capacity

Wireless Channel Capacity Fundamental Limit on Data Rates

Capacity: The set of simultaneously achievable rates {R1,…,Rn}

R3



R1



R2



R3



R2 R1



 



Impact of ISI on Capacity Impact of Multiple Antennas on Capacity



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 xk

H1(w)



y1k   h1i xk i w1k

i 1



m



w2k

H2(w)



y2 k   h2i xk i w2 k

i 1



m



   



Both H1 and H2 are finite IR filters of length m. The w1k and w2k are correlated noise samples. For 1


Circular Channel Model





Define the zero padded filters as:

~ n {hi }i 1  (h1 ,..., hm ,0,...,0)







The n-Block Circular Gaussian Broadcast Channel (n-CGBC) is defined based on circular convolution:

~ ~  y1k  h1i x(( k i )) w1k  xi  h1i  w1k

i 0 n 1



~ ~  y2 k  h2i x(( k i )) w2 k  xi  h2i  w2 k

i 0



n 1



0


where ((.)) denotes addition modulo n.



Equivalent Channel Model





Taking DFTs of both sides yields

~ ~ Y1 j  H1 j X j  W1 j ~ ~ Y2 j  H2 j X j  W2 j



0






Dividing by H and using additional properties of the DFT yields

Y1j  X   V1j j

Y2j  X   V2j j



~



0


where {V1j} and {V2j} are independent zero-mean Gaussian ~ random variables with  2  n( Nl (2j / n)/|H lj |2 , l  1,2. lj



Parallel Channel Model

V11



X1



+

+



Y11



V21



Y21



Ni(f)/Hi(f)



V1n



f



Xn



+

+



Y1n

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)



n 1



Optimal power allocation obtained by Hughes-Hartogs(’75).







The power constraint  E[ xi2 ]  nP on the original channel is n 1 i 0 converted by Parseval’s theorem to  E[( X i) 2 ]  n 2 P on the i 0 equivalent channel.



Capacity Region of Parallel Set





Achievable Rates (no common information)

 a j Pj    a j Pj 1    .5  log 1  , R1  .5  log   (1  a ) P    1j  j: 1 j  2 j j: 1 j  2 j j j 1j       (1  a j ) Pj  (1  a j ) Pj    .5  log 1  , R2  .5  log 1   a P    2j  j: 1 j  2 j j: 1 j  2 j j j 2j     2 0  a j  1,  Pj  n P 







Capacity Region  For 0
 



Let



(R1*,R2*)n,b



denote the corresponding rate pair.



1 Cn={(R1*,R2*)n,b : 0


R2



b R1



Limiting Capacity Region

  a j Pj  a ( f ) P ( f ) | H 1 ( f ) |2  1    .5 R1  .5  log 1  ) H ( f ) log (1  a j ) Pj   1 j ,    .5 N 0   f : H1 ( f 2 f :H1 ( f )  H 2 ( f )      (1  a ( f ))P ( f ) | H 2 ( f ) |2  (1  a ( f ))P ( f ) , R2  .5 ) H ( f ) log1  a ( f ) P ( f )  .5 N 0 / | H 2 ( f ) |2   .5 f :H ( f ) H ( f ) log1      .5 N 0     f : H1 ( f 2 1 2 0  a ( f )  1,



 P ( f )df  P



Optimal Power Allocation: Two Level Water Filling



Capacity vs. Frequency



Capacity Region



Gaussian Broadcast and Multiple Access Channels

Broadcast (BC): One Transmitter to Many Receivers.



Multiple Access (MAC): Many Transmitters to One Receiver.

x



x



h22(t)

x



x



h3(t)



h1(t)



h21(t)



Multiple Access Channel





Multiple transmitters





Transmitter i sends signal Xi with power Pi



 



Common receiver with AWGN of power N0B Received signal:

Y   Xi  N

i 1 M



X1 X2 X3



MAC Capacity Region





Closed convex hull of all (R1,…,RM) s.t.

   Ri  B log 1   Pi / N0 B, iS  iS 





S  {1,...,M }



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 Ĉ2

 Pi  Ci  B log 1  , i  1,2  N0 B 



SC w/ IC and time sharing or rate splitting Frequency division SC w/out IC



Ĉ1



C1



 P 1 ˆ  B log 1  C1  , N 0 B  P2  



 P2 ˆ  B log 1  C2  , N0 B  P  1 



Comparison of MAC and BC





Differences:

 



P



Shared vs. individual power constraints Near-far effect in MAC



P1







Similarities:





P2 Optimal BC “superposition” coding is also optimal for MAC (sum of Gaussian codewords)

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

h1 (n)



z1 ( n)

x +



h1 (n)



y1 ( n)



x1 ( n)

( P1 )



x



z (n)

+



x(n)

(P )



hM (n)



z M (n)

+



y(n)



hM (n)



x



yM (n)



xM (n)

( PM )



x



Broadcast Channel (BC)



Multiple-Access Channel (MAC)



The BC from the MAC

C MAC ( P1 , P2 ; h1 , h2 )  C BC ( P1  P2 ; h1 , h2 )



h1  h2



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 BC ( P; h1 , h2 ) 

0 P1  P







C MAC ( P1 , P  P1 ; h1 , h2 )



Sum-Power MAC

CBC ( P; h1 , h2 ) 



0 P  P 1 Sum CMAC ( P , P  P ; h1 , h2 )  CMAC ( P; h1 , h2 )  1 1



MAC with sum power constraint  Power pooled between MAC transmitters  No transmitter coordination

Same capacity region!



P

BC



MAC



P



BC to MAC: Channel Scaling

 









Scale channel gain by a, power by 1/a MAC capacity region unaffected by scaling Scaled MAC capacity region is a subset of the scaled BC capacity region for any a MAC region inside scaled BC region for any scaling



P1 a



a h1



MAC

+



P1  P2 a



a h1

h2



+



P2



h2



+



BC



The BC from the MAC

a  

h2 h1

a  0



Blue = Scaled BC Red = MAC



a



C MAC ( P1 , P2 ; h1 , h2 )   C BC (

a 0



P1



a



 P2 ; a h1 , h2 )



Duality: Constant AWGN Channels





BC in terms of MAC

0 P1  P



C BC ( P; h1 , h2 ) 







C MAC ( P1 , P  P1 ; h1 , h2 )







MAC in terms of BC



P1 C MAC ( P1 , P2 ; h1 , h2 )   C BC (  P2 ; ah1 , h2 ) a 0 a



What is the relationship between the optimal transmission strategies?



Transmission Strategy Transformations





Equate rates, solve for powers

R1M  log(1 

M R2  log(1 



h12 P M 1 ) M 2 h2 P2  

2 h2 P2M



 log(1 



2 h1 P B 1



2



)  R1B







Opposite decoding order

 



2



)  log(1 



2 h2 P2B 2 h2 P B   1



B )  R2 2



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



Duality: Minimum Rate Capacity

MAC in terms of BC

Blue = Scaled BC Red = MAC



BC region known  MAC region can only be obtained by duality





What other unknown capacity regions can be obtained by duality?



Broadcast MIMO Channel

(r  t ) 1



n1



t1 TX antennas r11, r21 RX antennas

y1  H1x  n1



H1

x

(r2  t )



n2



Perfect CSI at TX and RX

y2  H2 x  n 2



H2



n1 ~ N(0, I r1 ) n 2 ~ N(0, I r2 )



Non-degraded broadcast channel



MIMO Channel Model

n TX antennas h11 m RX antennas h21 h22 h13

h32 h23 h33 h12



x1

h31



y1 y2 y3



x2 x3



 y1   h11  h1n   x1   n1                 ,         ym  hm1  hmn   xn  nm        



y  Hx  n

n ~ N (0,  2 I )



Model applies to any channel described by a matrix (e.g. ISI channels)



What’s so great about MIMO?





Fantastic capacity gains (Foschini/Gans’96, Telatar’99)





Capacity of single-user system grows linearly with antennas when channel known perfectly at Tx and Rx

Rank ( H T QH ) i 1



C  max log | I  H T QH | max Pi : Pi  P B Q:Tr (Q ) P

i



log(1  pi li2 ) 







Can we get such gains in broadcast systems? Need some new techniques (dirty paper coding)







Dirty Paper Coding (Costa’83)





Basic premise











If the interference is known, channel capacity same as if there is no interference Accomplished by cleverly distributing the writing (codewords) and coloring their ink Decoder must know how to read these codewords



Dirty Paper Coding Clean Channel



Dirty Paper Coding Dirty Channel



Modulo Encoding/Decoding







Received signal Y=X+S, -1X1





S known to transmitter, not receiver

Set X so that Y[-1,1]=desired message (e.g. 0.5) Receiver demodulates modulo [-1,1]



Modulo operation removes the interference effects









-1



0



+1



…

-7 -5 -3 -1 0 +1 +3 +5



…

+7



S

-1



X

0 +1



Capacity Results





Non-degraded broadcast channel  Receivers not necessarily “better” or “worse” due to multiple transmit/receive antennas  Capacity region for general case unknown Pioneering work by Caire/Shamai (Allerton’00):  Two TX antennas/two RXs (1 antenna each)  Dirty paper coding/lattice precoding (achievable rate)









Computationally very complex











MIMO version of the Sato upper bound Upper bound is achievable: capacity known!



Dirty-Paper Coding (DPC) for MIMO BC





Coding scheme:

  



Choose a codeword for user 1 Treat this codeword as interference to user 2 Pick signal for User 2 using “pre-coding”

T R 2  log(det(I  H 2 S 2 H 2 ))







Receiver 2 experiences no interference:







Signal for Receiver 2 interferes with Receiver 1:

T  det(I  H1 (S1  S 2 ) H1 )   R1  log  T   det(I  H1S 2 H1 )  







Encoding order can be switched



Dirty Paper Coding in Cellular



Does DPC achieve capacity?





DPC yields MIMO BC achievable region.





We call this the dirty-paper region



 



Is this region the capacity region?

We use duality, dirty paper coding, and Sato’s upper bound to address this question



MIMO MAC with sum power





MAC with sum power:









Transmitters code independently Share power

Sum MAC



P



C





( P) 



Theorem: Dirty-paper BC region equals the dual sum-power MAC region



0 P  P 1







CMAC ( P , P  P ) 1 1



C



DPC BC



( P)  C



Sum MAC



( P)



Transformations: MAC to BC





Show any rate achievable in sum-power MAC also achievable with DPC for BC:

DPC BC

Sum MAC



DPC Sum C BC ( P )  C MAC ( P )







A sum-power MAC strategy for point (R1,…RN) has a given input covariance matrix and encoding order We find the corresponding PSD covariance matrix and encoding order to achieve (R1,…,RN) with DPC on BC  The rank-preserving transform “flips the effective channel” and reverses the order  Side result: beamforming is optimal for BC with 1 Rx antenna at each mobile



Transformations: BC to MAC





Show any rate achievable with DPC in BC also achievable in sum-power MAC:

DPC Sum C BC ( P )  C MAC ( P )

DPC BC



Sum MAC







We find transformation between optimal DPC strategy and optimal sum-power MAC strategy  “Flip the effective channel” and reverse order



Computing the Capacity Region

C



DPC BC



( P)  C



Sum MAC



( P)



Hard to compute DPC region (Caire/Shamai’00) “Easy” to compute the MIMO MAC capacity region

  







Obtain DPC region by solving for sum-power MAC and applying the theorem Fast iterative algorithms have been developed Greatly simplifies calculation of the DPC region and the associated transmit strategy



Sato Upper Bound on the BC Capacity Region

 Based on receiver cooperation

n1

H1

x



+

n2



y1



Joint receiver



H2



+



y2



 BC sum rate capacity  Cooperative capacity

sumrate CBC (P, H)







max 1 Sx 2



log | I  HΣ x HT |



The Sato Bound for MIMO BC

 





Introduce noise correlation between receivers BC capacity region unaffected





Only depends on noise marginals



Tight Bound (Caire/Shamai’00)





Cooperative capacity with worst-case noise correlation



inf max 1 sumrate CBC formula for worst-case noiseΣ 1/2HΣ x HT Σ 1/2 | (P, H)  log | I  covariance z z  Explicit Sz Sx 2





By Lagrangian duality, cooperative BC region equals the sum-rate capacity region of MIMO MAC



Sum-Rate Proof

DPC Achievable



C



DPC BC



( P )  C BC ( P )

sumrate



Sum DPC C MAC ( P )  C BC ( P )



Duality



DPC C BC ( P)  C BC ( P)



C BC ( P )  C

*Same result by Vishwanath/Tse for 1 Rx antenna



Coop BC



(P)

sumrate



C MAC ( P )  C

Sum MAC



Sato Bound



Obvious



Sum MAC



( P)



C



Coop BC



( P)  C



( P)

Compute from MAC



Lagrangian Duality



MIMO BC Capacity Bounds

Single User Capacity Bounds Dirty Paper Achievable Region



BC Sum Rate Point Sato Upper Bound



Does the DPC region equal the capacity region?



Full Capacity Region





DPC gives us an achievable region





 



Sato bound only touches at sum-rate point

We need a tighter bound to prove DPC is optimal



Recent results by Shamai et. al. (CISS, March 04) have found the full capacity region.



Summary





Shannon capacity gives fundamental data rate limits for wireless channels Broadcast channels with ISI can use OFDM with nearoptimality Duality and dirty paper coding are used to obtain the capacity of a broadcast MIMO channel.







