EE 6332, Spring, 2014 Wireless Communication Zhu Han Department of Electrical and Computer Engineering Class 11 Feb. 19 th, 2014

Outline 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 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) Achievable rate –Finding the maximizer is non trivial –For Rayleigh independent channel coefficients u Maximizing input is discrete with finite number of mass points u Mass at zero –Achievable rate computed numerically –Maximizing input distribution computed numerically –Not much to discuss—little analytical results

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

CSIR Proof

Ergodic Capacity The achievable rate when CSI at receiver but no CSI at transmitter The model Perfect channel state information at receiver

Ergodic 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

Example A flat fading (frequency nonselective) with independent identically distributed channel gain as CSIR no CSIT

Example The three possible signal to noise ratios Ergodic capacity

Example Average SNR The capacity of AWGN channel with the average SNR

CSI 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 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 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

Power Control

Water Filling Solution

Capacity with CSITR The maximized rate The threshold not a function of average power limit

CSITR Example A flat fading (frequency nonselective) with independent identically distributed channel gain as

Example The three possible signal to noise ratios Calculate the threshold If the weakest channel is not used a consistent threshold emerges Ergodic capacity

Example Average SNR The capacity of AWGN channel with the average SNR

Probability 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

Outage 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 CSITR Use CSITR to meet a target rate –Channel inversion –Minimize outage u Truncated channel inversion Probability of outage with CSITR Fading channel with CSITR

Power Control Outage minimization The solution for CSITR Truncation with channel inversion

Power Control Realization

Outage 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) 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 Selective The sum of rates The power distribution

Power 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 Fading Continuous transfer function Power distribution across spectrum

Techniques to Approach Capacity Coding Accurate model –Statistical –Deterministic Feedback –Power control –Rate control Multipath maximal ratio combing