UTA EE5362 PhD Diagnosis Exam (Spring 2011)

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1 EE5362 Spring 2 PhD Diagnosis Exam ID: UTA EE5362 PhD Diagnosis Exam (Spring 2) Instructions: Verify that your exam contains pages (including the cover shee. 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 2 from the 3 problems). Problem Points Scores Channel Coding 5 2 Signal Detection 5 3 Packet Acquisition 5 Total Score (choose 2 problems)

2 EE5362 Spring 2 PhD Diagnosis Exam ID 2 (5 Points) Channel Coding For the [4,2] linear block codec having parity check matrix = (a) ( points) Set up the entire standard array for the code C. The array must contain all possible binary 4-tuples as well as all possible syndromes. As in class, the entire row of your standard array must list all the codewords. The leftmost column should contain the coset leaders and the rightmost column should list the syndromes.

3 EE5362 Spring 2 PhD Diagnosis Exam ID 3 (b) ( points) When the transmitted codeword is vector t r = [] t c = [], given that the received what is the decoded codeword? What is the residue error? ow many message bits are in error? (c)( points) When the code is used only for the purposes for correcting error, what is the probability of decoding error (Pw) when the crossover probability of the binary symmetric channel (BSC) is ε?

4 EE5362 Spring 2 PhD Diagnosis Exam ID 4 (d)( points) What is the probability that ONLY the first message symbol will be in error (i.e., the second message bit no error) if the code is used solely for the purpose of error correction error when the crossover probability of the BSC is ε? (e)( points) What is the probability of undetected error if the code is used solely for the purpose of error detection error when the crossover probability of the BSC is ε?

5 EE5362 Spring 2 PhD Diagnosis Exam ID 5 2 Signal Detection (5 points) Consider the following four equally likely hypotheses 72E : r( u, = sin(2πf c t + φ) + n( u, T 8E : r( u, = sin(2πf c t + φ) + n( u, T 32E 2 : r( u, = sin(2πf c t + φ) + n( u, T 98E 3 : r( u, = sin(2πf c t + φ) + n( u, T where n( u, is the standard AWGN (i.e., K n ( τ ) = ( N / 2) δ ( τ ) ) and t [, T ]. Assume that the carrier frequency and phase rotation are known, and the narrowband assumption is valid. (a) (5points) Derive and plot the optimal decision boundaries.

6 EE5362 Spring 2 PhD Diagnosis Exam ID 6 (b) (5 points) Determine the probability of error P a (ε ) for the optimal decision boundaries derived in (a).

7 EE5362 Spring 2 PhD Diagnosis Exam ID 7 (c) (2 Points) A new 2-D modulation is constructed, and each dimension is drawn from the four signals in the above hypotheses, so there are totally 6 signal constellations for this new modulation. Based on the result of P (ε ) derived in (b), determine the probability of error, P b (ε ) for this new modulation if optimal decision boundaries for this modulation are used. a

8 EE5362 Spring 2 PhD Diagnosis Exam ID 8 3. (5 Points) Packet Acquisition You are working as a system engineer to design a packet detector for acquisition. Your job is to determine if a packet is present or not in a communication channel. Your sources have been given you some reliable information: you know that communication is carried out by BPSK modulation, and you also know T, N, along with the carrier frequency and phase. Your observation is the output of the correlator for a BPSK receiver. Your mission is to decide between the following two hypotheses: : R ( u) = N( u) (packet not presen : R ( u) = EB( u) + N( u) (packet presen where the random variable B(u) represent the random BPSK signal: Pr{ B ( u) = + } = Pr{ B( u) = } = 2 The noise at the output of the correlator (i.e., N (u) ) is Gaussian with zero mean and variance N / 2. The a priori probability of packet present on the channel (i.e., approximate bandwidth 3 utilization) is assumed to be 75 % (i.e., p ( ) = ). 4 (a) ( Points) Determine the probability density function of R(u) under either hypothesis: f R ( u) ( r ) = ( r ) = f R ( u) sketch these two pdfs on the same axis.

9 EE5362 Spring 2 PhD Diagnosis Exam ID 9 (b) ( points) Determine the Bayes decision rule (based on r, the realization of R (u)) and describe it below (Function of r, E, N ): Rule: (c) ( Points) This rule can be represented using the hyperbolic cosine (cosh) function: x x cosh( x) = ( e + e ). Represent this rule using cosh( ) function: 2 Rule: (Function of cosh( ), r, E, N,) The hyperbolic cosine (cosh) is an even function (i.e., cosh( x ) = cosh( x ) ), and its positive inverse ( cosh ) is cosh( x ) = y x = cosh ( y) Both cosh(x) and cosh are monotonically increasing function when x >. Using the property of cosh function and its inverse, the rule can be simplified to:

10 EE5362 Spring 2 PhD Diagnosis Exam ID r Specify the value of A which results in the minimum error probability: A = (Function of cosh ( ) > < A, E, N ) (d) (2 Points) Your supervisor wants you to evaluate the probability of detection P D. P D is the probability of detecting a packet when it is truly present. Determine this probability: P D = P( ε ) = (Function of E, N, and A ) P = (Function of γ = E ) D N Show your work here:

11 EE5362 Spring 2 PhD Diagnosis Exam ID

Optimum Soft Decision Decoding of Linear Block Codes

Optimum Soft Decision Decoding of Linear Block Codes {m i } Channel encoder C=(C n-1,,c 0 ) BPSK S(t) (n,k,d) linear modulator block code Optimal receiver AWGN Assume that [n,k,d] linear block code C is