hw19 - X k by viewing x t as the result of multiplying a...

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ENEE 241 02* HOMEWORK ASSIGNMENT 19 Due Thu 04/30 (i) (4 pts.) Using the time delay property, derive Fourier series coefficients { R k } of the periodic signal r ( t ) of period T 0 = 6 (seconds) shown below. r ( t ) 1 -1 -6 -5 -4 -3 1 2 3 6 7 8 t ... ... -8 -7 -2 -1 4 5 (ii) (4 pts.) Using the result of (i), determine the complex Fourier series coefficients { S k } of the periodic signal s ( t ) shown below. s ( t ) 1 -8 -7 -5 -4 -2 -1 0 1 2 4 5 7 8 t ... ... 2 (iii) (4 pts.) Use MATLAB to graph the output of an ideal lowpass filter with cutoff frequency 8 . 1 Hz when the input is s ( t ). Display two periods with 400 samples per period. Submit commands and graphs. (iv) (4 pts.) Using the result (i) and the modulation property, carefully derive the Fourier series coefficients { X k } of the periodic signal x ( t ) of period of period T 0 = 3 shown below (curved segments are sinusoidal). -5 -4 -2 -1 0 1 2 4 5 t x ( t ) 1 ... ... (v) (4 pts.) Derive an alternative expression for

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Unformatted text preview: { X k } by viewing x ( t ) as the result of multiplying a symmetric rectangular pulse train by a cosine, followed by a time delay. Solved Examples S 19.1. Express the Fourier series coeﬃcients of x ( t ) and y ( t ) in terms of those of s ( t ). s ( t ) ... ...-T 0 T t x ( t ) ... ...-T 0 T t y ( t ) ... ...-T-T /2 0 T /2 T t S 19.2. Determine the Fourier series coeﬃcients of the periodic signal s ( t ) of period T = 5 shown below. s ( t ) 1-1-5 -4 -3 0 1 2 5 6 7 t ... ... S 19.3. Use two diﬀerent methods to express the complex Fourier series coeﬃcients of y ( t ) (full-wave rectiﬁed sinusoid) in terms of those of x ( t ) (half-wave rectiﬁed sinusoid), as shown below. T-T /4 0 T /4 T t x ( t ) 1 ... ...-T /4 0 T /4 t y ( t ) 1 ... ......
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