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).