Multiuser Wireless Systems Overview

EE360: Multiuser Wireless Systems and Networks Lecture 2 Outline Announcements HW 0 due Monday (high-impact papers you decide) Bandwidth Sharing in Multiuser Channels FD, TD, CD, SD, Hybrid (OFDMA) Overview of Multiuser Channel Capacity Capacity of Broadcast Channels AWGN, Fading, and ISI Capacity of MAC Channels Duality between BC and MAC channels MIMO Channels

Review of Last Lecture Overview of multiuser wireless systems and networks Multiuser systems Cellular, Ad-Hoc, Cognitive, and Sensor Networks Random Access Techniques ALOHA CSMA-CD Reservation Protocols Random access works poorly in heavily loaded systems

Deterministic Bandwidth Sharing in Multiuser Channels

Deterministic Bandwidth Sharing Code Space Frequency Division Time Code Space Frequency Time Division Time Frequency Code Space Code Division Multiuser Detection Time Space Division (MIMO) Hybrid Schemes (OFDMA) What is optimal? Frequency Look to Shannon.

OFDMA and SDMA OFDMA Combines OFDM and frequency division multiplexing Different subcarriers assigned to different users SDMA (space-division multiple access) Different spatial dimensions assigned to different users Implemented via multiuser beamforming (e.g. zero- force beamforming) Benefits from multiuser diversity

Code Division via Multiple Access SS Interference between users mitigated by code cross correlation T b = t f + t f 2 c 2 ( x ) ( ) ( ) cos 2 ( ) ( ) ( ) ( ) cos( 2 ) cos( 2 ( )) t s t s t s t s t s t f t dt 1 1 1 2 2 2 1 c c c c c 0 T b = 5 . + = 5 . + c 5 . ( ) ( ) 5 . cos( 2 ) ( ) d d s t s t dt d d f 1 1 2 2 1 2 1 2 12 c c 0 In downlink, signal and interference have same received power In uplink, close users drown out far users (near-far problem)

Multiuser Detection In all CDMA systems and in TD/FD/CD cellular systems, users interfere with each other. In most of these systems the interference is treated as noise. Systems become interference-limited Often uses complex mechanisms to minimize impact of interference (power control, smart antennas, etc.) Multiuser detection exploits the fact that the structure of the interference is known Interference can be detected and subtracted out Better have a darn good estimate of the interference

Ideal Multiuser Detection - Signal 1 = A/D Signal 1 Demod A/D A/D A/D Iterative Multiuser Detection Signal 2 Signal 2 Demod - = Why Not Ubiquitous Today? Power and A/D Precision

Capacity Limits of Broadcast Channels

Multiuser Shannon Capacity Fundamental Limit on Data Rates Capacity: The set of simultaneously achievable rates {R1, ,Rn} with arbitrarily small probability of error R3 R2 R3 R2 R1 R1 Main drivers of channel capacity Bandwidth and received SINR Channel model (fading, ISI) Channel knowledge and how it is used Number of antennas at TX and RX Duality connects capacity regions of uplink and downlink

Broadcast Channel Capacity Region in AWGN Model One transmitter, two receivers with spectral noise density n1, n2: n1<n2. Transmitter has average power Pand total bandwidth B. Single User Capacity: Maximum achievable rate with asymptotically small Pe + 1 P = log C B i n B i Set of achievable rates includes (C1,0) and (0,C2), obtained by allocating all resources to one user.

Rate Region: Time Division Time Division (Constant Power) Fraction of time allocated to each user is varied ( 1 ( , 2 1 1 = = R C R U ) 0 ; C ) 1 2 Time Division (Variable Power) Fraction of time and power iallocated to each user is varied + 1 + 1 n n U = = log , 1 ( ) log ; 1 B 2 B R B R B . 1 1 2 1 2 + = 1 ( ) , 0 P 1 2

Rate Region: Frequency Division Frequency Division Bandwidth Bi and power Si allocated to each user is varied. + 1 + 1 P P U = = log , log ; 1 2 R B R B 1 1 2 2 n B n B 1 1 2 2 + = + = , P P P B B B 1 2 1 2 Equivalent to TD for Bi= iB and P = i i.

Superposition Coding Best user decodes fine points Worse user decodes coarse points

Code Division Superposition Coding Coding strategy allows better user to cancel out interference from worse user. + 1 P P U = = + + = log , log 1 ; 1 2 R B R B P P P 1 2 1 2 + n B n B S 1 2 1 DS spread spectrum with spreading gain G and cross correlation 12= =G: = G G B n G 1 / U + 1 B P B P = + + = log , log 1 ; 1 2 R R P P P 1 2 1 2 + / / n B G S G 2 1 By concavity of the log function, G=1 maximizes the rate region. DS without interference cancellation B P B P U = + = + + = log 1 , log 1 ; 1 2 R R P P P 1 2 1 2 + + / / / / G n B G P G G n B G P G 1 2 2 1

Capacity Limits of Fading Broadcast Channels

Broadcast and MAC Fading Channels Broadcast: One Transmitter to Many Receivers. Wireless Gateway Wired Network x g3(t) Multiple Access: Many Transmitters to One Receiver. x g2(t) x g1(t) R3 R2 R1 Goal: Maximize the rate region {R1, ,Rn}, subject to some minimum rate constraints, by dynamic allocation of power, rate, and coding/decoding. Assume transmit power constraint and perfect TX and RX CSI

Fading Capacity Definitions Ergodic (Shannon) capacity: maximum long-term rates averaged over the fading process. Shannon capacity applied directly to fading channels. Delay depends on channel variations. Transmission rate varies with channel quality. Zero-outage (delay-limited*) capacity: maximum rate that can be maintained in all fading states. Delay independent of channel variations. Constant transmission rate much power needed for deep fading. Outage capacity: maximum rate that can be maintained in all nonoutage fading states. Constant transmission rate during nonoutage Outage avoids power penalty in deep fades *Hanly/Tse, IT, 11/98

Two-User Fading Broadcast Channel h1[i] 1[i] Y1[i] x + X[i] Y2[i] x h2[i] + 2[i] At each time i: n={n1[i],n2[i]} n [i]= 1[i]/ h1[i] Y1[i] + X[i] Y2[i] + n2[i]= 2[i]/ h2[i]

Ergodic Capacity Region* Capacity region: ,where Cergodic = P C P ( ) ( ) P F P n ( ) j = + C R E B j M ( P P ) log 1 , 1 } j n M = + n B P [ 1 ) n n n ( ] j i j i i 1 M =1 The power constraint implies = E P n P ( ) n j j Superposition coding and successive decoding achieve capacity Best user in each state decoded last Power and rate adapted using multiuser water-filling: power allocated based on noise levels and user priorities *Li/Goldsmith, IT, 3/01

Zero-Outage Capacity Region* The set of rate vectors that can be maintained for all channel states under power constraint P = C P C P ( ) ( ) P F zero n N P n ( ) j = + C R B j M ( P P ) log 1 , 1 j M = i + n B P n n n ( [ 1 ) ] j i j i 1 Capacity region defined implicitly relative to power: For a given rate vector R and fading state n we find the minimum power Pmin(R,n) that supports R. R Czero(P) if En[Pmin(R,n)] P *Li and Goldsmith, IT, 3/01

Outage Capacity Region Two different assumptions about outage: All users turned off simultaneously (common outage Pr) Users turned off independently (outage probability vector Pr) Outage capacity region implicitly defined from the minimum outage probability associated with a given rate Common outage: given (R,n), use threshold policy IfPmin(R,n)>s* declare an outage, otherwise assign this power to state n. ) n = min P E P R ( , Power constraint dictates s* : min n P R n s : ( , ) * = p n ( ) Pr Outage probability: min n P R n s : ( , ) *

Independent Outage With independent outage cannot use the threshold approach: Any subset of users can be active in each fading state. Power allocation must determine how much power to allocate to each state and which users are on in that state. Optimal power allocation maximizes the reward for transmitting to a given subset of users for each fading state Reward based on user priorities and outage probabilities. An iterative technique is used to maximize this reward. Solution is a generalized threshold-decision rule.

Broadcast Channels with ISI ISI introduces memory into the channel The optimal coding strategy decomposes the channel into parallel broadcast channels Superposition coding is applied to each subchannel. Power must be optimized across subchannels and between users in each subchannel.

Broadcast Channel Model w1k m = m = = + y h x i 1 w H1( ) 1 1 k k k i 1 i xk w2k = + y h x 2 w H2( ) 2 2 k i k k i 1 i Both H1 and H2 are finite IR filters of length m. The w1k and w2k are correlated noise samples. For 1<k<n, we call this channel the n-block discrete Gaussian broadcast channel (n-DGBC). The channel capacity region is C=(R1,R2).

Circular Channel Model Define the zero padded filters as: {~} h i i 1 == n ( ,..., h 1 , ,..., ) 0 0 h m The n-Block Circular Gaussian Broadcast Channel (n-CGBC) is defined based on circular convolution: ~ y h x w x k i i 0 = ~ y h x w x k i i 0 = 1 n ~ = + = + h w 1 1 1 1 1 k i i k k i (( )) 0<k<n 1 n ~ = + = + h w 2 2 2 2 2 k i i k k i (( )) where ((.)) denotes addition modulo n.

Equivalent Channel Model Taking DFTs of both sides yields ~ ~ Y H X j j 1 1 = ~ ~ Y H X j j 2 2 = ~ + W 1 j j 0<j<n + W 2 j j Dividing by H and using additional properties of the DFT yields = + Y X V j j j 1 1 = + Y X V j j j 2 2 0<j<n where {V1j} and {V2j} are independent zero-mean Gaussian random variables with lj n N = ( / )/|~| , 2 = 2 2 ( 12 , . j n H l l lj

Parallel Channel Model V11 Y11 + X1 Y21 + V21 Ni(f)/Hi(f) V1n f Y1n + Xn Y2n + V2n

Channel Decomposition The n-CGBC thus decomposes to a set of n parallel discrete memoryless degraded broadcast channels with AWGN. Can show that as n goes to infinity, the circular and original channel have the same capacity region The capacity region of parallel degraded broadcast channels was obtained by El-Gamal (1980) Optimal power allocation obtained by Hughes-Hartogs( 75). 1 n The power constraint on the original channel is converted by Parseval s theorem to on the equivalent channel. 2 E x [ ] nP i 1 n = 0 i ) ] 2 2 E X [( n P i = 0 i

Capacity Region of Parallel Set Achievable Rates (no common information) P P j j j j + 5 . + + R 5 . log 1 log 1 , 1 + P 1 ( ) : : j j j j j j 1 1 j j j j 1 2 1 2 P P 1 ( ) 1 ( ) j j j j + 5 . + + R 5 . log 1 log 1 , 2 + P : : j j j j j j 2 2 j j j j 1 2 1 2 2 P n P 0 , 1 j j Capacity Region For 0< find { j}, {Pj} to maximize R1+ R2+ Pj. Let (R1*,R2*)n, denote the corresponding rate pair. Cn={(R1*,R2*)n, : 0< }, C=liminfnCn . R2 1 n R1

Optimal Power Allocation: Two Level Water Filling

Capacity vs. Frequency

Capacity Region

Capacity Limits of MAC Channels

Multiple Access Channel Multiple transmitters Transmitter i sends signal Xi with power Pi Common receiver with AWGN of power N0B Received signal: X1 M = i+ Y X N X3 X2 =1 i

MAC Capacity Region Closed convex hull of all (R1, ,RM) s.t. + log 1 / , 1 { ,..., } R B P N B S M 0 i i i S i S For all subsets of users, rate sum equals that of 1 superuser with sum of powers from all users Power Allocation and Decoding Order Each user has its own power (no power alloc.) Decoding order depends on desired rate point

Two-User Region Superposition coding w/ interference canc. Time division C2 SC w/ IC and time sharing or rate splitting 2 Frequency division SC w/out IC + P C1 1 = 2 , 1 = log 1 , i C B i i N B 0 P P = + = + log 1 , log 1 , 1 + 2 C B C B 1 2 + N B P N B P 0 2 0 1

Fading and ISI MAC capacity under fading and ISI determined using similar techniques as for the BC In fading, can define ergodic, outage, and minimum rate capacity similar as in BC case Ergodic capacity obtained based on AWGN MAC given fixed fading, averaged over fading statistics Outage can be declared as common, or per user MAC capacity with ISI obtained by converting to equivalent parallel MAC channels over frequency

Duality between Broadcast and MAC Channels

Comparison of MAC and BC P Differences: Shared vs. individual power constraints Near-far effect in MAC P1 Similarities: Optimal BC superposition coding is also optimal for MAC (sum of Gaussian codewords) P2 Both decoders exploit successive decoding and interference cancellation

MAC-BC Capacity Regions MAC capacity region known for many cases Convex optimization problem BC capacity region typically only known for (parallel) degraded channels Formulas often not convex Can we find a connection between the BC and MAC capacity regions? Duality

Dual Broadcast and MAC Channels Gaussian BC and MAC with same channel gains and same noise power at each receiver ( ) 1n z ( ) 1n h ( ) 1n h x ( ) 1n x (1P ( ) 1n y + x (n ) z ) (n ) y + (n ) zM (n (P ) ) x (n ) hM (n ) hM (n ) M P ) xM ( x x (n ) yM + Multiple-Access Channel (MAC) Broadcast Channel (BC)

The BC from the MAC + + C P P h h C P P h h ( , ; , ) ( ; , ) MAC BC 1 2 1 2 1 2 1 2 h h 1 2 P1=0.5, P2=1.5 P1=1, P2=1 Blue = BC Red = MAC P1=1.5, P2=0.5 MAC with sum-power constraint = = ) C P h h C P P P h h ( ; , ( , ; , ) P BC MAC 1 2 1 1 1 2 0 1 P

Sum-Power MAC = Sum MAC ( ; , ) ( , ; , ) ( ; , ) C P h h C P P P P h h C P h h 1 2 1 1 1 2 1 2 BC MAC 0 P 1 MAC with sum power constraint Power pooled between MAC transmitters No transmitter coordination P Same capacity region! BC MAC P

BC to MAC: Channel Scaling Scale channel gain by , power by 1/ MAC capacity region unaffected by scaling Scaled MAC capacity region is a subset of the scaled BC capacity region for any MAC region inside scaled BC region for anyscaling 1 P MAC 1 h 1 h + P+ + 1 P + 2 2 h + 2 h 2 P BC

The BC from the MAC h = = 2 Blue = Scaled BC Red = MAC h 1 0 P 1 = = + + C P P h h C P h h ( , ; , ) ( ; , ) MAC BC 1 2 1 2 2 1 2 0

Duality: Constant AWGN Channels BC in terms of MAC = = ) C P h h C P P P h h ( ; , ( , ; , ) P BC MAC 1 2 1 1 1 2 0 1 P MAC in terms of BC P = = + + 1 C P P h h C P h h ( , ; , ) ( ; , ) MAC BC 1 2 1 2 2 1 2 0 What is the relationship between the optimal transmission strategies?

Transmission Strategy Transformations Equate rates, solve for powers 2 1 B h P 2 1 M h P 1 M 1 B 1 = + = + = 1 + log( 1 ) log( 1 ) R R 2 2 M 2 h P 2 2 2 2 2 M 2 B h P h P 2 + M 2 B 2 = + = + = log( 1 ) log( 1 ) R R 2 2 2 2 B h P 1 Opposite decoding order Stronger user (User 1) decoded last in BC Weaker user (User 2) decoded last in MAC

Duality Applies to Different Fading Channel Capacities Ergodic (Shannon) capacity: maximum rate averaged over all fading states. Zero-outage capacity: maximum rate that can be maintained in all fading states. Outage capacity: maximum rate that can be maintained in all nonoutage fading states. Minimum rate capacity: Minimum rate maintained in all states, maximize average rate in excess of minimum Explicit transformations between transmission strategies