Implementing discrete-time bandpass QPSK Consider the QPSK modulator and an isolated rectangular pulse with amplitude A = 1 and duration Ts = 1 second

h(t) = A /√ Ts , 0 ≤ t ≤ Ts,

0, otherwise.

Suppose we have N = 128 as the upsampling factor.

A. Derive the expression for the discrete-time version of the pulse shape, denoted as h[n].

B. Derive the discrete-time Fourier Transform of h[n] and plot its magnitude in MATLAB as well. Indicate the peak and null values and their locations. Hint: use the formula X(Ω) = P∞ n=−∞ x[n]e −iΩn.

C. Suppose the output of the LUT0 and LUT1 at k = 0 are aI [0] = − √ 2 2 and aQ[0] = − √ 2 2 , respectively. If Ω0 = 2π×8 /128 = π/ 8 , derive the expression for sI [n] and sQ[n] from n = 0 up to n = 127.

D. Repeat B for s[n] = sI [n] + sQ[n].