Energy-Efficient Full-Duplex Massive MIMO Relayshome.iitk.ac.in/~rohitbr/pdf/spcom_invited_talk_2018.pdf · Single-hop massive MIMO Consider a single-hop multiple-access system with

Energy-Efficient Full-Duplex Massive MIMO Relays

Rohit Budhiraja

July 17, 2018

SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 1

Single-hop massive MIMO

Consider a single-hop multiple-access system with two users




Received signal y = g1x1 + g2x2 + n





Using maximal ratio combiner wH1 = 1

Ng1, we have

y1 = wH1 y = wH

1 g1x1 +wH1 g2x2 +wH

1 n






Ng1, we have

y1 = wH1 y = wH


1 n

=gH1 g1N

x1︸︷︷︸

desired signal

+gH1 g2N

x2︸︷︷︸

interference

+gH1 n

N︸︷︷︸

noise






Ng1, we have

y1 = wH1 y = wH


1 n

=gH1 g1N

x1︸︷︷︸

desired signal

+gH1 g2N

x2︸︷︷︸

interference

+gH1 n

N︸︷︷︸

noise

y1a.s.−−−−→

N→∞x1






Ng1, we have

y1 = wH1 y = wH


1 n

=gH1 g1N

x1︸︷︷︸

desired signal

+gH1 g2N

x2︸︷︷︸

interference

+gH1 n

N︸︷︷︸

noise

y1a.s.−−−−→

N→∞x1

Both interference and noise asymptotically vanish


Half-duplex: One-way relay (MAC phase)


Half-duplex: One-way relay (BC phase)


Half-duplex: Two-way relay (MAC phase)


Half-duplex: Two-way relay (BC phase)


Half-duplex: mathematical model for MAC phase

Received signal at the relay, yR =∑2K

k=1

√pkgkxk + zR




k=1

√pkgkxk + zR

Relay amplifies the received signal as




k=1

√pkgkxk + zR


xR = αWyR = αW

2K∑

k=1

√pkgkxk + αWzR




k=1

√pkgkxk + zR


xR = αWyR = αW

2K∑

k=1

√pkgkxk + αWzR

Relay MRC/MRT precoder is designed as W = F∗GH


Half-duplex: Mathematical model

Received signal at k′

th user is

yk′ = fTk′xR + zk′




th user is

yk′ = fTk′xR + zk′ = αfT

k′W

2K∑

k=1

√pkgkxk + αfT

k′WzR + zk′




th user is


k′W

2K∑

k=1

√pkgkxk + αfT

k′WzR + zk′

= αfTk′W

√pk gk xk

︸︷︷︸

desired signal




th user is


k′W

2K∑

k=1

√pkgkxk + αfT

k′WzR + zk′

= αfTk′W

√pk gk xk

︸︷︷︸

desired signal

+ αfTk′W

√pkgk′ xk′

︸︷︷︸

self-interference




th user is


k′W

2K∑

k=1

√pkgkxk + αfT

k′WzR + zk′

= αfTk′W

√pk gk xk

︸︷︷︸

desired signal

+ αfTk′W

√pkgk′ xk′

︸︷︷︸

self-interference

+ αfTk′W

2K∑

i 6=k,k′

√pigixi

︸︷︷︸

inter-pair interference




th user is


k′W

2K∑

k=1

√pkgkxk + αfT

k′WzR + zk′

= αfTk′W

√pk gk xk

︸︷︷︸

desired signal

+ αfTk′W

√pkgk′ xk′

︸︷︷︸

self-interference

+ αfTk′W

2K∑

i 6=k,k′

√pigixi

︸︷︷︸


+ αfTk′WzR

︸︷︷︸

amplified relay noise

+ zk′

︸︷︷︸

AWGN at user k′


Full-duplex: Two-way relay

Relay receive signal

yR(n) =

2K∑

k=1

√pkgkxk(n) + zR(n)

Signal received by kth user

yk(n) = fTk xR(n) + zk(n)


Full-duplex: Two-way relay with loop interference at relay


yR(n) =

2K∑

k=1

√pkgkxk(n) + GRRxR(n) + zR(n)


yk(n) = fTk xR(n) + zk(n)


Full-duplex: Two-way relay with loop and inter-user interference


yR(n) =2K∑

k=1

√pkgkxk(n) + GRRxR(n) + zR(n)


yk (n) = fTk xR(n) +∑

i ,k∈Uk

Ωk,i

√pixi (n) + zk(n)


Full-duplex: Two-way relay with self-interference suppression

At instant n, the relay transmit signal is xR(n) = αWyR(n − 1)




By iteratively substituting the value of yR , without any loop-interference cancellation, we have

xR(n) = f [x(n− 1) + x(n − 2) + · · ·+ zR(n − 1) + zR(n − 2) + · · · ]





xR(n) = f [x(n− 1) + x(n − 2) + · · ·+ zR(n − 1) + zR(n − 2) + · · · ]

With loop interference suppression, xR(n) is modelled as a Gaussian noise source xR(n)

yR(n) =

2K∑

k=1

√pkgkxk(n) + GRR xR(n) + zR(n), and





xR(n) = f [x(n− 1) + x(n − 2) + · · ·+ zR(n − 1) + zR(n − 2) + · · · ]


yR(n) =

2K∑

k=1

√pkgkxk(n) + GRR xR(n) + zR(n), and xR(n) = WyR(n − 1)





xR(n) = f [x(n− 1) + x(n − 2) + · · ·+ zR(n − 1) + zR(n − 2) + · · · ]


yR(n) =

2K∑

k=1

√pkgkxk(n) + GRR xR(n) + zR(n), and xR(n) = WyR(n − 1)

yk(n) = fTk WyR(n − 1) +∑

i ,k∈Uk

Ωk,i

√pixi (n) + zk(n)


Full-duplex two-way relay: signal detection

Receive signal after self-interference cancellation at the user Sk as

yk = αfTk W√pk′gk′ xk′

︸︷︷︸

desired signal





︸︷︷︸

desired signal

+ αfTk W

2K∑

i 6=k,k′

√pigixi

︸︷︷︸






︸︷︷︸

desired signal

+ αfTk W

2K∑

i 6=k,k′

√pigixi

︸︷︷︸


+∑

i ,k∈Uk

Ωk,i

√pixi

︸︷︷︸

self loop interferenceand inter-user interference





︸︷︷︸

desired signal

+ αfTk W

2K∑

i 6=k,k′

√pigixi

︸︷︷︸


+∑

i ,k∈Uk

Ωk,i

√pixi

︸︷︷︸


+ αfTk WGRRxR︸︷︷︸

amplified loop interference





︸︷︷︸

desired signal

+ αfTk W

2K∑

i 6=k,k′

√pigixi

︸︷︷︸


+∑

i ,k∈Uk

Ωk,i

√pixi

︸︷︷︸




+ αfTk WzR︸︷︷︸

amplified noise from relay

+ zk︸︷︷︸

AWGN at Sk





︸︷︷︸

desired signal

+ αfTk W

2K∑

i 6=k,k′

√pigixi

︸︷︷︸


+∑

i ,k∈Uk

Ωk,i

√pixi

︸︷︷︸




+ αfTk WzR︸︷︷︸

amplified noise from relay

+ zk︸︷︷︸

AWGN at Sk

We only exploit the knowledge of the E[fTk Wgk′

]in the detection

yk = α√pk′ E

[fTk Wgk′

]xk′

︸︷︷︸

desired signal

+ nk︸︷︷︸

effective noise

, where


Full-duplex two-way relay: spectral efficiency lower bound

Lower bound on the SE is

R lower =

2K∑

k=1

log2 (1 + SNRk) , where


Full-duplex two-way relay: spectral efficiency lower bound

Lower bound on the SE is

R lower =

2K∑

k=1

log2 (1 + SNRk) , where

SNRk =α2pk′

∣∣E[fTk Wgk′

]∣∣2

E [|nk |2]=

akpk′

2K∑

i=1

(

b(1)k,i +b

(2)k,iP

−1R +

∑

i ,k∈Uk

piP−1R b

(3)k,i

)

pi+(

d(1)k +d

(2)k PR+d

(3)k P−1

R

)


Energy efficiency metrics and insights (1)

Energy efficiency of k − k′

pair is defined as

log2 (1 + SNRk(pk ,PR))

µkpk + PR/2K + Pc




pair is defined as


µkpk + PR/2K + Pc

Pc is the circuit power used in transmitter and receiver components




pair is defined as


µkpk + PR/2K + Pc


µk ≥ 1 is the inverse of the power amplifier efficiency of transmit user k




pair is defined as


µkpk + PR/2K + Pc


µk ≥ 1 is the inverse of the power amplifier efficiency of transmit user k

Energy efficiency is a link-centric (or user-centric) performance metric

Global energy efficiency metric combines the individual EEs of different links



GEE is defined as2K∑

k=1


2K∑

k=1

pk + PR + Pc




k=1


2K∑

k=1

pk + PR + Pc

Network-centric GEE metric is not suited when different users have different EE priorities




k=1


2K∑

k=1

pk + PR + Pc

Network-centric GEE metric is not suited when different users have different EE priorities

User-centric weighted sum energy efficiency (WSEE) metric is defined as

2K∑

k=1

wk


pk + PR/2K + Pc


Global energy efficiency when SE (numerator) is optimized

GEE =

2K∑

k=1


2K∑

k=1

pk + PR + Pc

−10 0 10 20 30 400

1

2

3

4

5x 10

7

η (in dB)

GE

E (

bits/J

ou

le)

Figure: When SE (numerator of GEE) is optimized


Global energy efficiency optimization

GEE =

2K∑

k=1


2K∑

k=1

pk + PR + Pc

−10 0 10 20 30 401

2

3

4

5

6x 10

7

η (in dB)

GE

E (

bits/J

ou

le)

Figure: When GEE is optimized


GEE Maximization

GEE maximization problem is formulated as

P1 :Maximizepk ,PR

2K∑

k=1


2K∑

k=1

pk + PR + Pc

s.t.


GEE Maximization


P1 :Maximizepk ,PR

2K∑

k=1


2K∑

k=1

pk + PR + Pc

s.t. 0 ≤ PR ≤ PmaxR , 0 ≤ pk ≤ Pmax, ∀k ∈ K

2K∑

k=1

pk + PR ≤ Pmaxt


GEE Maximization


P1 :Maximizepk ,PR

2K∑

k=1


2K∑

k=1

pk + PR + Pc


2K∑

k=1

pk + PR ≤ Pmaxt

Constraints in P1 are convex but numerator of objective is non-concave


GEE Maximization


P1 :Maximizepk ,PR

2K∑

k=1


2K∑

k=1

pk + PR + Pc


2K∑

k=1

pk + PR ≤ Pmaxt

Constraints in P1 are convex but numerator of objective is non-concave

Approximate it as concave – problem becomes concave-convex fractional program


Solution of concave-convex fractional program

Proposition

Consider a concave-convex fractional program (CCFP) g(x) = u(x)/v(x), with u being non-negative,differentiable and concave,



Proposition

Consider a concave-convex fractional program (CCFP) g(x) = u(x)/v(x), with u being non-negative,differentiable and concave, while v being positive, differentiable and convex.



Proposition

Consider a concave-convex fractional program (CCFP) g(x) = u(x)/v(x), with u being non-negative,differentiable and concave, while v being positive, differentiable and convex. Then the function g(x) ispseudo-concave and a stationary point x∗ of g(x) is its global maximizer



Proposition

Consider a concave-convex fractional program (CCFP) g(x) = u(x)/v(x), with u being non-negative,differentiable and concave, while v being positive, differentiable and convex. Then the function g(x) ispseudo-concave and a stationary point x∗ of g(x) is its global maximizer– Problem of maximizing g(x) is equivalent to finding the positive zero of D(λ), which is defined as

D(λ) , Maxx

u(x)− λv(x)



Proposition

Consider a concave-convex fractional program (CCFP) g(x) = u(x)/v(x), with u being non-negative,differentiable and concave, while v being positive, differentiable and convex. Then the function g(x) ispseudo-concave and a stationary point x∗ of g(x) is its global maximizer– Problem of maximizing g(x) is equivalent to finding the positive zero of D(λ), which is defined as

D(λ) , Maxx

u(x)− λv(x)– Positive zero of D(λ) is found using Dinkelbach’s algorithm


WSEE maximization

WSEE maximization problem is formulated as

P2 : Maximizep

2K∑

k=1

wkEEk =

2K∑

k=1

wk

log2(1 + SNR(pk ))

pk + Pc,k

(2a)

s.t.


WSEE maximization


P2 : Maximizep

2K∑

k=1

wkEEk =

2K∑

k=1

wk

log2(1 + SNR(pk ))

pk + Pc,k

(2a)

s.t. 0 ≤ pk ≤ Pmax, ∀k ∈ K (2b)

2K∑

k=1

pk + PR ≤ Pmaxt , ∀k ∈ K (2c)

Rk ≥ Rk , ∀k ∈ K (2d)


WSEE maximization


P2 : Maximizep

2K∑

k=1

wkEEk =

2K∑

k=1

wk

log2(1 + SNR(pk ))

pk + Pc,k

(2a)

s.t. 0 ≤ pk ≤ Pmax, ∀k ∈ K (2b)

2K∑

k=1


Rk ≥ Rk , ∀k ∈ K (2d)

Numerator of WSEE for each k , can be approximated as a pseudo-concave (PC) function


WSEE maximization


P2 : Maximizep

2K∑

k=1

wkEEk =

2K∑

k=1

wk

log2(1 + SNR(pk ))

pk + Pc,k

(2a)

s.t. 0 ≤ pk ≤ Pmax, ∀k ∈ K (2b)

2K∑

k=1


Rk ≥ Rk , ∀k ∈ K (2d)


WSEE therefore becomes a sum of PC functions – not guaranteed to be a PC functionDinkelbachs algorithm thus cannot be used to optimize it


WSEE maximization


P2 : Maximizep

2K∑

k=1

wkEEk =

2K∑

k=1

wk

log2(1 + SNR(pk ))

pk + Pc,k

(2a)

s.t. 0 ≤ pk ≤ Pmax, ∀k ∈ K (2b)

2K∑

k=1


Rk ≥ Rk , ∀k ∈ K (2d)


WSEE therefore becomes a sum of PC functions – not guaranteed to be a PC functionDinkelbachs algorithm thus cannot be used to optimize it

WSEE maximization problem for full duplex relays is an open problem


WSEE optimization for half-duplex relays

Epigraph form of the problem P2 is as follows

P3 :Maximizep,g

2K∑

k=1

wkgk (3a)

s.t. gk ≤ log2(1 + SNRk(p))

pk + Pc,k

, ∀k ∈ K (3b)

0 ≤ pk ≤ Pmax, ∀k ∈ K (3c)

2K∑

k=1

pk + PR ≤ Pmaxt (3d)

Rk ≥ Rk , ∀k ∈ K (3e)




P3 :Maximizep,g

2K∑

k=1

wkgk (3a)


pk + Pc,k

, ∀k ∈ K (3b)

0 ≤ pk ≤ Pmax, ∀k ∈ K (3c)

2K∑

k=1


Rk ≥ Rk , ∀k ∈ K (3e)

where SNRk(p)=Nβ2

kβk′ pk

2K∑

i 6=k′(β

iβk′ βk

+β2iβi′ )pi+σ2

nrβk′ βk




P3 :Maximizep,g

2K∑

k=1

wkgk (3a)


pk + Pc,k

, ∀k ∈ K (3b)

0 ≤ pk ≤ Pmax, ∀k ∈ K (3c)

2K∑

k=1


Rk ≥ Rk , ∀k ∈ K (3e)

where SNRk(p)=Nβ2

kβk′ pk

2K∑

i 6=k′(β

iβk′ βk

+β2iβi′ )pi+σ2

nrβk′ βk

Constraint in (3b) is non-convex; linearly approximate it using Taylor series


Global energy efficiency results

−10 0 10 20 30 400

1

2

3

4

5

6x 10

7

η (in dB)

GE

E (

bits/J

oule

)

Optimal

Equal

Figure: GEE versus η = Pmaxt /σ2


Effect of weights on weighted sum energy efficiency (1)

0 200 400 600 800 10000

0.5

1

1.5

2

2.5x 10

7

Number of relay antennas, N

EE

(bits/J

oule

)

User−1

User−2

User−3

User−4

Figure: EE of each user versus N, with different weights: Λ1 : w1 = 0.15,w2 = 0.30,w3 = 0.40,w4 = 0.15


Effect of weights on weighted sum energy efficiency (2)

0 200 400 600 800 10000

0.5

1

1.5

2

2.5x 10

7

Number of relay antenna, N

EE

(bits/J

oule

)

User−1

User−2

User−3

User−4

Figure: EE of each user versus N, with different weights: Λ2 : w1 = 0.40,w2 = 0.15,w3 = 0.15,w4 = 0.30


Thank you

Relevant references

[1] Ekant Sharma, Rohit Budhiraja et. al. “Full-Duplex Massive MIMO Multi-Pair Two-Way AF Relaying: Energy Efficiency

Optimization ”, IEEE Trans. Communications, accepted, to appear, 2018.

[2] Ekant Sharma, Swadha Siddhi Chauhan, and Rohit Budhiraja “Weighted Sum Energy Efficiency Optimization for Massive

MIMO Two-Way Half-Duplex AF Relaying”, IEEE Wireless Communications Letters, accepted, to appear, 2018.


Documents

Energy-Efficient Full-Duplex Massive MIMO Relayshome.iitk.ac.in/~rohitbr/pdf/spcom_invited_talk_2018.pdf · Single-hop massive MIMO Consider a single-hop multiple-access system with