Part A

Consider a point-to-point communication link where a base station (BS) communicates with an user directly. In this point-to-point link, it is assumed that the signal transmission is subject to path loss and log-normal shadowing. Some parameters of such a point-to-point link are given as follows:
The carrier frequency is f_c = 1.5 GHz, the reference distance is d₀ = 1 m, and the unitless constant K is determined from the free space path gain formula at this d₀ given by K dB = 20 log₁₀ ( λ/π4d₀), where  λ is the wavelength of carrier frequency. The distance between the BS and the user is denoted as d and the path loss exponent between them is denoted as η  . The variance of log-normal shadowing is σ_ψ² = 15 dB.
Based on the parameters clarified above, please complete the following problems:

a) Describe “What are path loss and shadowing in wireless propagation?” In the description, please describe the reasons of these two phenomenons and the impact of these phenomenons on wireless transmission.

b) If the transmit power is denoted as PT and the minimum power requirement is denoted as Pth, what is the outage probability of the transmission over the combined path loss and log-normal shadowing model?

c) find out the outage probabilities for the following systems: i) d = 1 km,   η = 3, PT = 100 dB, and Pth = -40 dB; ii) d = 1:5 km,  η= 2:5, PT = 115 dB, and Pth = -10 dB; iii) d = 800 m, η= 2, PT = 95 dB, and Pth = -5 dB. If 1% is a typical outage probability target in wireless system designs, which system needs to be re-designed? How to re-design this system?

Part B

Consider a point-to-point communication link where a base station (BS) communicates with an user directly. In this point-to-point link, it is assumed that the signal transmission is subject to path loss and Rayleigh fading.

Please complete the following problems:

a) Describe “What is Rayleigh fading?”

b) If the reference distance is normalized as d0 = 1 and the variance of the noise at the user is 1, use Matlab to plot the simulated outage probabilities versus transmit power PT in a log-log scale for the following systems: i) d = 0.8, η = 3.5, and Pth = 0 dB; ii) d = 1.2, η=2.5, and Pth = 5 dB. Please adopt the range of the transmit power from -10 dB to 20 dB and the range of the outage probability from 10⁰ to 10³ in the plotted figure. Also, please include the analytical result for the outage probability in the figure and compare your simulations with the analytical result.
NB: In Rayleigh fading, the analytical outage probability is given by Pout = 1-e^Pth/γ , where γ is the average received SNR.

Part C

Consider a dual-hop communication system. In such a system, the direct transmission between the BS and the user is not available; therefore, a relay station with CSI-based gain amplify-and-forward (AF) relaying protocol is deployed between them to help their transmission. For this system, the following assumptions are made:

i)The distance between the BS and the user is normalized to one, i.e., d_SD = 1. The relay is placed between them such that d_SR + d_RD = 1, where d_SR is distance between the BS and the relay, and d_RD is the distance between the relay and the user.

ii) The total transmit power P_T are dynamically allocated to the BS with P_S = nP_T and to the relay with PR = (1-n) PT , where 0 < n < 1. The variances of the noises at the relay and at the receiver are 1.

iii) The path loss exponent between the BS and the relay is ηSR = 2 and the path loss exponent between the relay and the user is ηRD = 3.
Based on the parameters clarified above, please complete the following problems:

a) If the relay station is placed at halfway between the BS and the user, i.e., d_SR = d_RD = 0.5, and the total transmit power is equally allocated between the BS and the user, i.e., PS = PR = 0.5P_T , plot the simulated outage probabilities versus transmit power P_T in a log-log scale. In this plot, the outage probability is defined the probability that the end-to-end SNR γeq = (γSR γRD) /(γSR+ γRD+1) drop below an SNR threshold γth. Here, γth = 0 dB is adopted. Also, please include the analytical result for the outage probability in the figure and compare your simulations with the analytical result. Confirm whether the analytical result is accurate and describe the reason.
Please adopt the range of the transmit power from -5 dB to 25 dB and the range of the outage probability from 10⁰ to 10^-3 in the plotted figure. NB: In Rayleigh fading, the analytical outage probability for dual-hop transmission is given by:

P_out   1 - 2γ_th/√γ‾SRγ‾RD e -γ_th ( 1/γ‾SR + 1/γ‾RD ) K1 ( 2γ_th √γ‾SRγ‾RD )
where γ‾SR is the average received SNR between the BS and the relay and γ‾RD is the average received SNR between the relay and the user.

b) If the relay station is placed at halfway between the BS and the user, i.e., dSR = dRD = 0.5. Assume that the total transmit power is P_T=20 dB. Plot the analytical and simulated outage probabilities versus power allocation factor n, n ε (0, 1), in a log-log scale for three SNR thresholds: i) γ_th = -5 dB, ii) γ_th = 0 dB, and iii) γ_th = 5 dB. Based on the figure, please find the optimal power allocation factor for each SNR threshold. In this plot, please adopt the end-to-end SNR as γ_eq = (γSRγRD/γSR+γRD).

c) If the total transmit power is allocated between the BS and the user as P_S = 0.6P_T and PR = 0.4PT . Assume that the SNR threshold is γ_th = 0 dB. Plot the analytical and simulated outage probabilities versus d_SR, d_SR ε (0,1), in a log-log scale for three total transmit power: i) P_T = 20 dB, ii) P_T = 25 dB, and iii) P_T = 30 dB. Based on the figure, please find the optimal BS-relay distance for each total transmit power. In this plot, please adopt the end-to-end SNR as γ_eq = (γSRγRD/γSR+γRD).

NB: The end-to-end SNR adopted in b) and c) is different from that adopted in a).

Part D

Consider a cooperative communication system where the direct transmission between the BS and the user is available. In such a system, a relay station with CSI-based gain amplify-and-forward (AF) relaying protocol is deployed between them to improve their transmission. The destination adopts cooperative selection diversity (CSD) to combine the received signals such that the signal with the higher quality between the direct link and the relay link is selected. For this system, the following assumptions are made:

i) The distance between the BS and the user is normalized to one, i.e., d_SD = 1. The relay is placed between them such that d_SR + d_RD = 1, where d_SR is distance between the BS and the relay, and d_RD is the distance between the relay and the user.

ii) The total transmit power P_T are dynamically allocated to the BS with P_S = nP_T and to the relay with P_R = (1-n) PT , where 0 < n < 1. The variances of the noises at the relay and at the receiver are 1.

iii) The path loss exponent between the BS and the relay is ηSR = 2, the path loss exponent between the relay and the user is ηRD = 3, and the path loss exponent between the BS and the user is  ηSD = 4. Based on the parameters clarified above, please complete the following problems:

The relay station is placed between the BS and the user with d_SR = 0.55 and d_RD = 0.45. The total transmit power is allocated between the BS and the user such that P_S = 0.65P_T and P_R = 0.35P_T . Plot the three simulated and analytical outage probabilities versus transmit power P_T in a log-log scale: i) Outage probability for the direct path only; ii) Outage probability for the relay path only; and iii) Outage probability for CSD. In this plot, please adopt the end-to-end SNR for the relay path as  γ_eq = (γSRγRD/γSR+γRD). and adopt γth = 0 dB. Also, please include the analytical result for the outage probability in the figure.
Compare these three outage probabilities and comment on the diversity gains of three outage probabilities based on the plots. Please adopt the range of the transmit power from 0 dB to 20 dB and the range of the outage probability from 10⁰ to 10^-4 in the plotted figure.