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solution0 - ECE 101 Linear Systems, Fall 2009 Problem Set...

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Unformatted text preview: ECE 101 Linear Systems, Fall 2009 Problem Set #0 Solutions September 24, 2009 (send comments/questions to psiegel@ucsd.edu) 1.48 Using Eulers relation, we have: z = r e j = r cos + jr sin = x + jy Then z 1 through z 5 are: (a) z 1 = x- jy (b) z 2 = q x 2 + y 2 (c) z 3 =- x- jy =- z (d) z 4 =- x + jy (e) z 5 = x + jy = z (recall, e j is periodic with period 2 ) Plots: 1 1.49 Once in polar form re j , the magnitude is given by r and angle given by . (a) 2 e j/ 3 (b) 5 e j (c) 5 2 e j 5 / 4 (d) 5 e jtan- 1 (4 / 3) = 5 e j 53 . 13 o (e) 8 e- j (f) 4 2 e j 5 / 4 (g) 2 2 e- j 5 / 12 (h) e- j 2 / 3 (i) e j/ 6 Plot: 1.51 By Eulers relation, we have (i) e j = cos + j sin and (ii) e- j = cos - j sin (a) Summing (i) + (ii) yields cos = 1 2 ( e j + e- j ). (b) Subtracting (i) - (ii) yields sin = 1 2 j ( e j- e- j )....
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solution0 - ECE 101 Linear Systems, Fall 2009 Problem Set...

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