HW1sol_FA08 - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN...

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Unformatted text preview: UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Department of Electrical and Computer Engineering ECE 410 Digital Signal Processing Homework 1 Solution Prof. Bresler, Prof. Jones Wednesday, September 3, 2008 Problem 1 (a) 5 negationslash- 120 ◦ + j (8- 4 j ) = 5 parenleftBigg- 1 2- √ 3 2 j parenrightBigg + j (8- 4 j ) = 3 2 +- 5 √ 3 + 16 2 j = 1 . 5 + 3 . 6699 j = radicalbig 1 . 5 2 + 3 . 6699 2 e j tan- 1 ( 3 . 6699 1 . 5 ) = 3 . 9646 e j 1 . 1828 (b) 15 negationslash 45 ◦ 3 negationslash 60 ◦ + 2 negationslash- 120 ◦ = 15 negationslash 45 ◦ 3 parenleftBig 1 2 + √ 3 2 j parenrightBig + 2 parenleftBig- 1 2- √ 3 2 j parenrightBig = 15 negationslash 45 ◦ 1 2 + √ 3 2 j = 15 negationslash 45 ◦ 1 negationslash 60 ◦ = 15 negationslash- 15 ◦ = 15 e j- . 2618 = 14 . 4889- 3 . 8823 i (c) (2 + 5 j )(9- 4 j ) 5 negationslash 135 ◦ = 38 + 37 j 5 negationslash 135 ◦ =- 38 + 37 j 5 negationslash 45 ◦ =- 38 + 37 j 5 parenleftbigg 1 √ 2 + 1 √ 2 j parenrightbigg =- √ 2 10- 15 √ 2 2 j =- . 1414- 10 . 6066 j = 10 . 6075 e- j 1 . 5841 1 Problem 2 (a) u [- n + 5] + u [ n + 3] = braceleftBigg 1 n <- 3 , n > 5 2- 3 ≤ n ≤ 5 (b) n ( u [ n ]- u [ n- 10]) = braceleftBigg n < , n ≥ 10 n ≤ n < 10 (c) u [- n ] u [ n- 4] = 0 ∀ n (d) u [ n ] u [- n + 4] = braceleftBigg n < , n > 4 1 0 ≤ n ≤ 4 Refer Figure 1 2 Figure 1: Problem 2 Solution Problem 3 (a) integraldisplay ∞-∞ ( t 2 + 2 t + 4) δ ( t- 2) dt = t 2 + 2 t + 4 | t =2 = 12 (b) integraldisplay ∞ 3 ( t 2 + 2 t + 4) δ ( t- 2) dt = 0 , since 2 / ∈ [3 , ∞ ] 3 (c) integraldisplay 3-∞ ( t 2 + 2 t + 4) u ( t + 1) dt = integraldisplay 3- 1 t 2 + 2 t + 4...
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HW1sol_FA08 - UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN...

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