Multiuser Channel Capacity

Mohammad RezaeianResearch supervisor Dr. Alex GrantInstitute for Telecommunications Research

University of South Australia

April 10, 2002

Overview

� Multiuser channels

� Capacity for memoryless multiuser channels

� Characterizations of interference in multiuser channels

� The limiting characterization of capacity for some multiuser channels

� A new limiting expression for the capacity of the interference channel.

1

Multiuser communication systems with a common channel

Information transmission in communication systems is limited by the randomness characteristics

of system. This limited capability can be shared by users.

Transmtter Receiver

noise

Multiple access channel Broadcast channel

Point to point communication model

Interference Network

2

Why channel capacity is important

Source Channel Decoder DestinationEncoder

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Two basic quality measures for communication systems areEfficiency (transmission rate) and reliability (vanishing ! ).One fundamental question: ” Is reliability achieved only by reducing the rate?”

Shannon theory showed that the answer is negative.

Reliability can be improved by more information processing (in encoder and decoder) as long asrate is below the channel capacity.

Channel capacity is a benchmark showing whether we can improve system reliability by more

information processing.

3

Multiuser channel capacity

Channel capacity is the limit on code rates beyond which the tradeoff betweenreliability and complexity fails.

Receiver 1

Receiver 2

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Code rate ��� � � � � � � � � � �� � .

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Capacity Region

For a� user channel, capacity is the boundary of a region in� �� . The region is

called the capacity region.

4

Capacity Region analysis

Capacity region is the set of all approachable rates.

An approachable rate is a rate that for increasing block length code to infinitythere exist codes for which probability of error approach zero.

Finding a formula for the capacity region requires proving two propositions for a region.

� Direct part: To prove that all points inside the region are approachable.

� Converse part: To prove any approachable rate has to be inside the region.

A region is

Inner bound for the capacity region if only the direct part is proved

Outer bound for the capacity region if only the converse part is proved

Finding the capacity is a formidable task due to requirement of simultaneousjustification of direct and converse parts of coding theorem.

5

Computation of capacity region

Capacity region by a sequence of iner bounds Capacity region by a sequence of outer bounds

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Limiting

expressions

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� �

Capacity region

One-off computationPSfrag replacements

Single letter

expression

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� �

Capacity region

Finding the boundary of regions is an optimization problem, cf: � �� �� �� ��� �� �

6

Why are single letter descriptions important?� Discrete channels: computational preference. Single letters descriptions

need only one optimization.

Compare � �� �� � ��� �� � (single letter) and � � � �� � � �� �� �� � � �� � � � � (limiting ex-

pression) for single user case.

Channel

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� Continuous channels: Direct conversion of capacity formula for discretechannel to capacity for equivalent continuous channels, only possible for

single letter formulas.PSfrag replacements

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Why does a channel capacity appear as a limiting expression?

MEMORY (Dependency between noise instances)

A single letter formula defines capacity in terms of input - output mutual information in one timeinstance. Knowing channel statistics in one channel use (single letter transmission) is sufficientfor capacity calculation.

For a single user channel, single letter capacity formula is only for memoryless channel.

� � � � ��� � � � �� � � � � ��� � � � � � � � �� � � � � �

Memoryless channel

x y

x y

Channel with memory

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Does memoryless assumption for multiuser channels reduce capacity re-gion to a single letter formula?

A Big Question.

Among different models of multiuser channel only for multiple access channelsingle letter capacity region has been found.

Attempts for other models have a 35 years chronicle.

Multiple access channel Broadcast channel Interference channel

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My research on the multiuser channel capacity

A few dead end attempts for finding single letter capacity region for interferencechannel. The last one was a converse proof for the best single letter inner bound.

� Three new limiting expressions for the capacity region. Two of these expres-sions are shown to have a faster convergence to the capacity region.

� Categorization of interference in multiuser channels based on single userchannel decomposition.

� Providing evidences that there may not exist a single letter formula for thecapacity region of interference channel (A scheme proof).

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Two fundamentally different types of Interference

For a multiuser channel the disturbing factors for the decoding process are:

internal noise + interference.

Receiver 1

Receiver 2

User 3

User 2

User 1

T1

T2

multiple access interference

non intended user interference

Interferences affecting the link between user 1 and receiver 1.

In a multiuser channel each receiver decodes a subset of transmitters

We show that the multiple access interference approach a memoryless characteristics with suf-

ficiently large code length, but non intended user interference remains as a noise with memory.

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Two basic models of multiuser channel

Multiple access interference

Noise

+Interference channelNon intended user interference

Multiple access channelNoise

+

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Mobile communication system with high intercell interference

Mobile communication system with no

intercell interference

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Memoryless assumption for the two basic multiuser channel model

Multiple access channel Interference channel

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" $ ) * !-5 $ .6 ( 5 & '" & ) * !-5 & .6 ( 5 $ '" $ � " & ) * !-5 $ % 5 & .6 '1 Has not been found

We say does not exist

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Communication links of a multiuser channel

Even though we have assumed that channel is memoryless, communication links are not memoryless.

Each link in the above multiuser channels can be modeled by the following single user channel model

We call this model,

satisfies

is the channel state , is a time variation parameter

State Conditioned memoryless channel (SCMC).

Main property: Given the state, channel is memoryless.

SCMC is a discrete channel in which

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State conditioned Memoryless Channel, a Lemma

Lemma - If a complete knowledge about channel state is acquired by the channel input andoutput, the channel statistics approach (for long channel use) to a memoryless characteristics.

-

can be approximated by a memoryless link

For the interference channel, the link -

is always modeled by a channel with memory

For multiple access channel, the link

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Memoryless internal noise is not enough for single letter capacity of mul-tiuser channels

A receiver

A communication

link

A network

All transmitters

A receiver

A communication

All transmitters

A network

link

noise with memory

0

assimilating memoryless noise

Total Interference is recognised at time

The Interference is not recognised

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An analogy for limiting expressions in calculus

The solution of differential equation

� ��� � a polynomial of� , of any degree has finite terms of calculation, but

� ��� � �� has an infinite term sequence of calculation (a limiting expression).

Attempt for finding a finite degree polynomial satisfying this differential equationis useless.

� By the method of contradiction we can show that the solution is a limitingexpression (Supposing that � is polynomial of � , contradicts this differentialequation).

� The solution ( � � � � ) is calculated by truncating the limiting expression.Theconvergence speed of the limiting expressions are important for computa-tional efficiency.

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For interference channels, capacity is inherently a limiting expression

Single letter descriptions of the capacity region are of the following form

� IC� � � �� �� � � � �

� � � (1)

where is a finite set of random variables at least contains� � � � � �� � �� � and is a jointdistribution on , satisfying �� � �� � � � � � � � � given by channel.

Suggestion: A single letter capacity (1) implies:

� IC� � � � � � ��� ��� � � � �

�� � � � � � �� ��� ��� � � �� � � � � � � � � ��� � �� ��� � ��� � � � (2)

where� � � is a subset of � for which the conditional probabilities � � � � � � and � � � � � �

are memoryless-like links.

Using previous lemma, (2) necessities the detection of the non intended user for rates inside the

capacity region, which contradicts the definition of interference channel.

Therefore the capacity region cannon be described by functions of probabilitydistributions on finite number of random variables.

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For interference channels, capacity is inherently a limiting expression

Single letter descriptions of the capacity region are of the following form

� IC � � � �� �� � � � �

� � �

where� is joint distribution on the finite set of random variables .

We have given evidence (Chapter 5 of the dissertation) that this description ofcapacity region contradicts a fundamental property of capacity region for theinterference channel.

Therefore the capacity region cannon be described by functions of probabilitydistributions on finite number of random variables.

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A new limiting expression

Capacity region

Single letter inner bound

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� � � � ��� � �� � ��� The largest single letter inner bound

The previous limiting expression is derived by multiple letter extension of the Shannon inner bound.

We have shown that the multiple letter extension of the Han and Kobayashi inner bound also gives the capacityregion. Theorem - � � � �� � � � �� � � IC.

Because for any � , � � � � � �� ��� � � �� , the new limiting expression converge faster to the capacity

region.

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Multiple letter extension, the concept

The Han and Kobayashi inner bound specifies relation between � ��� � , and� � for which there exists code with

arbitrary small probability of error.Encoder Decoder

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A code for interference channel

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Apply the Han and kobayashi rule to 3 -th extension of channel.

Each letter contains 3 -times more information.

Divide the achievable rates by 3 , the result willbe an achievable rate for the original channel

By this process some extra achievable rate canbe obtained that are not inside the Han andKobayashi of the original channel.

If we proceed with this channel extension, we can obtain

all the achievable rates, ie: the capacity region.

Encoder Decoder

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Conclusion� Although the memoryless assumption simplifies the derivation of capacity

region for single user channel, for multiuser channel this is not always true.

� The interference of non intended users that is not decoded error free by thereceived signal, induces memory in the communication links of multiuserchannel.

� By virtue of the memory characteristics of links of a multiuser channel withnon intended user interference, the capacity region for such channels is alimiting expression. Finding a single letter capacity for such channels is anuseless effort.

� We have improved the limiting expression for the capacity region of the in-terference channel to a faster approaching expression.

Slides are available at www.itr.unisa.edu.au/ � mjreza

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