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Proakis J. (2002) Communication Systems Engineering - Solutions Manual (299s) Episode 10

Tham khảo tài liệu 'proakis j. (2002) communication systems engineering - solutions manual (299s) episode 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | The structure of this optimal receiver is shown in the next figure. The optimal receivers derived in this problem are more costly than those derived in the text since N is usually less than M the number of signal waveforms. For example in an M-ary PAM system N 1 always less than M. Problem 7.21 1 The optimal receiver see Problem 7.20 computes the metrics C r sm f r t sm t dt - sm t 2dt N0 lnP sm J tt 2 x 2 and decides in favor of the signal with the largest C r sm . Since s1 t s2 t the energy of the two message signals is the same and therefore the detection rule is written as s1 - . - W.CM. Nữ. P S2 _ Nữ P2 L W 4ln P -S 4 p S2 2 If s1 t is transmitted then the output of the correlator is i r t s1 t dt i s1 t 2dt n t s1 t dt J Jữ J0 Es n where Es is the energy of the signal and n is a zero-mean Gaussian random variable with variance 2 Ẹ . E n T n v s1 T s1 v dTdv ữữ rT rT s1 t s1 v E n T n v dTdv ữữ Nữ rT Í T -pp- s1 T s1 v ỗ t v dTdv 2 Jo Jo N L S1 t 2dT Nữ E 2 Es Hence the probability of error P e s1 is P e s1 x e N0Es dx Q 1 Ĩ2N ữ. P2 -4 ln 4 l Es P1 fN ln pl Es 1 nN Es 178 Similarly we find that P e s2 Q yiiz I 1y N ln - The average probability of error is P e PiP e si P2P e s2 pi Q IN - 1 2N0 1 - Pi ln 4 V Es Pi 1 - pi Q I T 3 In the next figure we plot the probability of SNR N. As it is observed the probability signals. error as a function of p i for two values of the of error attains its maximum for equiprobable Problem 7.22 1 The two equiprobable signals have the same energy and therefore the optimal receiver bases its decisions on the rule s 1 Ỉ r t si t dt i r t s2 t dt tt J tt s2 2 If the message signal si t is transmitted then r t si t n t and the decision rule becomes r si t n t si t - S2 t dt J tt i si t si t - s2 t dt i n t si t - S2 t dt J tt J tt .tt si Si t si t - s2 t dt n 0 J tt S2 where n is a zero mean Gaussian random variable with variance ơ2n i i si T - S2 t si v - S2 v E n r n v drdv J tt J tt 179 fT fT.N0 0 Jo si t - S2 tJJ si v - S2 v -2ỗ