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HW3 (S 2007) - odic signal provided in Figure 2 Figure 2 A...

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ELCT 321 Digital Signal Processing Homework Assignment # 3 (Chapter 3: Spectrum Representation) by Dr. Yong-June Shin Assigned: February 13, 2007 Due: February 20, 2007 1. (25 Points) A signal composed of sinusoids is give by the equation x ( t ) = 1 + 2 sin(2 π (100) t + π/ 4) + 2 cos(2 π (200) t - π/ 4) (a) Using Euler’s formula, rewrite the signal x ( t ) as a sum of complex exponential signals. (b) Sketch the spectrum of this signal in separate plots of magnitude and phase in frequency domain. (c) Let us assign y ( t ) = 2 · x ( t - 1 800 ). Sketch the the spectrum of y ( t ) in separate plots of magnitude and phase in frequency domain. Please explain how and why the the magnitude and phase spectrum are different for x ( t ) and y ( t ). 2. (25 Points) Consider following frequency domain magnitude and phase plot: Figure 1 Spectrum Representation 1

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(a) Determine the time domain representation of the signal as sum of sinusoidal function. (b) Is x ( t ) a periodic signal? If so, what is the period? 3. (25 Points) Determine the Fourier series representation of the peri-
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Unformatted text preview: odic signal provided in Figure 2. Figure 2 A Periodic Waveform in Time Domain (a) Determine Fourier series coeﬃcients of x ( t ) (b) Please simplify the Fourier series expression by use of inverse Euler equation, the Fourier series will be represented in terms of trigonometric functions. 4. (25 Points) Determine the Fourier series representation of the peri-odic signal provided in Figure 3. T/2 -T/2 -T/4 T/4 1 t Figure 3 A Periodic Waveform in Time Domain (a) Determine Fourier series coeﬃcients of x ( t ) (b) Please simplify the Fourier series expression by use of inverse Euler equation, the Fourier series will be represented in terms of trigonometric functions. 2 (c) Explain how the Fourier series expansions in Figure 3 and Figure 2 are related. (d) Determine Fourier series expansion of following waveform pro-vided in Figure 4 by use of the results of Figure 3. Figure 4 A Periodic Waveform in Time Domain 3...
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