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Lecture 7 Multi- Input Multi- Output Lightwave Channels Polarization- Multiplexed Channel Lecture 7 Space- Multiplexed Channel Electrical Channels-2 Types of Electrical Channels Linear Electrical Channels in the Lightwave

SLIDE 1

Lecture 7 Multi- Input Multi- Output Lightwave Channels

Polarization- Multiplexed Channel Space- Multiplexed Channel

Types of Electrical Channels

Linear Electrical Channels in the Lightwave Amplitude

Lecture 7

Electrical Channels-2

ECE243b Lightwave Communications - Spring 2019 Lecture 7 1

SLIDE 2

Lecture 7 Multi- Input Multi- Output Lightwave Channels

Polarization- Multiplexed Channel Space- Multiplexed Channel

Types of Electrical Channels

Linear Electrical Channels in the Lightwave Amplitude

MIMO Channels

Multiple independent datastreams, called subchannels can often be supported by the same physical medium. This kind of channel is called a multi-input multi-output (MIMO) channel. The number of subchannels and the properties of each subchannel depend on the number of spatial and polarization modes that physical medium supports. In general the output of each subchannel depends on more than one input, and the number of subchannels does not necessarily correspond to the maximum number of subchannels that the physical channel can support.

ECE243b Lightwave Communications - Spring 2019 Lecture 7 2

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Lecture 7 Multi- Input Multi- Output Lightwave Channels

Polarization- Multiplexed Channel Space- Multiplexed Channel

Types of Electrical Channels

Linear Electrical Channels in the Lightwave Amplitude

Example - Polarization Channels

(a) (b) (c)

Polarization beamsplitter Polarization combiner

Modulator (E/O)

Photo- detection (O/E) O/E O/E O/E O/E E/O E/O E/O Lightwave Channel Electrical Channel

ECE243b Lightwave Communications - Spring 2019 Lecture 7 3

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Lecture 7 Multi- Input Multi- Output Lightwave Channels

Polarization- Multiplexed Channel Space- Multiplexed Channel

Types of Electrical Channels

Linear Electrical Channels in the Lightwave Amplitude

Types of Lightwave MIMO Channels (b) (a)

ECE243b Lightwave Communications - Spring 2019 Lecture 7 4

SLIDE 5

Lecture 7 Multi- Input Multi- Output Lightwave Channels

Polarization- Multiplexed Channel Space- Multiplexed Channel

Types of Electrical Channels

Linear Electrical Channels in the Lightwave Amplitude

Two-by-two MIMO Channel

Now consider a multi-input multi-output lightwave channel that has two inputs and two outputs. Let sin(t) be a block with components s1(t) and s2(t) that are the input lightwave waveforms for each subchannel. Let sout(t) be the corresponding output block. The linear channel response can be written as sout(t) = h(t) ⊛ sin(t) (1) . =

h11(t) ⊛ s1(t) + h12(t) ⊛ s2(t)

h21(t) ⊛ s1(t) + h22(t) ⊛ s2(t)

(2) In this expression h(t) representes the impulse response of the multi-input multi-output channel with hij(t) being the complex-baseband impulse response for the ith output subchannel from the jthe input subchannel.

ECE243b Lightwave Communications - Spring 2019 Lecture 7 5

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Lecture 7 Multi- Input Multi- Output Lightwave Channels

Polarization- Multiplexed Channel Space- Multiplexed Channel

Types of Electrical Channels

Linear Electrical Channels in the Lightwave Amplitude

Wavelength-dependent Delay

The output block Sout(f)in the frequency domain is Sout(f) = H(f)Sin(f) (3) =

H11(f)S1(f) + H12(f)S2(f)

H21(f)S1(f) + H22(f)S2(f)

(4) The transfer function Hij(f) is the Fourier transform of the complex-baseband impulse response hij(t). The matrix H consisting of the transfer functions Hij(f) is called the channel matrix. The on-diagonal elements of the channel matrix describe the output lightwave signal in each subchannel from the corresponding input subchannel. The off-diagonal elements of the channel matrix describe the linear redistribution

f the lightwave signal energy between the subchannels

linear interchannel interference.

ECE243b Lightwave Communications - Spring 2019 Lecture 7 6

SLIDE 7

Lecture 7 Multi- Input Multi- Output Lightwave Channels

Polarization- Multiplexed Channel Space- Multiplexed Channel

Types of Electrical Channels

Linear Electrical Channels in the Lightwave Amplitude

Memoryless Channel Matrix

For a memoryless channel, there is no frequency dependence of the channel matrix and we can write sout(t) = Hsin(t) (5) with the elements of the channel matrix H describing the coupling between the subchannels.

ECE243b Lightwave Communications - Spring 2019 Lecture 7 7

SLIDE 8

Lecture 7 Multi- Input Multi- Output Lightwave Channels

Polarization- Multiplexed Channel Space- Multiplexed Channel

Types of Electrical Channels

Linear Electrical Channels in the Lightwave Amplitude

Polarization-Multiplexed Channel

A common multi-input, multi-output lightwave channel is a polarization-multiplexed channel For this channel, random coupling between the polarization modes introduces a combination of polarization-mode dispersion and polarization-dependent loss. Consider a fiber segment that has only first-order polarization-dependent group delay and no polarization-dependent loss. For a short segment of fiber, the principal polarization states have a fixed

rientation within the segment with constant polarization-dependent group delay.

ECE243b Lightwave Communications - Spring 2019 Lecture 7 8

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Lecture 7 Multi- Input Multi- Output Lightwave Channels

Polarization- Multiplexed Channel Space- Multiplexed Channel

Types of Electrical Channels

Linear Electrical Channels in the Lightwave Amplitude

Channel Matrix for Static Segment

The transfer function for each principal polarization state, denoted by H+(f) and H−(f), can be written as H+(f) = H(f)e−iπfτ (6) H−(f) = H(f)eiπfτ, (7) where H(f) = H0e−i2π2β2f2L is the polarization-independent, wavelength-dependent delay given in (??), and ±τ/2 is the differential group delay for each principal polarization state. The channel matrix H(f) for a segment of fiber for which the principal states of polarization are fixed has the form in principal polarization basis as H(f) = H(f)

e−iπfτ

eiπfτ

H(f)P(f), (8) where P(f) is the part of the channel matrix that is polarization dependent.

ECE243b Lightwave Communications - Spring 2019 Lecture 7 9

SLIDE 10

Lecture 7 Multi- Input Multi- Output Lightwave Channels

Polarization- Multiplexed Channel Space- Multiplexed Channel

Types of Electrical Channels

Linear Electrical Channels in the Lightwave Amplitude

Channel Matrix for Span of Many Segments

For a fiber span that consists of many segments, the differential group delay varies from segment to segment with the overall delay modeled as a random variable τ that has a maxwellian probability density function. The corresponding random channel matrix H(f), defined in the principal polarization-state basis, is modified to read H(f) = H(f)

e−iπfτ

eiπfτ

H(f)P(f) (9) where P(f) is the random part of the channel matrix that is polarization dependent. The polarization basis defined by a polarization beamsplitter at a lightwave receiver is not usually aligned with the principal polarization axes.

ECE243b Lightwave Communications - Spring 2019 Lecture 7 10

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Lecture 7 Multi- Input Multi- Output Lightwave Channels

Polarization- Multiplexed Channel Space- Multiplexed Channel

Types of Electrical Channels

Linear Electrical Channels in the Lightwave Amplitude

Space-Multiplexed Channel

A space-multiplexed channel consists of spatially-separated cores within the same cladding structure. Analyzed in isolation, the modes for each core can be determined using the methods presented in Chapter 3. If the cores are sufficiently close, then the modes in a set of cores can couple. Weak coupling between a pair of cores can be analyzed using coupled-mode theory. For a long segment of fiber, there are many small perturbations that can cause coupling. Over the complete length of the segment, the random coupling can be modeled as a zero-mean, circularly-symmetric gaussian random variable with a variance σ2

ij for each quadrature signal component.

ECE243b Lightwave Communications - Spring 2019 Lecture 7 11

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Lecture 7 Multi- Input Multi- Output Lightwave Channels

Polarization- Multiplexed Channel Space- Multiplexed Channel

Types of Electrical Channels

Linear Electrical Channels in the Lightwave Amplitude

For many modulation formats, a real-baseband waveform sI(t) for the in-phase signal component is modulated onto a coherent lightwave carrier cos(2πfct). A separate real-baseband waveform sQ(t) for the quadrature signal component is modulated onto an orthogonal coherent lightwave carrier sin(2πfct). Ignoring scaling constants, the two modulated signals are combined to produce a passband lightwave signal

s(t)

= sI(t) cos 2πfct − sQ(t) sin 2πfct = As(t) cos 2πfct + φs(t) = Re[s(t)ei2πfct] (10) The equivalent complex-baseband signal is s(t) = sI(t) + isQ(t) = As(t)eiφs(t). (11)

ECE243b Lightwave Communications - Spring 2019 Lecture 7 12

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Lecture 7 Multi- Input Multi- Output Lightwave Channels

Polarization- Multiplexed Channel Space- Multiplexed Channel

Types of Electrical Channels

Linear Electrical Channels in the Lightwave Amplitude

+

cos(2πfct)

−90o

sQ(t) sI(t) sI(t) cos(2πfct) −sQ(t) sin(2πfct)

+

Baseband Signal for the I-component Baseband Signal for the Q-component

sI(t) + isQ(t)

Complex Baseband Signal

2

ECE243b Lightwave Communications - Spring 2019 Lecture 7 13

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Lecture 7 Multi- Input Multi- Output Lightwave Channels

Polarization- Multiplexed Channel Space- Multiplexed Channel

Types of Electrical Channels

Linear Electrical Channels in the Lightwave Amplitude

Phase Synchronous Demodulation

At the receiver, a demodulated electrical signal r(t) that is linear with respect to the lightwave signal amplitude s(t) requires a linear demodulation process using a nonlinear square-law photodetector. This kind of demodulator can be realized using a local phase reference L(t), called a local oscillator (LO) signal. provides a bias that preserves the magnitude As(t) and the phase φs(t) of the incident lightwave signal through the square-law photodetection process. The lightwave signal s(t) and the local oscillator signal L(t) are combined using balanced photodetection

ECE243b Lightwave Communications - Spring 2019 Lecture 7 14

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Lecture 7 Multi- Input Multi- Output Lightwave Channels

Polarization- Multiplexed Channel Space- Multiplexed Channel

Types of Electrical Channels

Linear Electrical Channels in the Lightwave Amplitude

Types of Demodulation

The type of demodulation depends on the frequency fLO of the local oscillator as compared to the carrier frequency fc. If fLO = fc, then the receiver implements homodyne demodulation. If fLO = fc, then the receiver implements heterodyne demodulation

frequency difference fc − fLO called the intermediate frequency fIF.

Considering only a single polarization, the passband local oscillator lightwave signal is

L(t)

= ALO cos(2πfLOt + φLO). (12) This signal has a fixed frequency, fLO, a constant amplitude, ALO, and an adjustable phase, φLO. The complex-baseband local oscillator signal is sLO(t) = Le−i2πfIFt, (13) where L = ALOeiφLO is the complex amplitude and the term e−i2πfIFt

ECE243b Lightwave Communications - Spring 2019 Lecture 7 15

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Lecture 7 Multi- Input Multi- Output Lightwave Channels

Polarization- Multiplexed Channel Space- Multiplexed Channel

Types of Electrical Channels

Linear Electrical Channels in the Lightwave Amplitude

Heterodyne Demodulation

(a) fIF B (b) −fIF −fc −fc fc fc

Lightwave signal before noise filtering Electrical signal after heterodyne demodulation

n(t) Optical Electrical

Lightwave signal after noise filtering Electrical Demodulation

r(t)

r(t) = rI(t) + irQ(t)

s(t)

sLO(t)

Balanced Photodetector

s(t) + no(t)

ECE243b Lightwave Communications - Spring 2019 Lecture 7 16

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Lecture 7 Multi- Input Multi- Output Lightwave Channels

Polarization- Multiplexed Channel Space- Multiplexed Channel

Types of Electrical Channels

Linear Electrical Channels in the Lightwave Amplitude

Heterodyne Demodulation

The prefiltered signal plus noise and the local oscillator signal are combined in a balanced photodetector that uses a 180-degree hybrid coupler. The output passband electrical signal r(t) is

r(t)

=

1 4

|s1(t) − is2(t)|2
Photodetector

One − |s1(t) + is2(t)|2

Photodetector

Two

(14) where s1(t) and s2(t) are the two input lightwave signals, and the responsivity R is set to one. Set s1(t) = s(t) + no(t) and set s2(t) = sLO(t) = iALOe−i2πfIFt.

ECE243b Lightwave Communications - Spring 2019 Lecture 7 17

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Lecture 7 Multi- Input Multi- Output Lightwave Channels

Polarization- Multiplexed Channel Space- Multiplexed Channel

Types of Electrical Channels

Linear Electrical Channels in the Lightwave Amplitude

Terms for Only One Photodetector

First only single photodetector Setting the second term in (14) to zero, the electrical signal is

r(t)

=

1 4

s(t) + no(t) + ALOe−i2πfIFt

2 ,

(15) where s(t) = As(t)eiφs(t). The demodulated passband electrical signal r(t) can be grouped into three terms

r(t)

=

rs−LO(t) +

rn−LO(t) + rbase(t). (16)

ECE243b Lightwave Communications - Spring 2019 Lecture 7 18

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Lecture 7 Multi- Input Multi- Output Lightwave Channels

Polarization- Multiplexed Channel Space- Multiplexed Channel

Types of Electrical Channels

Linear Electrical Channels in the Lightwave Amplitude

FirstTerm for Only One Photodetector

The first term rs−LO(t) is the mixing of the signal with the local oscillator

rs−LO(t)

=

1 4

s(t)ALOei2πfIFt + s∗(t)ALOe−i2πfIFt

=

1 2 Re

As(t)ALOei(2πfIFt+φs(t)) =

is(t)iLO cos

2πfIFt + φs(t) , (17) where is(t) = A2

s(t)/2 and iLO = A2

LO/2 . ECE243b Lightwave Communications - Spring 2019 Lecture 7 19

SLIDE 20

Lecture 7 Multi- Input Multi- Output Lightwave Channels

Polarization- Multiplexed Channel Space- Multiplexed Channel

Types of Electrical Channels

Linear Electrical Channels in the Lightwave Amplitude

Second Term for Only One Photodetector

The second term rn−LO(t) is the mixing of the noise with the local oscillator

rn−LO(t)

=

1 4

no(t)ALOei2πfIFt + n∗
(t)ALOe−i2πfIFt

=

1 2 ALO Re

no(t)ei(2πfIFt+φn(t)) =

1 2 ALO

nIF(t), (18) The demodulated passband noise process nIF(t) is centered at fIF. If the optical prefilter affects only the spontaneous emission power without affecting the statistical properties of the noise and the local oscillator is deterministic, then the demodulated electrical noise process has the same statistical properties as the incident spontaneous emission before the prefilter.

ECE243b Lightwave Communications - Spring 2019 Lecture 7 20

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Lecture 7 Multi- Input Multi- Output Lightwave Channels

Polarization- Multiplexed Channel Space- Multiplexed Channel

Types of Electrical Channels

Linear Electrical Channels in the Lightwave Amplitude

Third Term for Only One Photodetector

The third term rbase(t) has the baseband electrical signal components rbase(t) =

1 2

|s(t) + no(t)|2 + |L|2

. (19) If the local oscillator is not used so that L = 0, then both rs−LO(t) and

rn−LO(t) are zero.

For this case, the demodulated electrical signal is the baseband signal

1 2 |s(t) + no(t)|2 generated from direct photodetection.

ECE243b Lightwave Communications - Spring 2019 Lecture 7 21

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Lecture 7 Multi- Input Multi- Output Lightwave Channels

Polarization- Multiplexed Channel Space- Multiplexed Channel

Types of Electrical Channels

Linear Electrical Channels in the Lightwave Amplitude

Output Using Both Photodetectors

Using both photodetectors, the demodulated electrical signal is the difference in the outputs of the two separate photodetectors. For this case, the signal rs−LO(t) given in (17) and the signal rn−LO(t) given in (18) are doubled. The expected value of the baseband term given in (19) is zero.

Therefore, the expected values of the baseband terms cancel.

The fluctuation of the baseband terms does not cancel — produces shot noise. The demodulated electrical signal from the balanced photodetector is

r(t)

= 2

is(t)iLO cos

2πfIFt + φs(t) +ALO nIF(t) (20) = ALORe s(t) + no(t) ei2πfIFt . (21)

ECE243b Lightwave Communications - Spring 2019 Lecture 7 22