Revision of Lecture Eight

Transcription

1 Revision of Lectue Eight Baseband equivalent system and equiements of optimal tansmit and eceive filteing: (1) achieve zeo ISI, and () maximise the eceive SNR Thee detection schemes: Theshold detection eceive, matched filte eceive and coelation eceive They ae equivalent and all based on the pinciple of maximising the eceive SNR Detecto is impotant, as evey communication system has one So you should know detecto schematic diagams and how they wok Last component of modem eceive: demappe Note that maximising eceive SNR is diectly linked to minimum detection eo, and in next two lectues we analyse pefomance of the system Specifically, we analyse bit eo atio of the system in AWGN and fading channels 114

2 Pefomance in AWGN In AWGN channel, sampled quadatue (I o Q) component of eceive signal is k = k + n k = g 0 x k + n k x k : tansmitted symbol at k, g 0 : known CSI (fo coheent system) and may be assumed g 0 = 1 With optimal tansmit & eceive filteing as well as channel G c (f) = 1, Note that Z G Rx (f) = G Tx (f) and G Tx (f)g Rx (f) is equied Nyquist filte G Rx (f) df = 1 Thus, eceived noise sample n k has powe P N = σ n = N 0 Z Z G Tx (f) G Rx (f) df = 1 G Rx (f) df = N 0 Let ā be aveage (I o Q) symbol powe, eceived signal sample k has powe P Rx = ā Z G Tx (f) G Rx (f) df = ā Detection eo pobability depends on noise pobability density function, which is Gaussian with vaiance σ n, and eceive output SNR= P Rx/P N Maximising eceive output SNR in AWGN leads to minimising detection eo pobability 115

3 BPSK Bit Eo Rate BPSK tansmitte: bit 0 a = +d, bit 1 a = d As all detectos ae equivalent, we assume theshold detecto Received sample is = a + n, a {±d} and n N(0, σ ) As the decision bounday is = 0, the theshold decision ule is > 0 ba = d, 0 ba = d ^ a= d 0 ^ a=+d d +d Using Bayes theoem, the eo pobability o BER is given by P e = P(ba a) = P(a = d ba = d) + P(a = d ba = d) = P(a = d)p(ba = d a = d) + P(a = d)p(ba = d a = d) As tansmitted bit is equally likely to be 0 o 1, the two a pio pobabilities ae P(a = d) = P(a = d) = 1 Given a = d, the decision ba = d means that = d + n > 0 o noise value n > d, and the conditional pobability P(ba = d a = d) equals to P(n > d) = Z d 1 πσ exp x σ! dx = 1 π Z d/σ! exp y dy = Q (d/σ) 116

4 BPSK BER: Result Intepetation of conditional eo pobability o eo Q-function P(ba = d a = d): Gaussian tail aea ove theshold = 0 Similaly, the othe conditional eo pobability As aveage signal powe E s = 1 (d +d ) = d and noise powe N 0 = σ, BPSK BER is P e = 1 Q (d/σ) + 1 Q (d/σ) =Q (d/σ) = Q SNR 0s 1 E s A N 0 P(bx = d x = d) = Q (d/σ) Note the eo Q-function apidly deceases as the SNR inceases Maximising eceive SNR leads to minimising eo pobability o bit eo ate Bit Eo Rate d vaiance d σ Es/No (db) BPSK 117

5 4QAM Bit Eo Rate 4QAM o QPSK: I and Q components ae both BPSK, and aveage signal powe is E s = d, noise powe σ = N 0 Let = I + j Q be eceived signal sample, then decision ule is 10 Q d 00 bits i q I, Q > 0 i, q = 0 I, Q 0 i, q = 1 Applying BPSK esult to both I and Q yields P e,i = Q (d/σ) and P e,q = Q (d/σ) Aveage eo ate fo 4QAM is then 0s P e = 1 (P e,i + P e,q ) = Q (d/σ) = E s N 0 q In compaison to BPSK, who has P e = Q E s N 0 Fo same bit ate R b, 4QAM bandwidth is half but equies highe signal powe (facto of o 3 db) to achieve same level of BER When plot as functions of E b N 0, how two BER cuves will look like? whee E b is aveage bit enegy 1 A Bit Eo Rate d d -d Es/No (db) BPSK QPSK I 118

6 4-ay Constellation BER 4-ay constellation: bits pe symbol and Gay coding b 1 b : 01, 00, 10, 11 3d, d, d, 3d Most significant bit b 1 and least significant bit b have diffeent immunities to noise, and ae called class-one and class-two bits, espectively Intinsically, as though C1 and C bits ae tansmitted though two diffeent sub-channels C1 bit decision ule: > 0 b 1 = 0, 0 b 1 = 1 C1 bit BER: b =1 eo 1 P e,1 = 4 Q (d/σ n) + 4 Q (3d/σ n) = 1 Q (d/σ n) b =0 1 eo + 1 Q (3d/σ n) C1 bits b 1 ae at a potection distance of d fom the decision bounday ( = 0) fo 50% of the time, and thei potection distance is 3d fo the othe 50% of the time 119

7 4-ay Constellation BER (continue) C bit decision ule: > d o d b = 1, d < d b = 0 Fo symbol 3d: when noise value n > d, > d and decision eo occus but when noise value n > 5d, > d and decision is coect again, thus the conditional eo ate is Q (d/σ n ) Q(5d/σ n ) Fo symbol d, it is Q (d/σ n ) + Q (3d/σ n ) C bit eo ate P e, = 4 4 Q (d/σ n) + 4 Q (3d/σ n) 4 Q (5d/σ n) b = 1 b = 1 b = b = 1 C decision bounday Note that P e, P e,1 Aveage eo ate of 4-ay constellation: b = 1 eo P e = 1 (P e,1 + P e, ) = 3 4 Q d σ n «+ 1 «3d Q σ n 1 «5d 4 Q σ n b = 0 eo 10

8 16QAM Bit Eo Rate 16QAM: I and Q ae both 4-ay C1 bit decision: I, Q > 0 i 1, q 1 = 0 I, Q 0 i 1, q 1 = 1 C bit decision: I, Q > d o I, Q d i, q = 1 Q d d d I Gay encoded 3d: 01 d: 00 -d: 10-3d: 11 bits: i 1 q 1 i q C1 C d < I, Q d i, q = d C decision bounday Signal powe E s = 10d and noise powe σ n = N 0 d/σ n = p E s /5N 0, and 16QAM BER: P e = 1 (P e,i + P e,q ) = 3 «d 4 Q + 1 «3d σ n Q 1 «5d 4 Q = 3 4 Q q E s /5N 0 «σ n σ n + 1 «3 qe Q s /5N 0 1 «4 Q 5 qe s /5N 0 11

9 8-ay Constellation BER 8-ay: b 1 b b 3 Gay coded. 3 classes of bits, C1 b 1 has the highest immunity to noise, and C3 b 3 has the lowest, as though these thee classes of bits wee tansmitted though 3 diffeent sub-channels One C1 decision bounday, two C decision boundaies, and fou C3 decision boundaies C1 decision ule: > 0 b 1 = 0 0 b 1 = 1 b 1b b3 Gay coded C1 decision bounday C decision bounday C3 decision bounday Thus, conditional eo ate fo symbol 7d is Q (7d/σ n ) and so on C1 bit eo ate: -7d -5d 5d 7d P e,1 = 1 4 (Q (d/σ n) + Q (3d/σ n ) + Q (5d/σ n ) + Q (7d/σ n )) C decision: > 4d o 4d b = 1, 4d < 4d b = 0 P e, = 1 Q (d/σ n) + 1 Q (3d/σ n) Q (5d/σ n) Q (7d/σ n) 1 4 Q (9d/σ n) 1 4 Q (11d/σ n) 1

10 8-ay Constellation BER (continue) C3 decision: > 6d o 6d o d < d b 3 = 1 6d < d o d < 6d b 3 = 0 Hence, C3 eo pobability P e,3 = Q (d/σ n ) Q (3d/σ n) 3 4 Q (5d/σ n) 1 Q (7d/σ n) + 1 Q (9d/σ n) Q (11d/σ n) 1 4 Q (13d/σ n) Note P e,3 P e, and P e, P e,1 Aveage eo of 8-ay constellation is: P e = 1 3 (P e,1 + P e, + P e,3 ) o P e = 7 1 Q (d/σ n) + 1 Q (3d/σ n) 1 1 Q (5d/σ n) Q (9d/σ n) 1 1 Q (13d/σ n) Since the aveage symbol enegy of 8-ay is E s = 1d and σ n = N 0/, p P e = 7 1 Q Es /1N Q 3 p E s /1N Q 5 p E s /1N Q 9 p E s /1N Q 13 p E s /1N 0 13

11 64QAM Bit Eo Rate 64QAM: i 1 q 1 i q i 3 q 3 Thee classes of bits I & Q ae identical to 8-ay C decision bounday C3 decision bounday C1 decision bounday d 3d 5d 7d

12 64QAM BER (continue) C1 decision: I, Q > 0 i 1, q 1 = 0, I, Q 0 i 1, q 1 = 1 C decision: I, Q > 4d o I, Q 4d i, q = 1 4d < I, Q 4d i, q = 0 C3 decision: I, Q > 6d o I, Q 6d o d < I, Q d i 3, q 3 = 1 6d < I, Q d o d < I, Q 6d i 3, q 3 = 0 Aveage eo of 64QAM is: P e = 7 1 Q (d/σ n) + 1 Q (3d/σ n) 1 1 Q (5d/σ n) Q (9d/σ n) 1 1 Q (13d/σ n) Noting the aveage symbol enegy of 64QAM is E s = 4d, P e = 7 «qe 1 Q s /1N «Q 3 qe s /1N 0 1 «1 Q 5 qe s /1N «9 qe 1 Q s /1N 0 1 «1 Q 13 qe s /1N 0 15

13 Geneal Comments Fo geneic high-ode QAM, deive appoximate bit eo ate as Q `p SNR equivalent In theoy, one can define the uppe bound which is lage than tue BER and the lowe bound which is smalle than tue BER By making the two bounds vey tight, ensue the appoximation is vey accuate In geneal, union bound is deived, which is vey accuate appoximation to epesent the tue bit eo ate In pactice, union bound is often used to epesent the BER of high-ode QAM BER lowe bound SNR (db) appoximation uppe bound One may just simply use Monte Calo simulation to evaluate the system s bit eo ate Sending lage numbe of bits, and simply counting the bit eos: if total numbe of bits sent is N b, and eo counts is N e BER N e N b How accuate this appoximation? You must ensue at least a few hundeds of eo counts! Systems like fibe netwoks, BER< , may have to build pototype to evaluate, o Impotance sampling simulation may be adopted to evaluate these systems 16

14 Summay Implication of optimal Tx and Rx filte design: the aea unde G Rx (f) is unity Decision theoy: eo pobability, Bayes theoem, a pioi pobability, conditional pobability, Q-function 4QAM (I & Q ae -ay o BPSK): decision ule, BER deivation 16QAM (I & Q ae 4-ay): C1 and C bits, two vitual sub-channels and diffeent noise immunity, decision ules, BER deivation 64QAM (I & Q ae 8-ay): C1, C and C3 bits, thee vitual sub-channels and diffeent noise immunity, decision ule, BER deivation Fo 56QAM o highe, simplified appoximation athe than exact deivation is used fo BER calculation 17