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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
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 2
Single-hop massive MIMO
Consider a single-hop multiple-access system with two users
Received signal y = g1x1 + g2x2 + n
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 2
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
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 2
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
=gH1 g1N
x1︸ ︷︷ ︸
desired signal
+gH1 g2N
x2︸ ︷︷ ︸
interference
+gH1 n
N︸︷︷︸
noise
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 2
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
=gH1 g1N
x1︸ ︷︷ ︸
desired signal
+gH1 g2N
x2︸ ︷︷ ︸
interference
+gH1 n
N︸︷︷︸
noise
y1a.s.−−−−→
N→∞x1
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 2
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
=gH1 g1N
x1︸ ︷︷ ︸
desired signal
+gH1 g2N
x2︸ ︷︷ ︸
interference
+gH1 n
N︸︷︷︸
noise
y1a.s.−−−−→
N→∞x1
Both interference and noise asymptotically vanish
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 2
Half-duplex: One-way relay (MAC phase)
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 3
Half-duplex: One-way relay (BC phase)
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 4
Half-duplex: Two-way relay (MAC phase)
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 5
Half-duplex: Two-way relay (BC phase)
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 6
Half-duplex: mathematical model for MAC phase
Received signal at the relay, yR =∑2K
k=1
√pkgkxk + zR
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 7
Half-duplex: mathematical model for MAC phase
Received signal at the relay, yR =∑2K
k=1
√pkgkxk + zR
Relay amplifies the received signal as
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 7
Half-duplex: mathematical model for MAC phase
Received signal at the relay, yR =∑2K
k=1
√pkgkxk + zR
Relay amplifies the received signal as
xR = αWyR = αW
2K∑
k=1
√pkgkxk + αWzR
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 7
Half-duplex: mathematical model for MAC phase
Received signal at the relay, yR =∑2K
k=1
√pkgkxk + zR
Relay amplifies the received signal as
xR = αWyR = αW
2K∑
k=1
√pkgkxk + αWzR
Relay MRC/MRT precoder is designed as W = F∗GH
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 7
Half-duplex: Mathematical model
Received signal at k′
th user is
yk′ = fTk′xR + zk′
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 8
Half-duplex: Mathematical model
Received signal at k′
th user is
yk′ = fTk′xR + zk′ = αfT
k′W
2K∑
k=1
√pkgkxk + αfT
k′WzR + zk′
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 8
Half-duplex: Mathematical model
Received signal at k′
th user is
yk′ = fTk′xR + zk′ = αfT
k′W
2K∑
k=1
√pkgkxk + αfT
k′WzR + zk′
= αfTk′W
√pk gk xk
︸ ︷︷ ︸
desired signal
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 8
Half-duplex: Mathematical model
Received signal at k′
th user is
yk′ = fTk′xR + zk′ = αfT
k′W
2K∑
k=1
√pkgkxk + αfT
k′WzR + zk′
= αfTk′W
√pk gk xk
︸ ︷︷ ︸
desired signal
+ αfTk′W
√pkgk′ xk′
︸ ︷︷ ︸
self-interference
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 8
Half-duplex: Mathematical model
Received signal at k′
th user is
yk′ = fTk′xR + zk′ = αfT
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
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 8
Half-duplex: Mathematical model
Received signal at k′
th user is
yk′ = fTk′xR + zk′ = αfT
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
+ αfTk′WzR
︸ ︷︷ ︸
amplified relay noise
+ zk′
︸︷︷︸
AWGN at user k′
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 8
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)
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 9
Full-duplex: Two-way relay with loop interference at relay
Relay receive signal
yR(n) =
2K∑
k=1
√pkgkxk(n) + GRRxR(n) + zR(n)
Signal received by kth user
yk(n) = fTk xR(n) + zk(n)
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 10
Full-duplex: Two-way relay with loop and inter-user interference
Relay receive signal
yR(n) =2K∑
k=1
√pkgkxk(n) + GRRxR(n) + zR(n)
Signal received by kth user
yk (n) = fTk xR(n) +∑
i ,k∈Uk
Ωk,i
√pixi (n) + zk(n)
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 11
Full-duplex: Two-way relay with self-interference suppression
At instant n, the relay transmit signal is xR(n) = αWyR(n − 1)
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 12
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) + · · · ]
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 12
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) + · · · ]
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
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 12
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) + · · · ]
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) = WyR(n − 1)
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 12
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) + · · · ]
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) = WyR(n − 1)
yk(n) = fTk WyR(n − 1) +∑
i ,k∈Uk
Ωk,i
√pixi (n) + zk(n)
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 12
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
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 13
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
+ αfTk W
2K∑
i 6=k,k′
√pigixi
︸ ︷︷ ︸
inter-pair interference
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 13
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
+ αfTk W
2K∑
i 6=k,k′
√pigixi
︸ ︷︷ ︸
inter-pair interference
+∑
i ,k∈Uk
Ωk,i
√pixi
︸ ︷︷ ︸
self loop interferenceand inter-user interference
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 13
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
+ αfTk W
2K∑
i 6=k,k′
√pigixi
︸ ︷︷ ︸
inter-pair interference
+∑
i ,k∈Uk
Ωk,i
√pixi
︸ ︷︷ ︸
self loop interferenceand inter-user interference
+ αfTk WGRRxR︸ ︷︷ ︸
amplified loop interference
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 13
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
+ αfTk W
2K∑
i 6=k,k′
√pigixi
︸ ︷︷ ︸
inter-pair interference
+∑
i ,k∈Uk
Ωk,i
√pixi
︸ ︷︷ ︸
self loop interferenceand inter-user interference
+ αfTk WGRRxR︸ ︷︷ ︸
amplified loop interference
+ αfTk WzR︸ ︷︷ ︸
amplified noise from relay
+ zk︸︷︷︸
AWGN at Sk
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 13
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
+ αfTk W
2K∑
i 6=k,k′
√pigixi
︸ ︷︷ ︸
inter-pair interference
+∑
i ,k∈Uk
Ωk,i
√pixi
︸ ︷︷ ︸
self loop interferenceand inter-user interference
+ αfTk WGRRxR︸ ︷︷ ︸
amplified loop interference
+ 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
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 13
Full-duplex two-way relay: spectral efficiency lower bound
Lower bound on the SE is
R lower =
2K∑
k=1
log2 (1 + SNRk) , where
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 14
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
)
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 14
Energy efficiency metrics and insights (1)
Energy efficiency of k − k′
pair is defined as
log2 (1 + SNRk(pk ,PR))
µkpk + PR/2K + Pc
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 15
Energy efficiency metrics and insights (1)
Energy efficiency of k − k′
pair is defined as
log2 (1 + SNRk(pk ,PR))
µkpk + PR/2K + Pc
Pc is the circuit power used in transmitter and receiver components
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 15
Energy efficiency metrics and insights (1)
Energy efficiency of k − k′
pair is defined as
log2 (1 + SNRk(pk ,PR))
µkpk + PR/2K + Pc
Pc is the circuit power used in transmitter and receiver components
µk ≥ 1 is the inverse of the power amplifier efficiency of transmit user k
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 15
Energy efficiency metrics and insights (1)
Energy efficiency of k − k′
pair is defined as
log2 (1 + SNRk(pk ,PR))
µkpk + PR/2K + Pc
Pc is the circuit power used in transmitter and receiver components
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
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 15
Energy efficiency metrics and insights (2)
GEE is defined as2K∑
k=1
log2 (1 + SNRk(pk ,PR))
2K∑
k=1
pk + PR + Pc
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 16
Energy efficiency metrics and insights (2)
GEE is defined as2K∑
k=1
log2 (1 + SNRk(pk ,PR))
2K∑
k=1
pk + PR + Pc
Network-centric GEE metric is not suited when different users have different EE priorities
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 16
Energy efficiency metrics and insights (2)
GEE is defined as2K∑
k=1
log2 (1 + SNRk(pk ,PR))
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
log2 (1 + SNRk(pk ,PR))
pk + PR/2K + Pc
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 16
Global energy efficiency when SE (numerator) is optimized
GEE =
2K∑
k=1
log2 (1 + SNRk(pk ,PR))
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
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 17
Global energy efficiency optimization
GEE =
2K∑
k=1
log2 (1 + SNRk(pk ,PR))
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
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 18
GEE Maximization
GEE maximization problem is formulated as
P1 :Maximizepk ,PR
2K∑
k=1
log2 (1 + SNRk(pk ,PR))
2K∑
k=1
pk + PR + Pc
s.t.
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 19
GEE Maximization
GEE maximization problem is formulated as
P1 :Maximizepk ,PR
2K∑
k=1
log2 (1 + SNRk(pk ,PR))
2K∑
k=1
pk + PR + Pc
s.t. 0 ≤ PR ≤ PmaxR , 0 ≤ pk ≤ Pmax, ∀k ∈ K
2K∑
k=1
pk + PR ≤ Pmaxt
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 19
GEE Maximization
GEE maximization problem is formulated as
P1 :Maximizepk ,PR
2K∑
k=1
log2 (1 + SNRk(pk ,PR))
2K∑
k=1
pk + PR + Pc
s.t. 0 ≤ PR ≤ PmaxR , 0 ≤ pk ≤ Pmax, ∀k ∈ K
2K∑
k=1
pk + PR ≤ Pmaxt
Constraints in P1 are convex but numerator of objective is non-concave
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 19
GEE Maximization
GEE maximization problem is formulated as
P1 :Maximizepk ,PR
2K∑
k=1
log2 (1 + SNRk(pk ,PR))
2K∑
k=1
pk + PR + Pc
s.t. 0 ≤ PR ≤ PmaxR , 0 ≤ pk ≤ Pmax, ∀k ∈ K
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
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 19
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,
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 20
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, while v being positive, differentiable and convex.
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 20
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, 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
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 20
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, 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)
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 20
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, 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
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 20
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.
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 21
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. 0 ≤ pk ≤ Pmax, ∀k ∈ K (2b)
2K∑
k=1
pk + PR ≤ Pmaxt , ∀k ∈ K (2c)
Rk ≥ Rk , ∀k ∈ K (2d)
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 21
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. 0 ≤ pk ≤ Pmax, ∀k ∈ K (2b)
2K∑
k=1
pk + PR ≤ Pmaxt , ∀k ∈ K (2c)
Rk ≥ Rk , ∀k ∈ K (2d)
Numerator of WSEE for each k , can be approximated as a pseudo-concave (PC) function
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 21
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. 0 ≤ pk ≤ Pmax, ∀k ∈ K (2b)
2K∑
k=1
pk + PR ≤ Pmaxt , ∀k ∈ K (2c)
Rk ≥ Rk , ∀k ∈ K (2d)
Numerator of WSEE for each k , can be approximated as a pseudo-concave (PC) function
WSEE therefore becomes a sum of PC functions – not guaranteed to be a PC functionDinkelbachs algorithm thus cannot be used to optimize it
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 21
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. 0 ≤ pk ≤ Pmax, ∀k ∈ K (2b)
2K∑
k=1
pk + PR ≤ Pmaxt , ∀k ∈ K (2c)
Rk ≥ Rk , ∀k ∈ K (2d)
Numerator of WSEE for each k , can be approximated as a pseudo-concave (PC) function
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
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 21
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)
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 22
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)
where SNRk(p)=Nβ2
kβk′ pk
2K∑
i 6=k′(β
iβk′ βk
+β2iβi′ )pi+σ2
nrβk′ βk
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 22
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)
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
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 22
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
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 23
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
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 24
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
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 25
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.
SPCOM 2018 (Rohit Budhiraja, IITK) Energy-Efficient Full-Duplex Massive MIMO Relays 26