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### Tele3113 wk11tue

1. 1. TELE3113 Analogue and Digital Communications – Detection Theory Wei Zhang w.zhang@unsw.edu.au School of Electrical Engineering and Telecommunications The University of New South Wales6 Oct. 2009 TELE3113 1
2. 2. Digital Signal Detection At the receiving end of the digital communication system: AWGN n(t) Sampled at t=kTs 1 if y(kTs)>λ si(t) Receive Decision r(t) filter y(t) y(kTs) device 0 if y(kTs)<λPolar NRZ Signaling  s (t ) = + A 0≤t ≤T for 1 Threshold si (t ) =  1 λ s2 (t ) = − A 0≤t ≤T for 0 Noise power spectral density: Sn(ω)=η/26 Oct. 2009 TELE3113 2
3. 3. Digital Signal Detection Suppose there are M possible signal symbols: {si} for i=1,…,M r We can represent these symbols in vector form si r r Similarly the noise vectors, n and the received signal vectors, r Thus r r r ri = s i + n ϕ2 r r n n r s1 r r s2 r1 r r n s3 r ϕ1 s4 r n ϕ36 Oct. 2009 TELE3113 3
4. 4. Digital Signal Detection In each time interval, the signal detector makes a decision based on r the observation of the vector r so that the probability of correct decision is maximized. Consider a decision rule based on the posterior probabilities r r P(signal si was transmitted | received vector r ) for i = 1,2 ,K ,M r r = P( si | r ) The decision is based on selecting the signal corresponding to the maximum set of posterior probabilities. r r r r r r f (r | si ) P( si ) Choose s i to maximize: P( si | r ) = r r f (r ) where f(r ) = M r r r ∑ f(r | s m ) P( s m ) m =1 r r where P ( s i ) is probability of si being transmitted and r f(r ) is the pdf function of the received signal vector r . This kind of decision is called maximum a posteriori probability (MAP) criterion6 Oct. 2009 TELE3113 4
5. 5. Digital Signal Detection MAP criterion r r r r r f (r | si ) P( si ) r M r r r P( si | r ) = r where f(r ) = ∑ f(r | s m ) P( s m ) f (r ) m =1 r r where f(r | si ) is called the likelihood function. r If the M symbols are equally probable; i.e. P ( si ) = 1 / M for all i, the decision r r rule based on finding the signal thatrmaximizes P( si | r ) is equivalent to r finding the signal that maximizes f(r | si ). r r The decision based on the maximum of f(r | si ) over M signal symbols is called the maximum-likelihood (ML) criterion6 Oct. 2009 TELE3113 5
6. 6. Digital Signal Detection r r r Recall: r = si + n For AWGN, the noise {nk} components are uncorrelated Gaussian variables which are statistically independent E[nk ] = 0 (zero mean) , E[rk ] = E[sik + nk ] = sik η η Variance σ n = E[n 2 ] − (E[n]) = → σ r2 = σ n = 2 2 2 2 2 Thus {rk} are statistically independent Gaussian variables r r N f(r | si ) = Π f(rk | sik ) where N is number of base vectors k =1 1 2 2 /( 2σ n ) and f(rk | sik ) = e −( rk − sik ) 2π σ n 1 − ( rk − sik ) 2 / η = e πη Take natural logarithm on both sides, gives r r −N 1 N ln(πη ) − ∑ (rk − sik ) 2 ln f(r | si ) =6 Oct. 2009 2 TELE3113 η k =1 6
7. 7. Digital Signal Detection r r −N 1 N ln(πη ) − ∑ (rk − sik ) 2 ln f(r | si ) = 2 η k =1 r r r With ln(•) is a monotonic function, the maximum of f(r | si ) over s i is r equivalent to finding the signal s i that minimizes the Euclidean distance: N r r D(r , si ) = ∑ (rk − sik ) 2 k =1 r r So, the ML decision criterion (maximize f(r | si ) over M signal symbols, i.e. r i=1,…M) reduces to finding the signal s i that is the closest in distance to r the received signal vector r . Example: 3 signal symbols. Note the decision regions formed by the perpendicular bisectors of any two signal symbols.6 Oct. 2009 TELE3113 7
8. 8. Digital Signal Detection r Detection error will occur when the received signal vector r falls into the decision region of other signal symbols. This is due to the presence of strong random noise. Consider there are two signal symbols s1 and s2 , which are spaced d apart. The decision boundary is their perpendicular bisector. r r r r As, r = s i + n , the uncertainty of the received signal vector r is r r r mainly contributed by the random noise n (= (r − s i ) ), which is Gaussian-distributed, around the signal symbol. decision boundary noise distribution s1 d s26 Oct. 2009 TELE3113 8
9. 9. Digital Signal Detection Assume s1 is sent, at the receiver, the probability of rr 1 −( r r r − s1 )2 /η With f(r |s1 ) = e detection error is: πη r r rr rr r rr 1 −( r P ( s 2 is detected s1 is sent ) f(r |s1 ) = f((r -s1 )|s1 ) = f(n|s1 ) = e n )2 /η r r r r r πη = P ( r − s1 > r − s 2 s1 ) ∞ rr = ∫ f (n |s1 ) dn noise decision boundary d/ 2 distribution ∞ 1 ∫ 2 /η = e −n dn d /2 πη ∞ 1 2n ∫ 2 Using Q ( x) = e−y /2 dy and let y = 2π x η ∞ s1 d s2 r r 1 − y2 / 2 P ( s2 is detected s1 is sent ) = ∫ 2π e dy d / 2η  d  = Q   2η   6 Oct. 2009 TELE3113 9
10. 10. Digital Signal Detection r For a signal symbol set: {si } for i = 1,...M Detection error probability is M r r Pe = ∑ P[erroneous detection|si sent ]P[ si ] i =1 r r M M  s k − si  r ≤ ∑∑ Q   P[ si ] i =1 i ≠ k  2η    k =1 r If all signal symbols are equally probable, i.e. P[ si ] = 1 / M M r r Pe = ∑ P[erroneous detection|si sent ]P[ si ] i =1 r r 1 M M s k − si  ≤ ∑∑ Q  2η M i =1 i ≠ k      k =16 Oct. 2009 TELE3113 10
11. 11. Digital Signal Detection Calculation of error probabilities: ( (a) Antipodal signaling: s1 = + E ,0 ; s2 = − E ,0) ( ) Pe = P ( s 2 is detected | s1 ) P ( s1 ) + P ( s1 is detected | s 2 ) P ( s 2 ) 2 E    ≤Q   P ( s1 ) + Q 2 E  P ( s 2 )  2η   2η      s2 s1  2E  = Q  [P ( s1 ) + P ( s 2 )]  η  − E + E    2E  = Q  η   signal symbol energy=E   Example: for NRZ signaling which takes amplitude either +A or 0. For bit interval Tb, the energy per bit Eb=A2Tb.  A2T    Pe = Q  = Q Eb   η   η 6 Oct. 2009  TELE3113    11
12. 12. Digital Signal Detection ( (b) Orthogonal signaling: s1 = + E ,0 ; s2 = 0,+ E ) ( ) Pe = P( s 2 is detected | s1 ) P( s1 ) + P( s1 is detected | s 2 ) P ( s 2 ) + E s2  2E    ≤ Q  P( s1 ) + Q 2 E  P( s 2 ) s1  2η   2η      + E  E   E = Q   [P(s1 ) + P( s 2 )] = Q   η  η   (c) Square signaling: s1 = + ( ) ( ) ( E ,− E ; s2 = + E ,+ E ; s3 = − E ,+ E ; s4 = − E ,− E) ( ) 4 Pe = ∑ P ( si is not detected | si ) P ( si ) s3 + E s2 i =1  2 E        4 ≤ ∑ P ( si )Q  + Q 2 2 E  + Q 2 E   2η   2η   2η  i =1         − E + E  2E   E s4 − E s1  = 2Q  + Q 2    η6 Oct. 2009  η    TELE3113 12
13. 13. Digital Signal Detection Integrate-and-Dump detectorr(t)=si(t)+n(t)  s (t ) = + A 0≤t ≤T for 1 si (t ) =  1 s2 (t ) = − A 0≤t ≤T for 0 t 0 +T  a1 (t ) + no for 1 Output of the integrator: z (t ) = ∫ [si (t ) + n(t )]dt =  t 0 +T t0 a2 (t ) + no for 0 where a1 = ∫ Adt = AT t0 t 0 +T a2 = ∫ (− A)dt = − AT t0 t 0 +T no = ∫ n(t )dt t06 Oct. 2009 TELE3113 13
14. 14. Digital Signal Detection no is a zero-mean Gaussian random variable. t0 +T   t 0 +T  E{no } = E  ∫ n(t )dt  = ∫ E{n(t )}dt = 0  t0   t0    t 0 +T   2  { } σ no = Var{no } = E no = E  ∫ n(t )dt   2 2    t0      t o +T t 0 +T = ∫ ∫ E{n(t )n(ε )}dtdε t0 t0 t o +T t 0 +T η = ∫ ∫ δ (t − ε )dtdε t0 t0 2 t 0 +T η ηT = ∫ t0 2 dε = 2 1 1 pdf of no: f n (α ) = −α 2 /( 2σ no ) 2 2 e = e −α /(ηT ) o 2π σ no πηT6 Oct. 2009 TELE3113 14
15. 15. Digital Signal Detection  s1 (t ) = + A 0≤t ≤T for 1 As si (t ) =  s 0 (t ) = − A 0≤t ≤T for 0 s0 s1 We choose the decision threshold to be 0. 0 − AT + AT Two cases of detection error: (a) +A is transmitted but (AT+no)<0 no<-AT (b) -A is transmitted but (-AT+no)>0 no>+AT Error probability: Pe = P (no < − AT | A) P ( A) + P (no > AT | A) P (− A) − AT 2 ∞ 2 e −α /(ηT ) e −α /(ηT ) = P ( A) −∞ ∫ πηT dα + P (− A) ∫ AT πηT dα ∞ 2 e −α /(ηT )  2 A2T  dα [P( A) + P (− A)] ∞ 2 e −u / 2 = ∫ πηT Thus, Pe = Q   Q Q(x ) = ∫ du AT  η   x 2π ∞ 2 e −u / 2 2α  2 Eb  = ∫ T du Qu = 2π ηT = Q  η   Q Eb = ∫ A2 dt 2 A2T η   06 Oct. 2009 TELE3113 15
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