Digital Communication I: Modulation and Coding Course-Lecture 5 potx

Digital Communications I: Modulation and Coding Course Term - 2008 Catharina Logothetis Lecture Last time we talked about:  Receiver structure  Impact of AWGN and ISI on the transmitted signal  Optimum filter to maximize SNR  Matched filter and correlator receiver  Signal space used for detection  Orthogonal N-dimensional space  Signal to waveform transformation and vice versa Lecture Today we are going to talk about:  Signal detection in AWGN channels  Minimum distance detector  Maximum likelihood  Average probability of symbol error  Union bound on error probability  Upper bound on error probability based on the minimum distance Lecture Detection of signal in AWGN  Detection problem:  Given the observation vector , perform a mapping z from to an estimate of the transmitted ˆ m z the average probability of symbol, , such that mi error in the decision is minimized n mi Modulator si z Decision rule Lecture ˆ m Statistics of the observation Vector  AWGN channel model: z = si + n  Signal vector s i = (ai1 , , , aiN ) is deterministic  Elements of noise vector n = (n1 , n2 , , nN ) are i.i.d Gaussian random variables with zero-mean and variance N / The noise vector pdf is  n2  pn (n) = exp − ( πN ) N /  N     The elements of observed vector z = ( z1 , z , , z N ) are independent Gaussian random variables Its pdf is  z − si   pz ( z | s i ) = exp − N /2  N0  ( πN )   Lecture 5 Detection  Optimum decision rule (maximum a posteriori probability): ˆ Set m = mi if Pr(mi sent | z ) ≥ Pr(mk sent | z ), for all k ≠ i  where = 1, , rule ApplyingkBayes’ M gives: ˆ Set m = mi if pz (z | mk ) pk , is maximum for all k = i pz ( z ) Lecture Detection …  Partition the signal space into M decision regions, such that Z1 , , Z M Vector z lies inside region Z i if pz (z | mk ) ln[ pk ], is maximum for all k = i pz ( z ) That means ˆ m = mi Lecture Detection (ML rule)  For equal probable symbols, the optimum decision rule (maximum posteriori probability) is simplified to: ˆ Set m = mi if pz (z | mk ), is maximum for all k = i or equivalently: ˆ Set m = mi if ln[ pz (z | mk )], is maximum for all k = i which is known as maximum likelihood Lecture Detection (ML)…  Partition the signal space into M decision regions, Z1 , , Z M  Restate the maximum likelihood decision rule as follows: Vector z lies inside region Z i if ln[ pz (z | mk )], is maximum for all k = i That means ˆ m = mi Lecture Detection rule (ML)…  It can be simplified to: Vector z lies inside region Z i if z − s k , is minimum for all k = i or equivalently: Vector r lies inside region Z i if N ∑ z j akj − Ek , is maximum for all k = i j =1 where Ek is the energy of sk (t ) Lecture 10 Maximum likelihood detector block diagram 〈⋅,s1 〉 − E1 z Choose the largest ˆ m 〈⋅, s M 〉 − EM Lecture 11 Schematic example of the ML decision regions ψ (t ) Z2 s2 Z3 s3 s1 Z1 ψ (t ) s4 Z4 Lecture 12 Average probability of symbol error  Erroneous decision: For the transmitted symbol or m i equivalently signal vector , s error in decision occurs if an i the observation vector z does not fall inside region Z  i Probability of erroneous decision for a transmitted symbol or equivalently ˆ Pe (mi ) = Pr(m ≠ mi and mi sent) ˆ Pr(m ≠ mi ) = Pr(mi sent)Pr(z does not lie inside Z i mi sent)  Probability of correct decision for a transmitted symbol ˆ Pr(m = mi ) = Pr( mi sent)Pr(z lies inside Z i mi sent) Pc (mi ) = Pr(z lies inside Z i mi sent) = ∫ p (z | m )dz z i Zi Pe (mi ) = − Pc (mi ) Lecture 13 Av prob of symbol error …  Average probability of symbol error : M ˆ PE ( M ) = ∑ Pr (m ≠ mi )  i =1 For equally probable symbols: PE ( M ) = M M ∑ Pe (mi ) = − M i =1 = 1− M M ∑ P (m ) i =1 c i M ∑ ∫ p (z | m )dz i =1 Z i z i Lecture 14 Example for binary PAM pz (z | m2 ) pz (z | m1 ) s2 − Eb s1 ψ (t ) Eb  s1 − s /   Pe (m1 ) = Pe (m2 ) = Q  N /2     Eb PB = PE (2) = Q  N      Lecture 15 Union bound Union bound The probability of a finite union of events is upper bounded by the sum of the probabilities of the individual events    Let Aki denote that the observation vector z is closer to the symbol vector s k than s i , when s i is transmitted Pr( Aki ) = P2 (s k , s i ) depends only on s i and s k Applying Union bounds yields M Pe (mi ) ≤ ∑ P2 (s k , s i ) k =1 k ≠i PE ( M ) ≤ M Lecture M M ∑∑ P (s i =1 k =1 k ≠i k , si ) 16 Example of union bound Pe (m1 ) = ∫ p (r | m )dr r Z ∪Z3 ∪Z r Z2 ψ2 Z1 s2 s1 Union bound: ψ1 s3 s4 Z4 Z3 Pe (m1 ) ≤ ∑ P2 (s k , s1 ) k =2 A2 r ψ2 r ψ2 s2 r s2 s1 s3 P2 (s , s1 ) = s4 ∫ p (r | m )dr r A2 s2 s1 ψ1 ψ2 s1 ψ1 s3 P2 (s , s1 ) = A3 s4 ∫ p (r | m )dr r A3 Lecture ψ1 s3 P2 (s , s1 ) = A4 s4 ∫ p (r | m )dr r A4 17 Upper bound based on minimum distance P2 (s k , s i ) = Pr(z is closer to s k than s i , when s i is sent) =  d ik / u2 exp(− )du =Q  N /2 N0 πN 0  ∞ ∫ d ik     d ik = s i − s k PE ( M ) ≤ M  d /  ∑∑ P2 (s k , si ) ≤ (M − 1)Q N /    i =1 k =1   k ≠i M M Minimum distance in the signal space: d = d ik i ,k i≠k Lecture 18 Example of upper bound on av Symbol error prob based on union bound ψ (t ) s i = Ei = Es , i = 1, ,4 d i ,k = Es i≠k Es d = Es s3 − Es s2 d1, d 2,3 s1 d 3, d1, Es ψ (t ) s4 − Es Lecture 19 Eb/No figure of merit in digital communications  SNR or S/N is the average signal power to the average noise power SNR should be modified in terms of bit-energy in DCS, because:  Signals are transmitted within a symbol duration and hence, are energy signal (zero power)  A merit at bit-level facilitates comparison of different DCSs transmitting different number of bits per symbol Eb STb S W = = N N / W N Rb Rb W : Bit rate : Bandwidth Lecture 20 Example of Symbol error prob For PAM signals Binary PAM s2 s1 − Eb s4 −6 Eb ψ (t ) Eb 4-ary PAM s3 s2 E −2 b E b s1 Eb ψ (t ) T Lecture T t 21 ψ (t ) ... we talked about:  Receiver structure  Impact of AWGN and ISI on the transmitted signal  Optimum filter to maximize SNR  Matched filter and correlator receiver  Signal space used for detection... deterministic  Elements of noise vector n = (n1 , n2 , , nN ) are i.i.d Gaussian random variables with zero-mean and variance N / The noise vector pdf is  n2  pn (n) = exp − ( πN ) N /  N... z = ( z1 , z , , z N ) are independent Gaussian random variables Its pdf is  z − si   pz ( z | s i ) = exp − N /2  N0  ( πN )   Lecture 5 Detection  Optimum decision rule (maximum a posteriori