ECE-GY 6013. Digital Communications Midterm Exam Solutions, Fall 2022
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ECE-GY 6013. Digital Communications
Midterm Exam Solutions, Fall 2022
This document is the written component of the midterm. There is a also a MATLAB component. Answer all questions. Submit a PDF of your solutions to Gradescope.
1. Passband channel. The power gain of a passband channel, jHp(f)j 2 , is shown in the left of Fig. 1.
(a) Draw the power gain of the complex baseband equivalent channel, jH(f)j 2 , if the carrier frequency is fc = 2.505 GHz. Label all the relevant axes points. Note the carrier frequency is slightly ofset from the peak of the jHp(f)j 2 .
(b) The PSD, Sx(f), of the transmitted complex baseband signal x(t) is shown in the right of Fig. 1. Draw the PSD, Sg(f), of the received complex signal y(t). Label all the relevant axes points.
(c) What is the average power of the received signal y(t)? Leave your answer in A and G0 assuming A is in units of mW/MHz and G0 is dimensionless.
jHp(f)j 2 (linear)
Sx(f) (linear)
f (MHz)
2.4 2.5 2.6
-10
10
Figure 1: Problem 1: Left: Passband power gain (only the positive frequencies are shown). Right: Transmitted PSD.
2. Pulse shaping. Suppose that the complex baseband transmitted signal is
x(t) = s[n]p(t - nT),
n=-1
where s[n] are i.i.d. symbols with average symbol energy Es = Ejs[n]j 2 , and p(t) is the pulse shape
p(t) = max {0, 1 - .
(a) Draw p(t).
(b) What is the transmitted PSD, Sx(f)?
(c) Suppose that 1/T = 20 MHz. What is the minimum positive f such that Sx(f) is 12 dB below its maximum? You may use the MATLAB fzero function to ind roots.
3. Sampling. Consider a communication system that maps complex TX symbols s[n] to received samples r[n] via the following steps:
x(t) =n(Σ) s[n]ptx(t)
y(t) = h chan(t) * x(t)
v(t) = prx(t) * y(t)
r[n] = v(nT)
where
❼ The sample rate is 1/T = 100 MHz
❼ Prx(f) and Ptx(f) are ideal low passilters with cutof atf1 = 40 MHz and unit maximum gain.
❼ Hchan(f) has a frequency response:
Hchan(f) = A [1 - , jfj f2
with f2 = 80 MHz and A = (10)-4 .
(a) Draw G(f) = Prx(f)Ptx(f)Hchan(f). Label the axes and all the key points.
(b) Draw the discrete-time equivalent channel H(Ω) = R(Ω)/S(Ω). Label the axes and all key points.
(c) The TX sends s[n] = exp(2πif0nT) where f0 = 10 MHz. What is r[n]?
4. Signal spaces. Suppose a system transmits a signal:
x(t) = s[0]p0 (t) + s[1]p1 (t),
where s[n], n = 0, 1, are two complex symbols and the pulse shapes are:
p0 (t) = Rect(t/2T), p1 (t) = Rect((t - T)/2T).
(a) Draw p0 (t) and p1 (t).
(b) Find an orthonormal basis u0 (t) and u1 (t) that spans the signal space. (c) Find α0 and α1 such that p0 (t) = α0u0 (t) + α1u1 (t).
2023-10-23