Lecture 5 Capacity of Multiuser Channels I-Hsiang Wang - PowerPoint PPT Presentation

Lecture ¡5 Capacity ¡of ¡Multiuser ¡Channels I-Hsiang Wang ihwang@ntu.edu.tw 4/10, 2014

From ¡Single-­‑User ¡to ¡Multi-­‑User • In Lecture 3 we studied various techniques for multiple access and interference management in cellular systems • In Lecture 4 we learned about information theory and investigate the capacity of point-to-point channels • In this lecture we extend the information theoretic framework to multi-user channels • Present new techniques that emerge from the information theoretic study: - Success interference cancellation (SIC) - Superposition coding - Multi-user diversity - Opportunistic communication paradigm 2

Plot • Two scenarios: - Uplink channel (many-to-one) - Downlink channel (one-to-many) • Start with AWGN (no fading) - Uplink channel: successive interference cancellation (SIC) - Downlink channel: superposition coding • Fast Fading - CSIR only - Full CSI • Multi-user Diversity 3

Outline • Uplink/Downlink AWGN channel • Uplink/Downlink fading channel • Multi-user diversity • Opportunistic beamforming 4

Uplink/Downlink ¡ AWGN ¡Channel 5

Uplink ¡and ¡Downlink ¡Channel Uplink Downlink Rx: decodes Tx: encodes both users’ data both users’ data y x h 1 h 2 h 1 h 2 y 1 y 2 x 1 x 2 User 1 User 2 User 1 User 2 y 1 [ m ] = h 1 x [ m ] + w 1 [ m ] y [ m ] = h 1 x 1 [ m ] + h 2 x 2 [ m ] + w [ m ] y 2 [ m ] = h 2 x [ m ] + w 2 [ m ] • Channel gains are fixed over time and known to Tx & Rx • Uplink noise: CN (0 , σ 2 ) • Downlink noise at User k , k = 1,2 : CN (0 , σ 2 k ) 6

Capacity ¡Region • Point-to-point channel: Achievable ⟺ P e ( N ) → 0 as N → ∞ - Capacity C C 0 R Achievable Not Achievable if R < C if R > C • Multi-user channel R 2 - Each user has its own data C - Not Achievable Two data rates R 1 & R 2 if ( R 1 , R 2 ) ∉ C - Capacity region C - ( R 1 , R 2 ) is achievable ⟺ Achievable - both error probability → 0 if ( R 1 , R 2 ) ∈ C R 1 7

Capacity ¡Region ¡of ¡the ¡UL ¡Channel 8 8 9 R 1 ≤ log (1 + SNR 1 ) > > > < < = [ C Uplink = ( R 1 , R 2 ) ≥ 0 : R 2 ≤ log (1 + SNR 2 ) > > > R 1 + R 2 ≤ log (1 + SNR 1 + SNR 2 ) : : ; R 2 SNR k := | h k | 2 P k , k = 1 , 2 σ 2 log (1 + SNR 2 ) R 1 + R 2 ≤ log (1 + SNR 1 + SNR 2 ) C Uplink ⇣ ⌘ SNR 2 log 1 + 1+ SNR 1 R 1 ⇣ ⌘ SNR 1 log (1 + SNR 1 ) log 1 + 1+ SNR 2 8

Non-­‑Achievability ¡Outside ¡C uplink C Uplink • R k ≤ log(1+ SNR k ) : obvious, since log(1+ SNR k ) is the point-to-point capacity as if there is only one Tx • R 1 + R 2 ≤ log(1+ SNR 1 + SNR 2 ) : obvious, since the maximum received SNR from the two independent Tx is SNR 1 + SNR 2 , and therefore the total rate cannot exceed the capacity of the point-to-point channel with this SNR 9

Successive ¡Interference ¡Cancellation R 2 Achieving point A B User k encodes its data using a capacity log (1 + SNR 2 ) achieving AWGN channel code at rate R k , k =1,2 C Uplink Rx first decodes User 2’s data, treating ⇣ ⌘ User 1’s signal x 1 as Gaussian noise SNR 2 A log 1 + 1+ SNR 1 ⇣ ⌘ | h 2 | 2 P 2 = ⇒ R 2 = log 1 + | h 1 | 2 P 1 + σ 2 R 1 ⇣ ⌘ SNR 1 ⇣ ⌘ log (1 + SNR 1 ) log 1 + SNR 2 = log 1 + 1+ SNR 2 1+ SNR 1 can be achieved Rx: decodes Rx then subtracts x 2 from y and get a both users’ data y point-to-point channel for User 1 = ⇒ R 1 = log (1 + SNR 1 ) h 1 h 2 x 1 x 2 can be achieved User 1 User 2 Note: smaller R 2 can also be achieved 10

Equivalent ¡Point-­‑to-­‑Point ¡Channels • Equivalent channels - For User 2, the equivalent noise is h 1 x 1 + w , with variance | h 1 | 2 P 1 + σ 2 h 1 x 1 [ m ] + w [ m ] h 2 x 2 [ m ] y [ m ] - For User 1, after removing x 2 , Rx sees a clean point-to-point channel without interference w [ m ] h 1 x 1 [ m ] y [ m ] − h 2 x 2 [ m ] 11

Time ¡Sharing R 2 Similarly point B can be achieved B log (1 + SNR 2 ) To achieve a rate point on AB, say, q A + (1- q ) B, the system can C Uplink take the following two strategies with a prescribed portion of time: ⇣ ⌘ SNR 2 A log 1 + 1+ SNR 1 Strategy achieving A R 1 ⇣ ⌘ SNR 1 log (1 + SNR 1 ) log 1 + Decode User 2 first and then 1+ SNR 2 decode User 1; q of time Rx: decodes Strategy achieving B both users’ data y Decode User 1 first and then decode User 2; (1- q ) of time h 1 h 2 x 1 x 2 User 1 User 2 12

Comparison ¡with ¡Conventional ¡CDMA • For each user, treat the other user’s signal as noise - No successive interference cancellation (SIC) - Hence a single-user receiver, not a multi-user receiver • It is strictly suboptimal (achieving point C) R 2 B log (1 + SNR 2 ) C Uplink C ⇣ ⌘ SNR 2 A log 1 + 1+ SNR 1 R 1 ⇣ ⌘ SNR 1 log (1 + SNR 1 ) log 1 + 1+ SNR 2 13

UL ¡Orthogonal ¡Multiple ¡Access • Consider time-division access - User 1 uses the first α of the time - User 2 uses the rest (1– α ) of the time • Power constraint: - User 1 can now use power P 1 / α during its transmission - User 2 can now use power P 2 /(1– α ) during its transmission • Achievable rates: ( 1 + SNR 1 � � R 1 = α log α α ∈ [0 , 1] ⇣ ⌘ 1 + SNR 2 R 2 = (1 − α ) log 1 − α - When α = SNR 1 /( SNR 1 + SNR 2 ) , the sum capacity is achieved (i.e., R 1 + R 2 = log(1+ SNR 1 + SNR 2 ) is achieved) 14

Orthogonal ¡MA ¡is ¡Sum ¡Rate ¡Optimal 1 + SNR 1 ( ( ) � � R 1 = α log [ α ( R 1 , R 2 ) : ⇣ ⌘ 1 + SNR 2 R 2 = (1 − α ) log 1 − α α ∈ [0 , 1] Orthogonal multiple access can R 2 only achieve the optimal sum rate at a single point, when � � α = SNR 1 /( SNR 1 + SNR 2 ) log (1 + SNR 2 ) Fairness is an issue C Uplink D ( SNR 1 R 1 = SNR 1 + SNR 2 C sum D: SNR 2 ⇣ ⌘ R 2 = SNR 1 + SNR 2 C sum SNR 2 log 1 + 1+ SNR 1 C sum = log (1 + SNR 1 + SNR 2 ) R 1 ⇣ ⌘ SNR 1 log (1 + SNR 1 ) log 1 + 1+ SNR 2 15

K -­‑user ¡Uplink ¡Channel ¡Capacity • For a general K -user uplink channel - Capacity region: [ ( ! ) X X C Uplink = ( R 1 , . . . , R K ) ≥ 0 : R k ≤ log 1 + , ∀ S ⊆ [1 : K ] SNR k k ∈ S k ∈ S - Sum capacity: ! K X C sum SNR k Uplink = log 1 + k =1 • For example, 3-user uplink channel capacity region: R k ≤ log (1 + SNR k ) , k = 1 , 2 , 3 R 1 + R 2 ≤ log (1 + SNR 1 + SNR 2 ) R 2 + R 3 ≤ log (1 + SNR 2 + SNR 3 ) R 3 + R 1 ≤ log (1 + SNR 3 + SNR 1 ) R 1 + R 2 + R 3 ≤ log (1 + SNR 1 + SNR 2 + SNR 3 ) 16

Capacity ¡Region ¡of ¡the ¡DL ¡Channel ( ( ) R 1 ≤ log (1 + β SNR 1 ) [ C Downlink = ( R 1 , R 2 ) ≥ 0 : ⇣ ⌘ 1 + (1 − β ) SNR 2 R 2 ≤ log 1+ β SNR 2 β ∈ [0 , 1] R 2 SNR k := | h k | 2 P , k = 1 , 2 σ 2 k log (1 + SNR 2 ) WLOG assume SNR 1 ≥ SNR 2 C Downlink Maximum sum rate is achieved when β = 1 C sum = Downlink = log (1 + SNR 1 ) ⇒ R 1 log (1 + SNR 1 ) Note: proof of non-achievability outside this region is beyond the scope of this course 17

Superposition ¡Coding • Tx sends x = x 1 + x 2 , where for k =1,2 Tx: encodes both users’ data x - User k ’s data is encoded onto x k - Power of x 1 = β P ; power of x 2 = (1– β ) P h 1 h 2 y 1 y 2 • User 1 has a better received SNR User 1 User 2 - User 1’s channel is better than User 2 - User 1 can decode whatever User 2 can decode • Single-user decoding at User 2: - Decode x 2 by treating x 1 as noise - ⇣ ⌘ 1 + (1 − β ) SNR 2 ⟹ can achieve R 2 = log 1+ β SNR 2 • SIC Decoding at User 1: - First decode x 2 by treating x 1 as noise, and remove it from y 1 - Then decode x 1 ¡⟹ can achieve R 1 = log (1 + β SNR 1 ) 18

Comparison ¡with ¡Conventional ¡CDMA • Conventional CDMA: the same as before except that User 1 does not do SIC • Strictly suboptimal • Exercise: how to choose β such that all DL users have the same received SINR? 19

DL ¡Orthogonal ¡Multiple ¡Access • Consider time-division access - User 1 uses the first ¡ α ¡ of the time with power P - User 2 uses the rest (1– α ) of the time with power P • Achievable rates: ( R 1 = α log (1 + SNR 1 ) α ∈ [0 , 1] R 2 = (1 − α ) log (1 + SNR 2 ) - Strictly suboptimal (except the two corner points when one of the users is shut down) 20

K -­‑user ¡Downlink ¡Channel ¡Capacity • WLOG assume SNR 1 ≥ SNR 2 ≥ … ≥ SNR K • Capacity region: 8 9 ✓ ◆ β k SNR k : ( R 1 , . . . , R K ) ≥ 0 : R k ≤ log 1 + , < = [ 1+ P k − 1 C Downlink = j =1 β j SNR k ∀ k ∈ [1 : K ] ; β 1 ,..., β K ≥ 0 β 1 + ··· + β K =1 - β k denotes the portion of power allocated to User k ’s codeword • Sum capacity: achieved by sending only to the best user C sum Downlink = log (1 + SNR 1 ) 21

Uplink/Downlink ¡ Fading ¡Channel 22

Setting Uplink Downlink Rx: decodes Tx: encodes both users’ data both users’ data y x h 1 h 2 h 1 h 2 y 1 y 2 x 1 x 2 User 1 User 2 User 1 User 2 y 1 [ m ] = h 1 [ m ] x [ m ] + w 1 [ m ] y [ m ] = h 1 [ m ] x 1 [ m ] + h 2 [ m ] x 2 [ m ] + w [ m ] y 2 [ m ] = h 2 [ m ] x [ m ] + w 2 [ m ] • Fast fading: ∀ k , { h k [ m ]} is stationary and ergodic • Symmetry: ∀ k , { h k [ m ]} is identically distributed • We shall focus on ergodic sum capacity 23