Convolution

# Convolution - ECE 3250 DISCRETE-TIME CONVOLUTION Fall 2008...

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ECE 3250 DISCRETE-TIME CONVOLUTION Fall 2008 We’ve been modeling discrete-time signals as functions x : Z F , where F denotes R or C . I introduced the notation F Z to stand for the set of all F -valued discrete-time signals. If you want, you can view the typical element x of F Z as a two-sided inﬁnite sequence of numbers from F , i.e. . . . , x ( - 3) , x ( - 2) , x ( - 1) , x (0) , x (1) , x (2) , . . . x stands for a signal and x ( n ) stands for the value of the signal x at time n . Given two signals x 1 and x 2 in F Z , the convolution of x 1 and x 2 , if it exists, is the signal x F Z with speciﬁcation (1) x ( n ) = X k = -∞ x 1 ( k ) x 2 ( n - k ) , n Z . Some notations I’ll be using for the convolution of x 1 and x 2 are x 1 * x 2 and Conv( x 1 , x 2 ). Alternative terminologies for the convolution of x 1 and x 2 are “the convolution of x 1 with x 2 ” and “ x 1 convolved with x 2 .” The convolution of x 1 and x 2 exists if and only if the sum in (1) converges for every n Z . Note that the sum is a series that’s potentially “inﬁnite in both directions.” We haven’t encountered such series before, so suppose that { a n } is a two-sided inﬁnite sequence of numbers from F , i.e., . . . a - 2 , a - 1 , a 0 , a 1 , a 2 , . . . We say that the doubly inﬁnite series P n = -∞ a n converges if and only if the two one-sided series X n =0 a n and - 1 X n = -∞ a n = X m =1 a - m both converge. In this case, the limit of the doubly inﬁnite series is the sum of the limits of the two one-sided series. Let’s begin with an elementary observation about convolution. If x 1 * x 2 exists, then the sum in (1) converges for every n Z . Change the index of summation as follows: x 1 * x 2 ( n ) = X k = -∞ x 1 ( k ) x 2 ( n - k ) = X m = -∞ x 1 ( n - m ) x 2 ( m ) = X k = -∞ x 1 ( n - k ) x 2 ( k ) . Setting m = n - k yields the middle equality. To get the last, re-name the “dummy index of summation” m as k . The bottom line is that on the right-hand side of equation (1), it doesn’t matter where we put the k and where we put the ( n - k ) — the result is the same. One could dignify this observation by saying something along the lines of, “convolution, deﬁned by (1), is a commutative operation in the sense that if x 1 * x 2 exists, then x 1 * x 2 = x 2 * x 1 .” That’s ﬁne, but it’s a bit unnecessary in my view. A slightly less elementary observation about convolution is that it is an associative operation in the sense that if x 1 * ( x 2 * x 3 ) exists, then so does ( x 1 * x 2 ) * x 3 , and vice versa, and both convolutions are the same. Proving this fact is an exercise in summation manipulation. I’ll be a bit casual about interchanging orders of summation here. It turns out that the interchanges are legal since all the sums converge. Assuming that x 1 * ( x 2 * x 3 ) 1

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2 exists, we have x 1 * ( x 2 * x 3 )( n ) = X k = -∞ x 1 ( k )( x 2 * x 3 ( n - k )) = X k = -∞ x 1 ( k )
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Convolution - ECE 3250 DISCRETE-TIME CONVOLUTION Fall 2008...

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