Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation Lecture 15 with Shot Noise Chapter 10 Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation 1 ECE243b Lightwave Communications - Spring 2019 Lecture 15
Approximate Forms Based on Q Lecture 15 The evaluation of p e based on the value of Q only requires the mean Approximate values s 0 and s 1 and the variances σ 2 0 and σ 2 1 . Forms Based on Q For multilevel intensity modulation (See Lecture 13) Demodulation � Q � with Phase L − 1 Noise = p e erfc √ (1) L 2 Demodulation with Shot Noise For intensity modulation σ 2 = N 0 / 2 and Q for an additive noise channel Four- is Dimensional Signal Con- stellations d min = Q 2 σ , (2) Dual- Polarization Modulation and De- The minimum distance d min of the photodetected signal constellation is modulation 2 W / ( L − 1 ) where W is the photocharge Substituting this expression into (2) and then into (1) gives � � L − 1 W p e = erfc √ (3) L ( L − 1 ) 2 σ For L greater than two, required to maintain the same p e is larger than for binary case by a factor of L − 1 . 2 ECE243b Lightwave Communications - Spring 2019 Lecture 15
Expressions Including Lightwave Amplifier Noise Lecture 15 Approximate Forms Based on 130 Q Total Noise Demodulation 140 with Phase Noise Noise dBm Hz 150 Demodulation with Shot Noise-Noise Mixing 160 Noise Thermal Noise Four- 170 Dimensional Signal Con- stellations Signal-Noise 180 Signal Shot Noise + Mixing Dual- Spontaneous Emission Shot Noise Polarization 190 Modulation and De- 200 modulation 40 30 20 10 0 10 Amplified Signal Power dBm Figure: Electrical noise terms for the direct photodetection of an optically-amplified lightwave signal at λ = 1550 nm, for: G = 30 dB, n sp = 1 , an electrical noise figure F N = 5 dB, B = 0.1 nm, and an output resistance of R = 50 Ω . Signal - noise beat term dominates for typically operating characteristics. 3 ECE243b Lightwave Communications - Spring 2019 Lecture 15
Lightwave Amplifiers Based on Q Lecture 15 Approximate Signal-dependent variance of the conditional gaussian distribution f ( r | 1 ) Forms Based on for a mark is approximated as Q Demodulation σ 2 1 ≈ 2 W 1 W n (4) with Phase Noise Demodulation Term W 1 is the expected number of photoelectrons for a mark with Shot Noise Four- Term W n is the expected number of noise photoelectrons generated by Dimensional Signal Con- ideal photodetection of the noise generated by an ideal lightwave amplifier stellations Dual- W = G E where G is the gain of the lightwave amplifier, and E the Polarization Modulation expected number of photons at the input to the amplifier and De- modulation Using concept of noise figure (Chapter 7) F NP ≈ 2 N sp / G and W n ≈ N sp , the expected number of noise photoelectrons is W n ≈ F NP G / 2 Similarly, let W 1 = G E 1 . Substituting these expressions into (4) gives σ 2 1 ≈ F NP E 1 G 2 4 ECE243b Lightwave Communications - Spring 2019 Lecture 15
Lightwave Amplifiers Based on Q Lecture 15 Approximate Forms Based on Q Demodulation with Phase In the same way, when there is a background term E 0 , the variance σ 2 0 for Noise a transmitted space is Demodulation with Shot σ 2 0 ≈ F NP E 0 G 2 Noise Four- Using these expressions, the value of Q is Dimensional Signal Con- √ E 1 − √ E 0 stellations G ( E 1 − E 0 ) s 1 − s 0 � � Q = = = √ F NP , (5) Dual- σ 1 + σ 0 F NP E 1 G 2 + F NP E 0 G 2 Polarization Modulation which is a scaled form of Q for an ideal shot-noise-limited system and De- modulation Q reduced by √ F NP to account for the lightwave amplifier noise 5 ECE243b Lightwave Communications - Spring 2019 Lecture 15
Chain of Lightwave Amplifiers Lecture 15 Approximate Forms Now estimate the performance of a span of J fiber segments with Based on Q amplifier gain and attenuation balanced in each segment Demodulation with Phase Noise The noise at the output is J times as large as the noise in one amplified Demodulation fiber segment (Chapter 7) with Shot Noise Therefore Q is modified to read Four- Dimensional √ E 1 − √ E 0 Signal Con- stellations Q = √ JF NP (6) , Dual- Polarization Modulation where the expected values E 0 and E 1 are defined at the input to the total and De- modulation fiber span If E 1 is much greater than E 0 , then E 1 can be solved in terms of the desired value of Q so that JF NP Q 2 E 1 ≈ (7) 6 ECE243b Lightwave Communications - Spring 2019 Lecture 15
Demodulation with Phase Noise Lecture 15 Phase-synchronous demodulation requires that the phase of the carrier be Approximate Forms known— never known perfectly Based on Q Demodulation The estimated phase has a time-varying residual phase error φ e ( t ) , called with Phase Noise phase noise Demodulation with Shot Causes a corresponding time-varying rotation e i φ e ( t ) of the signal Noise Four- constellation in the complex plane Dimensional Signal Con- stellations In the presence of phase noise and in the absence of additive noise, the Dual- complex sample value r for a Nyquist pulse at the output of a matched Polarization Modulation filter p ( t ) is and De- modulation � T s ℓ | p ( t ) | 2 e i φ e ( t ) d t = (8) r 0 When phase noise is approximately constant over the symbol interval T , the phase noise φ e ( t ) random process can be treated as a random variable φ e for the phase error in each interval. 7 ECE243b Lightwave Communications - Spring 2019 Lecture 15
Phase Error Lecture 15 A phase error produces a random rotation of the constellation for each Approximate Forms symbol interval T . Based on Q Rotated Signal Error when Demodulation with a Phase Error | φ e | > π with Phase 2 Noise √ E Decision Demodulation Boundary with Shot Noise φ e √ √ E E Four- − Dimensional Signal Con- stellations √ E cos φ e Dual- Polarization (a) (b) Modulation and De- modulation Figure: (a) When the phase noise is slowly varying, it produces a random rotation of the signal constellation of a symbol interval T . (b) An error occurs � � � φ e � > π / 2 . for binary phase-shift keying when For binary phase-shift keying, the random rotation e i φ e reduces the euclidean distance d between the two signal points defined along the real line by cos φ e 8 ECE243b Lightwave Communications - Spring 2019 Lecture 15
p e with a Phase Error Lecture 15 Setting E = E b , where E b is the expected energy per bit, the conditional Approximate Forms probability p ( e | φ e ) of a detection error is Based on Q �� � Demodulation p ( e | φ e ) = 1 2 erfc E b / N 0 cos φ e with Phase Noise Demodulation with Shot The unconditioned probability of a detection error is determined by Noise averaging over the probability density function f ( φ e ) for the random Four- Dimensional phase error φ e Signal Con- stellations If the phase error after estimation is not too large, then,the probability Dual- Polarization density function f ( φ e ) for the phase error is well-approximated by a Modulation and De- zero-mean gaussian probability density function with a variance σ 2 modulation φ e The unconditioned probability of a detection error is � ∞ p e = p ( e | φ e ) f ( φ e ) d φ e −∞ � �� � ∞ − φ 2 e / 2 σ 2 1 = φe erfc (9) √ e E b / N 0 cos φ e d φ e 2 2 πσ φe −∞ 9 ECE243b Lightwave Communications - Spring 2019 Lecture 15
Phase Error Curves Lecture 15 Approximate Forms Based on Q 0 0 Log Probability of a Bit Error Log Probability of a Bit Error Demodulation (a) (b) o 40 with Phase o 20 - 2 - 2 Noise o 30 o 15 - 4 Demodulation - 4 with Shot o 20 o 10 Noise - 6 - 6 Four- - 8 o - 8 7.5 o 15 Dimensional Signal Con- - 10 o 10 - 10 stellations o 5 o 0 o 0 - 12 - 12 Dual- 0 5 10 15 20 0 5 10 15 20 Polarization E b /N 0 (dB) E b /N 0 (dB) Modulation and De- modulation Figure: (a) Probability of a detection error for binary phase-shift keying as a function of E b / N 0 for several values of the root-mean-squared phase error σ e expressed in degrees. (b) Probability of a bit error for quadrature phase-shift keying with phase noise. 10 ECE243b Lightwave Communications - Spring 2019 Lecture 15
Phase Error Floor Lecture 15 Approximate Forms Based on Q Detection error for binary phase-shift keying occurs when the phase Demodulation rotation is greater than π / 2 with Phase Noise Modeling the phase error in each sample as a gaussian random variable Demodulation with Shot with a variance σ 2 φ e , the p e in the absence of additive noise is Noise � Four- erfc � π / � � π / 2 Dimensional − φ 2 e / 2 σ 2 1 = 1 − φe d φ = 8 σ 2 Signal Con- p e √ e φe stellations 2 πσ φe − π / 2 Dual- Polarization This error-rate floor cannot be reduced by increasing the signal power Modulation and De- because the signal power does not affect the variance of the phase error modulation unless the phase is estimated from the received signal When the phase is estimated,(Chapter 12), increasing the signal power reduces the variance of the estimated phase error 11 ECE243b Lightwave Communications - Spring 2019 Lecture 15
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