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 Amplitude 1 ECE243b Lightwave Communications - Spring 2019 Lecture 7
MIMO Channels Lecture 7 Multi- Input Multi- Output Lightwave Channels Polarization- Multiple independent datastreams, called subchannels can often be supported by Multiplexed Channel the same physical medium. Space- Multiplexed Channel This kind of channel is called a multi-input multi-output (MIMO) channel . Types of Electrical Channels The number of subchannels and the properties of each subchannel depend on the Linear number of spatial and polarization modes that physical medium supports. Electrical Channels in the Lightwave In general the output of each subchannel depends on more than one input, and Amplitude the number of subchannels does not necessarily correspond to the maximum number of subchannels that the physical channel can support. 2 ECE243b Lightwave Communications - Spring 2019 Lecture 7
Example - Polarization Channels Lecture 7 Multi- Input (a) Multi- Output Photo- Lightwave Modulator Channels detection (E/O) Polarization- (O/E) Multiplexed Channel Space- Polarization Multiplexed (b) Channel beamsplitter Types of E/O O/E Electrical Channels O/E Linear Electrical Channels in Polarization the Lightwave (c) combiner Amplitude E/O O/E E/O O/E Lightwave Channel Electrical Channel 3 ECE243b Lightwave Communications - Spring 2019 Lecture 7
Types of Lightwave MIMO Channels Lecture 7 Multi- Input Multi- Output Lightwave Channels Polarization- Multiplexed Channel (a) Space- Multiplexed Channel Types of Electrical Channels Linear (b) Electrical Channels in the Lightwave Amplitude 4 ECE243b Lightwave Communications - Spring 2019 Lecture 7
Two-by-two MIMO Channel Lecture 7 Multi- Input Multi- Now consider a multi-input multi-output lightwave channel that has two inputs Output Lightwave and two outputs. Channels Polarization- Multiplexed Let s in ( t ) be a block with components s 1 ( t ) and s 2 ( t ) that are the input Channel lightwave waveforms for each subchannel. Space- Multiplexed Channel Let s out ( t ) be the corresponding output block. Types of Electrical Channels The linear channel response can be written as Linear Electrical Channels in s out ( t ) = h ( t ) ⊛ s in ( t ) (1) the Lightwave � h 11 ( t ) ⊛ s 1 ( t ) + h 12 ( t ) ⊛ s 2 ( t ) � Amplitude . = (2) . h 21 ( t ) ⊛ s 1 ( t ) + h 22 ( t ) ⊛ s 2 ( t ) In this expression h ( t ) representes the impulse response of the multi-input multi-output channel with h ij ( t ) being the complex-baseband impulse response for the i th output subchannel from the j the input subchannel. 5 ECE243b Lightwave Communications - Spring 2019 Lecture 7
Wavelength-dependent Delay Lecture 7 Multi- Input The output block S out ( f ) in the frequency domain is Multi- Output Lightwave S out ( f ) = H ( f ) S in ( f ) (3) Channels � H 11 ( f ) S 1 ( f ) + H 12 ( f ) S 2 ( f ) � Polarization- Multiplexed = (4) . Channel H 21 ( f ) S 1 ( f ) + H 22 ( f ) S 2 ( f ) Space- Multiplexed Channel The transfer function H ij ( f ) is the Fourier transform of the complex-baseband Types of Electrical impulse response h ij ( t ) . Channels Linear Electrical The matrix H consisting of the transfer functions H ij ( f ) is called the channel Channels in the matrix . Lightwave Amplitude 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 of the lightwave signal energy between the subchannels linear interchannel interference . 6 ECE243b Lightwave Communications - Spring 2019 Lecture 7
Memoryless Channel Matrix Lecture 7 Multi- Input Multi- Output Lightwave Channels Polarization- Multiplexed Channel Space- For a memoryless channel, there is no frequency dependence of the channel Multiplexed Channel matrix and we can write Types of Electrical s out ( t ) = H s in ( t ) (5) Channels Linear Electrical with the elements of the channel matrix H describing the coupling between the Channels in the subchannels. Lightwave Amplitude 7 ECE243b Lightwave Communications - Spring 2019 Lecture 7
Polarization-Multiplexed Channel Lecture 7 Multi- Input Multi- Output Lightwave Channels Polarization- A common multi-input, multi-output lightwave channel is a Multiplexed Channel polarization-multiplexed channel Space- Multiplexed Channel For this channel, random coupling between the polarization modes introduces a Types of combination of polarization-mode dispersion and polarization-dependent loss. Electrical Channels Linear Consider a fiber segment that has only first-order polarization-dependent group Electrical Channels in delay and no polarization-dependent loss. the Lightwave Amplitude For a short segment of fiber, the principal polarization states have a fixed orientation within the segment with constant polarization-dependent group delay. 8 ECE243b Lightwave Communications - Spring 2019 Lecture 7
Channel Matrix for Static Segment Lecture 7 Multi- Input The transfer function for each principal polarization state, denoted by H + ( f ) and Multi- Output H − ( f ) , can be written as Lightwave Channels H ( f ) e − i πfτ Polarization- H + ( f ) = (6) Multiplexed Channel H ( f ) e i πfτ , H − ( f ) = (7) Space- Multiplexed Channel where H ( f ) = H 0 e − i2 π 2 β 2 f 2 L is the polarization-independent, Types of Electrical wavelength-dependent delay given in ( ?? ), and ± τ / 2 is the differential group Channels delay for each principal polarization state. Linear Electrical Channels in the The channel matrix H ( f ) for a segment of fiber for which the principal states of Lightwave Amplitude polarization are fixed has the form in principal polarization basis as � e − i πfτ � 0 H ( f ) = H ( f ) e i πfτ 0 = H ( f ) P ( f ) , (8) where P ( f ) is the part of the channel matrix that is polarization dependent. 9 ECE243b Lightwave Communications - Spring 2019 Lecture 7
Channel Matrix for Span of Many Segments Lecture 7 Multi- Input Multi- Output For a fiber span that consists of many segments, the differential group delay Lightwave Channels varies from segment to segment with the overall delay modeled as a random Polarization- variable τ that has a maxwellian probability density function. Multiplexed Channel Space- The corresponding random channel matrix H ( f ) , defined in the principal Multiplexed Channel polarization-state basis, is modified to read Types of � e − i πfτ � Electrical 0 Channels H ( f ) = H ( f ) e i πfτ Linear 0 Electrical Channels in = H ( f ) P ( f ) (9) the Lightwave Amplitude 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. 10 ECE243b Lightwave Communications - Spring 2019 Lecture 7
Space-Multiplexed Channel Lecture 7 Multi- Input Multi- A space-multiplexed channel consists of spatially-separated cores within the same Output Lightwave cladding structure. Channels Polarization- Multiplexed Analyzed in isolation, the modes for each core can be determined using the Channel methods presented in Chapter 3. Space- Multiplexed Channel If the cores are sufficiently close, then the modes in a set of cores can couple. Types of Electrical Channels Weak coupling between a pair of cores can be analyzed using coupled-mode Linear Electrical theory. Channels in the Lightwave Amplitude 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 σ 2 ij for each quadrature signal component. 11 ECE243b Lightwave Communications - Spring 2019 Lecture 7
Linear Electrical Channels in the Lightwave Amplitude Lecture 7 Multi- Input For many modulation formats, a real-baseband waveform s I ( t ) for the in-phase Multi- Output signal component is modulated onto a coherent lightwave carrier cos ( 2 πf c t ) . Lightwave Channels Polarization- A separate real-baseband waveform s Q ( t ) for the quadrature signal component is Multiplexed modulated onto an orthogonal coherent lightwave carrier sin ( 2 πf c t ) . Channel Space- Multiplexed Channel Ignoring scaling constants, the two modulated signals are combined to produce a Types of passband lightwave signal Electrical Channels � s ( t ) = s I ( t ) cos 2 πf c t − s Q ( t ) sin 2 πf c t Linear A s ( t ) cos � 2 πf c t + φ s ( t ) � Electrical Channels in = the Lightwave Amplitude Re [ s ( t ) e i2 πf c t ] = (10) The equivalent complex-baseband signal is s ( t ) = s I ( t ) + i s Q ( t ) A s ( t ) e i φ s ( t ) . = (11) 12 ECE243b Lightwave Communications - Spring 2019 Lecture 7
Recommend
More recommend
