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Eq. (1.5) becomes
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which is known as the superposition summation.
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Shift-Invariance
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If a system has the property that a shift (delay) in the input by no results in a shift in the output by no, the system is said to be shift-invariant. More formally,
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Definition: Let y ( n ) be the response of a system to an arbitrary input x ( n ) . The system is said to be shift-invariant if, for any delay no, the response to x ( n - no) is y(n - no). A system that is not shift-invariant is said to be shift-~arying.~
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In effect, a system will be shift-invariant if its properties or characteristics do not change with time. To test for shift-invariance one needs to compare y(n - n o ) to T [ x ( n - no)]. If they are the same for any input x ( n ) and for all shifts no, the system is shift-invariant.
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EXAMPLE 1.3.3 The system defined by
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y(n) = x2(n)
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is shift-invariant, which may be shown as follows. If y(n) = x2(n)is the response of the system to x(n), the response of the system to
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x'(n) = x(n - no)
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Because y'(n) = y(n
- no), the system is shift-invariant. However, the system described by the equation
is shift-varying. To see this, note that the system's response to the input x(n) = S(n) is
whereas the response to x(n - 1) = S(n - 1 ) is
Because this is not the same as y(n - 1) = 2S(n - I ) , the system is shift-varying.
4 ~ o m authors refer to this property as rime-invorionce. However. because n does not necessarily represent "time:' shift-invariance is a bit e more general.
Linear Shin-Invariant Systems
SIGNALS AND SYSTEMS
[CHAP. 1
A system that is both linear and shift-invariant is referred to as a linear shifi-invariant (LSI) system. If h(n) is the response of an LSI system to the unit sample 6(n),its response to 6(n - k ) will be h(n - k). Therefore, in the superposition sum given in Eq. (1.6), hk(n)= h(n - k )
and it follows that
y(n) =
C *(k)h(n - k )
Equation ( 1 . 9 , which is known as the convolution sum, is written as
where * indicates the convolution operator. The sequence h(n),referred to as the unit sample response, provides a complete characterization of an LSI system. In other words, the response of the system to any input x(n) may be found once h(n) is known.
Causality
A system property that is important for real-time applications is causality, which is defined as follows:
Definition: A system is said to be causal if, for any no, the response of the system at time no depends only on the input up to time n = no.
For a causal system, changes in the output cannot precede changes in the input. Thus, if xl ( n ) = x2(n)for n 5 no, yl(n) must be equal to y2(n) for n 5 no. Causal systems are therefore referred to as nonanticipatory. An LSI system will be causal if and only if h(n) is equal to zero for n < 0.
EXAMPLE 1.3.4 The system described by the equation y ( n ) = x ( n ) + x ( n - 1 ) is causal because the value of the output at any time n = no depends only on the inputx(n) at time no and at time no - 1. The system described by y ( n ) = x ( n ) x(n+ I), on the other hand, is noncausal because the output at time n = no depends on the value of the input at time no 1.
Stability
In many applications, it is important for a system to have a response, y(n), that is bounded in amplitude whenever the input is bounded. A system with this property is said to be stable in the bounded input-bounded output (BIBO) sense. Specifically,
Definition: A system is said to be stable in the bounded input-bounded output sense if, for any input that is bounded, Ix(n)l IA < m, the output will be bounded,
For a linear shift-invariant system, stability is guaranteed if the unit sample response is absolutely summable:
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