Multiuser Channel Capacity Mohammad Rezaeian Research supervisor - - PowerPoint PPT Presentation

Multiuser Channel Capacity

Mohammad Rezaeian Research supervisor Dr. Alex Grant

Institute 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

• f 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 Destination Encoder

• ✁✄✂

message set

sequence

Rate =

The probability

averaged over

Two basic quality measures for communication systems are Efficiency (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 as rate 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 between reliability and complexity fails.

Receiver 1 Receiver 2

• ✁
• ✂
• ✄
• ☎

Code rate

.

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 infinity there 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 simultaneous justification 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 Limiting expressions

• ✁
• ✂

Capacity region One-off computation Single letter expression

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

• Continuous channels: Direct conversion of capacity formula for discrete

channel to capacity for equivalent continuous channels, only possible for single letter formulas.

7

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 time

• instance. Knowing channel statistics in one channel use (single letter transmission) is sufficient

for 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 channel single letter capacity region has been found. Attempts for other models have a 35 years chronicle.

Multiple access channel Broadcast channel Interference channel

9

My research on the multiuser channel capacity A few dead end attempts for finding single letter capacity region for interference

• channel. 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 user

channel decomposition.

• Providing evidences that there may not exist a single letter formula for the

capacity 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 channel

Non intended user interference

Multiple access channel

Noise

+

• ✁
• ✁
• ✂
• ✂

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

• ✁
• ✁
• ✂
• ✂

MAC

IC

MAC (Single letter)

IC (Single letter)

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

• ✁
• ✁
• ✂
• ✂

is not product form.

is not product form.

14

State conditioned Memoryless Channel, a Lemma Lemma - If a complete knowledge about channel state is acquired by the channel input and

• utput, 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

• ✁

15

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 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 equation is useless.

• By the method of contradiction we can show that the solution is a limiting

expression (Supposing that

is polynomial of

, contradicts this differential equation).

• The solution (

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

17

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 joint distribution 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 probability distributions on finite number of random variables.

18

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 of capacity region contradicts a fundamental property of capacity region for the interference channel. Therefore the capacity region cannon be described by functions of probability distributions on finite number of random variables.

18

A new limiting expression

Capacity region Single letter inner bound

• ✁✂
• ✄

• ✟
• ✟
• ✠
• ✠

IC

IC

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 capacity

• region. Theorem -

• IC.

Because for any

,

, the new limiting expression converge faster to the capacity region.

19

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

A code for interference channel

Apply the Han and kobayashi rule to

• th extension of channel.

Each letter contains

• times more information.

Divide the achievable rates by

, the result will be an achievable rate for the original channel By this process some extra achievable rate can be obtained that are not inside the Han and Kobayashi of the original channel. If we proceed with this channel extension, we can obtain all the achievable rates, ie: the capacity region.

Encoder Decoder

• ✻

• ❄

• ❂

• ❂

The

• th extension of channel

20

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 the

received signal, induces memory in the communication links of multiuser channel.

• By virtue of the memory characteristics of links of a multiuser channel with

non intended user interference, the capacity region for such channels is a limiting expression. Finding a single letter capacity for such channels is an useless 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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