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arXiv:0910.3768v1 [cs.IT] 20 Oct 2009

On the transmit strategy for the interference MIMO

relay channel in the low power regime

Anas Chaaban and Aydin Sezgin

Emmy-Noether Research Group on Wireless Networks

Institute of Telecommunications and Applied Information Theory

Ulm University, 89081, Ulm, Germany

Email: anas.chaaban@uni-ulm.de, aydin.sezgin@uni-ulm.de

Abstract—This paper studies the interference channel with

two transmitters and two receivers in the presence of a MIMO

relay in the low transmit power regime. A communication

scheme combining block Markov encoding, beamforming, and

Willems’ backward decoding is used. With this scheme, we get an

interference channel with channel gains dependent on the signal

power. A power allocation for this scheme is proposed, and the

achievable rate region with this power allocation is given. We

show that, at low transmit powers, with equal power constraints

at the relay and the transmitters, the interference channel with a

MIMO relay achieves a sum rate that is linear in the power. This

sum rate is determined by the channel setup. We also show that in

the presence of abundant power at the relay, the transmit strategy

is significantly simplified, and the MAC from the transmitters to

the relay forms the bottle neck of the system from the sum rate

point of view.

I. INTRODUCTION

The capacity of the interference channel (IC) is a thirty

years old problem in network information theory, that is of

practical importance as well. When more than one transmitter

and receiver want to communicate simultaneously, interference

limits their communication. The rate region for the simplest

case of two transmitters and two receivers has been thoroughly

studied, but the problem remains open for the general case.

Recently, some good achievements have been made in

characterizing the degrees of freedom and achievable rate

regions of interference networks. It was shown in [1], that

by using a simple Han-Kobayashi scheme [2], the capacity of

a two user interference channel can be achieved to within one

bit. For the general case of a K-user interference network, it

was shown in [3] that the degrees of freedom is given by K/2,

i.e. the capacity can be well characterized by

K

2log(1 + SNR) + o(SNR),

where the second term decreases for increasing SNR. From

a practical point of view, it is always interesting to analyze

the performance of suboptimal schemes. For instance, in [4],

the rate region of a K-user interference channel is analyzed

for the case in which the interference is treated as noise.

The optimality of treating interference as noise for the two-

user interference channel has been analyzed in [5],[6],[7],[8].

This work is supported by the German Research Foundation, Deutsche

Forschungsgemeinschaft (DFG), Germany, under grant SE 1697/3.

Power allocation strategies for the same system have been

analyzed in [9]. Game-theoretic aspects have been considered

in [10].

Another direction in the study of the IC is the interference

relay channel (IRC), where a relay is used to support the

communication between transmitters and receivers. This has

gained research interest since [11]. Recently, a communication

scheme that achieves full degrees of freedom at high SNR was

proposed in [12] for the interference channel with a MIMO

relay (IMRC). In this scheme, the transmitters communicate

with the relay in a MAC phase, then the relay broadcasts the

received data to the receivers. This is of practical interest, since

in practice, the relay does not have knowledge of the transmit

signals.

In this paper, we consider the IMRC with the communica-

tion scheme proposed in [12]. Namely, this scheme uses super-

position block Markov encoding, beamforming, and Willems’

backward decoding. In spite of its complexity, this scheme

transforms the IMRC to an IC, with channel gains dependent

on the signal power, which simplifies the study of the IMRC.

In [12], some power allocation strategies are considered, but

these power allocations are not optimal; they are of interest

for high transmit power P, where they were used to state the

degrees of freedom of the system. We extend the study to the

low P case, where we study the performance of this scheme,

and propose an (approximately) optimal power allocation.

We give the model of the IMRC in section II, and describe

the communication scheme in section III. Then we study its

performance at low P in section IV. A numerical example is

included in section V. Finally, we conclude with section VI.

II. SYSTEM MODEL

Figure (1) shows a model of the IMRC. Each transmitter

needs to communicate with its respective receiver, and the

relay tries to support this communication. We assume that the

transmitters and receivers are equipped with one antenna each,

and the relay is equipped with 2 antennas.

We denote by x1, x2, and xR the transmitted signals of

transmitter 1, 2 and the relay respectively, and by y1and y2the

received signals at receivers 1 and 2, respectively. We consider

zero mean, unit variance, additive white Gaussian noises at the

receivers and the relay denoted as z1, z2, and zR. So we can

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Relay

Tx1

Tx2 Rx2

Rx1

h22

h11

h21

h12

hR2

hR1

g2R

g1R

Fig. 1. A model for the interference relay channel

write the input-output relations as:

y1

y2

yR

=

=

h11x1+ h21x2+ hR1xR+ z1,

h12x1+ h22x2+ hR2xR+ z2,

g1Rx1+ g2Rx2+ zR,=

where for i,j ∈ {1,2}, i ?= j, hiidenotes the direct channel

gain from transmitter i to receiver i, hijthe cross channel gain

from transmitter i to receiver j, giR= [gi1gi2]Tthe channel

gain from transmitter i to the relay, and hRi= [hRi,1hRi,2]T

the channel gain from the relay to receiver i. The transmitters

have a power constraint P, and the relay has a power constraint

PR. We assume that the relay operates in full duplex mode,

and has global channel knowledge.

III. CODING SCHEME

The coding strategy considered is the one proposed in [12],

and we will briefly explain it in this section. We consider

transmission over a period of B blocks, where the sources

and the relay send sequences of B−1 messages. If a rate pair

(R1,R2) is achievable in a block, then this scheme achieves

a rate pair (R1B−1

B → ∞. This coding strategy at the transmitters and the relay

is sketched in Table I for the general case, and in the following,

we explain it in more details.

B,R2B−1

B), that approaches (R1,R2) as

A. Encoding at the Sources

We use super-position block Markov encoding at the sources

[11], i.e.

x1(b)

x2(b)

=u1(b) + u′

u2(b) + u′

1(b),

2(b),=

where for user i, i ∈ {1,2}, ui(b) is the codeword of the

message of block b, with power pi, and u′

is the codeword of the message of the previous block b −

1, with power p′

The transmitters use predefined messages φ1 and φ2 as the

messages of block 0, i.e. u1(0) and u2(0).

i(b) =

?

p′

piui(b−1),

i

i, such that pi ∈ [0,P[, and pi+ p′

i= P.

B. Decoding and Re-encoding at the Relay

Therelay uses

[13, Section 10.1]. Assuming that the decoding of messages

u1(b−1) and u2(b−1) was successful, the relay can subtract

them from the received signal, and then decode the messages

u1(b) and u2(b) using successive interference cancellation,

achieving rate constraints given by

the SDMAschemedescribed in

R1≤

R2≤

log?1 + ?g1R?2p1

log(det(I2+ GKpG∗)) =

?=

?=

RMAC

1

RMAC

2

RMAC

sum,

,

(1)

(2)

(3)

log?1 + ?g2R?2p2

,

R1+ R2≤

where G = [g1Rg2R], Kp= diag(p1,p2), and I2is the 2×2

identity matrix.

After decoding, the relay uses multimode beamforming to

transmit to the receivers, i.e. the relay constructs the signal

xR(b) = u′

R1(b)t1+ u′

R2(b)t2,

where t1 and t2 are unitary 2 × 1 beamforming vectors. In

our approach, t1 and t2 are chosen such that they reduce

interference at the receivers. Let ρ1,ρ2∈ [0,1] be the power

trade-off coefficients at the relay, i.e. the relay splits its power

to ρ1PRand ρ2PRfor u′

that ρ1+ ρ2= 1. So

R1(b) and u′

R2(b) respectively, such

u′

Ri(b) =

?

ρiPR

p′

i

u′

i(b),for i ∈ {1,2}.

C. Decoding at the destinations

The received signal at receiver i for block b can be written

as

yi(b) =hiiui(b) + (hii+

?

?

ρiPR

p′

i

hT

Riti)u′

i(b)

+ hjiuj(b) + (hji+

ρjPR

p′

j

hT

Ritj)u′

j(b) + zi,

with i ?= j, i,j ∈ {1,2}. In order to reduce interference, the

relay chooses the beamforming vectors t1and t2such that

h21+

?

?

ρ2PR

p′

2

hT

R1t2= 0,

(4)

h12+

ρ1PR

p′

1

hT

R2t1= 0.

Let us denote by t10 and t20 the vectors

?

?t10?2

?

ρ1PR

p′

1t1 and

ρ2PR

p′

2

t2respectively. Since t1and t2are unitary, it follows

=

ρ1PR

p′

ρ2PR

p′

1

,

(5)

?t20?2

=

2

.

With (4) and (5), we get a system of two equations with two

unknowns for each of the beamforming vectors. Notice that

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block b

x1

x2

xR

123 ...

...

...

...

B-1B

(φ1,u1(1))

(φ2,u2(1))

(φ1,φ2)

(u1(1), u1(2))

(u2(1), u2(2))

(u1(1), u2(1))

(u1(2),u1(3))

(u2(2),u2(3))

(u1(2),u2(2))

(u1(B − 2),u1(B − 1))

(u2(B − 2),u2(B − 1))

(u1(B − 2),u2(B − 2))

u1(B − 1)

u2(B − 1)

(u1(B − 1),u2(B − 1))

TABLE I

SKETCH OF THE SUPERPOSITION BLOCK MARKOV CODING SCHEME, HERE, φ1AND φ2ARE ARBITRARY INITIALIZATION MESSAGES KNOWN BY THE

TRANSMITTERS AND THE RELAY, AND (x,y) MEANS A SUPERPOSITION OF x AND y.

these equations do not have a unique solution. Equation (4)

tells us that the components of ti0 are linear with respect to

each other, while (5) tells us that the beamforming vector lies

on a circle, leading to two solutions. Solving for t10and t20,

we get for i ?= j, i,j ∈ {1,2},

?

−

where

ti0=

1

?hRj?2Ti0

hij

hRj,2−

1

?hRj?2

hRj,1

hRj,2Ti0

?

,

(6)

Ti0= nihRj,2

?

−h2

ij+ ?hRj?2ρiPR

P − pi

− hijhRj,1,

(7)

with ni∈ {−1,1}. This gives unitary t1and t2, and satisfies

(4). This choice of ti0 reduces interference seen by the

receivers, so then we can express yi(b) as

yi(b) = hiiui(b) + (hii+ hT

Riti0)u′

i(b) + hjiuj(b) + zi.

Now, the receivers can use Willems’ backward decoding [14]

to decode their signals. Starting from block B, receivers 1 and

2 have interference free signals and can decode u1(B−1) and

u2(B − 1) respectively. Then, in each block b, the receivers

subtract the already known signals u1(b) and u2(b) from their

received signals before attempting to decode u1(b − 1) and

u2(b − 1). Now we can express yi(b) as

yi(b) = (hii+ hT

Riti0)u′

i(b) + hjiuj(b) + zi.

(8)

As a result, the interference relay channel transforms into an

IC. To simplify the notation, we will use f11,f12,f21, and f22

to denote the new channel coefficients:

fii= hii+ hT

Riti0,fij= hij.

(9)

Now we can write the obtained IC input-output equations (8)

as

yi(b) = fiiu′

i(b) + fjiuj(b) + zi,

where fiiand fjidepend on the channel coefficients, pi, P,

PRand ρi.

IV. PERFORMANCE AT LOW TRANSMIT POWER P

We aim in this section to analyze the performance of the

given scheme at low transmit power P. Denote the optimal

power allocation at the transmitters for a fixed power allocation

ρi as ˜ p1 and ˜ p2, and denote the rate region achieved by

this power allocation as Rρ. Then we have the following

proposition.

Proposition 1: The rate region R of the IMRC with the

considered scheme, at low P is given by

R = ch

?

ρ∈[0,1]

Rρ

,

where ch(S) denotes the convex hull of S.

A. Treating interference as noise

Let us assume for the moment being, that we fix a choice

of t10and t20, and we consider a fixed power allocation at the

relay, i.e. fixed niand ρi. Since pi< P, we can approximate

f11 and f22 as linear functions of p1 and p2 respectively as

follows (see details in appendix A)

fii≈ µii+ νiipi

P,

(10)

where we drop the arguments of f(0)

This approximation is needed for solving our optimization

problem, due to the fact that the argument of the square root

in (7) is not concave in pi, and hence can not be optimized

using standard convex optimization tools (e.g. [15]).

The receivers in the obtained IC treat interference as noise,

resulting in rates bounded by

ii, and f(1)

ii

for readability.

R1≤ log

?

?

1 +?f11?2(P − p1)

1 + ?f21?2p2

1 +?f22?2(P − p2)

1 + ?f12?2p1

?

?

= RIC

1,

(11)

R2≤ log

= RIC

2.

(12)

B. Power allocation at low P for sum rate maximization

Up to this point, the expressions are not low-P-specific.

From this point on, we restrict ourself to low P. We still

consider fixed niand ρi. Let us write the rate region for this

scenario as

R1

R2

≤

≤

≤

min(RMAC

min(RMAC

RMAC

sum.

1

,RIC

,RIC

1),

2),

2

R1+ R2

It is required to find powers pithat maximize this region. In

the following proposition, we will specify this rate region at

low P for fixed arbitrary ρi and ni, the proof is shown in

Appendix B.

Proposition 2: The rate region of the IMRC, with the

coding scheme described in section III, with fixed niand ρi

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can be approximated at low P as

R1

≤

?g1R?2ˆ p1

ln2

?g2R?2ˆ p2

ln2

,

(13)

R2

≤

,

where

ˆ p1=λ1+?λ2

ˆ p2=λ2+?λ2

11)−?µ11?2−?g1R?2, and λ2= 2ℜ(µ22ν∗

?µ22?2− ?g2R?2.

Notice that the rate bounds in proposition 2 are linear in ˆ p1

and ˆ p2, which are functions of n1 and n2, so we have the

following corollary.

Corollary 1: The rate region in proposition 2 is maximized

for a fixed arbitrary ρiby choosing powers

1+ 8?µ11?2ℜ(µ11ν∗

4ℜ(µ11ν∗

2+ 8?µ22?2ℜ(µ22ν∗

4ℜ(µ22ν∗

11)

11)

P,

22)

22)

P,

λ1= 2ℜ(µ11ν∗

22)−

˜ p1

=max

n1∈{−1,1}ˆ p1,

max

n2∈{−1,1}ˆ p2.˜ p2

=

Plugging these powers in (13), we get the region Rρ.

C. Special Case: PR≫ P

In this subsection, we introduce a special case, which has

the advantage of significantly simplifying the transmit strategy.

Namely, we consider the case of abundant power at the relay,

i.e. PR≫ P. In this case, we can approximate ti0as

where ni∈ {−1,1}. It follows that the coefficients of the IC

become

ti0≈

nihRj,2

?hRj?

−nihRj,2

?

?

ρ1PR

P−pi

ρ1PR

P−pi

?hRj?

,

f11≈ n1det(H)

?hR2?

?

?

ρ1PR

P − p1,

ρ2PR

P − p2.

f22≈ n2det(H)

?hR1?

Substituting in (11) and (12), we get the following for RIC

and RIC

1

2:

RIC

1

≈ log

?

?

1 +

det2(H)ρ1PR

?hR2?2(1 + ?f21?2p2)

det2(H)ρ2PR

?hR1?2(1 + ?f12?2p1)

?

?

= RAP

1

,

RIC

2

≈ log 1 += RAP

2

.

If PRis high enough, then the rates with abundant relay power

RAP

1

and RAP

2

are greater than the rates at the MAC side of

the IMRC RMAC

1

and RMAC

2

Consequently, the sum rate is determined by the MAC side

of the IMRC, i.e. by RMAC

1

respectively for all p1and p2.

and RMAC

2

, and the optimal

0 0.20.4 0.60.81

−8

−6

−4

−2

0

2

4

6

8

p1/P

Components of t10

t10, 1stcomponent, Exact

t10, 2ndcomponent, Exact

t10, 1stcomponent, Approximate

t10, 2ndcomponent, Approximate

Fig. 2.

p1for P = 0.1. The plot also shows our approximation for the components

of t10.

Components of the beamforming vector t10plotted as a function of

0 0.20.40.60.81

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

p1/P

rMAC

1

rIC

1

RMAC

1

RIC

1

ˆ p1delivering max. R1

Fig. 3.

the power that delivers optimal R1. It also shows plots of the exact expressions

RMAC

1

and RIC

1

(for arbitrary small p2< P).

Plots of the functions rMAC

1

and rIC

1, showing their intersection at

power allocation for maximizing the sum rate in this case is

p1= p2= P.

Remark 1: The expressions in section III are defined for

pi∈ [0,P[, however, they can be easily modified to include

pi= P.

As a result, at high PR, the transmitters do not need to

use super-position block Markov encoding. Each transmitter

sends ui(b) in block b, the relay decodes ui(b), and then

sends them delayed at the next block b + 1 while still using

multimodal beamforming. In this case, we achieve RMAC

log(1 + ?giR?2P).

V. NUMERICAL EXAMPLE

Consider the channel with parameters

i

=

h11= h22= 1.2

g1R= [0.6 1.2]T

hR1= [0.5 1]T

,h12= h21= 0.5,

g2R= [1 0.5]T,

hR2= [1 2]T,

,

,

and assume PR = P = 0.1.

relay, i.e. ρ = 0.5, the components of beamforming vector

t1, and its approximation are shown in figure (2). Figure (3)

For equal power split at the

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−40 −200 2040

0

0.2

0.4

0.6

0.8

1

P(dB)

ˆ p1/P

Exhaustive search solution

Low P approximation

Fig. 4.

problem for p1 (normalized to P).

Exhaustive search and approximate solution of the power allocation

−40 −200 2040

0

0.5

1

1.5

2

P(dB)

Normalized Sum Rate

Exhaustive search solution

Low P approximate solution

p1= p2= P/2

p1= p2=

√P

Fig. 5.Normalized sum rates for different power allocation strategies.

shows a plot of rMAC

we choose ˆ p1= 0.0583, then we maximize min(rMAC

allowing us to achieve maximum R1.

Figure (4) shows the exhaustive search (numerical) solution

of the power allocation problem (ˆ p1) and the approximate

solution, both normalized to P. Finally, in figure (5), we

show the sum rates (normalized to log(1+?hii?2P)) for four

different power allocations:

• Optimal power allocation for maximum sum rate (exhaus-

tive search),

• Approximate power allocation as in corollary 1,

• Equal power allocation with p1= p2= P/2, and

• Equal power allocation with p1= p2=

Notice that at low P, our approximation (dashed line) is close

to the maximal sum rate, and that it is constant in that region.

Notice also that the maximum sum rate approaches one for

large P. The power allocation p1= p2=√P and p1= p2=

P/2 give a normalized sum rate approaching zero and one

respectively at high P which confirms results in [12].

1

and rIC

1. Notice that in this example, if

1

,rIC

1),

√P for P ≥ 1.

VI. CONCLUSION

As a result of this work, we have obtained an approximation

for the optimal power allocation, that maximizes the sum rate

for the given scheme. If we consider the special case of PR=

P, then we obtain a sum rate that is linear in P at low P.

It follows that the normalized sum rate is a constant at low

P, given by the channel parameters and the power split at the

relay.

Using super-position block Markov encoding at the sources,

beamforming at the relay, and Willems’ backward decoding at

the receivers, the IMRC transforms into an IC. We have given

the channel gains of this IC, as functions of the parameters of

the system, including the powers.

Of practical interest is the case where the relay power is

much greater than the transmit power. In this case, we have

shown that the encoding at the transmitters becomes simpler,

since there is no need to perform super-position block Markov

encoding. Furthermore, the MAC from the transmitters to the

relay forms the bottle neck for the system from the sum rate

point of view in this case.

Given the obtained IC, the question of the optimality of

treating interference as noise at the receivers arises. It would

be interesting to find conditions on this channel that allow us

to optimally treat interference as noise. This work can also be

extended to the high power regime, where an optimal power

allocation that maximizes the sum rate at high transmit power

needs to be found.

APPENDIX A

APPROXIMATIONS FOR pi≪ P

Since pi < P, we can approximate

?1 +pi

approximated as

1

P−pi

in (6) as

1

P

using Taylor series, the square root term in (7) can be also

P

?

using Taylor series to the first order. Moreover,

?

ρiPR

P

?hRj?2− h2

ij+

ρiPR

P?hRj?2

ρiPR

P?hRj?2− h2

2P

?

ij

pi.

Remark 2: Note that this approximation is precise only

when pi≪ P, in our case, we only know that pi< P, so this

is a rough approximation.

After substituting in (6) and (9), we get the following expres-

sions for f11and f22

f11

≈

≈

µ11(n1,ρ1) + ν11(n1,ρ1)p1

µ22(n2,ρ2) + ν22(n2,ρ2)p2

P,

P,

f22

Page 6

where

µ11(n1,ρ1)=h11−hR1,2h12

+n1det(H)

?hR2?2

n1ρ1PRdet(H)

2PS1

h22−hR2,2h21

+n2det(H)

?hR1?2

−n2ρ2PRdet(H)

2PS2

hR2,2

?

S1− n1h12hR2,1

hR2,2

?

,

ν11(n1,ρ1)=

,

µ22(n2,ρ2)=

hR1,2

?

−S2+ n2h21hR1,1

hR1,2

?

,

ν22(n2,ρ2)=

,

with H = [hR1 hR2], Si =

i,j ∈ {1,2}.

?

ρiPR

P?hRj?2− h2

ij, i ?= j,

APPENDIX B

LOW P APPROXIMATIONS

In the following, we state the proof of Proposition 2. We

consider low P, i.e. P → 0, and since pi< P, it follows that

pi→ 0, i ∈ {1,2}. Equations (1) and (2) can be respectively

approximated at low P as

RMAC

1

≈

?g1R?2p1

ln(2)

?g2R?2p2

ln(2)

= rMAC

1

,

(14)

RMAC

2

≈

= rMAC

2

.

Equation (3) can be re-written as

RMAC

sum = log(αp1p2+ βp1+ γp2+ 1),

(15)

where

α = ?g11?2?g22?2+?g21?2?g12?2−g12g21g∗

β = ?g11?2+ ?g12?2= ?g1R?2,

γ = ?g21?2+ ?g22?2= ?g2R?2,

and this can be approximated at low P as

11g∗

22−g11g22g∗

12g∗

21,

RMAC

sum ≈?g1R?2p1+ ?g2R?2p2

ln(2)

.

(16)

Notice that the bound RMAC

be considered for low P. Now, equations (11) and (12) can

be approximated as

sum

is redundant and needs not to

RIC

1

≈

1

ln(2)(?µ11?2P + (2ℜ(µ11ν∗

−2ℜ(µ11ν∗

1

ln(2)(?µ22?2P + (2ℜ(µ22ν∗

−2ℜ(µ22ν∗

11) − ?µ11?2)p1

11)p2

1/P) = rIC

1,

RIC

2

≈

22) − ?µ22?2)p2

22)p2

2/P) = rIC

2.

As a result of (14) and (17) we can write the rate region at

low P as

R1

R2

≤

≤

min(rMAC

min(rMAC

1

,rIC

1),

,rIC

2),

2

In order to maximize this rate region, we would like to

choose a power allocation that maximizes min(rMAC

and min(rMAC

2

,rIC

rMAC

1

−rIC

for p1= 0, rMAC

1

−rIC

0 admits a solution ˆ p1∈ [0,P]. Similarly, rMAC

admits a solution ˆ p2 ∈ [0,P]. After solving the resulting

quadratic equations, we get

1

,rIC

1)

2) over p1 and p2 respectively. Since

is a quadratic function of p1, and rMAC

> 0 for p1= P, then rMAC

11

−rIC

−rIC

− rIC

1

< 0

=

= 0

111

2

2

ˆ p1=λ1+?λ2

ˆ p2=λ2+?λ2

1+ 8?µ11?2ℜ(µ11ν∗

4ℜ(µ11ν∗

2+ 8?µ22?2ℜ(µ22ν∗

4ℜ(µ22ν∗

11)

11)

P,

22)

22)

P,

with λ1 = 2ℜ(µ11ν∗

2ℜ(µ22ν∗

(14) gives us the rate region achievable by this scheme at low

P.

11) − ?µ11?2− ?g1R?2and λ2 =

22)−?µ22?2−?g2R?2. Substituting these powers in

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