EE536 Spring 013 PhD Diagnosis Exam ID: UTA EE536 PhD Diagnosis Exam (Spring 01) Communications Instructions: Verify that your exam contains 11 pages (including the cover sheet). Some space is provided for you to show your work. If more space is needed, show your work on the back of the exam sheet. The point values listed on this exam serve only as a guideline. The Dept reserves the right to make modifications to the weighting of the problems. Calculator is okay. I Choose to work on Problems and (Choose only from the 3 problems). Problem Points Scores 1 Bayes Detection 50 Signal Detection 50 3 MLSE Equalization 50 Total Score (choose problems) 100
EE536 Spring 01 PhD Diagnosis Exam ID 1 Bayes Detection (50 Points) In wireless cellular communication, Rayleigh and Rician fading channels are two most popular channels, which can be treated as a binary hypothesis (Rayleigh under H 0 and Rician under H 1 ). If the magnitude of received signal envelop is R (u), then the pdf of R(u) is Rayleigh under H 0 and Rician under H 1, i.e., ( r r f R( u) r H 0 ) = exp( ) U( r) σ σ ( r r + E r E f R( u) r H1) = exp( ) I 0 ( ) U( r) σ σ σ where U (r) is the unit step function and I0( ) is the zero-order modified Bessel function of the first kind, and it s monotonically increasing for positive argument. Assume Rician fading happens (i.e., H 1 ) with probability 0 < p < 1, determine what s kind of fading channel you are communicating in based on received signal sample envelop r. (a) (5 points) The Bayes decision rule can be written as a simple threshold detection H determine T. r > < H 1 0 T
EE536 Spring 01 PhD Diagnosis Exam ID 3 (b) (5 points) What is the probability of error for the above rule when H 0 is true, i.e., pε H )? ( 0
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EE536 Spring 01 PhD Diagnosis Exam ID 6 Signal Detection (50 points) Let g(t) denote a rectangular pulse; i.e., g(t)=1 for t (0,1) and zero elsewhere. In order to transmit four equiprobable symbols, the following four signal waveforms are used in a 4-ary transmission system: s 1 (t)= 0, s (t)= g (t), s 3 (t)= g(t)cos(πt), s 4 (t)= [1+ cos(πt)] g (t). The signals are transmitted through an additive white Gaussian noise (AWGN) channel. Thus, the received signal will have the form r (t)= s(t)+ n(t), where s(t) is any of the four aforementioned symbol waveforms and n(t) is the Gaussian noise with power spectral density N o/ = 1. i) Find the constellation that corresponds to the aforementioned signaling scheme. ii) Determine the optimal decision rule that minimizes the probability of reception error. Be specific and simplify you answer. The derived decision rule should depend on the received signal r(t).
EE536 Spring 01 PhD Diagnosis Exam ID 7 iii) Find the symbol error probability (when using the optimal decision rule) in terms of the Gaussian tail function Q( x)= (1/ π) e u / du. x
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EE536 Spring 01 PhD Diagnosis Exam ID 9 3 MLSE Equalization (50 Points) You are working as a system engineer to design a channel equalizer for a fixed wireless LAN system with BPSK modulation (equal probability of + 1 or 1). The channel has i L +1 paths, which can be modeled as an FIR filter, ( z ) = g z i. Besides the L H i= 0 intersymbol interference (ISI) because of the channel, the received signal is also corrupted by the AWGN noise n (k). n (k) has zero mean and variance σ. You are asked to determine the transmitted signal s (k) based on the received signal r (k). Assume that the unique words are long enough for estimating the channel coefficients. You are asked to design a channel equalizer using MLSE. (a) (0 Points) Derive the MLSE Viterbi algorithm (for channel equalization) mathematically. gi
EE536 Spring 01 PhD Diagnosis Exam ID 10 1 (b) (15 Points) If the channel has 3 taps, H ( z) = 1 0.8z 0.5z, draw the state diagram and Trellis diagram (for 5 symbol time). (Please start from state (-1,-1) and go back to state (-1,-1).
EE536 Spring 01 PhD Diagnosis Exam ID 11 (c) (15 Points) For the channel in (b), if the received signal r (k) sequence is (1., -.4, 1.5, -3., 1.8), using MLSE viterbi algorithm to recover the transmitted signals. (You can work on the Trellis diagram in (b))
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