etection Fundamentals - 2003 Miterm - 15 ptsA random variablex1takes the 2 values1 with equal probability independently of a second randomvariablex2that takes the values2 also with equal probability. The two random variables are sumedtox=x1+x2, andxcan only be observed after zero-mean Gaussian noise of variance2=.1 is added,that isy=x+nis observed wherenis the noise.a. (1 pt) What are the values that the discrete random variablextakes, and what are their proba-bilities? (1 pt)b. (1 pt) What are the means and variances ofxandy?c. (2 pts) What is the lowest probability of error in detectingxgiven only an observation ofy? Drawcorresponding decision regions.

d. (1 pt) Relate the value ofxwith a table to the values ofx1andx2. Explain why this is called a"noisy DAC" channel.e. (1 pt) What is the (approximate) lowest probability of error in detectingx1given only an obser-vation ofy?f. (1 pt) What is the (approximate) lowest probability of error in detectingx2given only an obser-vation ofy?g. Suppose additional binary independent random variables are added so that the two bipolar valuesforxuare2u1,u= 1,...,U. Whichxuhas lowest probability of error for any AWG noise, andwhat is thatPe? (1 pt)h. ForU= 2, what is the lowest probability of error in detectingx1given an observation ofyand acorrect observation ofx2? (1 pt)i. ForU= 2, what is the lowest probability of error in detectingx2given an observation ofyand acorrect observation ofx1? (1 pt)j. What is the lowest probability of error in any of parts e through i if2= 0? What does this meanin terms of the DAC? (1 pt)k. Derive a general expression for the probability of error for all bitsu= 1,...,Uwherex=x1+x2+...+xUin AWGN with variance2for part g? (2 pts)1.34Honey Comb QAM - 2005 Miterm - 15 ptsThe QAM constellation in Figure 1.61 is used for transmission on an AWGN with symbol rate 10MHzand a carrier frequency of 100 MHz.012345Figure 1.61: Constellation for Problem 1.34.Each of the solid constellation symbol possibilities is at the center of a perfect hexagon (all sides areequal) and the distance to any of the closest sides of the hexagon isd2. The 6 empty points representa possible message also, but each is used only every 6 symbol instants, so that for instance, the pointlabelled 0 is a potential message only on symbol instants that are integer multiples of 6. The 1 pointcan only be transmitted on symbol instants that are integer multiples of 6 plus one, the 2 point only onsymbol instants that are integer multiples of 6 plus two, and so on. At any symbol instant, any of thepoints possible on that symbol are equally likely.

a. What is the number of messages that can be possibly transmitted on any single symbol? Whatareband b? (3 pts)b. What is the data rate? (1 pt)c. Draw the decision boundaries for time 0 of a ML receiver. ( 2 pts)d. What isdmin? (1 pt)e. What areExand Exfor this constellation in terms ofd? (3 pts)f. What is the number of average number nearest neighbors? (1 pt)g. Determine the NNUB expression that tightly upper bounds Pefor this constellation in terms ofSNR. (2 pts)h. Compare this constellation fairly to Cross QAM transmission. (1 pt)i. Describe an equivalent ML receiver that uses time-invariant decision boundaries and a constantdecision device with a simple preprocessor to the decision device. (1 pt).