fin - ECE 562 Fall 2011 December 1, 2011 Final (Max score =...

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Unformatted text preview: ECE 562 Fall 2011 December 1, 2011 Final (Max score = 130 pts) Assigned: December 1, 2011 Due: December 9, 2011 at 1:00 PM in room 106 CSL Note: This is an open-book, take-home exam. You can use your class notes, the text book, any of the references that I’ve put on reserve in the library, and the HW sets and solutions. The contents of the exam are confidential until 1 PM on December 9, 2011. Until that time, you must not show this examination to, or discuss its contents with, any person other than Prof. Veeravalli. This also applies to your solutions to the exam of course. You may use Matlab or similar software packages to do your numerical computations. 1. (15 pts) Signal Space and Performance in AWGN. Consider the signal set shown below: s1(t) √ s2(t) √ Es 0 0.5 1 1.5 2 0 0.5 t s3(t) √ 1 1.5 2 t 1 1.5 2 t s4(t) √ Es 0 0.5 Es 1 1.5 2 t Es 0 0.5 (a) Find an orthonormal basis for this signal set and express the signals in terms of this basis. (b) Find the measure of goodness ζ for the signal constellation. (c) Suppose this signal set is used for communication over an AWGN channel with noise PSD N0 . Assuming equally likely symbols, find the IUB and NNA approximations for the average probability of symbol error Pe in terms of γs = Es /N0 . (d) Now find an exact expression for Pe in terms of γs . c V.V. Veeravalli, 2011 1 2. (25 pts) V.29 Modem. The V.29 standard for 9.6 Kbit/s modems uses the following 16-ary signal constellation: 5 3 1 −5 −3 −1 1 3 5 −1 −3 −5 Assume equally likely symbols. (a) Carefully draw the MPE decision regions for this constellation (b) Find the IUB for Pe as a function of γs . (c) Find the NNA for Pe as a function of γs . (d) Find the NNA to Pe for a (square) 16-QAM constellation as a function of γs . (e) Comparing (c) and (d), which constellation is more power efficient at high SNR? (f) Determine whether you can label the signal points using three bits so that nearest neighbors differ by at most one bit (Gray coding). If so, find such a labeling. If not, state why not and find a labeling that minimizes the maximum number of bit transitions between neighbors. (g) For the labelings found in part (a), compute the nearest neighbor approximation for the average bit error probability Pb as a function of γb . c V.V. Veeravalli, 2011 2 3. (15 pts) QPSK with Errors and Erasures Decoding. Consider signaling using the QPSK constellation shown in the figure below. Instead of making a decision in favor of one of the symbols for every value of the demodulator output R, suppose the detector has the option of declaring an erasure for certain values of R. The decision regions for one such detector are shown, with the shaded region corresponding to erasure. (Note that the shaded regions extend out to infinity in all four directions.) 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 1111111111111111111111111111111 0000000000000000000000000000000 a d Assume equally likely symbols. Let α = a/d. Note that 0 ≤ α ≤ 1. (a) Find exact expressions for the average probability of error (Pe ) and the average probability of erasure (Ps ) as a function of γs and α. (b) Find good high SNR approximations for Pe and Ps . (c) Use the approximations of part (b) to find the value of α such that Ps = 2Pe , at a symbol SNR of 7 dB. You may use Matlab to solve for the value of α numerically. Remark: The motivation for setting Ps = 2Pe in part (c) is that a typical error correcting code can correct twice as many erasures as errors. c V.V. Veeravalli, 2011 3 4. (15 pts) Demodulation with phase uncertainty. Consider the 4-ary constellation shown below: Q s2 d s1 0 s4 s3 2d I 3d Suppose this signal set is used on an AWGN channel with phase offset φ that is unknown to the receiver. After demodulation we obtain the sufficient statistic R = sm ejφ + W when symbol m is sent, m = 1, 2, 3, 4, where W ∼ CN (0, N0 ). (a) Assuming that φ ∈ [0, 2π ], find mJML (r). ˆ (b) Carefully draw the decision regions for the JML detector. (c) Find a closed-form expression for maxm Pe,m for the JML detector. 5. (15 pts) Equalization. Consider the problem of BPSK communication on an ISI channel with: √ √ √ Es = 2, and h0 = 0.5, h1 = 0.3, h2 = − 0.2 Suppose the received sequence (z1 , z2 , . . . , z6 ) is given by (2,0.5,-1,0,-2,3). (a) Find the MLSE estimate of the transmitted symbols. (b) Find the estimate of the transmitted symbols produced by the matched filter equalizer. (c) Find the ZF equalizer (e.g., using Matlab), and find the estimate of the transmitted symbols produced by the ZF equalizer. 6. (20 pts) Selection diversity for fading channels. Consider BPSK signaling on an L-th order diversity channel, where the received signal at the output of channel is given by r (t) = ±α ejφ Eb g (t) + w (t) = 1, . . . , L. where {g (t)} are unit energy pulse shaping functions, and the processes {w (t)} are independent WGN processes with PSD N0 . We know from class that maximal-ratio combining (MRC) across the L channels is optimum. However, it may not always be practical to combine the signals in this optimal fashion. In some cases (e.g., the reverse link in the wireless cellular system), the receiver may base its decision only on the strongest of the L channels. This is c V.V. Veeravalli, 2011 4 known as selection diversity. Specifically, the receiver first determines the channel with the largest amplitude, i.e., λ = arg max α ∈{1,...,L} and then uses the statistic RSD = rλ (t), gλ (t)ejφλ to make a decision. A decision in favor of +1 (symbol 1) is made if Re(RSD ) > 0, and −1 (symbol 2) otherwise. (a) Assuming that the fading across the L channels is i.i.d. Rayleigh, show that distribution function of αλ is given by: Fαλ (x) = P{αλ < x} = 1 − e−x 2 L 1 { x≥ 0 } . 1 (b) For fixed {α }L=1 , show that the bit error probability for the selection diversity receiver is given by: Pb = Q(αλ 2γ b ) where γ b = Eb /N0 . (c) Use integration by parts to show that the average bit error probability with fading for the selection diversity receiver is given by √ ∞ γ 2 Fαλ (x) √ b e−γ b x dx Pb = π 0 (d) Show that the expression for Pb of part (c) can be written in closed-form as: 1 Pb = 2 L L k (−1)k k=0 γb , k + γb Hint: You may want to use the binomial expansion for Fαλ (x) as given in part (a), and the fact that a zero-mean Gaussian pdf integrates to 1/2 over [0, ∞). 7. (25 pts) Bit-by-bit noncoherent demodulation of orthogonal signals. Consider M -ary orthogonal signaling on a slow, flat fading channel. Assume that M = 2ν for some positive integer ν . The received signal corresponding to one symbol interval is given by √ r(t) = αejφ E gm (t) + w(t) when symbol m is sent on the channel. The functions {gm (t)} are orthonormal. From class we know that the statistics Rk = r(t), gk (t) , k = 1, . . . , M , are sufficient for optimum detection. The MPE decision rule for noncoherent detection is given by mMPE = arg max |Rk | = arg max |Rk |2 = arg max Yk , ˆ k c V.V. Veeravalli, 2011 k k (1) 5 where Yk = 1 |Rk |2 . N0 Each M -ary symbol corresponds to a sequence of ν bits. Suppose these bits come from the output of an error control encoder. Then the orthogonal demodulator will be followed by a decoder. The MPE decision rule of (1) can be used to produce hard decisions for each bit in the ν -bit sequence. However, it is desirable to produce soft outputs for each bit to enable soft decision decoding. (a) Let Y = [Y1 , . . . , YM ] . Assuming that the fading is Rayleigh, show that the likelihood function for symbol m, based on y , is given by: ym e− γ +1 pm (y ) = γ+1 1 e−yk 1 {y≥0} k =m where 1 {y≥0} equals 1 if all the components of y are nonnegative, and 0 otherwise, 1 and γ = E /N0 . (b) Based on Y , we would like to form a soft decision statistic for the -th bit in the ν -bit sequence that represents the M -ary symbol, = 1, 2, . . . , ν . Assume equal () priors on the symbols. Let pi (y ) denote the conditional pdf of Y , given that the -th bit equals i, i = 0, 1. The log-likelihood ratio for the -th bit is given by: () () L (y ) = ln p1 (y ) () . p0 (y ) Show that m∈S L (y ) = ln exp ym γ 1+γ ¯ m∈S () exp ym γ 1+γ ¯ where S = {m : -th bit of symbol m is 1}, and S is its complement. (c) The metric L( ) (y ) is the desired soft decision statistic for the -th bit. Argue that L( ) (y ) can be approximated as L( ) (y ) ≈ γ Λ( ) (y ) 1+γ where Λ( ) (y ) = max ym − max ym . m∈S ¯ m∈S (d) The approximate metric Λ( ) (y ), known as the dual-max metric, is independent of γ , and is used for soft-decision decoding in IS-95 based CDMA systems. Now, suppose instead that we use Λ( ) (y ) for making a hard decision on the -th bit as follows: () ˆ ,DM = 1 if Λ (y ) > 0 b 0 otherwise Show that ˆ ,DM is the same as the hard decision for the -th bit based on the b MPE decision rule of (1). c V.V. Veeravalli, 2011 6 ...
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