1. Slides prepared on 09 September, 2018
Convex-optimization decoding for
Massive MIMO
Christos Thrampoulidis
joint work with
Ehsan Abbasi, Weiyu Xu and Babak Hassibi
(Main) Reference:
“Symbol Error Rate Performance of Box-Relaxation Decoders in Massive MIMO”, C Thrampoulidis; Weiyu Xu, Babak Hassibi, IEEE Transactions of Signal Processing, 2018
Slides prepared on 09 September, 2018

2. 5G wireless technologies: Massive MIMO
34 billion
including IoT
[Ericsson Mobility Report November 2017]
guaranteed QoS to as many users as possible
energy and cost efficient while maintaining high QoS
limits of achievable data rates
(Some) challenges:
promising solution for 5G
hundreds antennas at each base station
Massive MIMO

3. Massive MIMO decoding y=Ax0+z Decoder x0,1 y1 x0,2 y2 y x0,3 ym x0,n. y1
Decoder y2 ym y
Channel model:
A has entries iid [flat-fading Rayleigh model]
z has entries iid Gaussian
Noise model:
Signal constellation: for simplicity, [BPSK signal]

4. Curse of dimensionality
y=Ax0+z
x0,1
x0,2
x0,3
x0,n . y1
ML decoder y2 y . ym
(BPSK signal)

5. Zero-forcing decoder y=Ax0+z x0,1 y1 x0,2 y2 y x0,3 ym x0,n. y1 y2 y . ym
Computationally efficient
Performance guarantees
(BPSK signal)

6. Notice: m>n y2 ym Computationally efficient Performance guarantees. ym
Computationally efficient
Performance guarantees
(BPSK signal)
Notice: m>n

7. Zero-forcing decoder y=Ax0+z x0,1 y1 x0,2 y2 y x0,3 ym x0,n. y1 y2 y . ym
Computationally efficient
Performance guarantees
(BPSK signal)
Does not exploit structure

8. Box-relaxation decoder
y=Ax0+z
x0,1
x0,2
x0,3
x0,n . y1
Box relaxation y2 y . ym
Accounts for structural information
No closed form but still efficient
Flexibility
Higher-order constellations
other modulation schemes, e.g. GSSK
(BPSK signal)

9. Performance guarantees?
y=Ax0+z
x0,1
x0,2
x0,3
x0,n . y1
Box relaxation y2 y . ym
What is the BER?
How many antennas needed?
(BPSK signal)
vs ZF? vs optimal?

10. Performance of box-relaxation
Theory
[Τhrampoulidis, Xu, Hassibi. IEEE Transactions on Signal Processing, 2018]

11. Precise!
(n=512, #iter=100)

12. Optimality gap
3dB

13. Implications on system design:
How many antennas?

14. BER of box-relaxation
y=Ax0+z . y
Box relaxation
Theorem.

15. Much more…
Higher-order constellations

16. Much more… Higher-order constellations Optimal relaxation threshold
[Attitalah,T.,Kammoun,Naffouri,Hassibi,Alouini, arXiv 2017]

17. Much more… Higher-order constellations Optimal relaxation threshold
Empirical distribution
[T., Xu, Hassibi, arXiv 2017]

18. Much more… Higher-order constellations Optimal relaxation threshold
Empirical distribution
LASSO with box constraint (e.g., GSSK)
Optimal tuning
Imperfect CSI
[Alrashdi,Attitalah,Naffouri, Alouini. GlobalSIP 2017]
[Attitalah,T.,Kammoun,Naffouri,Alouini. ICASSP, ISIT 2017]

19. Our Approach Challenge: Approach: NO closed-form expression!
Gaussian Comparisons
Challenge:
NO closed-form expression!
Approach:
Analyze a simpler, but “equivalent”, optimization problem!
(* paper for details)

20. Energy Efficiency
Use of low-end hardware (low-resolution Analog to Digital Converters (ADC) at the receiver side for reduced power consumption and reduced cost.
Q: Computationally efficient decoder with provable guarantees?
A: Use the Box-relaxation Decoder!

21. Nonlinear = linear y Box relaxation Theorem. Ax0+z 1-bit measurements:
q-bit measurements:
[Τhrampoulidis, Xu. IEEE SSP, 2018]

22. Performance guarantees
[Τhrampoulidis, Xu. IEEE SSP, 2018]

23. Implications to System Design
ADC
Box-dec A
Box-dec A
Lloyd-Max (LM) algorithm is optimal