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Problem 1: {3 + 2 + 3 + 2 = If), sampling) Consider the continuoustime signal STU?) = 3 + oos(1[}s't + 5} + sin[15a't)1 t E R. {a} Find the Fourier

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For the following systems with input f and (zero state) output y, circle all the properties that they possess, where time k elementof Integers.

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Problem 1: {3 + 2 + 3 + 2 = If), sampling)
Consider the continuous—time signal STU?) = 3 + oos(1[}s't + 5} + sin[15a't)1 t E R. {a} Find the Fourier transform X = fr. Hint: [IF ejwflfflm} = 21r6[*u:r — w”). {b} ‘Nhat is the Nyqujst Frequency mm in radians/s of 3:? +
{e} Write an expression for the Fourier transform of the ideal sampling of I with sam— pling period T5: = 2af£2irin}, i.e., 2::_m m[kfl)5[t — kill).
Hint: (3- ejmf‘tflc [t}]{w} = X(w — rwfi} and recall Poisson’s identityJ oo 2 fit — kill} = T Z e3”"w“tJ where res = Err/Ts k=—oo rz—oo {d} Find the Discrete—Time Fourier Transform, DO 2 sergeiflk k=—DD in terms of X from (a).

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