Chapter 5: Optimum Receiver for Binary Data Transmission EE456 -Digital Communications

Chapter 5: Optimum Receiver for Binary Data Transmission EE456 – Digital Communications Professor Ha Nguyen September 2015 EE456 – Digital Communications 1 Chapter 5: Optimum Receiver for Binary Data Transmission Block Diagram of Binary Communication Systems m(t ) ˆ (t ) m
     {bk } { } bˆ k            b k = 0 ↔ s1 (t ) b k = 1 ↔ s2 (t )   r( t ) w (t ) Bits in two different time slots are statistically independent. a priori probabilities: P [bk = 0] = P1 , P [bk = 1] = P2 . Signals s1 (t) and s2 (t) have a duration of Tb seconds and finite energies: R R E1 = 0Tb s21 (t)dt, E2 = 0Tb s22 (t)dt. Noise w(t) is stationary Gaussian, zero-mean white noise with two-sided power spectral density of N0 /2 (watts/Hz):   N0 N0 E{w(t)} = 0, E{w(t)w(t + τ )} = δ(τ ), w(t) ∼ N 0, . 2 2 EE456 – Digital Communications 2 Chapter 5: Optimum Receiver for Binary Data Transmission m(t ) ˆ (t ) m   !" .!*'' ! {bk } { } bˆ k &%$' ()$!*+ '', -+%$' (. /, b k = 0 ↔ s1 (t ) b k = 1 ↔ s2 (t ) #$!!% r( t ) w (t ) Received signal over [(k − 1)Tb , kTb ]: r(t) = si (t − (k − 1)Tb ) + w(t), (k − 1)Tb ≤ t ≤ kTb . Objective is to design a receiver (or demodulator) such that the probability of making an error is minimized. Shall reduce the problem from the observation of a time waveform to that of observing a set of numbers (which are random variables). EE456 – Digital Communications 3 Chapter 5: Optimum Receiver for Binary Data Transmission OOK (Rx) OOK (Tx) NRZ (Rx) NRZ (Tx) Can you Identify the Signal Sets {s1 (t), s2 (t)}? 1 0 −1 0 0.5 1 1.5 2 2.5 t/Tb 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 t/Tb 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 t/Tb 3 3.5 4 4.5 5 0.5 1 1.5 2 2.5 t/Tb 3 3.5 4 4.5 5 5 0 −5 0 1 0 −1 0 5 0 −5 0 EE456 – Digital Communications 4 Chapter 5: Optimum Receiver for Binary Data Transmission Geometric Representation of Signals s1 (t) and s2 (t) (I) Wish to represent two arbitrary signals s1 (t) and s2 (t) as linear combinations of two orthonormal basis functions φ1 (t) and φ2 (t). φ1 (t) and φ2 (t) form a set of orthonormal basis functions if and only if: φ1 (t) and φ2 (t) are orthonormal if: Z Tb φ1 (t)φ2 (t)dt = 0 (orthogonality), 0 Z 0 Tb φ21 (t)dt = Z 0 Tb φ22 (t)dt = 1 (normalized to have unit energy). If {φ1 (t), φ2 (t)} can be found, the representations are s1 (t) = s11 φ1 (t) + s12 φ2 (t), s2 (t) = s21 φ1 (t) + s22 φ2 (t). It can be checked that the coefficients sij can be calculated as follows: sij = Z 0 Tb si (t)φj (t)dt, i, j ∈ {1, 2}, An important question is: Given the signal set s1 (t) and s2 (t), can one always find an orthonormal basis functions to represent {s1 (t), s2 (t)} exactly? If the answer is YES, is the set of orthonormal basis functions UNIQUE? EE456 – Digital Communications 5 Chapter 5: Optimum Receiver for Binary Data Transmission Geometric Representation of Signals s1 (t) and s2 (t) (II) Provided that φ1 (t) and φ2 (t) can be found, the signals (which are waveforms) can be represented as vectors in a vector space (or signal space) spanned (i.e., defined) by the orthonormal basis set {φ1 (t), φ2 (t)}. φ 2 (t ) s2 (t ) s1 (t) = s11 φ1 (t) + s12 φ2 (t), s22 s12 d12 = d 21 E2 s1 ( t ) s2 (t) = s21 φ1 (t) + s22 φ2 (t), Z Tb sij = si (t)φj (t)dt, i, j ∈ {1, 2}, Ei = E1 s21 0 s11 φ1 (t ) Z 0 Tb 0 s2i (t)dt = s2i1 + s2i2 , i ∈ {1, 2}, d12 = d21 = s Z 0 R Tb 0 Tb [s2 (t) − s1 (t)]2 dt si (t)φj (t)dt is the projection of signal si (t) onto basis function φj (t). The length of a signal vector equals to the square root of its energy. It is always possible to find orthonormal basis functions φ1 (t) and φ2 (t) to represent s1 (t) and s2 (t) exactly. In fact, there are infinite number of choices! EE456 – Digital Communications 6 Chapter 5: Optimum Receiver for Binary Data Transmission Gram-Schmidt Procedure Gram-Schmidt (G-S) procedure is one method to find a set of orthonormal basis functions for a given arbitrary set of waveforms. √ s1 (t) 1 Let φ1 (t) ≡ p . Note that s11 = E1 and s12 = 0. E1 ′ s2 (t) 2 Project s (t) = p onto φ1 (t) to obtain the correlation coefficient: 2 E2 ρ= Z Tb 0 s2 (t) 1 √ φ1 (t)dt = √ E2 E1 E2 ′ 3 ′ Subtract ρφ1 (t) from s2 (t) to obtain φ2 (t) = Z Tb s1 (t)s2 (t)dt. 0 s2 (t) √ E2 − ρφ1 (t). ′ 4 Finally, normalize φ2 (t) to obtain: ′ φ2 (t) = = EE456 – Digital Communications ′ φ2 (t) φ (t) = p2 2 Tb  ′ 1 − ρ2 φ (t) dt 2 0   1 ρs1 (t) s2 (t) p √ − √ . E2 E1 1 − ρ2 qR 7 Chapter 5: Optimum Receiver for Binary Data Transmission Gram-Schmidt Procedure: Summary φ2 (t ) s2′ (t ) φ2′ (t ) s2 (t ) s 22 E2 0 = φ2 (t) = s21 = s22 = d21 = d 21 ρφ1 (t ) α E1 s 21 s1 (t ) −1 ≤ ρ = cos(α ) ≤ 1 EE456 – Digital Communications φ1 (t) φ1 (t ) s1 (t) √ , E1   ρs1 (t) s2 (t) 1 p √ − √ , E2 E1 1 − ρ2 Z Tb p s2 (t)φ1 (t)dt = ρ E2 , 0 p p 1 − ρ2 E2 , s Z Tb [s2 (t) − s1 (t)]2 dt 0 = p E1 − 2ρ E1 E2 + E2 . 8 Chapter 5: Optimum Receiver for Binary Data Transmission Example 1 s1 (t ) s 2 (t ) V 0 Tb 0 Tb 0 0 123 −V φ1 (t ) 1 Tb s1 (t ) s 2 (t ) 0 Tb 567 4 − E 0 φ1 (t ) E 89: (a) Signal set. (b) Orthonormal function. (c) Signal space representation. EE456 – Digital Communications 9 Chapter 5: Optimum Receiver for Binary Data Transmission Example 2 s1 (t ) s 2 (t ) V V Tb ; 0 0 ; Tb −V <=> φ 2 (t ) φ1 (t ) 1 Tb 1 Tb Tb 0 φ2 (t ) ? 0 s 2 (t ) E Tb ? s1 (t ) − 1 Tb @AB EE456 – Digital Communications 0 φ1 (t ) E 10 Chapter 5: Optimum Receiver for Binary Data Transmission Example 3 s1 (t ) s 2 (t ) V V 0 α C Tb Tb C 0 −V φ 2 (t, α ) α= I E 3E G− G 2 , 2 H α =0 (− F D D E Tb 2 s 2 (t ) ρ = increasing α , ρ = α= E ,0 Tb 4 ) EE456 – Digital Communications E = s1 (t ) 0 ( E ,0 ) 1 E Z Tb s2 (t)s1 (t)dt 0 1 V 2 Tb 2α −1 Tb h i V 2 α − V 2 (Tb − α) φ1 (t ) 11 Chapter 5: Optimum Receiver for Binary Data Transmission Example 4 s1 (t ) s 2 (t ) 3V V 0 Tb J 0 KLM φ1 (t ) 1 Tb J φ2 (t ) 3 Tb Tb 0 Tb 2 Tb Tb N 0 − Tb 2 N 3 Tb OPQ EE456 – Digital Communications 12 Chapter 5: Optimum Receiver for Binary Data Transmission φ 2 (t ) s2 (t ) E 2 E 60o 0 s1 (t ) φ1 (t ) 3E 2 ! √ √ 3 2 3 V t V dt = , Tb 2 0 0 # " √   1 3 s1 (t) 2 s2 (t) s (t) , φ2 (t) = − ρ s (t) − = √ √ √ 1 2 1 2 E E E (1 − 43 ) 2 √ 3√ 1√ E, s22 = E. s21 = 2 2 ρ= 1 E Z d21 = EE456 – Digital Communications Tb Z s2 (t)s1 (t)dt = Tb 0 2 E Z Tb /2 r 1 √  2 2 = 2 − 3 E. [s2 (t) − s1 (t)] dt 13 Chapter 5: Optimum Receiver for Binary Data Transmission Example 5 φ2 (t ) θ = 3π 2 ρ =0 s1 (t) = √ E s 2 cos(2πfc t), Tb s2 (t) = √ E s 2 cos(2πfc t + θ). Tb where fc = s1 (t ) 0 θ =π ρ = −1 θ k , k an integer. 2Tb φ1 (t ) locus of s2 (t ) as θ E varies from 0 to 2π . s 2 (t ) θ =π 2 ρ =0 EE456 – Digital Communications 14 Chapter 5: Optimum Receiver for Binary Data Transmission Obtaining Different Basis Sets by Rotation φ2 (t ) φ 2 (t ) s2 (t ) φ 2 (t ) s22 s22 s11 θ 0 s11 = s21 s12 s1 ( t ) φ1 (t ) s11 s21 s22 φ 2 (t ) s21 s12 φ1 (t ) s2 (t ) φ1 ( t ) s22 s1 (t ) θ s21 0 s12 s11 φ1 (t ) s12 s11 = s21  φ̂1 (t) φ̂2 (t)  =  cos θ − sin θ sin θ cos θ  φ1 (t) φ2 (t)  . (a) Show that, regardless of the angle θ, the set {φ̂1 (t), φ̂2 (t)} is also an orthonormal basis set. (b) What are the values of θ that make φ̂1 (t) perpendicular to the line joining s1 (t) to s2 (t)? For these values of θ, mathematically show that the components of s1 (t) and s2 (t) along φ̂1 (t), namely ŝ11 and ŝ21 , are identical. Remark: Rotating counter-clockwise for positive θ and clock-wise for negative θ. EE456 – Digital Communications 15 Chapter 5: Optimum Receiver for Binary Data Transmission Example: Rotating {φ1 (t), φ2 (t)} by θ = 60◦ to Obtain {φ̂1 (t), φ̂2 (t)}  φ̂1 (t) φ̂2 (t)  =  cos θ − sin θ sin θ cos θ  φ1 (t) φ2 (t)  ⇒  φ1 (t ) φ̂1 (t) = cos θ × φ1 (t) + sin θ × φ2 (t) φ̂2 (t) = − sin θ × φ1 (t) + cos θ × φ2 (t) φ2 (t ) 1 1 0 .5 0 1 0 1 −1 φ1 (t ) φ2 (t ) (1 + 3 ) 2 (1 − 3 ) 2 0 0 .5 1 (1 − 3 ) 2 − EE456 – Digital Communications 0 0 .5 1 (1 + 3 ) 2 16 Chapter 5: Optimum Receiver for Binary Data Transmission Gram-Schmidt Procedure for M Waveforms {si (t)}M i=1 s1 (t) φ1 (t) = qR φi (t) = qR ′ φi (t) = ρij = ∞ s2 (t)dt −∞ 1 , ′ φi (t) 2 , ∞  ′ φi (t) dt −∞ i = 2, 3, . . . , N, i−1 si (t) X √ − ρij φj (t), Ei j=1 Z ∞ si (t) √ φj (t)dt, j = 1, 2, . . . , i − 1. Ei −∞ If the waveforms {si (t)}M i=1 form a linearly independent set, then N = M . Otherwise N < M. EE456 – Digital Communications 17 Chapter 5: Optimum Receiver for Binary Data Transmission Example: Find a Basis Set for the Following M = 4 Waveforms s1 (t ) s3 ( t ) 1 1 0 1 2 0 2 3 2 3 −1 s4 ( t ) s 2 (t ) 1 1 0 1 −1 EE456 – Digital Communications 2 0 −1 18 Chapter 5: Optimum Receiver for Binary Data Transmission Answer Found by Applying the G-S Procedure φ3 ( t ) φ1 (t ) 1 2 0 1 1 0 2 2 3 −1 φ2 ( t ) 1 2 − 1 2 0 1 2   √ 2 s1 (t)  s2 (t)     =  √0  s3 (t)   √2 s4 (t) 2  EE456 – Digital Communications √0 2 0 0    0 φ1 (t) 0    φ2 (t)  1  φ3 (t) 1 19 Chapter 5: Optimum Receiver for Binary Data Transmission Representation of Noise with Walsh Functions x1(t) 5 0 −5 5 x2(t) 0 −5 2 φ1(t) 1 φ2(t) 0 2 0 −2 2 φ3(t) 0 −2 2 φ4(t) 0 −2 0 0.1 0.2 0.3 0.4 0.5 t 0.6 0.7 0.8 0.9 1 Exact representation of noise using 4 Walsh functions is not possible. EE456 – Digital Communications 20 Chapter 5: Optimum Receiver for Binary Data Transmission The First 16 Walsh Functions 0 0.1 0.2 0.3 0.4 0.5 t 0.6 0.7 0.8 0.9 1 Exact representation might be possible by using many more Walsh functions. EE456 – Digital Communications 21 Chapter 5: Optimum Receiver for Binary Data Transmission The First 16 Sine and Cosine Functions Can also use sine and cosine functions (Fourier representation). 1.5 −1.5 0 EE456 – Digital Communications 0.25 0.5 t 0.75 1 22 Chapter 5: Optimum Receiver for Binary Data Transmission Representation of Noise To represent noise w(t), need to use a complete set of orthonormal functions: Z Tb ∞ X w(t) = wi φi (t), where wi = w(t)φi (t)dt. 0 i=1 The coefficients wi ’s are random variables and understanding their statistical properties is imperative in developing the optimum receiver. Of course, the statistical properties of random variables wi ’s depend on the statistical properties of the noise w(t), which is a random process. In communications, a major source of noise is thermal noise, which is modelled as Additive White Gaussian Noise (AWGN): White: The power spectral density (PSD) is a constant (i.e., flat) over all frequencies. Gaussian: The probability density function (pdf) of the noise amplitude at any given time follows a Gaussian distribution. When w(t) is modelled as AWGN, the projection of w(t) on each basis function, R wi = 0Tb w(t)φi (t)dt, is a Gaussian random variable (this can be proved). For zero-mean and white noise w(t), w1 , w2 , w3 , . . . are zero-mean and uncorrelated random variables: o n 1 2 E{wi } = E R Tb E{wi wj } = E 0 w(t)φi (t)dt nRT 0 b = dλw(λ)φi (λ) R Tb 0 R Tb 0 E{w(t)}φi (t)dt = 0. ( o N0 , 2 dτ w(τ )φj (τ ) = 0, i=j . i 6= j Since w(t) is not only zero-mean and white, but also Gaussian ⇒ {w1 , w2 , . . .} are Gaussian and statistically independent!!! EE456 – Digital Communications 23 Chapter 5: Optimum Receiver for Binary Data Transmission Need to Review Probability Theory & Random Processes – Chapter 3 Slides EE456 – Digital Communications 24 Chapter 5: Optimum Receiver for Binary Data Transmission Observing a waveform ⇒ Observing a set of numbers t = Tb Tb ∫ ( • ) dt 0 r1 = si1 + w1 φ1 ( t ) t = Tb Tb ∫ ( • ) dt 0 r2 = si 2 + w 2 φ2 ( t ) t = Tb r (t ) = Tb ∫ ( • ) dt si (t ) + w (t ) r3 = 0 + w 3 0 AWGN φ3 ( t ) The decision can be based on the observations r1 , r2 , r3 , r4 , . . .. t = Tb Tb ∫ ( • ) dt 0 φn ( t ) EE456 – Digital Communications Choose φ1 (t) and φ2 (t) so that they can be used to represent the two signals s1 (t) and s2 (t) exactly. The remaining orthonormal basis functions are simply chosen to complete the set in order to represent noise exactly. rn = 0 + w n Note that rj , for j = 3, 4, 5, . . ., does not depend on which signal (s1 (t) or s2 (t)) was transmitted. 25 Chapter 5: Optimum Receiver for Binary Data Transmission Optimum Receiver The criterion is to minimize the bit error probability. Consider only the first n terms (n can be very very large), ~ r = {r1 , r2 , . . . , rn } ⇒ Need to partition the n-dimensional observation space into two decision regions, ℜ1 and ℜ2 . Observation space ℜ Decide a "1" was transmitted R if r falls in this region. ℜ1 Decide a "0" was transmitted R if r falls in this region. ℜ2 ℜ1 EE456 – Digital Communications 26 Chapter 5: Optimum Receiver for Binary Data Transmission P [error] = P [(“0” decided and “1” transmitted) or (“1” decided and “0” transmitted)]. = P [0D , 1T ] + P [1D , 0T ] = P [0D |1T ]P [1T ] + P [1D |0T ]P [0T ] = = P [~r ∈ ℜ1 |1T ]P2 + P [~r ∈ ℜ2 |0T ]P1 Z Z f (~ r |0T )d~ r f (~ r |1T )d~ r + P1 P2 ℜ2 ℜ1 = P2 Z ℜ−ℜ2 = P2 Z ℜ = P2 + f (~ r |1T )d~ r+ Z ℜ2 = P1 − f (~ r |1T )d~ r + P1 Z ℜ1 Z ℜ2 Z ℜ2 f (~ r |0T )d~ r [P1 f (~ r |0T ) − P2 f (~ r |1T )]d~ r [P1 f (~ r |0T ) − P2 f (~ r |1T )] d~ r [P1 f (~ r |0T ) − P2 f (~ r |1T )] d~ r. The minimum error probability decision rule is  P1 f (~ r |0T ) − P2 f (~ r |1T ) ≥ 0 ⇒ P1 f (~ r |0T ) − P2 f (~ r |1T ) < 0 ⇒ EE456 – Digital Communications decide “0” (0D ) . decide “1” (1D ) 27 Chapter 5: Optimum Receiver for Binary Data Transmission Pr[error] = t = Tb P2 + Tb ∫ ( • ) dt ∫ [ P f (r | 0 1 ℜ2 ) − P2 f ( r |1T )]dr g(r ) = P1 + ∫ [ P2 f ( r |1T ) − P2 f ( r | 0T )]dr r1 = si1 + w1 0 T ℜ1 φ1 ( t ) −g(r ) 0D t = Tb ⇒ Optimal decision rule: g ( r ) Tb ∫ ( • ) dt 0 r2 = si 2 + w 2 φ2 ( t ) t = Tb r (t ) = Tb ∫ ( • ) dt si (t ) + w (t ) r3 = 0 + w 3 0 AWGN, PSD g(r ) ℜn > 0 < ℜ1 ⇔ 0 D g (r ) > 0 φ3 ( t ) N0 2 > 0 < 1D t = Tb Tb ∫ ( • ) dt 0 φn ( t ) g(r ) < 0 rn = 0 + w n g (r ) = 0 ℜ 2 ⇔ 1D EE456 – Digital Communications 28 Chapter 5: Optimum Receiver for Binary Data Transmission Equivalently, 1D f (~ r |1T ) f (~ r |0T ) The expression R 0D P1 . P2 (1) f (~ r |1T ) is called the likelihood ratio. f (~ r |0T ) The decision rule in (1) was derived without specifying any statistical properties of the noise process w(t). Simplified decision rule when the noise w(t) is zero-mean, white and Gaussian: (r1 − s11 )2 + (r2 − s12 )2 1D R 0D   (r1 − s21 )2 + (r2 − s22 )2 + N0 ln P1 . P2 For the special case of P1 = P2 (signals are equally likely): (r1 − s11 )2 + (r2 − s12 )2 1D R 0D (r1 − s21 )2 + (r2 − s22 )2 . ⇒ minimum-distance receiver! EE456 – Digital Communications 29 Chapter 5: Optimum Receiver for Binary Data Transmission Minimum-Distance Receiver 1D (r1 − s11 )2 + (r2 − s12 )2 | {z } R 0D d2 1 1D d21 R r2 φ 2 (t ) 0D (r1 − s21 )2 + (r2 − s22 )2 {z } | d2 2 d22 s 2 (t ) d2 ( s21 , s22 ) r (t ) d1 ( r1, r2 ) s1 (t ) ( s11 , s12 ) φ1 (t ) 0 Choose s2 (t ) EE456 – Digital Communications Choose s1 ( t ) r1 30 Chapter 5: Optimum Receiver for Binary Data Transmission Correlation Receiver Implementation t = Tb Tb T (•)dt r1 Compute (r1 − si1 )2 + (r2 − si 2 )2 0 r (t ) = si (t ) + w (t ) − N 0 ln( Pi ) φ1 (t ) t = Tb Tb S (•)dt r2 Decision for i = 1, 2 and choose the smallest 0 φ 2 (t ) t = Tb Tb V (•)dt r1 Form 0 the r (t ) φ1 (t ) dot product t = Tb Tb U (•)dt r2 a a N0 E ln(P1 ) − 1 2 2 WXYYZ[ \X[ ]^_`[Z\ Decision r ⋅ si 0 φ 2 (t ) EE456 – Digital Communications N0 E ln( P2 ) − 2 2 2 31 Chapter 5: Optimum Receiver for Binary Data Transmission Receiver Implementation using Matched Filters t = Tb Tb ∫ (•)dt r1 0 r (t ) Decision φ1 (t ) t = Tb Tb ∫ (•)dt r2 0 φ 2 (t ) t = Tb r1 h1 (t ) = φ1 (Tb − t ) r (t ) h2 (t ) = φ 2 (Tb − t ) EE456 – Digital Communications Decision t = Tb r2 32 Chapter 5: Optimum Receiver for Binary Data Transmission Example 5.6 s1 (t ) s 2 (t ) 1.5 0.5 0 0.5 1 b cde φ1 (t ) 1 b 1 f −2 φ 2 (t ) 1 1 0 0.5 0 1 f 0.5 0 −1 ghi EE456 – Digital Communications 33 Chapter 5: Optimum Receiver for Binary Data Transmission φ2 (t ) s 2 (t ) 1 s1 (t ) 0.5 −1 EE456 – Digital Communications − 0.5 0 s1 (t) = s2 (t) = 0.5 1 φ1 (t ) 1 φ2 (t), 2 −φ1 (t) + φ2 (t). φ1 (t) + 34 Chapter 5: Optimum Receiver for Binary Data Transmission For each value of the signal-to-noise ratio (SNR), Matlab simulation was conducted for transmitting/receiving 500 equally-likely bits. (E1 +E2 )/2 N0 =0.99 (dB); P[error]=0.1 3 3 2 2 1 1 φ2 (t) φ2 (t) (E1 +E2 )/2 N0 0 −1 −2 −3 =6.17 (dB); P[error]=0.01 0 −1 −2 −1 EE456 – Digital Communications 0 φ1 (t) 1 2 3 −2 −3 −2 −1 0 φ1 (t) 1 2 3 35 Chapter 5: Optimum Receiver for Binary Data Transmission φ2 (t ) r2 φ2 (t ) r2 s 2 (t ) s 2 (t ) 1 1 jkl mno s1 (t ) s1 (t ) 0.5 0.5 Choose s1 (t ) Choose s2 (t ) −1 − 0.5 0 0.5 1 Choose s1 (t ) Choose s2 (t ) φ1 (t ) r1 −1 − 0.5 0 0.5 1 φ1 (t ) r1 φ2 (t ) r2 s 2 (t ) 1 pqr Choose s2 (t ) s1 (t ) 0.5 Choose s1 (t ) −1 − 0.5 0 0.5 1 φ1 (t ) r1 (a) P1 = P2 = 0.5, (b) P1 = 0.25, P2 = 0.75. (c) P1 = 0.75, P2 = 0.25. EE456 – Digital Communications 36 Chapter 5: Optimum Receiver for Binary Data Transmission Example 5.7 s2 (t) = φ1 (t) + φ2 (t), s1 (t) = φ1 (t) − φ2 (t). φ 2 (t ) φ1 (t ) 3 1 1 0 −1 EE456 – Digital Communications 1 2 s 0 1 s 37 Chapter 5: Optimum Receiver for Binary Data Transmission φ2 (t ) r2 1 s 2 (t ) Choose s 2 (t ) N 0 y P1 v lnw t 4 wx P2 tu φ1 (t ) 0 r1 1 Choose s1 (t ) −1 EE456 – Digital Communications s1 (t ) 38 Chapter 5: Optimum Receiver for Binary Data Transmission t = Tb Tb  (•)dt r (t ) r2 z{|}~~€{ 0 φ 2 (t ) T= 3 0 Tb ˆ r2 Tb ŒŽ‘’‘ ™ T= š›œ EE456 – Digital Communications N 0 ‡ P1 „ ‚ ln 4 † P2 ‚ƒ t = Tb 3 0 r2 < T ‹ choose s1 (t ) ‰~Š h2 (t ) r (t ) r2 ≥ T ‹ choose s2 (t ) r2 ≥ T  choose s2 (t ) r2 < T  choose s1 (t ) N 0 ˜ P1 • ln – “ 4 –— P2 “” 39 Chapter 5: Optimum Receiver for Binary Data Transmission Implementation with One Correlator/Matched Filter Always possible by choosing φ̂1 (t) and φ̂2 (t) such that one of the two basis functions is perpendicular to the line joining the two signals. φ2 (t ) φ1 (t ) s2 (t ) s22 s =s 11 φ 2 (t ) 21 s12 s22 s1 (t ) θ 0 s21 s11 φ1 (t ) s12 The optimum receiver is still the minimum-distance receiver. However the terms (r̂1 − ŝ11 )2 and (r̂1 − ŝ21 )2 are the same on both sides of the comparison and hence can be removed. This means that one does not need to compute r̂1 ! 1D (r̂1 − ŝ11 )2 + (r̂2 − ŝ12 )2 {z } | R 0D d2 1 1D r̂2 R 0D ŝ22 + ŝ12 2 | {z } midpoint of two signals EE456 – Digital Communications +  1D (r̂1 − ŝ21 )2 + (r̂2 − ŝ22 )2 ⇔ (r̂2 − ŝ12 )2 R (r̂2 − ŝ22 )2 {z } | 0D d2 2  N0 /2 ln ŝ22 − ŝ12 | {z  P1 P2 equal to 0 if P1 =P2  } ≡ T. 40 Chapter 5: Optimum Receiver for Binary Data Transmission t = Tb Tb ¥ (•)dt r (t ) r̂2 rˆ2 ≥ T ¨ 1D žŸ ¡¢£¢¤Ÿ£ rˆ2 < T ¨ 0 D 0 φˆ2 ( t ) Threshold T ¦¢§ t = Tb r (t ) r̂2 h (t ) = φˆ2 (Tb − t ) rˆ2 ≥ T ³ 1D ©ª«¬­®­¯ª® rˆ2 < T ³ 0D °±² φ̂2 (t) = Threshold T s2 (t) − s1 (t) ŝ22 + ŝ12 , T ≡ + √ 1 2 2 (E2 − 2ρ E1 E2 + E1 ) EE456 – Digital Communications  N0 /2 ŝ22 − ŝ12  ln  P1 P2  . 41 Chapter 5: Optimum Receiver for Binary Data Transmission Example 5.8 φ2 (t ) φˆ1 (t ) s2 ( t ) φˆ2 (t ) E sˆ11 = sˆ21 θ =π /4 E EE456 – Digital Communications φ̂1 (t) = φ̂2 (t) = s1 (t ) φ1 (t ) 1 √ [φ1 (t) + φ2 (t)], 2 1 √ [−φ1 (t) + φ2 (t)]. 2 42 Chapter 5: Optimum Receiver for Binary Data Transmission t = Tb Tb r (t ) rˆ2 ≥ T ¿ 1D r̂2 » (•)dt ´µ¶·¸¹¸ºµ¹ rˆ2 < T ¿ 0D 0 φˆ2 ( t ) Threshold T 2 Tb 0 Tb 2 Tb ¾ ¼¸½ t = Tb h(t ) r (t ) 2 Tb 0 r̂2 Tb 2 rˆ2 ≥ T Ë 1D ÀÁÂÃÄÅÄÆÁÅ rˆ2 < T Ë 0D Ê Threshold T ÇÈÉ EE456 – Digital Communications 43 Chapter 5: Optimum Receiver for Binary Data Transmission Receiver Performance To detect bk , compare r̂2 = T = ŝ12 +ŝ22 2 + N0 2(ŝ22 −ŝ12 ) Z kTb r(t)φ̂2 (t)dt to the threshold  b (k−1)T P1 . ln P 2 ÍÎÏÐÑÐÒÓ ÔÒÕÓÖ×ØÙ f (rˆ2 0T ) f (rˆ2 1T ) ŝ12 choose 0T ⇐ P [error] ŝ22 r̂2 Ì choose 1T = T P [(0 transmitted and 1 decided) or (1 transmitted and 0 decided)] = P [(0T , 1D ) or (1T , 0D )]. EE456 – Digital Communications 44 Chapter 5: Optimum Receiver for Binary Data Transmission ßàåæ f (rˆ2 0T ) ßàááàâãä çàèéâêåë ìàâíë ìàâíë f (rˆ2 1T ) çàèéâêåë îãéïåéâä ðññéòàåéâä ìàâíë óôõæ ìàâíë ßàåæ ðññéòàåéâä îãéïåéâä çàèéâêåë çàèéâêåë ßàááàâãä öõàä÷ó óôõæ öõàä÷ó ôæøõñéíéáë îãéïåêéñ ùâéøæää ôæøõñéíéáë îãéïåêéñ ùâéøæää ŝ12 ŝ 22 ÚÛÜÝ Þ r̂2 ÚÛÜÝ û ú choose 1T choose 0T ⇐ T P [error] = = P [0T , 1D ] + P [1T , 0D ] = P [1D |0T ]P [0T ] + P [0D |1T ]P [1T ] Z T Z ∞ P1 f (r̂2 |1T )dr̂2 f (r̂2 |0T )dr̂2 +P2 −∞ T {z } | {z } | Area B = EE456 – Digital Communications P1 Q T − ŝ12 p N0 /2 Area A ! " + P2 1 − Q T − ŝ22 p N0 /2 !# . 45 Chapter 5: Optimum Receiver for Binary Data Transmission Q-function 2   1 − λ2 e 2π ý               

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x 1 Q(x) ≡ √ 2π λ Area = Q ( x ) Z ∞ exp x − λ2 2 ! dλ. 0 10 −2 10 −4 Q(x) 10 −6 10 −8 10 −10 10 EE456 – Digital Communications 0 1 2 3 x 4 5 6 46 Chapter 5: Optimum Receiver for Binary Data Transmission Performance when P1 = P2 f (r2 0T ) 0 f (r2 1T ) s12 r2 s 22 =Q choose 0T ⇐ T= P [error] = Q ŝ22 − ŝ12 p 2 N0 /2 ! ( s22 − s12 2 N 0 /2 ) ⇒ choose 1T s12 + s22 2 =Q  distance between the signals 2 × noise RMS value  . Probability of error decreases as either the two signals become more dissimilar (increasing the distances between them) or the noise power becomes less. To maximize the distance between the two signals one chooses them so that they are placed 180◦ from each other ⇒ s2 (t) = −s1 (t), i.e., antipodal signaling. The error probability does not depend on the signal shapes but only on the distance between them. EE456 – Digital Communications 47 Chapter 5: Optimum Receiver for Binary Data Transmission Example 5.9 1 φ2 (t ) 1 0 t 2 s1 (t ) ⇔ 0T 1 1 −2 −1 0 1 2 −1 1T ⇔ s 2 (t ) 1 φ1 (t ) 0 t −1 −2 (a) Determine and sketch the two signals s1 (t) and s2 (t). EE456 – Digital Communications 48 Chapter 5: Optimum Receiver for Binary Data Transmission (b) The two signals s1 (t) and s2 (t) are used for the transmission of equally likely bits 0 and 1, respectively, over an additive white Gaussian noise (AWGN) channel. Clearly draw the decision boundary and the decision regions of the optimum receiver. Write the expression for the optimum decision rule. (c) Find and sketch the two orthonormal basis functions φ̂1 (t) and φ̂2 (t) such that the optimum receiver can be implemented using only the projection r̂2 of the received signal r(t) onto the basis function φ̂2 (t). Draw the block diagram of such a receiver that uses a matched filter. (d) Consider now the following argument put forth by your classmate. She reasons that since the component of the signals along φ̂1 (t) is not useful at the receiver in determining which bit was transmitted, one should not even transmit this component of the signal. Thus she modifies the transmitted signal as follows:   (M) s1 (t) = s1 (t) − component of s1 (t) along φ̂1 (t)   (M) s2 (t) = s2 (t) − component of s2 (t) along φ̂1 (t) (M) (M) Clearly identify the locations of s1 (t) and s2 (t) in the signal space diagram. What is the average energy of this signal set? Compare it to the average energy of the original set. Comment. EE456 – Digital Communications 49 Chapter 5: Optimum Receiver for Binary Data Transmission s 2 (t ) s1 (t ) 3 1 t 1 0 0 −1 −1 t −3 EE456 – Digital Communications 50 Chapter 5: Optimum Receiver for Binary Data Transmission 1 φ2 (t ) 0   !"# 0D 1D 2 1 t φˆ2 ( t ) s1M (t ) s1 (t ) ⇔ 0T 1 1 −2 −1 0 1 θ =− s2M (t ) 1T ⇔ s 2 (t ) EE456 – Digital Communications π 2 −1 4 −2 φˆ1 (t ) 1 φ1 (t ) 0 t −1 51 Chapter 5: Optimum Receiver for Binary Data Transmission  φ̂1 (t) φ̂2 (t)  =  cos(−π/4) − sin(−π/4) sin(−π/4) cos(−π/4)  φ1 (t) φ2 (t)  = " √1 2 √1 2 − √1 2 1 √ 2 # φ1 (t) φ2 (t)  . 1 φ̂2 (t) = √ [φ1 (t) + φ2 (t)]. 2 1 φ̂1 (t) = √ [φ1 (t) − φ2 (t)], 2 φˆ1 (t ) φˆ2 (t ) 2 1/ 2 1 t 0 0 1/ 2 t − 2 h (t ) = φˆ2 (1 − t ) t =1 r (t ) 2 0 EE456 – Digital Communications 1/ 2 1 rˆ2 ≥ 0 $ 0D rˆ2 < 0 $ 1D t 52 Chapter 5: Optimum Receiver for Binary Data Transmission Antipodal Signalling w (t ) PSD N0 2 t = kTb (1) Tb 2Tb 0 t ( −1) ( −1) hT (t ) = p( t ) x (t ) s2 (t ) = − s1 ( t ) = p ( t ) E1 = E2 = E + hR (t ) = κ p (Tb − t ) y (t ) 0 y ( k ) = ± (κ E ) + w ( k ) (0, N20 E ) The pulse shaping filter hT = p(t) defines the power spectrum density of the transmitted signal, which can be shown to be proportional to |P (f )|2 . The error performance, P [error] only depends on the energy E of p(t) √and noise PSD level N0 . Specifically, the distance between s1 (t) and s2 (t) is 2 E (you should show this for yourself, algebraically or geometrically). Therefore s ! 2E P [error] = Q . N0 For antipodal signalling, the optimum decisions are performed by comparing the samples of the matched filter’s output (sampled at exactly integer multiples of the bit duration) with a threshold 0. Of course such an optimum decision rule does not change if the impulse response of the matched filter is scaled by a positive constant. EE456 – Digital Communications 53 Chapter 5: Optimum Receiver for Binary Data Transmission Scaling the matched filter’s impulse response hR (t) does not change the receiver performance because it scales both signal and noise components by the same factor, leaving the signal-to-noise ratio (SNR) of the decision variable unchanged! √ In the above block diagram, hR (t) = κp(Tb − t). We have been using κ = 1/ E in√ order to represent the signals on the signal space diagram (which would be at ± E) and to conclude that the variance of the noise component is exactly N0 /2. For an arbitrary scaling factor κ, the signal component becomes ±κE, while the variance of the noise component is N20 κ2 E. Thus, the SNR is SNR = (±κE)2 Signal power 2E = N , = 0 κ2 E Noise power N0 2 (indepedent of κ!) In terms of the SNR, the error performance of antipodal signalling is P [error] = Q s 2E N0 ! =Q √  SNR In fact, it can be proved that the receive filter that maximizes the SNR of the decision variable must be the matched filter. It is important to emphasize that the matching property here concerns the shapes of the impulse responses of the transmit and receive filters. EE456 – Digital Communications 54 Chapter 5: Optimum Receiver for Binary Data Transmission Outputs of the Matched/Mismatched Filters (No-Noise Scenario) Clean received signals for rect and half-sine (HS) shaping filters Output of a matched filter: rect/rect matching Outputs of HS/HS matched filter (red) and HS/rect mismatched filter (pink) When the matched filter is used, sampling at exact multiples of the bit duration maximizes the power of the signal component in the decision variable, hence maximizing the SNR. A timing error (imperfect sampling) would reduce the power of the signal component, hence reducing the SNR, hence degrading the performance, i.e., increasing P [error]. When the receive filter is not matched to the transmit filter, the power of the signal component and the SNR are not maximized, even under perfect sampling! EE456 – Digital Communications 55 Chapter 5: Optimum Receiver for Binary Data Transmission Antipodal Baseband Signalling with Rectangular Pulse Shaping bk h(t ) = p(Tb − t ) EE456 – Digital Communications 56 Chapter 5: Optimum Receiver for Binary Data Transmission Antipodal Baseband Signalling with Half-Sine Pulse Shaping bk h(t ) = p (Tb − t ) EE456 – Digital Communications 57 Chapter 5: Optimum Receiver for Binary Data Transmission PSD Derivation of Arbitrary Binary Modulation Applicable to any binary modulation with arbitrary a priori probabilities, but restricted to statistically independent bits. sT (t ) s1 ( t − 2Tb ) s1 (t ) −2Tb −Tb 0 2Tb Tb 3Tb 4Tb t s2 (t − 3Tb ) sT (t) = ∞ X k=−∞ gk (t), gk (t) =  s1 (t − kTb ), s2 (t − kTb ), with probability P1 . with probability P2 The derivation on the next slide shows that: ∞ P 1 S1 X P1 P2 SsT (f ) = |S1 (f ) − S2 (f )|2 + Tb n=−∞ EE456 – Digital Communications  n Tb  + P 2 S2 Tb  n Tb  2 δ  f− n Tb  . 58 Chapter 5: Optimum Receiver for Binary Data Transmission sT (t) = E{sT (t)} + sT (t) − E{sT (t)} = v(t) + q(t) | {z } | {z } DC AC ∞ X v(t) = E{sT (t)} = k=−∞ Sv (f ) = ∞ X n=−∞ Sv (f ) = |Dn |2 δ ∞ X n=−∞  [P1 s1 (t − kTb ) + P2 s2 (t − kTb )] f− P1 S1 n Tb   n Tb 1 Tb , Dn =  + P2 S2 Tb   n Tb P1 S1  2 δ  n Tb  + P2 S2  n Tb  ,  .   n . f− Tb To calculate Sq (f ), apply the basic definition of PSD: Sq (f ) = lim T →∞ SsT (f ) = P1 P2 E{|GT (f )|2 } = ··· = |S1 (f ) − S2 (f )|2 . T Tb ∞ P1 S1 X P1 P2 |S1 (f ) − S2 (f )|2 + Tb n=−∞  n Tb  + P2 S2 Tb  n Tb  2 δ  f− n Tb For the special, but important case of antipodal signalling, s2 (t) = −s1 (t) = p(t), and equally likely bits, P1 = P2 = 0.5, the PSD of the transmitted signal is solely determined by the Fourier transform of p(t): |P (f )|2 SsT (f ) = Tb EE456 – Digital Communications 59 Chapter 5: Optimum Receiver for Binary Data Transmission Baseband Message Signals with Different Pulse Shaping Filters Information bits or amplitude levels 1 0.5 0 −0.5 −1 2 4 6 8 10 12 Output of the transmit pulse shaping filter − Rectangular 14 PSD - Rectangular Magnitude (dB) 2 1 0 −1 −2 2 4 6 8 10 Half-sine pulse shaping filter 12 Magnitude (dB) 1 0 −1 2 4 6 8 10 SRRC pulse shaping filter (β = 0.5) 12 Magnitude (dB) 1 0 −1 2 EE456 – Digital Communications 4 6 8 t/T b 10 12 −40 −60 14 −2 0 2 f × Tb PSD - Half-sine 0 −20 −40 −60 14 2 −2 −20 14 2 −2 0 −2 0 2 f × Tb PSD - SRRC (30 symbols long) 0 −20 −40 −60 −2 0 2 Normalized frequency, f × T b 60 Chapter 5: Optimum Receiver for Binary Data Transmission Building A Binary (Antipodal) Comm. System in Labs #4 and #5 low-rate sequence, index k high-rate sequences, index n ... ... ... t s[n ] = i[n ] * h[n ] = å a[k ] p(nT - kTsym ) data bits k S/P bits-tolevels mapping N N= Tsym T pulse shaping filter hT [n ] = p(nT ) i[n ] = i (nT ) = å a[k ]d (nT - kTsym ) k DAC correction filter DAC cos( wˆ c n ) NCO fˆc T wc = wˆ c Fs æ ö s(t ) = ç å a ( k ) p(t - kTs ) ÷ cos(wct ) è k ø Figure 1: Block diagram of the transmitter. EE456 – Digital Communications 61 Chapter 5: Optimum Receiver for Binary Data Transmission s (t ) ∑ x[ n − d ] x[ n] hR [n ] = p( − nT ) xc [ n ] k =n Tsym T cos( ω c n − θ ) θ fc T sin( ω c n − θ ) ωc d = θ x s [n ] Figure 2: Block diagram of the receiver. EE456 – Digital Communications 62