Homework3

# Homework3 - (c Assume that we added another signal...

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ELCT 321 Digital Signal Processing Homework Assignment # 3 (Spectrum Representation) by Dr. Yong-June Shin Assigned: September 20, 2011 Due: September 27, 2011 1. (25 Points) A signal composed of sinusoids is give by the equation x ( t ) = 1 + sin(2 π (50) t + π/ 4) + cos(2 π (100) 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 of signal x ( t ): 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?
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Unformatted text preview: (c) Assume that we added another signal component so that y ( t ) = x ( t )-2sin(2 π √ 3 t-π/ 2). Please sketch magnitude and phase plot of y ( t ) in frequency domain. Is y ( t ) periodic signal? If so, please determine period. Otherwise, please explain the reasons. 3. (25 Points) Consider the periodic waveform in time domain in Figure 2 Figure 2 A Periodic Waveform in Time Domain (a) Determine DC component of the signal x ( t ) (b) Determine Fourier series coeﬃcients of x ( t ) 4. (25 Points) Determine the Fourier series representation of the peri-odic signal provided in Figure 3 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...
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