hw04su06_soln

hw04su06_soln - SOLUTIONS TO ECE 2025 SUMMER 2006 PROBLEM...

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S OLUTIONS TO ECE 2025 S UMMER 2006 P ROBLEM S ET N O . 4 PROBLEM 4.1*: The sample rate is 10 kHz, so any sinusoid with frequency less than 5 kHz will get through unscathed. Here x ( t ) is the sum of 3 sinusoids, only the first of which will get through unscathed. The others will be aliased somehow. x [ n ] = x () = x = cos( n π /5) + cos( n + 6 π /13) + cos( n 3 π /7) = cos(0.6 π n π /5) + cos(1.6 π n + 6 π /13) + cos(2.8 π n 3 π /7) = cos(0.6 π n π /5) + cos( 0.4 π n + 6 π /13) + cos(0.8 π n 3 π /7) = cos(0.6 π n π /5) + cos(0.4 π n 6 π /13) + cos(0.8 π n 3 π /7). The spectrum for x [ n ] is shown below: y ( t ) = x [ f s t ] = cos(0.6 π f s t π /5) + cos(0.4 π f s t 6 π /13) + cos(0.8 π f s t 3 π /7) = cos(6000 π t π /5) + cos(4000 π t 6 π /13) + cos(8000 π t 3 π /7) n f s ---- n 10000 -------------- 6000 π 10000 --------------- 16000 π 10000 ------------------ 28000 π 10000 FOLDING ALIASING (subtracted 2 π n from last 2 cos arguments) 0.4 π 0.6 π 0.8 π 0.8 π 0.6 π 0.4 π e j 3 π /7 e j π /5 e j 6 π /13 1 2 -- 1 2 1 2 e j 3 π /7 e j π /5 e j 6 π /13 1 2 1 2 1 2 ω ˆ FOLDED ALIASED UNSCATHED

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x [ n ] = x ( ) = 7 + 3cos( n ) 2cos( n ) = 7 + 3cos(8 π n ) 2cos(12 π n ) = 7 + 3 2 = 8. The discrete-time signal is a constant for all time. Passing this to an ideal D-C converter
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hw04su06_soln - SOLUTIONS TO ECE 2025 SUMMER 2006 PROBLEM...

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