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EE 6332, Spring, 2017 Wireless Communication Zhu Han Department of Electrical and Computer Engineering Class 11 Feb. 22 nd , 2017
EE 6332, Spring, 2017
Wireless Communication
Zhu Han
Department of Electrical and Computer Engineering
Class 11
Feb. 22nd, 2017
OutlineOutline Capacity in AWGN (Chapter 4.1)
– Entropy– Source: independent Gaussian distribution– Channel capacity: R<=C=Wlog(1+SNR)
Capacity of flat fading channel (Chapter 4.2) Capacity of frequency-selective fading channel (Chapter 4.3)
Discrete time modelDiscrete time model A simple discrete time model
where h is a complex Gaussian distributed fading coefficient Information about channel
1. Channel distribution information (CDI) at transmitter and receiver
2. Channel state information at receiver (and CDI)3. Channel state information at transmitter and receiver (and CDI)
Case 1 Channel Distribution Information (CDI)Case 1 Channel Distribution Information (CDI) Achievable rate
– Finding the maximizer is non trivial– For Rayleigh independent channel coefficients
Maximizing input is discrete with finite number of mass points Mass at zero
– Achievable rate computed numerically– Maximizing input distribution computed numerically– Not much to discuss—little analytical results
Channel State Information (CSI)Channel State Information (CSI) State of the channel S (a function of h )
– Known to the receiver as V– Known to the transmitter as U
Channel state as a part of channel output
since fading (or more precisely CSI at receiver) is independent of the channel input
Ergodic CapacityErgodic Capacity The achievable rate when CSI at receiver but no CSI at
transmitter
The model
Perfect channel state information at receiver
ErgodicErgodic The achievable rate is not a variable in time
– If channel gain changes instantaneously the rate does not change
The rate is achieved over a long long codebook across different realizations of the channel– Long long decoding delay
Fading does not improve Ergodic capacity
The key to the proof is Jensen’s inequality
ExampleExample A flat fading (frequency nonselective) with independent
identically distributed channel gain as
CSIR no CSIT
ExampleExample The three possible signal to noise ratios
Ergodic capacity
ExampleExample Average SNR
The capacity of AWGN channel with the average SNR
CSI at Transmitter and ReceiverCSI at Transmitter and Receiver
The mutual information
Capacity when there is CSI at transmitter and receiver The original definition is not applicable Define fading channel capacity
CSITR Ergodic CapacityCSITR Ergodic Capacity A result for multi-state channel due to Wolfowitz
capacity for each state Applied to CSITR
Channel state information at transmitter and receiver
Power adjusted with constraint
Achievable Rate with CSITRAchievable Rate with CSITR Constraint optimization
Solving via differentiation
The solution is power control Temporal water filling Variable rate and variable power
– Different size code books– Multiplexing encoders and decoders
Water Filling SolutionWater Filling Solution
Capacity with CSITRCapacity with CSITR The maximized rate
The threshold not a function of average power limit
CSITR ExampleCSITR Example A flat fading (frequency nonselective) with independent
identically distributed channel gain as
ExampleExample The three possible signal to noise ratios
Calculate the threshold
If the weakest channel is not used a consistent threshold emerges
Ergodic capacity
ExampleExample Average SNR
The capacity of AWGN channel with the average SNR
Probability of OutageProbability of Outage Achieving ergodic channel capacity
– Codewords much be longer than coherence time
Slow fading channels have long coherence times Ergodic capacity more relevant in fast fading cases A burst with signal to noise ratio Probability of outage
Capacity with outage– Information sent over a burst– Limited decoding delay– Nonzero probability of decoding error
OutageOutage The minimum required channel gain depends on the target rate. When instantaneous mutual information is less than target rate
depends on the channel realization Probability of outage (CSIR)
Fading channel (CSIR)
Outage with CSITROutage with CSITR Use CSITR to meet a target rate
– Channel inversion– Minimize outage
Truncated channel inversion Probability of outage with CSITR
Fading channel with CSITR
Power ControlPower Control Outage minimization The solution for CSITR
Truncation with channel inversion
Power Control RealizationPower Control Realization
Outage CapacityOutage Capacity Target probability of outage Fixed power The outage capacity
Frame Error Rate– An appropriate performance metric– In many examples, probability of outage is a lower bound to
Frame Error Rate
Frequency Selective (Chapter 4.3) Frequency Selective (Chapter 4.3) Input output relationship
Consider a time invariant channel CSI is available at transmitter and receiver Block frequency selective fading
An equivalent parallel channel model
CSITR: Frequency SelectiveCSITR: Frequency Selective The sum of rates
The power distribution
Power ControlPower Control The power distribution threshold
Spectral water filling Variable rate and variable power across channels
– Different size code books– Multiplexing encoders and decoders
Achievable Rate
Frequency Selective FadingFrequency Selective Fading Continuous transfer function
Power distribution across spectrum
Techniques to Approach CapacityTechniques to Approach Capacity Coding Accurate model
– Statistical– Deterministic
Feedback– Power control– Rate control
Multipath maximal ratio combing
HW3: 4.3, 4.5, 4.6, 4.8, 4.11, due 3/6/17
