Consider the pulses x 1 [n] = u[n] u[n 20] and x 2 [n] = u[n] u[n 10], and their product x[n] = x 1 [n] x 2 [n]. (a) Plot the three pulses. Could you say that x[n] is a down-sampled version of x 1 [n]? what would be the down-sampling rate? Find X 1 (e j ). (b) Find directly the DTFT of x[n] and compare it to X 1 (e j/M ), where M is the down-sampling rate found above. If we down-sample x 1 [n] to get x[n], would the result be affected by aliasing? Use MATLAB to plot the magnitude DTFT of x 1 [n] and x[n].

Question: Consider the pulses x 1 [n] = u[n] u[n 20] and x 2 [n] = u[n] u[n 10], and their product

Consider the pulses x₁[n] = u[n] − u[n − 20] and x₂[n] = u[n] − u[n − 10], and their product x[n] = x₁[n] x₂[n].

(a) Plot the three pulses. Could you say that x[n] is a down-sampled version of x¹[n]? what would be the down-sampling rate? Find X₁(e^jω).

(b) Find directly the DTFT of x[n] and compare it to X₁(e^jω/M), where M is the down-sampling rate found above. If we down-sample x₁[n] to get x[n], would the result be affected by aliasing? Use MATLAB to plot the magnitude DTFT of x₁[n] and x[n].

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