ECE5884 Wireless Communications – Quiz 6 29 August 2022
1. The constellation for M -PAM is Ai = (2i − 1 − M )d for i = 1, 2, ⋯, M . A wire- less communication system uses the 4-PAM constellation signalling, where 4-PAM can be represented as in Figure 1. We have equiprobable signaling.
Figure 1: 4-PAM constellation diagram.
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(a) The constellation mapping is usually done by Gray encoding. Propose a suitable Gray encoding for the above 4-PAM constellation.
(b) What is the minimum distance, dmin, between constellation points?
(c) If the average energy of the 4-PAM constellation is Es, write dmin in
terms of Es.
(d) Draw decision boundaries.
(e) By assuming the additive white Gaussian noise (AWGN) channel, write down a set of equations for the received signal at time t for all possible Ais of the 4-PAM constellation. The AWGN, n(t), follows a circularly symmetric complex Gaussian distribution, i.e., n(t) ∼ CN (0, N0), where N0 is the average noise power. (See Slides#10-11 of the lecture note) . You may write the set of equations as follows:
⎪⋯ + n(t),
for A1. for A2 for A3. for A4
⎪⋯ + n(t),
⎪⋯ + n(t),
⎪⋯ + n(t), ⎩
(f) Derive an expression for the symbol error rate (SER) in terms of Es and N0. Please provide all important steps in the derivation.
(g) (MATLAB) We define the average symbol SNR as γs = Es . Based N0
on the Maximum Likelihood (ML) decoding criteria, verify the SER expression derived in (f) by using a MATLAB simulation program. You may plot SER vs γs where γs varies from -10dB to 16dB. You may generate an SER vs γs plot as in Figure 2 .
Average SER vs average SNR for 4-PAM
-10 -5 0 5 10 15
4-PAM AWGN Simulation
4-PAM AWGN Analytical
Figure 2: SER vs γs for 4-PPAM.
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