HW 07_examples - 1 ME360 - Solved Examples for Homework #7...

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1 ME360 - Solved Examples for Homework #7 Problem 1: Sketch a few periods of the following periodic signal described over one period, and find the exponential Fourier series coefficient. Derive analytical expressions for, and plot, the single and double-sided magnitude, phase, and power spectra ; (a) x ( t )= e 2 t , 0 t 2 , with T =2 . Solution: x(t) = exp(-2t) t 2 4 6 1 T = 2 Figure 1. x(t) = exp(-2t) f o = 1 T = 1 2 ω o πf o = π X s ( k 1 T R T x ( t ) e jkω o t dt = 1 2 R 2 0 e 2 t e jkπt dt = 1 2 R 2 0 e (2+ jkπ ) t dt = 1 2 e (2+ jkπ ) t 2+ jkπ ¯ ¯ ¯ 2 0 ¸ = 1 2 h e 4 j 2 πk +1 2+ i = 1 2 h 1 e 4 e j 2 πk 2+ i = 1 e 4 4+ j 2 πk (since e ± j 2 πk =1 ) Analytical expressions for magnitude and phase spectra: X s ( k 1 e 4 4+ j 2 πk = 1 e 4 4+ j 2 πk · 4 j 2 πk 4 j 2 πk = ( 4 4 e 4 ) + j ( e 4 2 πk 2 πk ) 4 2 +(2 πk ) 2 = RE z }| { 4 4 e 4 4 2 +(2 πk ) 2 + j IM z }| { Ã e 4 2 2 4 2 ) 2 ! Magnitude: | X s ( k ) | = RE 2 + 2 = ¯ ¯ ¯ 1 e 4 4+ j 2 πk ¯ ¯ ¯ = | 1 e 4 | | 4+ j 2 πk | = | 1 e 4 | 4 2 +(2 πk ) 2 Phase: φ rad =tan 1 ¡ RE ¢ 1 ³ e 4 2 πk 2 πk 4 4 e 4 ´ 1 μ 2 πk ( 1 e 4 ) 4(1 e 4 ) = tan 1 ¡ πk 2 ¢ k | X s ( k ) | φ rad φ deg | X s ( k ) | 2 -5 0.0310 1.4441 82.74 0.0010 -4 0.0386 1.4130 80.96 0.0015 -3 0.0509 1.3617 78.02 0.0026 -2 0.0744 1.2626 72.34 0.0055 -1 0.1318 1.0039 57.52 0.0174 0 0.2454 0 0 0.0602 1 0.1318 -1.0039 -57.52 0.0174 2 0.0744 -1.2626 -72.34 0.0055 3 0.0509 -1.3617 -78.02 0.0026 4 0.0386 -1.4130 -80.96 0.0015 5 0.0310 -1.4441 -82.74 0.0010
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2 Single-sided: c o = X (0) ,c k =2 | X s ( k ) | ,p k | X s ( k ) | 2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 One-sided Magnitude for x(t)=exp(-2t) k magnitude 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1.5 -1 -0.5 0 One-sided Phase for x(t)=exp(-2t) k Phase (rad) Figure 2. One-sided Magnitude and Phase Spectra -5 -4 -3 -2 -1 0 1 2 3 4 5 0 0.05 0.1 0.15 0.2 0.25 Two-sided Magnitude for x(t)=exp(-2t) k -5 -4 -3 -2 -1 0 1 2 3 4 5 -1.5 -1 -0.5 0 0.5 1 1.5
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This note was uploaded on 04/18/2010 for the course ME 360 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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HW 07_examples - 1 ME360 - Solved Examples for Homework #7...

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