Homework 3 Due

Tufts University Electrical and Computer Engineering

EE108 – Wireless Communications Professor Mai Vu

Homework 3 Due: April 7, 2015 (Q1-Q4), April 16, 2015 (Q5-Q8)

Part A: To be turned in on the due date. 1. Simulate and plot the capacity of a Rayleigh fading channel vs. the average SNR for the following cases: (a) CSI is known at both the transmitter and the receiver, (b) CSI is known at the receiver only, (c) Channel inversion power control is used, (d) Maximum outage capacity, (e) AWGN channel capacity with the same average SNR as the Rayleigh channel. Simulate for average transmit SNR, γ = σP2 , in the range 0 − 30 dB in steps of 5 dB and generate n 10,000 channel realizations for each value of SNR. To generate a Rayleigh fading channel vector of length N , use the following Matlab code: 1 h = √ ∗ (randn(1, N ) + i ∗ randn(1, N )); 2 2. Use the outage probability expressions derived in class to plot the outage probability vs. SNR for Selective Combining, and Maximum Ratio Combining, diversity techniques. Pick a fixed value for the threshold SNR, γ0 . These expressions are also available in the book by Goldsmith. Plot for the number of diversity branches M = 1, 2, 4, 8, and 12, and SNR in the range 0 − 30 dB in steps of 5 dB. Comment on the diversity and array gains as M increases. 3. Use simulation to plot the bit error probability vs. SNR for Selective Combining, Maximum Ratio Combining, and Equal Gain Combining diversity techniques using QPSK modulation. The simulation system is shown in the figure below, assuming a block fading channel model with block length N as a parameter of choice. You should generate enough random input bits bk and the corresponding channel realizations to obtain a smooth error probability plot. Perform the simulation for the number of diversity branches M = 1, 2, 4 and 8, and for SNR in the range 0 − 20dB in steps of 5dB. Compare with the outage plots in Problem 2 and comment.

1



4. Plot the pdf pγΣ (γ) for the selection-combiner SNR in Rayleigh fading with M branch diversity for M = 1, 2, 4, 8, and 10, where iM −1 Mh pγΣ (γ) = e−γ/¯γ . 1 − e−γ/¯γ γ¯ Assume each branch has an average SNR of 10 dB. Your plot should be linear on both axes and should focus on the range of linear γ values of 0 ≤ γ ≤ 60. Discuss how the pdf changes with increasing M and why that leads to lower probability of error. 5. Consider a frequency-selective fading channel with total bandwidth 12 MHz and coherence bandwidth Bc = 4 MHz. Divide the total bandwidth into 3 subchannels of bandwidth Bc , and assume that each subchannel is a Rayleigh flat-fading channel with independent fading on each subchannel. Assume the subchannels have average gains E[|H1 (t)|2 ] = 1, E[|H2 (t)|2 ] = .5, and E[|H3 (t)|2 ] = .125. Assume a total transmit power of 30 mW, and a receiver noise spectral density of .001µW per Hertz. (a) Find the optimal two-dimensional (frequency and time) water-filling power adaptation for this channel and the corresponding Shannon capacity, assuming both transmitter and receiver know the instantaneous value of Hj (t), j = 1, 2, 3. (b) Find the optimal frequency domain water-filling power adaptation for this channel and the corresponding Shannon capacity, assuming only the receiver knows the instantaneous channel and the transmitter knows the average channel gain on each subchannel. (c) Compare the capacity of parts (a) and (b) with that obtained by allocating an equal average power of 10 mW to each subchannel without water-filling in either frequency or time dimension. 6. Assume a Rayleigh fading channel, where the transmitter and receiver have CSI and the distribution of the fading SNR p(γ) is exponential with mean γ¯ = 10dB. Assume a channel bandwidth of 10 MHz. (a) Find the cutoff value γ0 and the corresponding power adaptation that achieves Shannon capacity on this channel. (b) Compute the Shannon capacity of this channel. (c) Compare your answer in part (b) with the channel capacity in AWGN with the same average SNR. (d) Compare your answer in part (b) with the Shannon capacity where only the receiver knows γ[i]. (e) Compare your answer in part (b) with the zero-outage capacity and outage capacity with outage probability .05. (f) Repeat parts b, c, and d (i.e. obtain the Shannon capacity with perfect transmitter and receiver side information, in AWGN for the same average power, and with just receiver side information) for the same fading distribution but with mean γ¯ = −5dB. Describe the circumstances under which a fading channel has higher capacity than an AWGN channel with the same average SNR and explain why this behaivor occurs. 7. Consider a wireless channel with CSI only at the receiver with L i.i.d. Rayleigh faded diversity branches. (a) Compute the ergodic capacity of the fast fading channel (note there is no transmit power control). You can leave the capacity expression in an expectation form. What is the capacity with maximal ratio combining? Comment on the difference. (b) Provide an expression for the outage probability of the slow fading channel based on the ergodic capacity from part a). 2



(c) Suppose that the channel is frequency-selective with L resolvable paths. Consider a suboptimal scheme which transmits one symbol every L symbol times and uses maximal ratio combining at the receiver to detect each symbol. Find the outage and ergodic capacity performance achievable by this scheme if the transmitted symbols are ideally coded and the outputs are maximal-ratio combined. Calculate the loss in performance (with respect to the optimal outage and ergodic capacity performance) in using this scheme for a GSM system with two paths operating at average SNR of 15 dB. In what regime do we not lose much performance by using this scheme? 8. Perform simulation for the system using Alamouti code shown in the figure below, where bk is a random input bit stream. Plot the bit error rate of the Alamouti code for SNR in the range 0 − 25dB in steps of 5dB using QPSK modulation. Repeat with 16-QAM and superpose the error curves on the same plot. Assume a Rayleigh block fading channel. Your simulations should generate enough input bits and channel samples to obtain smooth error curves.

3



Part B: For study purposes only. ¯ the received 1. Consider a flat-fading channel of bandwidth 20 MHz where for a fixed transmit power S, SNR is one of six values: γ1 = 20 dB, γ2 = 15 dB, γ3 = 10 dB, γ4 = 5 dB, and γ5 = 0 dB and γ6 = −5 dB. The probability associated with each state is p1 = p6 = .1, p2 = p4 = .15, p3 = p5 = .25. Assume only the receiver has CSI. (a) Find the Shannon capacity of this channel. (b) Plot the capacity versus outage for 0 ≤ pout < 1 and find the maximum average rate that can be correctly received (maximum Co). ¯ the received SNR is one of four 2. Consider a flat-fading channel where for a fixed transmit power S, values: γ1 = 30 dB, γ2 = 20 dB, γ3 = 10 dB, and γ4 = 0 dB. The probability associated with each state is p1 = .2, p2 = .35, p3 = .3, and p4 = .2. Assume both transmitter and receiver have CSI. (a) Find the optimal power control policy S(i)/S for this channel and its corresponding Shannon capacity per unit Hertz (C/B). (b) Find the channel inversion power control policy for this channel and associated zero-outage capacity per unit bandwidth. (c) Find the truncated channel inversion power control policy for this channel and associated outage capacity per unit bandwidth for 3 different outage probabilities: pout = .1, pout = .01, and pout (and the associated cutoff γ0 ) equal to the value that achieves maximum outage capacity. 3. Consider a cellular system where the power falloff with distance follows the formula Pr (d) = Pt (d0 /d)α , where d0 = 100m and α is a random variable. The distribution for α is p(α = 2) = .4, p(α = 2.5) = .3, p(α = 3) = .2, and p(α = 4) = .1. Assume a receiver at a distance d = 1000 m from the transmitter, with an average transmit power constraint of Pt = 100 mW and a receiver noise power of .1 mW. Assume both transmitter and receiver have CSI. (a) Compute the distribution of the received SNR. (b) Derive the optimal power control policy for this channel and its corresponding Shannon capacity per unit Hertz (C/B). (c) Determine the zero-outage capacity per unit bandwidth of this channel. (d) Determine the maximum outage capacity per unit bandwidth of this channel. 4. Time-Varying Interference: This problem illustrates the capacity gains that can be obtained from interference estimation, and how a malicious jammer can wreak havoc on link performance. Consider the following interference channel.

4



The channel has a combination of AWGN n[k] and interference I[k]. We model I[k] as AWGN. The interferer is on (i.e. the switch is down) with probability .25 and off (i.e. the switch is up) with probability .75. The average transmit power is 10 mW, the noise spectral density is 10−8 W/Hz, the channel bandwidth B is 10 KHz (receiver noise power is NoB), and the interference power (when on) is 9 mW. (a) What is the Shannon capacity of the channel if neither transmitter nor receiver know when the interferer is on? (b) What is the capacity of the channel if both transmitter and receiver know when the interferer is on? (c) Suppose now that the interferer is a malicious jammer with perfect knowledge of x[k] (so the interferer is no longer modeled as AWGN). Assume that neither transmitter nor receiver have knowledge of the jammer behavior. Assume also that the jammer is always on and has an average transmit power of 10 mW. What strategy should the jammer use to minimize the SNR of the received signal? R∞ √ γ / x. Find the average Pb for a 5. Consider a fading distribution p(γ) where 0 p(γ)e−xγ dγ = .01¯ BPSK modulated signal where the receiver has 2-branch diversity with MRC combining, and each branch has an average SNR of 10 dB and experiences independent fading with distribution p(γ). 6. Compare the average probability of bit error for BPSK under MRC and under EGC assuming twobranch diversity with i.i.d. Rayleigh fading on each branch and average branch SNR γ¯ = 10 dB. 7. Compute the average BER using BPSK of a channel with two-branch transmit diversity under the Alamouti scheme, assuming the branch SNR is 10 dB. 8. Find the outage probability of QPSK modulation at Ps = 10−3 for a Rayleigh fading channel with SC diversity for M = 1 (no diversity), M = 2, and M = 3. Assume branch SNRs γ¯1 = 10 dB, γ¯2 = 15 dB, and γ¯3 = 20 dB. Note: For problems 6, 7 and 8, equations for the error probability of different combining techniques for specific modulations (BPSK, QPSK) are available in Goldsmith’s text. In exam, we will provide these equations if needed. For the midterm exam, you are expected to be able to derive the output SNR expression for each studied diversity technique and to derive its mean (average SNR) and distribution and/or outage probability where possible (in simple cases such as Rayleigh fading).

5