Georgia Institute of Technology School of Electrical and Computer Engineering ECE6604 Personal & Mobile Communications Final Exam Spring 2010 Tuesday May 6, 11:30am - 2:20pm • Attempt all questions. • All questions are of equal value. • Open book, open notes, exam. 1a) 5 marks: The LCR at the normalized threshold ? for a 2-D isotropic scattering channel can be expressed as v 2 LR = 2? fm ? e?? , where ?= ? and Rrms = R ?p = R Rrms E[? 2 ] is the rms envelope level. i) Find the normalized threshold level ? o at which the LCR reaches its maximum value. i) Explain why the LCR at ? decreases as ? deviates from ? o . 1b) 5 marks: Consider a cellular system with a carrier frequency of 2 GHz. Suppose that the user is in a vehicle travelling at 60 km/h. Assuming that the channel is characterized by 2D isotropic scattering, ? nd i) the LCR at the normalized level ? = ? 3 dB. ii) the AFD at the normalized level ? = ? 3 dB. 2) The power delay pro? le for a WSSUS channel is given by ? gg (? ) = 0. 5[1 + cos(2?? /T )] , 0, 0 ? ? ? T /2 otherwise a) 3 marks: Find the channel frequency correlation function. ) 4 marks: Calculate the mean delay and rms delay spread. c) 3 marks: If T = 0. 1 ms, determine whether the channel exhibits frequencyselective fading to the GSM system. 3) Cellular CDMA systems use soft hando? , where the transmissions to/from multiple base stations are combined to give a macro-diversity. Here we consider the e? ects of path loss and shadowing and ignore multipathfading. Suppose that the received signal power corresponding to the link with the ith base-station, ? pi , has the probability density function p? pi (x) = v dBm) (x ? ?? pi (dBm) ) 1 exp ? 2 2?? 2??? 2 . where ?? pi (dBm) = E[? pi (dBm) ] The ? pi are assumed to be statistically independent. a) 5 marks: The reverse link uses selection combining such that the best basestation is always selected. In this case, ? s p (dBm) An outage occurs if ? s p = max ? p1 (dBm) ? ? th (dBm) , . . . , ? pL (dBm) (dBm) . What is the probability of outage? b) 5 marks: The forward link uses coherent combining such that ? mr(dBm) = ? p1 p (dBm) + . . . + ? pL (dBm) Again, an outage occurs if ? mr(dBm) ? ?th (dBm) .
What is the probability of p outage if ?? p1 (dBm) = ?? p2 (dBm) = · · · = ?? pL (dBm) ? 4) Consider the reception of a signal in the presence of a single co-channel interferer and neglect the e? ect of AWGN. The received signal power, C , and interference power, I , due to Rayleigh fading have the exponential distributions 1 ? x/C ? ?e C 1 ? pI (y ) = ? e? x/I I pC (x) = ? ? where C and I are the average received signal power and interference power, respectively. a) 5 marks: Assuming that C and I are independent random variables, ? d the probability density function for the carrier-to-interference ratio ? = C . I Hint: If X and Y are independent random variables, then the probability density function of U = X/Y is pU (u) = pXY (v, v/u)|v/u2 |dv . b) 5 marks: Now suppose that the system uses 2-branch selection diversity. The branches are independent and balanced (i. e. , the distribution pU (u) is the same for each branch. What is the probability density function of ? at the output of the selective combiner? 5) Suppose that a system uses selection diversity.
The branches experience independent Rayleigh fading. However, the average received bit energy-to-noise ratio on each diversity branch is di? erent, such that ?i = 2? i ? o ? i = 1, . . . , L a) 5 marks: Find the probability density function of the bit energy-to-noise ratio at s the output of the selective combiner, denoted by ? b . b) 5 marks: If DPSK modulation is used, write down an expression for the probability of bit error. Obtain a closed-form expression if possible; otherwise leave your expression in integral form.
