Consider the discrete-time signal x[n] = cos (2n/7). (a) The discrete-time signal can be compressed by getting

Consider the discrete-time signal x[n] = cos (2πn/7).

(a) The discrete-time signal can be compressed by getting rid of some of its samples (down-sampling). Consider the down-sampling by 2. Write a script to obtain and plot z[n] = x[2n]. Plot also x[n] and compare it with z[n], what happened? Explain.

(b)  The expansion for discrete-time signals requires interpolation. However, a first step of this process is the so-called up-sampling. Up-sampling by 2, consists in defining a new signal y[n] such that y[n] = x[n/2] for n even, and y[n] = 0 otherwise. Write a script to perform up-sampling on x[n]. Plot the resulting signal y[n] and explain its relation with x[n].

(c) If x[n] resulted from sampling a continuous-time signal x(t) = cos (2πt) using a sampling period T_s and with no frequency aliasing, determine T_s. How would you sample the analog signal x(t) to get the down-sampled signals z[n]? That is, choose values for the sampling period T_s to get z[n] directly from x(t). Can you choose T_s to get y[n] from x(t) directly? Explain.