Transcript Systematic Design of Space-Time Trellis Codes for Wireless

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