Power Allocation for Interference Channels

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Krishna Chaitanya A, Utpal Mukherji, Vinod Sharma Department of Electrical Communication Engineering,

Indian Institute of Science, Bangalore-560012 Email: {akc, utpal, vinod}@ece.iisc.ernet.in

Abstract—We propose power allocation algorithms for increas- ing the sum rate of two and three user interference channels.

The channels experience fast fading and there is an average power constraint on each transmitter. Our achievable strategies for two and three user interference channels are based on the classification of the interference into very strong, strong and weak interferences. We present numerical results of the power allocation algorithm for two user Gaussian interference channel with Rician fading with mean total power gain of the fadeΩ = 3 and Rician factor κ= 0.5and compare the sum rate with that obtained from ergodic interference alignment with water-filling.

We show that our power allocation algorithm increases the sum rate with a gain of 1.66dB at average transmit SNR of 5dB.

For the three user Gaussian interference channel with Rayleigh fading with distribution CN(0,0.5), we show that our power allocation algorithm improves the sum rate with a gain of 1.5dB at average transmit SNR of 5dB.

Index Terms—Interference channel, Power allocation, Aug- mented Lagrangian method, Ergodic interference alignment.

I. INTRODUCTION

Interference channels were first introduced by Shannon in [1]. Since interference is ubiquitous in wireless communication and is in fact the limiting factor, such channels are being actively studied recently. Although the capacity region of a general interference channel is still not known, for very strong and strong interference channels this region was determined by Carleial and Sato in [2] and [3], respectively. Though we do not know the capacity region for weak interference channels, it is known that treating interference from a weak interferer as noise is sum rate optimal [4]. In this paper, we investigate the gains in sum rate due to power control, for several different interference channels. We show that power control can often provide significant gains.

For a 2-user interference channel the best known achievable rate region was given by Han and Kobayashi in [5], and a tighter outer bound was given by Etkin et al. in [6]. In [7], Tuninetti considered the ergodic fast fading interference channel with the assumption that all terminals know all fading gains from every source to every destination. With a power constraint across the channel states for each user, Tuninetti proposed a power allocation policy for Gaussian interference channels assuming a Han-Kobayashi achievable scheme. In the present paper, we perform power allocation by first classifying the fading into strong-strong, strong-weak and weak-weak categories and then optimizing the sum rate in each category.

Consequently, we obtain higher sum rate than in previous studies.

A significant result for multiuser interference channels with atleast three users, shown by Cadambe and Jafar in [8] and Nazer et al. in [9], is that atleast half of the interference free capacity can be achieved for each user via interference alignment. In the case of an interference channel with more than two users, if the interference is under a threshold, it is shown by Annapureddy and Veeravalli in [10] that treating interference as noise achieves sum capacity. Thus ergodic alignment may not be optimal in the low interference regime, but when the interference to noise ratio is close to the signal to noise ratio, ergodic interference alignment is the best solution known so far. As the number of users in the system increases, interference increases and ergodic interference alignment is shown to be close to the outer bound in [11] by Nazer et al.

In [12], Deng et al. consider power allocation for Gaussian interference channels with total transmit power constraint. The channel fading coefficients are fixed. With the objective of maximizing the sum rate for a fixed channel with total transmit power constraint, they consider various power allocation policies such as equal power allocation, a greedy algorithm, water-filling and iterative water-filling, for the two and multi user interference channels. In all these schemes, interference is treated as noise irrespective of the channel state. In the present paper, we also consider power allocation for three user interference channels. The optimization problem we consider is different from that in [12]. We consider a fast fading channel, with average power constraint on each transmitter to model distinct sources of power for each transmitter. The achievable strategy we follow for the three user interference channel is again based on the classification of the channel as in the case of two user interference channels.

This paper is organized as follows. Section II considers a two user channel and describes our power control policy to increase sum rate. Section III considers the three user case.

Section IV compares our policies numerically with existing policies. Section V concludes the paper.

II. POWERALLOCATION FOR2-USERINTERFERENCE

CHANNEL

We consider a two user interference channel as shown in Fig 1. Lethij denote a generic channel gain from transmitter i to receiver j,i, j = 1,2. Thus if X_i is transmitted by the ith transmitter,Y_j, received by thejth receiver is

Yj =hijXi+hjjXj+Zj, i6=j,

whereZj is the AWGN noise with variance 1. The channel is flat, fast fading, i.e., all the channel gains form independent

978-1-4673-5952-8/13/$31.00 c2013 IEEE

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bbb

⊕

X1

X2

Z1∼ CN(0,1)

Z2∼ CN(0,1) Y1

Y2

h11

h₁₂ h21

h₂₂

Fig. 1. Interference Channel

and identically distributed (i.i.d) sequences in time. The channel coefficients h11, h12, h21 andh22 are also assumed to be mutually independent. We assume complete causal knowledge of channel states at all transmitters and receivers. Throughout the paper we assume unit noise variance at each receiver.

The interference channel shown in Fig.1 is called very strong,strongandweakunder respective conditions

I(X₁;Y₁|X2)≤I(X₁;Y₂), I(X₂;Y₂|X1)≤I(X₂;Y₁);

I(X1;Y₁|X2)≤I(X1;Y₂|X2), I(X2;Y₂|X1)≤I(X2;Y₁|X1);

I(X1;Y1|X2)> I(X1;Y2|X2), I(X2;Y2|X1)> I(X2;Y1|X1).

We know the capacity region of very strong and strong interference channels from [2], [3]. The capacity region of very strong interference channels is

R1≤I(X1;Y1|X2), R2≤I(X2;Y2|X1). (1) The capacity region for the strong interference channel is

R1 ≤I(X1;Y1|X2), R2≤I(X2;Y2|X1), R1+R2 ≤min{I(X1, X2;Y1), I(X1, X2;Y2)}. (2) The sum capacity for weak interference channels is also known from [4].

We now further classify the interference channel as strong- weakif

I(X1;Y1|X2) ≤ I(X1;Y2|X2),

I(X2;Y2|X1) > I(X2;Y1|X1), (3) andvery strong-weak if

I(X₁;Y₁|X2)≤I(X₁;Y₂), I(X₂;Y₂|X1)> I(X₂;Y₁|X1). (4) Consistent with the terminology introduced above, we call the conventional very strong and strong interference channels as very strong-very strong and strong-strong interference channels respectively. We also refer to the conventional weak interference channel as weak-weak interference channel.

Optimal rates of very strong-very strong and strong-strong interference channels are achieved using successive decoding.

Also treating interference as noise is sum rate optimal for weak-weak interference channels. We use these facts in our power control scheme for the above two classes of interference channel.

In a strong-weak interference channel (3), we use a strategy that treats the interference from the weak interferer as noise and decodes the interfering message from the strong interferer. If receiver 1 experiences the weak interferer, receiver 1 treats interference as noise and hence the rate bound at receiver 1 isR1≤I(X1;Y1). Receiver 2 uses the successive decoding scheme: it first decodes the interferer’s message,X1, treating its own message as noise and subtracts the decoded message and decodes its own message in the second step.

The rate bounds at receiver 2 are R1 ≤ I(X1;Y2), R2 ≤ I(X2;Y2|X1).Thus an achievable rate region for strong-weak interference channel is

R1≤min{I(X1;Y1), I(X1;Y2)}, R2≤I(X2;Y2|X1). (5) Since a very strong-weak interference channel is also a strong- weak interference channel, the region in (5) is also achievable for a very strong-weak interference channel. The rate region resulting after exchanging indices 1 and 2 in (5), gives an achievable region of weak-strong and weak-very strong interference channels.

For the Gaussian interference channel shown in Fig.1, with average power constraint Pi at user i, i = 1,2, these rate regions can be expressed as follows. For the strong-weak interference channel, i.e.,|h11|²≤ |h12|²and|h22|²>|h21|², the rate region in (5) can be expressed as

R1 < min{log(1 + |h11|²P1

1 +|h21|²P2),log(1 + |h12|²P1

1 +|h22|²P2

)},

R2 < log(1 +|h22|²P2). (6)

In the case of a weak-weak interference channel, we treat interference as noise at both receivers and hence an achievable region is given as

R1<log(1 + |h11|²P1

1 +|h21|²P2

), R2<log(1 + |h22|²P2

1 +|h12|²P1

).

Similarly one can write achievable rate regions for the other cases.

A. Power Allocation

We propose a power allocation scheme for fast fading Gaussian interference channels. For a given channel state h = (h11, h12, h21, h22), the channel will be of one of the types mentioned above. We know an achievable region for each type of channel. Let R(h, P1(h), P2(h)) denote the achievable sum rate for a given statehand power allocation policyP1(h)andP2(h)(for a given power allocation policy the ‘type’ of the channel can change than for no power control case). Assuming that the channel is known at both transmitter and receiver, the achievable sum rate of the fast fading channel is then given by

R=E[R(h, P1(h), P2(h))] (7) where expectation is taken over the joint distribution ofh,and R(h, P1(h), P2(h))is defined as

R(h, P1(h), P2(h))

=log(1 +|h11|²P1(h)) +log(1 +|h22|²P2(h))

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if his of the Very strong-Very strong interference type, R(h, P1(h), P2(h))

=min

log(1 +|h11|²P1(h) +|h21|²P2(h)), log(1 +|h22|²P2(h) +|h12|²P1(h))

if his of the strong-strong interference type, R(h, P1(h), P2(h)) =log

1 + |h11|²P1(h) 1 +|h21|²P2(h)

+ log

1 + |h22|²P2(h) 1 +|h12|²P1(h)

if his of the weak-weak interference type, R(h, P1(h), P2(h)) =min{log

1 + |h11|²P1(h) 1 +|h21|²P2(h)

+ log 1 +|h22|²P2(h)

,log 1 +|h22|²P2(h) +|h12|²P1(h) } if h is of the strong-weak or very strong-weak interference type and

R(h, P1(h), P2(h)) =min{log

1 + |h22|²P2(h) 1 +|h12|²P1(h)

+ log 1 +|h11|²P1(h)

,log 1 +|h11|²P1(h) +|h21|²P2(h) } if h is of the weak-strong or weak-very strong interference type.

Our optimization problem is

minimize −E[R(h, P1(h), P2(h))]

subject to E[P1(h)] =P1,E[P2(h)] =P2.

Assuming Rayleigh fading or Rician fading for the channel states, it is difficult to get a closed form expression for the ex- pectations in the above problem. So, for channel states whose distributions are continuous, we use Monte-Carlo simulation to generate channel state samples and numerical methods to solve the optimization problem. We begin by takingNsamples of the channel state{h₁,h₂, . . . ,h_N}, and we denote the2N power variables in the sum of sum rate over the channel states byP1(h₁), . . . , P1(h_N), P2(h₁), . . . , P2(h_N). We denote the N variables P1(h₁), . . . , P1(h_N) by P₁; similarly we write P₂ to denote the N variables P2(h₁), . . . , P2(h_N). The objective function and constraints are defined as

f(P₁,P₂) , −

N

X

i=1

piR(h_i, P1(h_i), P2(h_i))

g1(P₁) ,

N

X

i=1

piP1(h_i)−P1

g2(P₂) ,

N

X

i=1

piP2(h_i)−P2

where p_i =_N¹ fori= 1,2, . . . , N. The optimization problem is

minimize f(P₁,P₂) subject to g1(P

1) = 0, g2(P

2) = 0. (8)

Algorithm 1 Augmented Lagrangian method to find Power allocation

Initializeλ⁽¹⁾₁ , λ⁽¹⁾₂ ,P

1(0)

,P

2(0)

. fork= 1→ ∞do

[P

1(k)

,P

2(k)] =

Steepest Descent(λ^(k)₁ , λ^(k)₂ ,P₁^(k⁻¹⁾,P₂^(k⁻¹⁾) if |g1(P₁)|< ǫand|g2(P₂)|< ǫthen

break else

λ^(k+1)₁ =λ^(k)₁ +cg1(P₁) λ^(k+1)₂ =λ^(k)₂ +cg2(P

2) k=k+ 1

end if end for

sum rate =L(P

1(k)

,P

2(k)

, λ^(k)₁ , λ^(k)₂ ).

function STEEPEST DESCENT(λ1, λ2,P₁,P₂) Fixα, δ, ǫ

loop

fori= 1→N do P₁^(i,δ)=P₁+δe_i

∂L

∂P1(hi)≈ ^L⁽^P¹^(i,δ)^,^P²^,λ¹^,λ²_δ⁾^−L⁽^P¹^,^P²^,λ¹^,λ²⁾

P₂^(i,δ)=P₂+δe_i

∂L

∂P2(hi)≈ ^L^(P¹^,P²^(i,δ)^,λ¹^,λ²_δ⁾^−L^(P¹^,P²^,λ¹^,λ²⁾ P1(h

i) =P1(h

i)−α_∂P^∂^L

1(hi)

P2(h

i) =P2(h

i)−α_∂P^∂^L

2(hi)

end for Till|_∂P^∂^L

1(hi)|< ǫand|_∂P^∂^L

2(hi)|< ǫfor eachi.

end loop returnP₁,P₂ end function

We approach the solution of optimization problem (8) using the augmented Lagrangian method [13] presented in Algorithm 1. The augmented Lagrange function with penalty factorc is

L(P

1,P

2, λ1, λ2) =f(P

1,P

2) +λ1g1(P

1)+

λ2g2(P

2) +1 2ch

g1(P

1)² +

g2(P

2)²i . We make the following remarks regarding Algorithm 1.

1) The classification of the channel into very strong-very strong and strong-strong depends on the power allocation. In Algorithm 1, we classify the channel in each iteration depending on the power allocation in that iteration and calculate the new power allocation using the appropriate sum rate expression.

2) The sum rate expressions in the cases of strong-weak, very strong-weak, weak-strong and weak-very strong interference are a minimum of two logarithmic functions and hence the objective function f(P₁,P₂) need not

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be differentiable at some points (P₁,P₂). Even in the case of weak-weak interference channel, though the sum rate is differentiable, the optimal powers are solutions of a system of two polynomial equations of degree 7.

Therefore, we use numerical methods to evaluate the gradient of augmented Lagrangian with respect to power variables as in Algorithm 1.

3) We use the steepest descent method to optimize the augmented Lagrange function in each iteration. Since this is a non-convex optimization problem, the algorithm may converge to a local minimum. We execute the Algorithm 1 for different initial power allocations and choose the best power allocation.

4) In Algorithm 1,ǫ >0 is the stopping threshold.

5) In the steepest descent algorithm used in Algorithm 1, e_idenotes unit vector of lengthN whosei^th component is1and all other components are0,δis used to approx- imate the gradient of the augmented Lagrange function with respect to power variables, andα is the step size in the variables in the negative gradient direction.

III. POWER ALLOCATION FOR3-USERINTERFERENCE CHANNEL

We now apply a similar power allocation algorithm to a 3-user interference channel. It is well known that ergodic interference alignment will achieve half the interference free rate for this channel. In ergodic interference alignment, the effect of the interference is completely removed by using the cancellation strategy proposed in [9], which states that out of n time slots, use ⁿ₂ slots for transmitting the message and remaining ⁿ₂ slots for cancellation of interference. Therefore, the rate of transmission for each transmitter-receiver pair will effectively depend only on the direct link gain. The sum rate is thus given by

R=

3

X

i=1

1

2log(1 + 2|hii|²Pi(h)).

As each user’s rate depends only on the direct link gain, it is quite natural to use water-filling to get the optimal sum rate for fast fading interference channels. We now define a coding strategy and apply power allocation so as to obtain an improvement in the sum rate compared to ergodic interference alignment with water-filling.

Our coding scheme jointly uses successive decoding and treating interference as noise depending on the type of the interference. Each receiver does the following to determine the decoding order and finds the users to treat as noise. A receiver first looks at the type of interference caused from each of the other transmitters. For example, receiver 1 treats the interference from transmitter 2 as strongif

I(X2;Y1|X1, X3)≥I(X2;Y2|X1, X3), andvery strong if

I(X2;Y1)≥I(X2;Y2|X1, X3),

Receiver1treats interference from the2^ndtransmitter asweak interference if

I(X2;Y1|X1, X3)< I(X2;Y2|X1, X3).

Of course, transmitters2and3can be interchanged to find how to treat the third transmitter. Similarly, other receivers classify their interferers as weak, strong or very strong. In the next step, each receiver uses the successive decoding strategy for strong and very strong interferers and treats the interference from weak interferers as noise. Very strong interferers are decoded first. If there are more than one very strong or strong interferers, the interferer with the higher cross link gain is decoded first. Each receiver adds a rate constraint onR_j if the message from the j^th transmitter is decoded at that receiver.

The sum rate can be found from all the rate constraints arising from all the receivers.

We illustrate the calculation of the achievable rate region for an example channel. Let us assume a 3-user interference channel with

By following the decoding strategy defined earlier, we need to decode X2 and X3 at receiver 1. To decode X3, treatX1 as noise at receiver2, and treatX1 andX2 as noise at receiver 3. Hence, we have the following rate constraints.

At receiver1:

R1 < I(X1;Y1|X2, X3), R2< I(X2;Y1), R3 < I(X3;Y1|X2).

At receiver2:

R2< I(X2;Y2|X3), R3< I(X3;Y2).

At receiver3:

R3 < I(X3;Y3).

Here in this example we have assumed a higher interference due to transmitter 2 than due to transmitter 3 at receiver 1.

The achievable rate region is thus given by R1 < I(X1;Y1|X2, X3),

R2 < min{I(X2;Y1), I(X2;Y2|X3)},

R₃ < min{I(X3;Y₁|X2), I(X3;Y₂), I(X3;Y₃)}.

Finding the rate expressions is cumbersome as there are many possible cases. We aim to optimize sum rate under average transmit power constraints. As in the case of two user interference channel, here also we use Monte-Carlo simulation to generate channel state samples and Lagrange multiplier method to approach the optimal powers to maximize sum

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0 5 10 15 20 25 0

2 4 6 8 10 12

Constrained Average transmit SNR in dB

Sum Rate in bits per channel use

Sum rate achieved by proposed power allocation Sum rate achieved by power allocation in [7]

Sum rate achieved by proposed scheme without power allocation Sum rate achieved by ergodic interference alignment with water−filling

Fig. 2. Sum rate comparison of proposed power allocation scheme for 2-user Rician faded interference channel with water-filling on ergodic interference alignment.

rate. The algorithm for Lagrange multiplier method is similar to Algorithm 1 with c = 0 and λ1, λ2 are updated in each iteration in a way similar to update of power variables in the steepest descent method, i.e., we find the gradient of the Lagrangian with respect to λ1 andλ2 and update them in negative gradient direction with a step size of α.

IV. RESULTS

We fix N = 1000, ǫ = 10⁻³, δ = 10⁻⁴, c = 4000 and variable step size α= 10⁻³ at the start of Algorithm 1. We decrease the step size by a factor of 10 if |_∂P^∂^L

1(hi)| < 100ǫ and|_∂P^∂^L

2(hi)|<100ǫ for eachi. In Fig 2, we show sum rates for the two-user fast fading interference channel with direct link and cross link gains Rician distributed with parameters κ= 0.5,Ω = 3. We compare the sum rate with water-filling on ergodic interference alignment. We see a gain of 1.66dB at 5dB and a gain of 1.96dB at 10dB. For Rayleigh fading, when the interference channel is in weak-weak interference regime, with high probability, we observe a marginal improvement in the sum rate. The gain in sum rate is substantial if probability of strong or very strong interference is high.

Next, we consider a 3-user interference channel with both direct link and cross link gains Rayleigh distributed with parameter σ² = 0.5. In this case, we fix N = 1000, ǫ = 10⁻³, δ= 10⁻⁴, c= 0 and variable step sizeα= 10⁻⁴at the start of Algorithm 1. We decrease the step size by a factor of 10 if |_∂P^∂^L

1(hi)|< 100ǫ and |_∂P^∂^L

2(hi)| <100ǫ for each i. We compare the sum rates calculated using the proposed power allocation algorithm for a 3-user interference channel with sum rates calculated using ergodic interference alignment with and without water-filling in Fig 3. We see a gain of about 1.5dB at an SNR of 5dB. However, water-filling itself does not give much improvement.

0 1 2 3 4 5 6 7 8 9 10

1 1.5 2 2.5 3 3.5 4 4.5 5

Constrained Average transmit SNR in dB

Sum Rate in bits per channel use

Sum rate achieved by proposed power allocation

Sum rate achieved by ergodic interference alignment with water−filling Sum rate achieved by ergodic interference alignment without water−filling

Fig. 3. Sum rate comparison of proposed power allocation scheme for 3-user Rayleigh faded interference channel with water-filling on ergodic interference alignment.

V. CONCLUSIONS

We have shown the importance of power allocation in increasing the sum rate for interference channels. In the case of the two user interference channel, the improvement in sum rate over water-filling on ergodic interference alignment is significant if the probability of strong interference is high.

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