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winter 07 final

# winter 07 final - 1 Pre-warping To partially combat the...

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Unformatted text preview: 1. Pre-warping To partially combat the effect of ,u-law quantization on large QAM signal sets, in V.34 a non-linear encoder is used that warps the constellation. An approximate relation between the inputs x and outputs x’ is 0.3125 ||x “2 6E avg a) If the minimum distance between signal points is 2a, compute the average energy for 16-QAM without pre—warping. b) Compute (I) for the inner, edge and comer points and state their new coordinates after warping. ' c) Compute the new average energy after warping. (1) By approximately what factor has the energy-normalized minimum distance between the comer and edge points been improved? Why would this quantity dominate P(e) with ,u-law quantization? x'= @x where (I) =1+ 2. Noise in a PLL a) Determine the effect on the error probability for coherent BPSK in an AWGN channel of a persistent phase error of (9. b) For a PLL, the phase error is actually approximately a Gaussian random variable. Write, but do not evaluate, an expression for error probability in terms of the probability density function for the phase noise, p(€). c) Why are large QAM constellations more sensitive to phase errors than BPSK? 3. Finite-Length DFE Consider the following channel: To this is added AWGN with variance No. A DFE with two forward ﬁlter coefﬁcients co and 0-1 will be used, along with a feedback ﬁlter with one coefﬁcient C]. a) Determine the minimum mse solution for the coefﬁcient set. You may assume that No<<1 to simplify your calculations b) Determine the mse and compare the result to that which would be obtained with a ﬂat channel and account for the difference. 4. MLSE For a particular channel, the MMSE solution for a DFE has the result that the feedback ﬁlter has two coefﬁcients where both values are 0.5. Assume binary PAM transmission with nominal signal values +1 and —1. a) Create a state transition table for MLSE, consisting of the input, the current state, the next state, and the output corresponding to the transition between states b) Compute the squared minimum distance between sequences in the trellis, assuming steady state conditions. c) What is the comparable squared minimum distance for steady state that would be obtained with a DFE? Why is this problematic? 5. Single carrier vs. DMT transmission A ve slowl changin; channel is robed with the followin; f-ﬂﬂ [WE a) Assuming brickwall transmit ﬁltering such that the transmit power is unity, sketch the folded spectrum for fc=6 and NT =10 Hz. b) Assuming an inﬁnite length MMSE DFE, what is the maximum achievable bit rate with QAM for P(e)=10'6, assuming No=1 and our choices are 4, 16, 32, and 64 QAM? c) Now assuming that DMT is used, with the same QAM choices for the individual tones. What is the maximum bit rate under the same transmit power constraint, and how does it compare to the DFE? d) What factors enter into a decision between using DMT and single carrier transmission? ...
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