ECE 564/645 - Digital Communication Systems (Spring 2014) Final Exam Friday, May 2nd, 8:00-10:00am, Marston 220

Transcription

1 ECE 564/645 - Digital Commuicatio Systems (Sprig 014) Fial Exam Friday, May d, 8:00-10:00am, Marsto 0 Overview The exam cosists of four (or five) problems for 100 (or 10) poits. The poits for each part of each problem are give i brackets - you should sped your two hours accordigly. The exam is closed book, but you are allowed three page-sides of otes. Calculators are ot allowed. I will provide all ecessary blak paper. Testmaship Full credit will be give oly to fully justified aswers. Givig the steps alog the way to the aswer will ot oly ear full credit but also maximize the partial credit should you stumble or get stuck. If you get stuck, attempt to eatly defie your approach to the problem ad why you are stuck. If part of a problem depeds o a previous part that you are uable to solve, explai the method for doig the curret part, ad, if possible, give the aswer i terms of the quatities of the previous part that you are uable to obtai. Start each problem o a ew page. Not oly will this facilitate gradig but also make it easier for you to jump back ad forth betwee problems. If you get to the ed of the problem ad realize that your aswer must be wrog (e.g. a egative probability), be sure to write this must be wrog because... so that I will kow you recogized such a fact. Academic dishoesty will be dealt with harshly - the miimum pealty will be a F for the course.

2 Some potetially useful iformatio cos(θ) = 1 (e jθ + e jθ) si(θ) = 1 j (e jθ e jθ) si(a ± b) = si(a) cos(b) ± cos(a) si(b) cos(a ± b) = cos(a) cos(b) si(a) si(b) cos(a) cos(b) = 1 [cos(a b) + cos(a + b)] si(a) si(b) = 1 [cos(a b) cos(a + b)] si(a) cos(b) = 1 [si(a b) + si(a + b)] cos (a) = 1 [1 + cos(a)] si (a) = 1 [1 cos(a)] si(a) cos(a) = 1 si(a)

3 1. Huffma Codig: [1] (a) A source produces a idepedet ad idetically distributed (IID) sequece from X = {A, B, C}, with P (X i = A) = 0.4, P (X i = B) = 0.4, P (X i = C) = 0.. A optimal lossless source coder takes symbols N = at a time ad compresses this source at miimum rate. Fid such a source code ad its rate (i output bits per iput symbol). [8] (b) Suppose I have a source that produces a radom variable X from a alphabet X = {A, B, C, D, E, F }, ad cosider codig this sigle radom variable with a optimal lossless source coder (i.e. Huffma codig with N = 1). My fried kows the probabilities of each of the letters (which he does ot tell me) ad desigs such a Huffma code. He the tells me that P (X = A) = 0.3, but ot ay of the other probabilities, ad A may or may ot be the most likely symbol. Give upper ad lower bouds o the legth of the bit sequece assiged to A by the Huffma code based o this iformatio. Be sure to justify your aswer. (Hit: Aalyze the Huffma codig algorithm.). Cosider the waveform chael: s(t) r(t) (t) where (t) is additive white Gaussia oise with power spectral desity N 0, r(t) is the received waveform, ad s(t) = s i (t) whe message m i is to be set durig time t (0, 1). Suppose there are M = 4 possible equally likely messages ad the correspodig sigals are: [7] (a) Calculate the itegrals: 0 cos(πt)dt 0 si(πt)dt 0 si(πt) cos(πt)dt 0 cos (πt)dt s 1 (t) =, 0 t 1 s (t) = 1 + cos(πt), 0 t 1 s 3 (t) = 1 + si(πt), 0 t 1 s 4 (t) = 1 + si(πt), 0 t 1 [13] (b) Fid a orthoormal basis {φ i (t) : i = 1,,..., N} for this sigal set ad give the vector represetatio of each sigal i this basis. [5] (c) Specify the MAP receiver by: Drawig a (simple) block diagram showig a method of obtaiig r j (the compoet of r(t) alog the basis fuctio φ j (t)) from r(t). Idicatig how a sigal is chose based o (r = (r 1,..., r N ) T ), (You ca do this with a picture or a descriptio with equatios.) [10] (d) Fid the Uio Boud to the probability of error of the MAP receiver i terms of uits of your sigal space. Idetify the term i your sum that will domiate performace at moderate-to-high sigal-to-oise ratios. Covert this domiat summad oly to be i terms of E b ad N 0.

4 3. Cosider the followig (uusual) biary amplitude-shift keyig (ASK) system for trasmittig a bit b 0, equally likely to be 0 or 1, i t (0, 1). For b 0 = 0, we let s(t) = 1 4 p(t), ad for b 0 = 1, we let s(t) = 3 4p(t), where p(t) = { 1 0 t 1 0 otherwise The sigal s(t) is trasmitted across a chael modeled as the followig: s(t) (t) r(t) where (t) is additive white Gaussia oise with power spectral desity N 0 ad r(t) is the received waveform. [5] (a) Fid the receiver for processig r(t), t (0, 1) to obtai a estimate for the trasmitted bit that miimizes the probability of a bit error. [5] (b) Fid the probability of a bit error i terms of the average eergy per symbol E s ad N 0. [5] (c) How much better or worse (i db of Es N 0 ) is this system tha a biary phase-shift keyed (BPSK) system operatig o a AWGN chael? Suppose ow that chael codig is employed i the system; that is, the iformatio bits {I k } are iput to the ecoder to produce the bits {b k } for modulatio. The parity check matrix of the code is give by: [15] (d) Fid the followig: H = A geerator matrix G for the code The weight eumerator polyomial for the code: A(x) = i=0 A i x i, where A i is the umber of weight-i codewords. The rate r of the code. The sydromes ad the correspodig coset leaders. [7] (e) Now suppose the bits from the ecoder are trasmitted with the biary amplitude shift-keyed (ASK) system o a AWGN chael as described i the first part of the problem. Fid the exact probability of a codeword error i terms of E s ad N 0 whe hard-decisio decodig is doe at the receiver; that is, the demodulator outputs its best estimate of each bit b k ad the decoder uses six such bit decisios to decode the codeword. The, covert your aswer to be i terms of the eergy per iformatio bit (i.e. eergy per bit at the iput to the chael coder) E b ad N 0.

5 [8] (f) Now suppose the bits from the ecoder are trasmitted with the biary amplitude shift-keyed (ASK) system o a AWGN chael as described i the first part of the problem. Fid the Uio Boud to the probability of a codeword error i terms of E s ad N 0 whe optimal soft-decisio decodig is doe at the receiver; that is, the decoder chooses the most likely codeword trasmitted based o the received waveform over the correspodig six symbol periods. The, covert your aswer to be i terms of the eergy per iformatio bit (i.e. eergy per bit at the iput to the chael coder) E b ad N [645 oly] Whe decodig a liear block code, we ca employ the stadard array, ad we kow a error patter is correctible if ad oly if it is declared a coset leader. [10] (a) Cosider a t-error correctig (, k) liear block code. (Recall, that for a code to be t-error correctig, it must correct every patter of t or fewer errors.) Show that the followig equatio must be true: k log ( ) t where k =! k!( k)!. The, show how this implies a upper boud o the umber of codewords i a code for a give t ad. [10] (b) Without usig a repetitio code, fid a example of a liear block code (give G, H, or the codewords) with t 1 (you get to choose t as log as t 1) such that: k = log ( ) t