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LINEAR TIME-INVARIANT SYSTEMS
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Fig. 2-2 Zero-state and zero-input responses.
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C. Causality: In order for the linear system described by Eq. (2.25) to be causal we must assume the condition of initial rest (or an initially relaxed condition). That is, if x( t) = 0 for t I t,,, then assume y(t) = 0 for t 5 to (see Prob. 1.43). Thus, the response for t > to can be calculated from Eq. (2.25) with the initial conditions
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where Clearly, at initial rest y,,(t) D. Time-Invariance: For a linear causal system, initial rest also implies time-invariance (Prob. 2.22).
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E. Impulse Response:
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The impulse response h(t) of the continuous-time LTI system described by Eq. (2.25) satisfies the differential equation
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with the initial rest condition. Examples of finding impulse responses are given in Probs. 2.23 to 2.25. In later chapters, we will find the impulse response by using transform techniques.
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RESPONSE OF A DISCRETE-TIME LTI SYSTEM AND CONVOLUTION SUM Impulse Response:
The impulse response (or unit sample response) h [ n ] of a discrete-time LTI system (represented by T) is defined to be the response of the system when the input is 6[n], that is,
LINEAR TIME-INVARIANT SYSTEMS
[CHAP. 2
B. Response to an Arbitrary Input:
From Eq. ( 1.51) the input x [ n ] can be expressed as
Since the system is linear, the response y [ n ] of the system to an arbitrary input x [ n ] can be expressed as
x [ k ] T { S [ n k]} -
Since the system is time-invariant, we have
h[n-k ]
= T { S [ n- k
(2.33)
Substituting Eq. (2.33) into Eq. (2.321, we obtain
Y [ ~ I= k =C
x [ k l h [ n- k l
Equation (2.34)indicates that a discrete-time LTI system is completely characterized by its impulse response h [ n ] .
C. Convolution Sum:
Equation (2.34) defines the convolution of two sequences x [ n ] and h [ n ]denoted by
sum. Thus, again, we have the Equation (2.35) is commonly called the con~~olution fundamental result that the output of any discrete-time LTI system is the concolution of the input x [ n ] with the impulse response h [ n ] of the system. Figure 2-3 illustrates the definition of the impulse response h [ n ] and the relationship of Eq. (2.35).
system
hlnl v l n l = x [ n l * hlnj
xlnl
Fig. 2-3 Discrete-time LTI system.
D. Properties of the Convolution Sum:
The following properties of the convolution sum are analogous to the convolution integral properties shown in Sec. 2.3.
CHAP. 21
LINEAR TIME-INVARIANT SYSTEMS
I . Commutative:
x [ n ]* h [ n ]= h [ n ]* x [ n ] 2. Associative: { ~ [ nh,[n]}* *] h2[nl =+I
3. Distributive:
(2.36) (2.37) (2.38)
* ( h , [ n l* h , [ n I l
x [ n ]* { h , [ n ]+ h , [ n ] ]=+I ] E. Convolution Sum Operation:
* h , [ n l + x b ]* h , [ n l
Again, applying the commutative property (2.36) of the convolution sum to Eq. (2.351, we obtain
which may at times be easier to evaluate than Eq. (2.35). Similar to the continuous-time case, the convolution sum [Eq. (2.391 operation involves the following four steps:
1. The impulse response h [ k ] is time-reversed (that is, reflected about the origin) to
obtain h [ - k ] and then shifted by n to form h [ n - k ] = h [ - ( k - n ) ] which is a function of k with parameter n. 2. Two sequences x [ k ] and h [ n - k ] are multiplied together for all values of k with n fixed at some value. 3. The product x [ k ] h [ n k ] is summed over all k to produce a single output sample y[nI. 4. Steps 1 to 3 are repeated as n varies over -GO to GO to produce the entire output y[n]. Examples of the above convolution sum operation are given in Probs. 2.28 and 2.30.
F. Step Response:
The step response s [ n ]of a discrete-time LTI system with the impulse response h [ n ]is readily obtained from Eq. (2.39) as
From Eq. (2.40) we have
h [ n ]= s [ n ]- s [ n - l ]
(2.41)
Equations (2,401 and (2.41) are the discrete-time counterparts of Eqs. (2.12) and (2,131, respectively.
PROPERTIES OF DISCRETE-TIME LTI SYSTEMS Systems with or without Memory:
Since the output y [ n ]of a memoryless system depends on only the present input x [ n ] , then, if the system is also linear and time-invariant, this relationship can only be of the
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