Isye 2027

# So sometimes we write x instead of x y and we write

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Unformatted text preview: = X + Y. Solution: We have fS = fX ∗ fY , and we will use Figure 4.15 to help work out the integral in (4.16). As shown, the graph of the function fY (c − u) is a rectangle over the interval [c − 1, c]. f (u) f (c!u) fX (u) X Y f (c!u) Y u c !1 0 c u c !1 1 0 1 c Figure 4.15: Calculating the convolution of two rectangle functions. To check this, note that if u = c then fY (c − u) = fY (0), and as u decreases from c to c-1, c − u increases from zero to c, so fY (c − u) = 1 for this range of u. The product fX (u)fY (c − u) is equal to one on [0, 1] ∪ [c − 1, c] and is equal to zero elsewhere. As shown in Figure 4.15, if 0 &lt; c ≤ 1, then the overlap is the interval [0, c], and therefore fX ∗ fY (c) = c. If 1 &lt; c &lt; 2, the overlap is the interval [c − 1, 1] which has length 2 − c, and therefore fX ∗ fY (c) = 2 − c. Otherwise, the value of the convolution is zero. Summarizing, 0&lt;c≤1 c 2−c 1&lt;c≤2 fS (c) = 0 else. so the graph of fS has the triangular shape shown in Figure 4.16. f (c) S 1 c 0 1 2 Figure 4.16: The convoluti...
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