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EEET 3028 2021 Digital Communications Assignment Task 1: Signals & Spectra, Filtering, and Analogue Modulation Final Report and Evaluation Your report should consist of: ? Solutions to the problems (including mathematical derivations)

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EEET 3028 2021 Digital Communications

Assignment Task 1: Signals & Spectra, Filtering, and Analogue Modulation

Final Report and Evaluation
Your report should consist of:
? Solutions to the problems (including mathematical derivations).
? A working copy of the MATLAB code.
? All the plots generated in MATLAB, with appropriate title description. (You can add
the titles to the plots in the report document after copying them to your document, if
that is easier).
Evaluation will be based on the components above. Hand-written solutions to the prob-
lems are to be scanned into a pdf le, to which you will attach the plots and the MATLAB code.
Note: Hand-written answers/mathematical derivations will be accepted only if done neatly
and clearly. You may scan them and submit electronically.

Part 1: Signal and Spectrum (30 marks)
1. An lter has the following transfer function
H(f) =
(
1, |f| ≤ f0
e−b(|f|−f0)), |f| > f0
for some b > 0 and f0 > 0.
(a) Plot H(f) for f0 = 10 and b = 3. What is the type of the lter? What is the lter
bandwidth? (4 marks)
(b) Find the impulse response of the lter. (10 marks)
(c) We would like to translate H(f) to construct a bandpass lter at center frequency
fc ? f0. Find the transfer function Hc(f) and impulse response hc(t) of the bandpass
lter. (4 marks)
(d) An implementation of the lter is obtained by approximating h(t) by
ˆh
c(t) = hc(t − td)rect

t − td
2td

.
Explain why the approximation is necessary and discuss the tradeos for choosing
td. (6 marks)
(e) Plot the impulse response and transfer function of the lter designed in question 1d.
Use the following parameters:
? b=0.5
? fc = 100 Hz.
? f0 = 10 Hz.
? Sampling frequency 5fc.
? td = 2
f0
.
(6 marks)
Part 2: Probability and Stochastic Processes (20 marks)
1. Let X be a discrete random variable independent of X with the following probability
mass function pX(x) = 1
3 , x = 0, 1, 2. Let Z follows an exponential distribution with
fZ(z) = 2e−2z, z ≥ 0. Finally, let Y = X + Z.
(a) Find the marginal distribution fY (y). (3 marks)
(b) Find the mean and variance of Y . (3 marks)
(c) Find the conditional probability Pr{Y < 2|X > 0}. (4 marks)
2. Consider a White Gaussian process X(t) with power spectrum density SX(f) = σ2.
Denote y(t) = h(t) ⊗ x(t), where h(t) is dened in question 1b.

(a) Express the autocorrelation function Ry(τ ) = E[y(t)y(t − τ )] in terms of h(τ ).
(b) Determine the spectral density function Sy(f).
(c) Determine the variance of σ2
y = E[y2(t)].
(10 marks)
Part 3: AM Radio broadcast (30 marks)
Consider an AM radio system, where channels are located in the frequency band 1kHz−2kHz.
The message bandwidth is 90Hz, and the channels are separated by 20Hz guard bands. Assume
the transmit power is 5W.
1. How many channels can be tted into the band if AM signal is transmitted? (3 marks)
2. How many channels can be tted into the band if SSB signal is transmitted? (3 marks)
3. Assume that the message is sinusoidal at frequency 40Hz and amplitude 2, i.e.
m(t) = 2 cos(80πt).
What is the transmit signal in time-domain, and its spectrum for
(a) AM transmission with modulation index μ = 0.8,
(b) upper-side band transmission.
(6 marks)
Ignoring noise and pathloss, we assume that the receive signal is equal to the transmit signal.
We consider a super-heterodyne receiver at intermediate frequency (IF) 400Hz. Assume that
AM signal with modulation index μ = 0.8 is transmitted.
4. Assume that the receiver is tune into a channel at center frequency fc = 2.3kHz. Deter-
mine the specication of the RF lter (passband and stopband) to avoid image frequency
at IF. (6 marks)
5. Determine the passband and stopband of the IF lter. (4 marks)
6. Determine a suitable demodulator. Draw the block diagram of the receiver, and specify
the requirement for the blocks (for e.g. the low pass lter). (5 marks)
7. Propose an alternative demodulator, and explain why you chose the demodulator in
question 6.

Part 4: AM simulations (20 marks)
In this task, you will simulate the above AM system, without the intermediate frequency step.
We consider the carrier frequency fc = 100Hz. The modulation index is μ = 0.8. The message
is
m(t) = [sinc2(10(t − ta)) − sinc2(10(t − 1.5ta)]rect

t
2ta

where ta = 0.5 second. Using a sampling time of Ts = 1/(5fc), simulate the system for 2ta
seconds with the following steps:
? Generate and plot the message m(t).
? Generate and plot the modulated signal xc(t).
? Plot the message and the modulated signal.
? Compute and plot the spectrum of the modulated carrier.
? Perform envelope detection method using half-wave rectier and the lter in (1d) for
lowpass ltering. Justify your selection of the lowpass lter parameters f0, b, td.
? Plot the output of the rectier output and its magnitude spectrum.
? Plot the demodulated message. Describe and explain any discrepancy between the re-
ceived message and m(t).
Marking scheme:
? working code and plots: 15 marks;
? interpretation: 5 marks