invertible; x ( t ) = i y ( r ) ( b ) Not invertible
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dy(0 Invertible; x(t ) = dt ( d ) Invertible; x [ n ]= y [ n ] - y[n - 11 (el Not invertible
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Linear Time-Invariant Systems
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2.1 INTRODUCTION
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Two most important attributes of systems are linearity and time-invariance. In this chapter we develop the fundamental input-output relationship for systems having these attributes. It will be shown that the input-output relationship for LTI systems is described in terms of a convolution operation. The importance of the convolution operation in LTI systems stems from the fact that knowledge of the response of an LTI system to the unit impulse input allows us to find its output to any input signals. Specifying the input-output relationships for LTI systems by differential and difference equations will also be discussed.
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RESPONSE OF A CONTINUOUS-TIME LTI SYSTEM AND THE CONVOLUTION INTEGRAL
A. Impulse Response:
The impulse response h(t) of a continuous-time LTI system (represented by T) is defined to be the response of the system when the input is 6(t), that is,
Response to an Arbitrary Input:
From Eq. (1.27) the input x ( t ) can be expressed as
Since the system is linear, the response y( t of the system to an arbitrary input x( t ) can be expressed as
Since the system is time-invariant, we have Substituting Eq. (2.4) into Eq. (2.31, we obtain
CHAP. 21
LINEAR TIME-INVARIANT SYSTEMS
Equation (2.5) indicates that a continuous-time LTI system is completely characterized by its impulse response h( t 1.
C. Convolution Integral:
Equation (2.5) defines the convolution of two continuous-time signals x ( t ) and h ( t ) denoted by
Equation (2.6) is commonly called the convolution integral. Thus, we have the fundamental result that the output of any continuous-time LTI system is the convolution of the input x ( t ) with the impulse response h ( t ) of the system. Figure 2-1 illustrates the definition of the impulse response h ( t ) and the relationship of Eq. (2.6).
Fig. 2-1 Continuous-time LTl system.
D. Properties of the Convolution Integral:
The convolution integral has the following properties. I . Commutative: ~ ( t h ( t ) = h ( t )* ~ ( t ) *)
2. Associative:
{ x P )* h l ( 4 * h , ( t ) = x ( t )* { h l ( f * h 2 ( 4 )
3. Distributive:
x ( t ) * { h , ( t ) ) h N = x ( t )* h l ( t ) + x ( t )* h , ( t ) + E. Convolution Integral Operation: Applying the commutative property (2.7) of convolution to Eq. (2.61, we obtain
(2.10) h ( r ) x ( t- r ) d r -m which may at times be easier to evaluate than Eq. (2.6). From Eq. ( 2 . 6 ) we observe that the convolution integral operation involves the following four steps:
( t )= h
)* x
1. The impulse response h ( ~ is time-reversed (that is, reflected about the origin) to ) obtain h( - 7 ) and then shifted by t to form h ( t - r ) = h [ - ( r - t ) ]which is a function of T with parameter t. 2. The signal x ( r ) and h ( t - r ) are multiplied together for all values of r with t fixed at some value.
LINEAR TIME-INVARIANT SYSTEMS
[CHAP. 2
3. The product x ( ~ ) h (- T ) is integrated over all T to produce a single output value t At). 4. Steps 1 to 3 are repeated as I varies over - 03 to 03 to produce the entire output y( t ).
Examples of the above convolution integral operation are given in Probs. 2.4 to 2.6.
F. Step Response:
The step response s(t) of a continuous-time LTI system (represented by T) is defined to be the response of the system when the input is 4 1 ) ; that is,
In many applications, the step response d t ) is also a useful characterization of the system. The step response s ( t ) can be easily determined by Eq. (2.10); that is,
Thus, the step response s(t) can be obtained by integrating the impulse response h(t). Differentiating Eq. (2.12) with respect to t, we get
Thus, the impulse response h ( t ) can be determined by differentiating the step response dl).
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