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 of system. This limited capability can be shared by users. noise Transmtter Receiver Point to point communication model Multiple access channel Broadcast channel Interference Network 2
✌ ✙ ✞ ✠ ✂ ✓ ✔ ✞ ✞ ✘ ✒ ✆ ✣ ✚ ✂ ✛ ✌ ✂ ✢ ✂ ✆ ✒ ✠ ✜ ✠ ✦ ✥ ✂ ✚ � ✢ ☎ ✆ ✜ ✡ ✑ ☛ ✆ ☞ ✡ ✑ ✎ ✂ ✏ ✤ ✡ Why channel capacity is important message set ☞✍✌ Source Encoder ✝✟✞ Channel Decoder Destination ✁✄✂ 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 Capacity Region Code rate . ✆✞✝ 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 Limiting expressions Capacity region Capacity region by a sequence of iner bounds Capacity region by a sequence of outer bounds Finding the boundary of regions is an optimization problem, cf: ✌✎✍ ✏✒✑ ✡☞☛ Single letter expression One-off computation Capacity region 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 y x Channel with memory y x 8
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. Broadcast channel Interference channel Multiple access 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). 10
Two fundamentally different types of Interference For a multiuser channel the disturbing factors for the decoding process are: internal noise + interference. multiple access interference T 1 In a multiuser channel each receiver decodes a subset of transmitters User 1 Receiver 1 non intended user interference User 2 Receiver 2 Interferences affecting the link between user 1 and receiver 1. User 3 T 2 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. 11
� ✂ ✄ ✂ ✄ ✁ ✄ ✂ � � ✁ � ✁ Two basic models of multiuser channel Noise Noise + + Multiple access channel Interference channel Multiple access interference Non intended user interference Mobile communication system with no intercell interference Mobile communication system with high intercell interference 12
✴ ✧ ✭ ✻ ✩ ✪ ✧ ✫ ✬ ✭ ✩ ✣ ✮ ✩ ✙ ✯ ✳ ✩ ✳ ✴ ✩ ✢ ✢ ✧ ✩ ✧ ✩ ✳ ✴ ✬ ✧ ✫ ✮ ✬ ✜ ✩ ✫ ✳ ✺ ✖ ✗ ✘ ✙✚ ✛ ✬ ✫ ✩ ✧ ✦ ✧ ✩ ✪ ✧ ✫ ✬ ✭ ✩ ✹ ✮ ✯ ✫ ✩ ✳ ✻ ✭ ✺ ✫ ✤ ✢ ✮ ✬ ✩ ✙ ✯ ✫ ✫ ✳ ✴ ✫ ✶ ✹ ✔ ✯ ✕ ✔ ✮ ✕ ✸ ✜ ✢ ✩ ✮ ✧ ✓ ✑ ☎ ✳ ✑ ✫ ☛ ✑ ☞ ✌ ✎ ✑ ✍ ☎ ✺ ☛ ☞ ✝ ✌ ✪ ✏ ✍ ☞ ✄ ✶ � ✁ � ✁ � ✂ � ✂ ✁ ✌ ✄ ✂ ✄ ☎ ✬ ✻ ☛ ☞ ✠ ☛ ✠ ✩ ✣ ✔ ✫ ✕ ✖✗✘ ✙✚ ✛ ✜ ✢ ✢ ✌ ✧ ✑ ✩ ✪ ✧ ✫ ✬ ✭ ✧ ✕ ✮ ✔ ✒ ✍ ✎ ✏ ✑ ☎ ✩ ☛ ✑ ☞ ✌ ✍ ✑ ✯ ☛ ✑ ☞ ✓ ✌ ✑ ✬ Multiple access channel Memoryless assumption for the two basic multiuser channel model MAC (Single letter) MAC ✆✞✝ ✟✡✠ ✣✥✤ ✆✞✒ ✦★✧ ✟✡✓ ✦✲✺ ✦✲✺ ✦✲✺ ✙✵✯ ✙✰✯ ✦✲✱ ✦✲✱ ✬✷✶ We say does not exist Has not been found IC (Single letter) IC ✣✥✤ ✆✞✝ ✦★✧ Interference channel ✆✞✒ ✟✡✠ ✟✡✓ ✦✲✱ ✦✲✱ 13
✥ ✤ ✠ ✠ ✌ ✌✜ ✠ ✕ ✆ ✡ ☛ ✕ ✣ ✣ ✆ ✓ ✝ ✢ ✡ ☛ ✕ ✆ ✓ ✡ ☛ ✓ ✕ ✘ ✝ ✔ ✘ ✚ ✄ ✕ ✜ ✌ � � ✁ � ✁ � ✏ � ✂ ✂ ✁ ✠ ☎ ✄ ✆ Communication links of a multiuser channel Even though we have assumed that channel is memoryless, communication links are not memoryless. is not product form. is not product form. ✝✟✢ Each link in the above multiuser channels can be modeled by the following single user channel model State Conditioned memoryless channel (SCMC) We call this model, . SCMC is a discrete channel in which satisfies ✝✟✢ ✡✧✦ ✕✗✖ ✝✟✞ ✡☞☛ ✌✎✍ ✌✎✞ ✝✟✞ ✡☞✙ ✠✒✑ ✌✛✚ is the channel state , is a time variation parameter Main property: Given the state, channel is memoryless. 14
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