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# EE 359: Wireless Communications - Fall 2005 SECOND EXAM This exam is open book and notes, and calculators and the textbook are needed. Graded exams

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EE 359: Wireless Communications - Fall 2005 SECOND EXAM This exam is open book and notes, and calculators and the textbook are needed. Graded exams and solutions will be available within the next week. Good luck! Problem 1 (25 points): Short Answer. (a) Name one advantage and one disadvantage of having a DSSS system whose spreading codes have an autocorrelation approximately equal to a delta function? (b) Consider a MIMO system with 10 transmit antennas and 6 receive antennas with matrix H describing the antenna gains from each transmit to each receive antenna. If the SVD of H has 4 nonzero singular values, then how many singular values are zero and what is the rank of H ? (c) Why are precoding and frequency equalization ineﬀective techniques for treating subcarrier fading in OFDM systems? (d) Consider an adaptive system with instantaneous SNR at the transmitter ˆ γ that is not necessarily equal to the true SNR γ . Assume ˆ γ has the same distribution as γ ,so p γ )= p ( γ ). For the variable-rate variable-power MQAM scheme described in Section 9.3 of the text, suppose that the transmit power and rate is adapted relative to ˆ γ instead of γ . Will the average transmitted data rate and power be larger, smaller, or the same as under perfect channel estimates (ˆ γ = γ ), and why? Problem 2 (25 points): MIMO-OFDM Systems MIMO techniques can be applied to systems with ISI as well as ﬂat fading. In this problem we illustrate how to combine MIMO and OFDM. Consider a 2x2 MIMO system, where the channel from transmit antenna i to receive antenna j , i, j =1 , 2, has discrete-time FIR h ij [ n ]= α ij δ [ n ]+ β ij δ [ n - 1], assuming a sampling time equal the symbol time T s . Assume a cyclic preﬁx of length equal to the delay spread is appended on the symbol sequence transmitted over each antenna. Also assume that received symbols in a given block that are aﬀected by ISI in a previous block are discarded. (a) What is the delay spread associated with the MIMO channel, and how many OFDM subchannels are needed to insure a ﬂat-fading MIMO system? (b) Let x 1 [ i ]and x 2 [ i ] denote, respectively, the symbol transmitted on the 1st and 2nd transmit antenna at time i . Similarly, let y 1 [ i ]and y 2 [ i ] denote, respectively, the symbol received on the 1st and 2nd receive antenna at time i , and let n 1 [ i ]and n 2 [ i ] denote, respectively, the noise sample on the 1st and 2nd receive antenna at time i . Assuming 5 OFDM subchannels, write the input-output relationship of your MIMO-OFDM system in matrix form y = Hx + n ,where y [ i ]=[ y 1 [ i ] y 1 [ i - 1] ... y 1 [ i - 4] y 2 [ i ] ... y 2 [ i - 4]] T and x [ i ]=[ x 1 [ i ] x 1 [ i - 1] ... x 1 [ i - 4] x 2 [ i ] ... x 2 [ i - 4]] T ,and n [ i ]=[ n 1 [ i ] n 1 [ i - 1] ... n 1 [ i - 4] n 2 [ i ] ... n 2 [ i - 4]] T .No t e that this is a generalization of the matrix representation of OFDM given in (12.24). (c) Suppose the matrix H obtained in part (d) has nonzero singular values σ 1 =1.5, σ 2 =1.2, and σ 3 =.8. Assuming ρ =10dBand B = 1 Hz, ﬁnd the capacity of the MIMO-OFDM system under optimal power and rate adaptation and under beamforming.
Problem 3 (25 points): Adaptive Loading in OFDM. Consider an OFDM system with 4 subchannels that experience AWGN but no fading. The total transmit power for the system is P = 100 mW and the subchannel bandwidth is 1 MHz. With 100 mW transmitted on each subchannel, the received SNR on each subchannel is γ 1 = 5 dB, γ 2 =15dB , γ 3 = 7 dB, and γ 4 = 10 dB. Assume MQAM modulation satisﬁes the BER approximation BER . 2 e - 1 . 5 γ/ ( M - 1) for any M . (a) Find the optimal power and rate per subchannel for this system to maximize data rate, and ﬁnd this maximum data rate, assuming a target BER of 10 - 3 . Assume MQAM constellations with no restriction on constellation size. (b) Using MQAM constellations restricted to either no transmission or M =2 k ,k =1 , 2 , 3 , ... , ﬁnd the maximum-rate constellation per subchannel and corresponding total data rate under a target BER of 10 - 3 assuming a constant power of 25 mW in each subchannel. (c) Suppose each subchannel is allocated a constant power of 25 mW and the constellations for each subchannel obtained in part (b) are used. Suppose further that there is mismatch between the estimated channel and actual channel, so that with equal power allocation the true channel has γ 1 = 6 dB, γ 2 =12dB , γ 3 = 8 dB, and γ 4 = 9 dB. Find the probability of bit error given the mismatch. Is it larger or smaller than the target, and why? Problem 4 (25 points): Spread Spectrum and RAKE Receivers. Consider a spread spectrum system with spreading code autocorrelation ρ c ( τ )= 1 | τ |≤ T c / 3 . 3 T c / 3 < | τ |≤ 2 T c / 3 0e l s e Suppose a DSSS signal with this spreading code is transmited over a channel with impulse response h ( t )= αδ ( t )+ βδ ( t - τ ). (a) For a single-branch spread spectrum receiver synchronized to the LOS signal component with a timing oﬀset Δ t such that | Δ t |≤ T c / 6, ﬁnd all values of τ such that the received signal experiences no ISI for all possible values of Δ t . (b) Assume τ>T c . Find the outage probability for a 2 branch RAKE receiver under selection combining assuming DPSK modulation with a target BER of 10 - 4 . Assume the ﬁrst branch is perfectly synchronized to the LOS component and the second branch is synchronized to the multipath component with a timing oﬀset of T c / 2. Assume independent Rayleigh fading on each branch, and a branch SNR/bit after despreading assuming perfect synchronization of 20 dB for the LOS component and 15 dB for the multipath component. (c) Again assume τ>T c . Find the average probability of error for BPSK modulation under MRC for a two-branch RAKE receiver, where the ﬁrst branch is perfectly synchronized to the LOS component and the second branch is perfectly synchronized to the multipath component. Assume i.i.d. Rayleigh fading and an average SNR of 12 dB on each branch.

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