ECE633F09_HW1solutions

ECE633F09_HW1solutions - ECE633 Signals and Systems I Fall...

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ECE633 Signals and Systems I, Fall 2009 – Homework 1 Solutions 1 From: Linear Systems and Signals, 2 nd ed . B. P. Lathi, Oxford University Press, 2005. B.1 Given a complex number wxj y =+ , the complex conjugate of w is defined in rectangular coordinates as * wx j y =− . Use this fact to derive complex conjugation in polar form. () 22 1 exp tan / j y j y xM e θ = 2 *2 1 2 2 1 exp tan / exp tan / j y j y x x y j y x M e −− = B.2 Express the following numbers in polar form (a) 1 j + (d) /4 2 jj ee π + (b) 4 3 j −+ (e) 1 j e + (c) ( ) 14 3 j j +− + (f) ( ) ( ) 1/ 4 3 j j + (a) 1 11t a n1 / 1 +∠ = 0.785 2 45 2 1.414 ∠° = (b) 2 21 43 t a n 3 / 4 −≈ 2.498 5 143.1 5 j e (c) from (a) and (b), 2 45 5 143.1 ∠° ∠ °= 3.283 5 2 188.1 7.071 j e (d) cos sin 2 cos sin 44 4 4 ππ ⎡⎤ ⎛⎞ ⎛ ⎞ ++ + ⎜⎟ ⎜ ⎟ ⎢⎥ ⎝⎠ ⎝ ⎠ ⎣⎦ 31 cos sin 2 cos sin 3cos sin 4422 j j + =−= 2 1 t a n 1 / 3 2 0.322 5 18.43 2.236 j e ∠− °≈ (e) () () () ( ) 2 2 1 sin 1 cos 1 sin 1 1 1 cos 1 sin 1 tan 1c o s1 j = + 2 sin 1 2 2cos 1 tan 1.755 28.65 1.755 o j e ° + (f) from (c), 2 45 / 5 143.1 1.712 2 98.1 0.2828 5 j e

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ECE633 Signals and Systems I, Fall 2009 – Homework 1 Solutions 2 B.3 Express the following numbers in Cartesian form: (a) /4 3 j e π (b) 1/ j e (c) () ( ) 14 3 j j +− + (d) 2 jj ee + (e) 1 j e + (f) 1/ 2 j (a) 3c o s s i n 44 j ππ ⎡⎤ ⎛⎞ += ⎜⎟ ⎢⎥ ⎝⎠ ⎣⎦ 33 2.12 2.12 22 +≈ + (b) () () cos 1 sin 1 j ej =− + = () ( ) cos 1 sin 1 0.540 0.841 −≈ (c) ()() 2 41 31 4 4 3 3 j j −+ + + = −++ = 7 j (d) cos sin 2 cos sin 4 4 ⎛ ⎞ ++ + ⎜ ⎟ ⎝ ⎠ cos sin 2 cos sin 3cos sin j =+ + 2.121 0.707 (e) cos 1 sin 1 1 j = 1 cos 1 sin 1 1.540 0.841 ++≈ + (f) ( ) ln 2 ln 2 2 cos ln 2 sin ln 2 cos ln 2 sin ln 2 j j j j j == = −= = cos ln 2 sin ln 2 0.769 0.639 B.4 For complex constant w , prove: (a) * Re / 2 ww w (b) * Im / 2 w j (a) ( ) ( ) * /2 2 Re x j y x j y x x w + + = = = (b) () ( )( ) ( ) ( ) ( ) ( ) * / 2 2 / 2 Im j y j jy j y w + = = =
ECE633 Signals and Systems I, Fall 2009 – Homework 1 Solutions 3 B.5 Given wxj y =− , determine: (a) () Re w e (b) Im w e (a) () ( ) ( ) ( ) Re Re Re cos wx j yx j y x ee e e e y −− == = = ( ) cos x ey (b) Im Im Im sin j j y x e e e y = = ( ) sin x B.6

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ECE633F09_HW1solutions - ECE633 Signals and Systems I Fall...

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