sec3solns - ECE 2200 Signals and Information Section 3...

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Unformatted text preview: ECE 2200 Signals and Information Section 3 (Week 4 Feb 15-18) Page 1 1. The continuous-time sinusoid x ( t ) = 4 . 5 sin(16 . 6 t ) is to be sampled at f s to produce the discrete-time sinusoid x [ k ] = M cos( k + ) (i) For f s = 15 Hz, determine M , , and . Has x ( t ) been undersampled or oversampled? (ii) For f s = 7 . 5 Hz, determine M , , and . Has x ( t ) been undersampled or oversampled? (iii) For f s = 22 . 5 Hz, determine M , , and . Has x ( t ) been undersampled or oversampled? Solution (read Section 4-1 Sampling): x ( t ) = 4 . 5 sin(16 . 6 t ) = 4 . 5 cos 2 8 . 3 t- 2 This means that the discrete-time signal x [ k ] = x ( kT s ) = x ( k/f s ) is parameterized as M = 4 . 5, = 2 8 . 3 /f s and =- / 2. (i) M = 4 . 5, = 2 8 . 3 / 15 (undersampled, 15 Hz < 2 8 . 3 Hz ) and =- / 2 (ii) M = 4 . 5, = 2 8 . 3 / 7 . 5 (undersampled, 7 . 5 Hz < 2 8 . 3 Hz ) and =- / 2 (iii) M = 4 . 5, = 2 8 . 3 / 22 . 5 (oversampled, 22 . 5 Hz > 2 8 . 3 Hz ) and =- / 2 0.05 0.1 0.15 0.2 0.25 0.3 0.35 4 2 2 4 0.05 0.1 0.15 0.2 0.25 0.3 0.35 4 2 2 4 0.05 0.1 0.15 0.2 0.25 0.3 0.35 4 2 2 4 (i) f s = 15 Hz (ii) f s = 7 . 5 Hz (iii) f s = 22 . 5 Hz t (sec) Continuous Waveform: x ( t ) = 4 . 5 sin(16 . 6 t ) Figure 1: A continuous-time 8 . 3 Hz sinusoid sampled at (top) f s = 15 Hz, (middle) f s = 7 . 5 Hz and f s = 22 . 5 Hz. ECE 2200 Signals and Information Section 3 (Week 4 Feb 15-18) Page 2 2. Consider the discrete-time sinusoid y [ n ] = 11 . 8 cos(0 . 21 n- / 5) generated by sampling y ( t ) = M cos(2 t + ) at a sampling rate of 4200 Hz. Determine three different continuous-time sinusoids y ( t ) with frequencies all less than 5 . 8 kHz that could have produced y [ n ]. Specify your three possibilities with unique triples of M , and . Solution (read Section 4-1 Sampling): the sampled signal y [ n ] is defined as y [ n ] = y ( n/f s ) = M cos(2 ( n/f s ) + ) = M cos( n + ) M = 11 . 8 = 2 /f s + 2 =- / 5 + 2 for = 0 , 1 , 2 ,... , which can be repeated for various choices of the parameter : = 0 : = 2 / 4200 + 2 (0) = 0 . 21 = 441 , =- 5 =- 1 : 1 = 2 1 / 4200 + 2 (- 1) = 0 . 21 1 = 4641 , 1 =- 5- 2 =- 2 : 2 = 2 2 / 4200 + 2 (- 2) = 0 . 21 2 = 8841 , 2 =- 5- 4 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10 3 10 5 5 10 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10 3 10 5 5 10 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10 3 10 5 5 10 = 441 , =- 5 = 4641 , =- 5- 2 = 8841 , =- 5- 4 Continuous Waveform: y ( t ) = 11 . 8 cos(2 t + ) t (sec) Figure 2: Plots of y [ n ] = 11 . 8 cos ( . 21 n- 5 ) for various parameterizations of y ( t ). ECE 2200 Signals and Information Section 3 (Week 4 Feb 15-18)...
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This note was uploaded on 03/11/2010 for the course ECE 2200 taught by Professor Johnson during the Spring '05 term at Cornell University (Engineering School).

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sec3solns - ECE 2200 Signals and Information Section 3...

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