Lecture 15 with Shot Noise Chapter 10 Four- Dimensional Signal - - PowerPoint PPT Presentation

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Lecture 15

Chapter 10

ECE243b Lightwave Communications - Spring 2019 Lecture 15 1

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Approximate Forms Based on Q

The evaluation of pe based on the value of Q only requires the mean values s0 and s1 and the variances σ2

0 and σ2 1.

For multilevel intensity modulation (See Lecture 13)

pe = L − 1 L erfc

Q

√ 2

• (1)

For intensity modulation σ2 = N0/2 and Q for an additive noise channel is Q = dmin 2σ , (2) The minimum distance dmin of the photodetected signal constellation is 2W /(L − 1) where W is the photocharge Substituting this expression into (2) and then into (1) gives

pe = L − 1 L erfc

• W

(L − 1) √ 2σ

• (3)

For L greater than two, required to maintain the same pe is larger than for binary case by a factor of L − 1.

ECE243b Lightwave Communications - Spring 2019 Lecture 15 2

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Expressions Including Lightwave Amplifier Noise

40 30 20 10 10 Amplified Signal Power dBm 200 190 180 170 160 150 140 130 Noise dBm Hz

Signal Shot Noise + Spontaneous Emission Shot Noise Thermal Noise Total Noise Signal-Noise Mixing Noise-Noise Mixing

Figure: Electrical noise terms for the direct photodetection of an optically-amplified lightwave signal at λ = 1550 nm, for: G = 30 dB, nsp = 1, an electrical noise figure FN = 5 dB, B = 0.1 nm, and an output resistance of R = 50Ω.

Signal - noise beat term dominates for typically operating characteristics.

ECE243b Lightwave Communications - Spring 2019 Lecture 15 3

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Lightwave Amplifiers Based on Q

Signal-dependent variance of the conditional gaussian distribution f(r|1) for a mark is approximated as σ2

1 ≈ 2W1Wn

(4) Term W1 is the expected number of photoelectrons for a mark Term Wn is the expected number of noise photoelectrons generated by ideal photodetection of the noise generated by an ideal lightwave amplifier W = GE where G is the gain of the lightwave amplifier, and E the expected number of photons at the input to the amplifier Using concept of noise figure (Chapter 7) FNP ≈ 2Nsp/G and Wn ≈ Nsp, the expected number of noise photoelectrons is Wn ≈ FNPG/2 Similarly, let W1 = GE1. Substituting these expressions into (4) gives σ2

1 ≈ FNPE1G2

ECE243b Lightwave Communications - Spring 2019 Lecture 15 4

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Lightwave Amplifiers Based on Q

In the same way, when there is a background term E0, the variance σ2

0 for

a transmitted space is σ2

0 ≈ FNPE0G2

Using these expressions, the value of Q is

Q = s1 − s0 σ1 + σ0 = G(E1 − E0)

• FNPE1G2 +
• FNPE0G2

= √E1 − √E0 √FNP , (5)

which is a scaled form of Q for an ideal shot-noise-limited system Q reduced by √FNP to account for the lightwave amplifier noise

ECE243b Lightwave Communications - Spring 2019 Lecture 15 5

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Chain of Lightwave Amplifiers

Now estimate the performance of a span of J fiber segments with amplifier gain and attenuation balanced in each segment The noise at the output is J times as large as the noise in one amplified fiber segment (Chapter 7) Therefore Q is modified to read Q = √E1 − √E0 √JFNP , (6) where the expected values E0 and E1 are defined at the input to the total fiber span If E1 is much greater than E0, then E1 can be solved in terms of the desired value of Q so that E1 ≈ JFNPQ2 (7)

ECE243b Lightwave Communications - Spring 2019 Lecture 15 6

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Demodulation with Phase Noise

Phase-synchronous demodulation requires that the phase of the carrier be known— never known perfectly The estimated phase has a time-varying residual phase error φe(t), called phase noise Causes a corresponding time-varying rotation eiφe(t) of the signal constellation in the complex plane In the presence of phase noise and in the absence of additive noise, the complex sample value r for a Nyquist pulse at the output of a matched filter p(t) is r =

T

sℓ|p(t)|2eiφe(t)dt (8) When phase noise is approximately constant over the symbol interval T, the phase noise φe(t) random process can be treated as a random variable φe for the phase error in each interval.

ECE243b Lightwave Communications - Spring 2019 Lecture 15 7

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Phase Error

A phase error produces a random rotation of the constellation for each symbol interval T.

Error when

(a) (b) √ E − √ E √ E

|φe| > π 2

φe √ E cos φe

Decision Boundary Rotated Signal with a Phase Error

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 for binary phase-shift keying when

• φe
• > π/2.

For binary phase-shift keying, the random rotation eiφe reduces the euclidean distance d between the two signal points defined along the real line by cos φe

ECE243b Lightwave Communications - Spring 2019 Lecture 15 8

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

pe with a Phase Error

Setting E = Eb, where Eb is the expected energy per bit, the conditional probability p(e|φe) of a detection error is p(e|φe) = 1

2erfc

• Eb/N0 cos φe
• The unconditioned probability of a detection error is determined by

averaging over the probability density function f(φe) for the random phase error φe If the phase error after estimation is not too large, then,the probability density function f(φe) for the phase error is well-approximated by a zero-mean gaussian probability density function with a variance σ2

φe

The unconditioned probability of a detection error is

pe =

• ∞

−∞

p(e|φe)f(φe)dφe = 1 2 √ 2πσφe

• ∞

−∞

e

−φ2 e/2σ2 φe erfc

• Eb/N0 cos φe
• dφe

(9)

ECE243b Lightwave Communications - Spring 2019 Lecture 15 9

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Phase Error Curves

Log Probability of a Bit Error

7.5

• 10
• 20
• (b)

Eb/N0 (dB)

5 10 15 20

• 12
• 10
• 8
• 6
• 4
• 2

15

• 5
• Log Probability of a Bit Error

Eb/N0 (dB) (a)

• 10
• 20
• 15
• 5

10 15 20

• 12
• 10
• 8
• 6
• 4
• 2

30

• 40
• Figure: (a) Probability of a detection error for binary phase-shift keying as a function
• f Eb/N0 for several values of the root-mean-squared phase error σe expressed in
• degrees. (b) Probability of a bit error for quadrature phase-shift keying with phase

noise.

ECE243b Lightwave Communications - Spring 2019 Lecture 15 10

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Phase Error Floor

Detection error for binary phase-shift keying occurs when the phase rotation is greater than π/2 Modeling the phase error in each sample as a gaussian random variable with a variance σ2

φe, the pe in the absence of additive noise is pe = 1 − 1 √ 2πσφe

• π/2

−π/2

e

−φ2 e/2σ2 φe dφ

= erfc π/ 8σ2

φe

• This error-rate floor cannot be reduced by increasing the signal power

because the signal power does not affect the variance of the phase error 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

ECE243b Lightwave Communications - Spring 2019 Lecture 15 11

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Demodulation with Shot Noise

The amount of shot noise depends on the signal model and the type of demodulation When a real signal is homodyne-demodulated, the shot noise is equivalent to half a photon When a real or complex signal is heterodyne-demodulated, the shot noise is equivalent to one photon To include shot noise, the expression for the power density spectrum from spontaneous emission N0 = 2erLOWn is augmented by the expression for the power density spectrum from shot noise Nshot = 2erLO (Chapter 7) N0 = Nshot + Nspe = 2erLO (1 + Wn) .

ECE243b Lightwave Communications - Spring 2019 Lecture 15 12

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Demodulation with Shot Noise

For binary phase-shift keying with homodyne demodulation, the average energy per bit Eb is equal to the pulse energy Ep Using this expression gives

pe = 1 2 erfc

• 2Wb

2Wn + 1

• (homodyne BPSK),

(10)

where Wb is the expected number of photoelectrons in a bit and Wn is the expected number of noise photoelectrons Neglecting the one in the denominator due to shot noise, and using Eb ≡ Wb and N0 ≡ Wn recovers pe for BPSK Alternatively, neglecting the spontaneous emission term Wn compared to

• ne gives

pe ≈ 1

2erfc

• 2Wb
• ≈

1 √2πWb e−2Wb(homodyne shot-noise limit)

This expression is the shot-noise limit as the spontaneous emission goes to zero For ideal photodetection with Eb equal to Wb, the effect of the shot noise is the equivalent of an incident level of half a photon

ECE243b Lightwave Communications - Spring 2019 Lecture 15 13

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Heterodyne Demodulation with Shot Noise

For the heterodyne demodulation of either a real or complex signal, a different expression is obtained because of the presence of an image mode.

fc −fc −fLO fLO fIF

Image mode Image mode

−fIF

Shifted negative frequencies

Sum

2fIF

Signal plus image mode quantum noise Frequency

Shifted positive frequencies

Figure: Heterodyne demodulation mixes the vacuum state fluctuations from an unoccupied image mode with the signal mode. This produces a term at the intermediate frequency that contains fluctuations from both modes.

ECE243b Lightwave Communications - Spring 2019 Lecture 15 14

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Heterodyne Demodulation with Shot Noise

The quantum-optics analysis of the image mode noise, (Chaper 15) shows that vacuum fluctuations in the image mode are mixed with the lightwave signal in a heterodyne demodulator This replaces the one by a two in the denominator Setting Ep equal to Eb, gives pe = 1 2erfc

• Wb

Wn + 1

• .

(heterodyne BPSK) (11) The shot-noise-limited probability of a detection error is determined by setting Wn equal to zero and is pe ≈

1 2erfc

• Wb
• (heterodyne shot-noise limit)

(12) ≈ 1 √πWb e−Wb.

ECE243b Lightwave Communications - Spring 2019 Lecture 15 15

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Displacement Detection of a Binary Signal

A method of binary demodulation/detection based on the dual wave/particle nature of a lightwave signal, called a displacement receiver Consider a binary signal constellation that transmits the complex lightwave pulse ±p(t) at a frequency fc using a coherent carrier The expected number of photoelectrons per bit Wb over a symbol interval T is the same for both signs At the receiver, a local oscillator signal sLO(t) is added to the incident pulse as shown in Figure 5.

Detected Symbol Photon counting

±p(t)

+

sLO(t)

Figure: A functional block diagram of a displacement receiver.

The local oscillator signal is equal to p(t) in amplitude, phase, polarization, and time The sum of the two pulses sLO(t) ± p(t) is the input to a photon-counting receiver.

ECE243b Lightwave Communications - Spring 2019 Lecture 15 16

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Displacement Detection of a Binary Signal

When −p(t) is transmitted, the sum of the local oscillator signal and the incident signal is zero When p(t) is transmitted, the sum of the local oscillator signal and the incident signal is 2p(t) and 4Wb photoelectrons are counted over time T The expected number of photoelectrons per bit after adding the local

• scillator and subsequent photon counting is (4Wb + 0)/2 = 2Wb

Is twice as large as ideal photon counting for an equiprobable prior when a matched local oscillator is not used The probability of a detection error for ideal photon counting is modified to read pe =

1 2e−4Wb,

(13) which is an increase in the exponent by a factor of two.

ECE243b Lightwave Communications - Spring 2019 Lecture 15 17

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Mixing Lightwave Attributes

The matching of the lightwave local oscillator signal sLO(t) to the pulse waveform p(t) is based on the wave-optics concept of phase

Shifts or displaces the antipodal signal constellation to a nonnegative signal constellation amenable to photon counting

Photon counting is based on the photon-optics concept of discrete energy A comprehensive analysis of a displacement receiver requires a signal model that encompasses both wave optics and photon optics. Using this model, demodulation/detection methods based on the complete set of properties of a lightwave signal can achieve a lower probability of a detection error compared to any demodulation method based solely on wave optics or photon optics alone. The displacement receiver is one method within this larger class of admissible methods.

ECE243b Lightwave Communications - Spring 2019 Lecture 15 18

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Binary Modulation Formats

Eb (dB) Log pe

Homodyne Photon counting (equi- probable prior) Heterodyne 2 4 6 8 10 12 14 20 15 10 5 Displacement receiver

Figure: The probability of a detection error for several shot-noise-limited binary modulation formats where the number of received photons Eb is equal to the number

• f received photoelectrons Wb for ideal photodetection.

ECE243b Lightwave Communications - Spring 2019 Lecture 15 19

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Four-Dimensional Signal Constellations

A polarization-multiplexed channel modulates a separate complex-baseband waveform onto each of the two polarization modes The two complex numbers, each in a signal constellation in the complex plane, can be regarded as a single four-component point in a four-dimensional euclidean space R4. The signal constellation is then a set of points in R4 The corresponding four-dimensional signal constellation is a set of L = N2 points where N is the number of signal points in each two-dimensional constellation For example, if QPSK is used for each polarization, then N=4 and there are 42 = 16 possible points in the four-dimensional constellation

ECE243b Lightwave Communications - Spring 2019 Lecture 15 20

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Four-Dimensional Signal Constellations

Each four-dimensional signal point can be described by a block signal s given by s = (sAI, sAQ, sBI, sBQ), (14) where the first subscript indexes the polarization component and the second subscript indexes the signal component If QPSK is used for each polarization, then the block signal s is any of the sixteen elements of the finite-dimensional signal constellation Thus, s ∈

• E/4

±1, ±1, ±1, ±1 . (15) The sixteen points in this set all lie on the surface of a hypersphere Each point is equidistant from the origin and equidistant from adjacent signal points This distance, which is the minimum distance dmin is given by dmin =

• (E/4)

1 − (−1)2 = √ E. (16)

ECE243b Lightwave Communications - Spring 2019 Lecture 15 21

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Four-Dimensional Signal Constellations

This suggests a four-dimensional generalization of PSK with L points lying on the surface of a hypersphere. For the same expected symbol energy E, the minimum distance for this four-dimensional constellation is a factor of √ 2 smaller than the minimum distance for a two-dimensional QPSK constellation using a single polarization Is a factor of √ 2 smaller than the distance 2 √ E for a one-dimensional BPSK constellation, for which E = Eb The reduction in the minimum distance as the dimension of the signal constellation increases is offset by the exponential increase in the number

• f possible signal points as the dimension of the signal constellation

increases.

ECE243b Lightwave Communications - Spring 2019 Lecture 15 22

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Four-Dimensional Signal Constellations

The energy efficiency of dual-polarization QPSK modulation is E = d2

min log2 L

4E = E log2 16 4E = 1 This value is equal to the energy efficiency of BPSK because dual-polarization QPSK modulation is equivalent to four orthogonal BPSK signals The probability of a detection error pe is pe = 1 − pc = 1 − 1 − pe(BPSK)4, (17) where the probability of correct decision pc is the product of the probabilities of four independent correct decisions Using a four-dimensional modulation format provides design flexibility to balance bandwidth efficiency and energy efficiency

ECE243b Lightwave Communications - Spring 2019 Lecture 15 23

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Four-Dimensional Signal Constellations

The signal constellation given in (15) suggests a smaller four-dimensional modulation format

s′ ∈

• E/2

(±1, ±1, 0, 0), (0, 0, ±1, ±1) (18)

This format has the same expected energy as before with half the number

• f points

This set corresponds to QPSK using either the eA polarization or the eB polarization for each symbol The minimum distance for this format is

dmin =

• (E/2)

1 − (−1)2 = √ 2E,

and is √ 2 larger than the distance for the constellation defined in (15) This distance is equal to the minimum distance of a single-polarization QPSK constellation It differs from single-polarization QPSK because there are now eight possible points instead of four because both polarizations are partially used The energy efficiency for this format is

E = d2

min log2 L

4E = 2E log2 8 4E = 3/2,

ECE243b Lightwave Communications - Spring 2019 Lecture 15 24

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Dual-Polarization Modulation

Linearly Polarized Coherent Source

+

• +

+

• +

−90o

Polarization combiner Beamsplitter

−90o

ˆ eA ˆ eB sAI(t) sAQ(t) sBQ(t) sBI(t)

cos(2πfct) ˆ eB cos(2πfct) ˆ eA

• sAI(t) + isAQ(t)
• ˆ

eA

• sBI(t) + isBQ(t)
• ˆ

eB Figure: Block diagram of a dual-polarization lightwave modulator.

ECE243b Lightwave Communications - Spring 2019 Lecture 15 25

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Dual-Polarization Demodulation

Filter

[s(t) + n(t)]a ˆ ea [s(t) + n(t)]b ˆ eb

Signal Local

• scillator

Reciever polarization basis

Le−i2πfIFt ˆ ea Le−i2πfIFt ˆ eb

Polarization Alignment

Direct to baseband Homodyne demodulation Sample

Estimated transmitter polarization basis

+

Noise

Direct to baseband Homodyne demodulation

raI raQ rbI rbQ zBI zBQ zAQ zAI z′

AI

z′

AQ

z′

BI

z′

BQ

Figure: Block diagram of a dual-polarization lightwave demodulator.

ECE243b Lightwave Communications - Spring 2019 Lecture 15 26

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Dual-Polarization Demodulation

At the receiver, the two polarization components, ea and eb, are defined by the two polarization beamsplitters shown in the figure

The first polarization beamsplitter projects the noisy received lightwave signal onto each of the two polarization components The second polarization beamsplitter projects the local oscillator onto the same set of two polarization components

For a lossless lightwave channel, the polarization axes at the receiver are rotated with respect to the polarization axes at the transmitter Rotation described by a unitary transformation (rotation on surface of sphere.) Each sample rij of the four demodulated signals is a linear combination

• f the transmitted I and Q components in the two transmitted

polarization components.

ECE243b Lightwave Communications - Spring 2019 Lecture 15 27

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Dual-Polarization Demodulation

For a channel that is dispersionless with fixed polarization modes over a suitable time interval, the appropriate discrete-time channel model for the received noisy block sample z before processing is z = Hs + n, (19) where the channel matrix H depends on space, but not time This is called a space-time separable channel For a dispersionless single-spatial-mode channel with no polarization dependent loss the channel matrix H is a normal channel matrix Techniques to estimate the matrix H are discussed in Chapter 12.

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Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Dual-Polarization Demodulation

For a memoryless space-time separable channel, the block sample r used for detection is generated in two separate steps—one for polarization and

• ne for time.

+ n(t) Block Matched Filter s(t) =

• s1(t)

s2(t)

• Polarization

Alignment Vector Channel Sample H Detection statistic z r z′ H−1 y

Figure: A block diagram of a system that uses dual-polarization modulation.

The first step is polarization alignment that estimates the complex sample value z′ in each transmitted polarization mode ( eA, eB) using a linear combination of the raw sample values z in the polarization modes ( ea, eb) defined at the receiver.

ECE243b Lightwave Communications - Spring 2019 Lecture 15 29

Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Dual-Polarization Demodulation

For each block sample z, the polarization transformation is the inverse H−1 of the channel matrix. The output block sample z′ after the polarization alignment is given by

z′ = H−1z = H−1 (Hs + n) = s + H−1n. (20)

When the channel matrix is unitary, H−1 = H†, and the covariance matrix W of the complex noise samples after the polarization alignment is

W =

• H†n

H†n† = H† nn† H = N0H†H = N0I. (21)

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Lecture 15 Approximate Forms Based on Q Demodulation with Phase Noise Demodulation with Shot Noise Four- Dimensional Signal Con- stellations Dual- Polarization Modulation and De- modulation

Dual-Polarization Demodulation

The second part of the demodulator transforms the sequence of block sample values {z′

k} into the detection statistic {rk} using a matched

filter for each component Yields a pair of complex numbers for the block detection statistic —one complex number for each polarization component in each symbol interval k Given this statistic, the detection process may:

decide each component of the four-dimensional block symbol separately, use joint detection using the components of the block based on prior knowledge of how each block symbol is encoded.

ECE243b Lightwave Communications - Spring 2019 Lecture 15 31