Modulation and coding course- lecture 5

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Modulation and coding course- lecture 5

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Receiver structure - Impact of AWGN and ISI on the transmitted signal - Optimum filter to maximize SNR - Matched filter and correlator receiver

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Nội dung Text: Modulation and coding course- lecture 5

  1. Digital Communications I: Modulation and Coding Course Period 3 - 2007 Catharina Logothetis Lecture 5
  2. 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 5 2
  3. 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 5 3
  4. Detection of signal in AWGN Detection problem: Given the observation vector z , perform a mapping from z to an estimate m of the ˆ transmitted symbol, mi , such that the average probability of error in the decision is minimized. n mi si z ˆ m Modulator Decision rule Lecture 5 4
  5. Statistics of the observation Vector AWGN channel model: z = s i + n Signal vector s i = (ai1 , ai 2 ,..., aiN ) is deterministic. Elements of noise vector n = (n1 , n2 ,..., nN ) are i.i.d Gaussian random variables with zero-mean and variance N 0 / 2 . The noise vector pdf is 1 ⎛ n2⎞ pn (n) = exp⎜ − ⎟ (πN 0 )N / 2 ⎜ N 0 ⎟ ⎝ ⎠ The elements of observed vector z = ( z1 , z 2 ,..., z N ) are independent Gaussian random variables. Its pdf is 1 ⎛ z − si 2 ⎞ pz ( z | s i ) = exp⎜ − ⎟ (πN 0 )N / 2 ⎜ N 0 ⎟ ⎝ ⎠ Lecture 5 5
  6. Detection Optimum decision rule (maximum a posteriori probability): Set m = mi if ˆ Pr( mi sent | z ) ≥ Pr(mk sent | z ), for all k ≠ i where k = 1,..., M . Applying Bayes’ rule gives: Set m = mi if ˆ pz (z | mk ) pk , is maximum for all k = i pz ( z ) Lecture 5 6
  7. Detection … Partition the signal space into M decision regions, Z1 ,..., Z M such that Vector z lies inside region Z i if pz (z | mk ) ln[ pk ], is maximum for all k = i. pz ( z ) That means m = mi ˆ Lecture 5 7
  8. Detection (ML rule) For equal probable symbols, the optimum decision rule (maximum posteriori probability) is simplified to: Set m = mi if ˆ pz (z | mk ), is maximum for all k = i or equivalently: Set m = mi if ˆ ln[ pz (z | mk )], is maximum for all k = i which is known as maximum likelihood. Lecture 5 8
  9. 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 m = mi ˆ Lecture 5 9
  10. 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 1 ∑ z j akj − 2 Ek , is maximum for all k = i j =1 where Ek is the energy of sk (t ). Lecture 5 10
  11. Maximum likelihood detector block diagram 〈⋅,s1 〉 1 − E1 Choose z 2 ˆ m the largest 〈⋅, s M 〉 1 − EM 2 Lecture 5 11
  12. Schematic example of ML decision regions ψ 2 (t ) Z2 s2 Z1 s3 s1 Z3 ψ 1 (t ) s4 Z4 Lecture 5 12
  13. Average probability of symbol error Erroneous decision: For the transmitted symbol mi or equivalently signal vector s i , an error in decision occurs if the observation vector z does not fall inside region Z i. Probability of erroneous decision for a transmitted symbol Pe (mi ) = Pr(m ≠ mi and mi sent) ˆ or equivalently Pr(m ≠ mi ) = Pr(mi sent)Pr(z does not lie inside Z i mi sent) ˆ Probability of correct decision for a transmitted symbol Pr(m = mi ) = Pr(mi sent)Pr(z lies inside Z i mi sent) ˆ Pc (mi ) = Pr(z lies inside Z i mi sent) = ∫ p (z | m )dz z i Pe (mi ) = 1 − Pc (mi ) Zi Lecture 5 13
  14. Av. prob. of symbol error … Average probability of symbol error : M PE ( M ) = ∑ Pr (m ≠ mi ) ˆ i =1 For equally probable symbols: M M 1 1 PE ( M ) = M ∑ Pe (mi ) = 1 − M i =1 ∑ P (m ) i =1 c i M 1 = 1− M ∑ ∫ p (z | m )dz i =1 Z i z i Lecture 5 14
  15. Example for binary PAM pz (z | m2 ) pz (z | m1 ) s2 s1 ψ 1 (t ) − Eb 0 Eb ⎛ s1 − s 2 / 2 ⎞ Pe (m1 ) = Pe (m2 ) = Q⎜ ⎟ ⎜ N /2 ⎟ ⎝ 0 ⎠ ⎛ 2 Eb ⎞ PB = PE (2) = Q⎜ ⎜ N ⎟ ⎟ ⎝ 0 ⎠ Lecture 5 15
  16. 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 M M 1 Pe (mi ) ≤ ∑ P2 (s k , s i ) PE ( M ) ≤ ∑∑ P (s 2 k , si ) k =1 M i =1 k =1 k ≠i k ≠i Lecture 5 16
  17. Example of union bound ψ2 Z2 r Z1 Pe (m1 ) = ∫ p (r | m )dr r Z 2 ∪Z3 ∪Z 4 1 s2 s1 ψ1 Union bound: s3 s4 4 Z3 Pe (m1 ) ≤ ∑ P2 (s k , s1 ) Z4 k =2 ψ2 ψ2 ψ2 A2 r r r s2 s1 s2 s1 s2 s1 ψ1 ψ1 ψ1 s3 s4 s3 s4 s3 s4 A3 A4 P2 (s 2 , s1 ) = ∫ p (r | m )dr A2 r 1 P2 (s 3 , s1 ) = ∫ p (r | m )dr A3 r 1 P2 (s 4 , s1 ) = ∫ p (r | m )dr r 1 A4 Lecture 5 17
  18. 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) ∞ 1 u2 ⎛ d ik / 2 ⎞ = ∫ exp(− )du =Q⎜ ⎟ πN 0 N0 ⎜ N /2 ⎟ d ik ⎝ 0 ⎠ d ik = s i − s k 1 M M ⎛ d min / 2 ⎞ PE ( M ) ≤ M ∑∑ P2 (s k , si ) ≤ (M − 1)Q⎜ N / 2 ⎟ ⎜ ⎟ i =1 k =1 k ≠i ⎝ 0 ⎠ d min = min d ik Minimum distance in the signal space: i ,k i≠k Lecture 5 18
  19. Example of upper bound on av. Symbol error prob. based on union bound s i = Ei = Es , i = 1,...,4 ψ 2 (t ) d i ,k = 2 Es i≠k Es s2 d min = 2 Es d 2,3 d1, 2 s3 s1 ψ 1 (t ) − Es Es d 3, 4 d1, 4 s4 − Es Lecture 5 19
  20. 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 Rb : Bit rate = = N 0 N / W N Rb W : Bandwidth Lecture 5 20
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