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Solutions_sampling

Solutions_sampling - Problem 1 Suppose that a discrete-time...

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Problem 1 Suppose that a discrete-time signal [ ] x n is given by the formula: [ ] 2.2cos(0.3 /3) x n n π π = - and that it was obtained by sampling a continuous-time signal 0 ( ) cos(2 ) x t A f t π φ = + at a sampling rate of 6000samples/sec s f = . Determine three different continuous-time signals that could have produced [ ] x n . These continuous-time signals should all have a frequency less than 8kHz . Write the mathematical formula for all three. Solution: [ ] 2.2cos(0.3 /3) x n n π π = - Compare to 0 0 ( ) cos(2 ) 2 0.3 , s s s n n x A f f f f or or f π ϕ π π π π π π = + = 0.3 + 2 , 0.3 - 2 Solve: 0 0 2 0.3 0.3 ( ) 6000 0.15 900Hz 2 ( ) 2.2cos(1800 /3) s s f f f f x t t π π π π = = = × = = - Then 0 0 2 2.3 2.3 ( ) 6900Hz 2 ( ) 2.2cos(2 (6900) /3) s s f f f f x t t π π π π = = = = - Finally 0 0 2 1.7 1.7 ( ) 5100Hz 2 ( ) 2.2cos(2 ( 5100) /3) ( ) 2.2cos(2 (5100) /3) s s f f f f x t t x t t π π π π π π - = - = = - = - - = +

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200 600 -200 -600 f(Hz) 2 3 8 j e π - 2 1 8 j e π 2 3 8 j e π 2 1 8 j e π - Problem 2 (a) Draw the spectrum of ) 400 ( sin ) ( 3 t t x π = . Label the frequencies and complex amplitudes of each component. (b) Determine the minimum sampling rate that can be used to sample ) ( t x without any aliasing. Sketch the spectrum of the corresponding discrete time signal. Solution: (a) ( 29 t j t j t j t j t j t j e e e e j j e e t x π π π π π π 1200 400 400 1200 3 400 400 3 3 8 1 2 ) ( - - - - + - - = - = (b) z s high s H f f f 1200 2 For z s H f 1200 = , s f T s s ) 1200 / 1 ( / 1 = = Then ( 29 ( 29 ( 29 3 / 3 / 3 / 3 / 1200 400 400 1200 3 3 8 3 3 8 1 3 3 8 1 ) ( ] [ n j n j n j n j n j n j nT
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