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
Y1j X V1j j
Y2j X V2j j
~
0
where {V1j} and {V2j} are independent zero-mean Gaussian ~ random variables with 2 n( Nl (2j / 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 ) log1 a ( f ) P ( f ) .5 N 0 / | H 2 ( f ) |2 .5 f :H ( f ) H ( f ) log1 .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, iS iS
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
t1 TX antennas r11, r21 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, -1X1
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.