Final Project of ECE 611: Convex Optimization, 2016 Fall

1. Capacity Region for Gaussian Vector Multiple Access Channel
Hao Ye and Kai Zhang
Department of Electrical Engineering and Computer Science
The University of Tennessee Knoxville (UTK)
Nov. 2016
H. Ye and K. Zhang (UTK) Capacity Region for Gaussian Vector Multiple Nov. 2016 1 / 16

2. Outline
Mathematical Model
Motivation and Background
H. Ye and K. Zhang (UTK) Capacity Region for Gaussian Vector Multiple Nov. 2016 2 / 16
Discussion
Water-filling Algorithm

3. Motivation
Compute using statistical properties
(noisy channel coding theorem) ;
Well established capacity for single-user
channel;
1948, Claude E. Shannon: reliable
communication iff. rate below channel
capacity;
H. Ye and K. Zhang (UTK) Capacity Region for Gaussian Vector Multiple Nov. 2016 3 / 16
Multiuser:
Same medium, mutual interference;
Opportunities to cooperate.
Multiple access channel:
multiple transmitters;
joint receiver.

4. Multiple access channel
Multiple access channel (MAC): ;
A code:
Message sets: and
Encoder j=1,2:
Decoder:
achievable if codes such that ;
Capacity region : closure of the set of achievable rate pairs .
H. Ye and K. Zhang (UTK) Capacity Region for Gaussian Vector Multiple Nov. 2016 4 / 16
;

5. Simple bound on the capacity region
H. Ye and K. Zhang (UTK) Capacity Region for Gaussian Vector Multiple Nov. 2016 5 / 16
Discrete-time memoryless MAC:
Fix input distribution ;
Input signals independent , non-convex.
When not fixed (satisfy certain constraint): convex hull of union;
Channel capacity ;
w.r.t.

6. Simple bound on the capacity region
H. Ye and K. Zhang (UTK) Capacity Region for Gaussian Vector Multiple Nov. 2016 6 / 16
pentagonDiscrete-time memoryless MAC:
Fix input distribution ;
w.r.t.
Channel capacity ;
Input signals independent , non-convex.
When not fixed (satisfy certain constraint): convex hull of union;

7. Simple bound on the capacity region
H. Ye and K. Zhang (UTK) Capacity Region for Gaussian Vector Multiple Nov. 2016 7 / 16
pentagonDiscrete-time memoryless MAC:
Fix input distribution ;
w.r.t.
Channel capacity ;
Input signals independent , non-convex.
When not fixed (satisfy certain constraint): convex hull of union;

8. Input vector signals: , covariance matrices: ;
Gaussian multiple access channel
H. Ye and K. Zhang (UTK) Capacity Region for Gaussian Vector Multiple Nov. 2016 8 / 16
Output vector: ;
Power constraints: ;
Noise: , covariance matrix .
model:
is Gaussian;

9. Optimize sum rate
H. Ye and K. Zhang (UTK) Capacity Region for Gaussian Vector Multiple Nov. 2016 9 / 16
maximize
For a user Gaussian multiple access channel
the input distribution that maximize are Gaussian
distributions whose covariance matrices can be found by solving the
following optimization problem:
subject to
if and only if each is single-user optimum, treating other users as noise.
The covariance matrices are a solution to the above problem,

10. Single-user (K=1) channel
H. Ye and K. Zhang (UTK) Capacity Region for Gaussian Vector Multiple Nov. 2016 10 / 16
maximize
subject to
maximize
subject to
Water-filling: the optimal is a diagonal matrix, , such that:
Let
Signaling direction = eigen-modes of ; power allocation = water filling allocation.

11. Single-user water-filling
H. Ye and K. Zhang (UTK) Capacity Region for Gaussian Vector Multiple Nov. 2016 11 / 16
Water-filling: the optimal is a diagonal matrix, , such that:
maximize
subject to
maximize
subject to
Let
Signaling direction = eigen-modes of ; power allocation = water filling allocation.

12. Multi-user water-filling
H. Ye and K. Zhang (UTK) Capacity Region for Gaussian Vector Multiple Nov. 2016 12 / 16
Iterative water-filling:
The sum capacity is achieved = each user’s input distribution maximize its own rate.

13. Each and randomly generated from an i.i.d. Gaussian distribution ;
Numerical results
H. Ye and K. Zhang (UTK) Capacity Region for Gaussian Vector Multiple Nov. 2016 13 / 16
A two-user ( ) MAC is simulated:
Each user/receiver is equipped with 20 antennas (20 x 20);
The channel matrix is assumed to be known at both the transmitters and the receiver;
The total power constraint , error tolerance .
Takes 21 steps to converge,
Optimal value: 85.4983 .

14. H. Ye and K. Zhang (UTK) Capacity Region for Gaussian Vector Multiple Nov. 2016 14 / 16
converging process:
Numerical results

15. H. Ye and K. Zhang (UTK) Capacity Region for Gaussian Vector Multiple Nov. 2016 15 / 16
Capacity region (pentagon) of each step:
zoom in
(R1, R2)
F (48.62, 0)
E (48.62, 36.84)
c (36.38, 49.07)
e (47.12, 38.37)
g (36.48, 48.94)
… …
The optimal R1 + R2 will converge to one point on CD.
Numerical results
upper bound
lower bound
0
d
*
* C
D
F
E
b c
e
f g

16. Dual problem
H. Ye and K. Zhang (UTK) Capacity Region for Gaussian Vector Multiple Nov. 2016 16 / 16
Lagrangian: introduce , then
maximize
subject to
Dual problem:
maximize
subject to
Original problem:
Result of the dual problem: 85.4982 . (As expected since Slater’s condition is satisfied)

17. Thank you!
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