ECE5884 Wireless Communications – Assignment 3 10 Oct. 2022
1. ([15marks] Independent and identical (i.i.d.) random variables). Say we have independent random variables X and Y , and we know their probabilitydensityfunctions(PFDs)fX(t)=2te−t2 andfY(t)=2te−t2 which are Rayleigh distributions. For fixed values a > 0, b > 0 and c > 0, find cumulative distribution functions (CDFs) of following Z:
(a) [5marks] Z = aX2 bY2
(b) [5marks] Z = aX2 bY2+c
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(c) [5 marks] Verify above CDF expressions by using MATLAB simulations for a = 0.3; b = 0.5; c = 0.8.
2. ([5marks] Co-channel reuse factor). By using geometric arguments, show that the co-channel reuse factor for cellular deployments based on hexagonal cells is given by DR = √3N, where D is the co-channel reuse dis- tance between cells using the same set of carrier frequencies, R is the radius of the cells (for hexagonal cells, R is the distance from the center to the corner of a cell), and N is the reuse cluster size.
3. ([15 marks] Cellular systems). Consider a regular hexagonal cell deploy- ment, where the mobile stations (MSs) and base stations (BSs) use omnidi- rectional antennas. Suppose that we are interested in the uplink (from MS to BS) channel performance and consider only the first tier of co-channel interferers. Ignore the effects of shadowing and multipath fading, and as- sume that the propagation path loss is described by the simplified path-loss model (refer the class note).
(a) [10 marks] Determine the worst case carrier-to-interference ratio, CI , for cluster sizes N = 3, 4, 7 when the path loss exponent β = 3.5. You may refer Slide#19 of Week#10 lecture note to identify the worst-case locations of MSs.
(b) [5 marks] What are the minimum cluster sizes that are needed if thresh- old levels of the carrier-to-interference ratio at radio receivers are 5 dB and 10.5 dB?
4. ([25marks] Multi-antenna systems). A single-input multiple-output (SIMO) system consists of a single-antenna transmitter and N antennas receiver. The multipath channel between the transmitter and the ith re- ceiver antenna is denoted as hi where all hi, i = 1, ⋯, N are independent and
identically distributed (i.i.d.) complex Gaussian random variables with zero mean and unit variance, i.e., hi ∼ CN (0, 1). The complex-valued additive white Gaussian noise (AWGN) of the ith antenna is denoted as ni which fol- lows ni ∼ CN (0, N0). The average transmit power is P . The channel state information (CSI) is only available at the receiver.
(a) [3 marks] Write the end-to-end SNR at the receiver with maximal ratio combining (MRC) in terms of hi and γ ̄ where the average SNR is γ ̄ = NP .
(b) [7marks] Derive a closed-form expression for the SNR outage proba- bility, when the received SNR falls below a threshold γth. All important steps of the derivation should be provided (You may use Hint#1).
(c) [5 marks] By using asymptotic analysis, i.e., for high average SNR γ ̄ =
NP → ∞, derive the achievable diversity order and the array gain of this
MRC system. Does this system provide the full-diversity order? Justify your answer (You may use Hint#2).
(d) [7marks] Verify analytical outage probability expression in (b) and asymptotic analysis in (c) by using MATLAB simulations. You may plot outage probability vs average SNR γ ̄ where γ ̄ varies from -10 dB to 24 dB when γth = 5 dB for N = 2, N = 3 and N = 4.
(e) [3 marks] How much power do you save at 10−3 outage probability level whenNincreasesfromN=2toN=3whenγth=5dB.? Youmay provide the answer as a ratio between two power values.
Hint#1: We have L number of i.i.d., random variables {X1,⋯,Xi,⋯,XL} where each Xi follows an exponential distribution, i.e., Xi ∼ ae−ax for a > 0. Then, the sum of L i.i.d. exponential random variables Z = ∑Li=1 Xi is a Gamma random variable where its distributions can be given as
PDF: fZ(z) = Γ(L)zL−1e−az (1)
CDF: FZ(z) = 1 γ(L,ax) (2) Γ(L)
where γ(u,v) is the lower incomplete Gamma function, which can be im- plemented in MATHEMATICA as Gamma(u,0,v) and in MATLAB as gammainc(v, u,′ lower′) ∗ gamma(u).
Hint#2: You may use the following series expansion:
limγ[n,x]≈ n. (3)
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