ECE5884 Wireless Communications – Quiz 9 19 September 2022
1. ((Independent and identical (i.i.d.) variables) Say we have indepen- dent random variables X and Y , and we know their probability density functions (PFDs) fX (t) = 2t e−t2 and fY (t) = 2t e−t2 which are Rayleigh dis- tributions. For a fixed value a > 0, find cumulative distribution functions (CDFs) of Z where (please provide all steps of your derivations)
(a) Z = a(X + Y )2 (b) Z = aX2 + aY 2
(c) Z = max(aX2,aY 2)
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2. (Multi-antenna systems) A wireless system consists of a single-antenna transmitter and two antennas receiver. The multipath channel gain between the transmitter and the receiver antennas are h1 and h2 where their ∣h1∣ and ∣h2∣ follows i.i.d. Rayleigh distributions with the unit power gain, i.e., f∣hi∣(x) = 2x e−x2 . The average SNR of each branch (channel) is γ ̄.
(a) Write the end-to-end SNR at the receiver with maximal ratio combining and selection combining (SC).
(b) Derive closed-form expressions for the SNR outage probabilities for both MRC and SC, when the received SNR falls below a threshold γth. Please provide details of your derivation.
(c) Verify analytical outage probability expressions by using MATLAB sim- ulations. You may plot outage probability vs average SNR γ ̄ where γ ̄ varies from -10 dB to 16 dB.
(d) Find the diversity orders and array gains for both MRC and SC. You may use asymptotic analysis for γ ̄ → ∞.
(e) (Optional) Plot asymptotic expressions on the same plot in (c).
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