Reference no: EM132360781
Assignment Instructions
Instructions: Complete the following problems from
Part -1
Q1. Let an RX be connected to two antennas, for which the SNRs are independent and exponentially distributed using the same average SNR. RSSI-driven selection diversity is employed and the outage probability is Pout. We are interested in the fading margin.
o (a) Derive an expression in terms of Pout for the fading margin when only one antenna is used.
o (b) Derive an expression in terms of Pout for the fading margin when both antennas are used.
o (c) Use the two results above to calculate the diversity gain for an outage probability of 1%.
Q2. In order to reduce the complexity. a hybrid selection MRC scheme can be used instead of full MRC. If we have five antennas but only use the three strongest, what is the loss in terms of mean SNR compared with full MRC?
Q3. Consider an Nr-branch antenna diversity system using the MRC rule. All branches are subject to Rayleigh fading and the fading at one branch is independent of fading at all other branches. Derive the pdf of the SNR at the output of the combiner for the following cases:
• (a) The average SNR per branch is ¯γ, for all branches.
• (b) The average SNR for the ith branch is ¯γi. Let the ¯γi's be distinct.
Part -2
Q 1. Let us consider a linear cyclic (7,3) block code with generator polynomial G(x) = x4 + x3 + x2 + 1.
• (a) Encode the message U(x) = x2 + 1 systematically, using G(x).
• (b) Calculate the syndrome S(x) when we have received (probably corrupted) R(x) = x6 + x5 + x4 + x + 1.
• (c) A close inspection reveals that G(x) can be factored as G(x) = (x + 1 )T(x), where T(x) = x3 + x + 1 is a primitive polynomial. This implies, we claim, that the code can correct all single errors and all double errors, where the errors are located next to each other. Describe how you would verify this claim.
Q 2. Show that for a cyclic (N,K) code with generator polynomial G(x) there is only one codeword with degree N - K and that codeword is the generator polynomial itself. In a cyclic block code, a cyclic shift of a codeword results in another valid codeword.
Q 3. Assume that we have a linear (7,4) code, where the codewords corresponding to messages u = [1000], [0100], [0010], and [0001] are given as
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• (a) Determine all codewords in this code.
• (b) Determine the minimum distance, dmin, and how many errors, t, the code can correct.
• (c) The code above is not in systematic form. Calculate the generator matrix G for the corresponding systematic code.
• (d) Determine the parity check matrix H, such that HGT = 0.
• (e) Is this code cyclic? If so, determine its generator polynomial.
Q 4. Show that the following inequality (the Singleton bound) is always fulfilled for a linear (N,K) block code:
dmin ≤ N - K + 1
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