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barcode generator for ssrs An LSI system with unit sample response h(n) = anu(n)will be stable whenever la1 < 1, because in Software
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CONVOLUTION
The relationship between the input to a linear shiftinvariant system, x(n), and the output, y(n), is given by the convolution sum x(n) * h(n) = x(k)h(n  k ) Because convolution is fundamental to the analysis and description of LSI systems, in this section we look at the mechanics of performing convolutions. We begin by listing some properties of convolution that may be used to simplify the evaluation of the convolution sum. 1.4.1 Convolution Properties
Convolution is a linear operator and, therefore, has a number of important properties including the commutative, associative, and distributive properties. The definitions and interpretations of these properties are summarized below. Commutative Property
The commutative property states that the order in which two sequences are convolved is not important. Mathematically, the commutative property is From a systems point of view, this property states that a system with a unit sample response h(n)and input x ( n ) behaves in exactly the same way as a system with unit sample response x ( n )and an input h(n). This is illustrated in Fig. 15(a). Associative Property
The convolution operator satisfies the associative property, which is
From a systems point of view, the associative property states that if two systems with unit sample responses hl(n) and h2(n)are connected in cascade as shown in Fig. I 5(b), an equivalent system is one that has a unit sample response equal to the convolution of hl ( n )and h2(n): SIGNALS AND SYSTEMS
[CHAP. 1
(b) The associative property.
( c ) The distributive property.
Fig. 15. The interpretation of convolution properties from a systems point of view. Distributive Property The distributive property of the convolution operator states that
From a systems point of view, this property asserts that if two systems with unit sample responses h l ( n ) and h 2 ( n ) are connected in parallel, as illustrated in Fig. 15(c), an equivalent system is one that has a unit sample response equal to the sum of h 1 ( n ) and h2(n): 1A.2 Performing Convolutions
Having considered some of the properties of the convolution operator, we now look at the mechanics of performing convolutions. There are several different approaches that may be used, and the one that is the easiest will depend upon the form and type of sequences that are to be convolved. Direct Evaluation
When the sequences that are being convolved may be described by simple closedform mathematical expressions, In the convolution is often most easily performed by directly evaluating the sum given in Eq. ( I 7). performing convolutions directly, it is usually necessary to evaluate finite or infinite sums involving terms of the form anor n a n . Listed in Table 11 are closedform expressions for some of the more commonly encountered series. EXAMPLE 1.4.1
Let us perform the convolution of the two signals
CHAP. 11
SIGNALS AND SYSTEMS
Table 11 Closedform Expressions for Some Commonly Encountered Series
enan, , lal < I
With the direct evaluation of the convolution sum we find
Because u(k) is equal to zero for k < 0 and u(n  k ) is equal to zero for k > n , when n < 0 , there are no nonzero terms in the sum and y ( n ) = 0. On the other hand, if n 3 0, Therefore,

