Uplink Base Station Cooperation by Iterative Distributed ... - CiteSeerX

Uplink Base Station Cooperation by Iterative Distributed Interference Subtraction Michael Grieger, Patrick Marsch, Gerhard Fettweis Technische Universität Dresden, Vodafone Chair Mobile Communications Systems, 01069 Dresden, Germany Email: {michael.grieger, marsch, fettweis}@ifn.et.tu-dresden.de

Abstract—The capacity of today’s cellular mobile communications systems is mainly limited by inter-cell interference. Multicell joint transmission or joint detection schemes are means to overcome this limitation and to actively exploit signal propagation across cell borders rather than treating it as noise. A major downside to multi-cell signal processing is the additional backhaul rate that is required to exchange information among cooperating base stations. Hence, cooperation schemes are desired that efficiently utilize the available backhaul rate. In the cellular uplink, one promising option to mitigate interference and thus to significantly increase the efficiency of spectrum usage is the exchange of decoded messages among base station, an information that can be used to subtract interference prior to the decoding of further messages. In this work, this distributed interference subtraction scheme is extended by allowing multiple iterative information exchanges among base stations. Possible gains of this strategy in terms of achievable data rates and required backhaul rates are explored from an information theoretical point of view.

I. I NTRODUCTION In order to satisfy the ever-increasing demand for higher transmission rates in cellular mobile communication systems, an optimized use of the available infrastructure is required. To actively exploit signal propagation among cells rather than treating it as noise, it is widely accepted that future wireless systems will incorporate some form of multi-cell signal processing, a concept that requires the exchange of information among the involved base stations or among base stations and a central processing unit. Several strategies for this cooperation exist that, for example, differ regarding the type of exchanged information, complexity, achievable gains, and robustness against channel impairments as well as channel estimation errors. However, from the point of view of system operators, the main cost driver is the required backhaul traffic between base stations. Hence, in order to achieve small additional costs connected to an upgrade of the available backhaul network, the identification of backhaul efficient schemes is an important challenge at this early point of research on base station cooperation (BSC). For the uplink, proposed BSC schemes are based on the exchange of quantized or compressed receive signals such as in distributed antenna systems [1]–[3], or the exchange of decoded messages such as in distributed interference subtraction schemes [2], [3]. For practical systems where e.g. channel estimation and synchronization errors have to be taken into account, several authors proposed BSC schemes that are based

on the exchange of reliability information (e.g. soft bits) and soft interference cancellation techniques. This information is exchanged iteratively in order to decrease the transmission error probability for a fixed data rate [4], [5]. The main contribution of this paper is the investigation of a cooperation scheme that is based on an iterative exchange of decoded messages. An information theoretical point of view is chosen to establish bounds on data rates that are achievable when the backhaul is constrained in terms of capacity. Therefore, this work investigates if an iterative exchange of certainty (decoded messages) offers benefits in terms of achievable data rates and/or backhaul consumption compared to cooperation schemes where information is only exchanged in one direction. We proceed as follows. After defining the transmission model in Section II, an iterative BSC scheme is introduced that is based on superposition coding and an iterative exchange of decoded messages among base stations (BSs) in Section III. Subsequently, possible gains are examined that this iterative scheme offers compared to a non-iterative scheme. A major result is the derivation of the upper bound on the maximum sum rate that is achievable for an unlimited number of iterations in Section IV-A. The paper closes with conclusions. II. T RANSMISSION M ODEL The system under investigation is depicted in Figure 1. It consists of K = 2 single antenna mobile terminals (MTs) and M = 2 BSs with Nbs antennas each. In the following, K = {1, 2} refers to the set of MTs and M = {1, 2} to the set of BSs. To enable BSC, the BSs are connected through an error-free bidirectional backhaul link of total capacity RBh . A low mobility scenario is considered, where the channel can be assumed to be quasi-static such that channel realizations are constant for at least the transmission of a whole codeword of length Nc . The transmission is disturbed by additive i.i.d. Gaussian noise. Furthermore, the channel is frequency flat and the BSs and MTs are assumed to be fully synchronized in time and frequency. Both MTs transmit on the same time and frequency resource. Under these assumptions, the transmission of a vector symbol can expressed as        y1 h h1,2 x1 n + 1 , (1) = 1,1 n2 y2 h2,1 h2,2 x2 {z } | H

Nbs

Nbs

1

2

Backhaul

h1,1

h2,1

h2,2 h1,2

2

1

Fig. 1: System Setup where ym = CNbs , m ∈ M is the received vector at BS m, xk ∈ C is the symbol transmitted at MT k under a per MT power constraint Pmax , and H ∈ C[2Nbs ×K] is the deterministic channel matrix. The vector hm,k ∈ CNbs , represents the channel gains from MT k to BS m. The noise vector nm ∈ CNbs , m ∈ M is a realization of a zero-mean circularly symmetric complex Gaussian (ZMCSCG) random process 2 nm ∼ N C (0Nbs , Φnn ), where Φnn = E[nm nH m ] = σn I, and H 0 E[nm nm0 ] = 0, m 6= m . III. I TERATIVE D ISTRIBUTED I NTERFERENCE S UBTRACTION The proposed uplink BSC scheme, Iterative distributed interference subtraction (IDIS), is based on the iterative exchange of decoded messages that are used to mitigate interference. Each MT transmits a linear superposition of up to L messages that are mapped onto codewords of a
Nc , 2rk,l Nc code, where rk,l is the rate of the lth message of MT k (l ∈ {1 . . . L}, k ∈ K), and Nc is the number of coded symbols per codeword. Gaussian coding is employed, i.e. each code consists of a set of codewords {1, . . . , 2rk,l Nc }, drawn randomly and independently from the distribution N C(0, pk,l ), where pk,l is the power assigned to the transmission of message l of MT k. Each code further consists of an encoder, which is a mapping from user data to codeword indices as well as the corresponding decoder. The power assignment PL at each MT is chosen with respect to the constraint ≤ Pmax , and a specific power assignment is denoted l=1 pk,l  H by p = p1,1 , . . . , p1,L , p2,1 , . . . , p2,L . The symbol transmitted by MT k is the linear superposition xk =

L X

xk,l , k ∈ K.

(2)

l=1

This superposition coding approach is motivated by the desired increase of cooperation flexibility, because it allows the successive exchange of decoded messages, each exchange reducing the remaining interference. The decoding of the messages is coordinated as follows. Each message of a MT is decoded at a distinct BS; messages of MT 1 are decoded at BS 1, messages of MT 2 are decoded at BS 2. The messages are decoded and forwarded in an

alternating manner, i.e. if a certain message is decoded at BS 1, the next message is decoded at BS 2. Furthermore, all but the last decoded message are forwarded to the other base station. We are only interested in the user rate, which is the sum of the rates of all messages transmitted by one MT. Hence, we can assume without loss of generality that messages are decoded in the order 1, 2, . . . , L. Based on the previous assumptions, the user rates solely depend on the power assignment and on the choice of the BS that decodes the first message which is denoted by π ∈ M. For example, we choose π = 1 such that in the fist step the first message of MT 1 is decoded at BS 1 and forwarded to BS 2. By employing interference subtraction and assuming error-free decoding (Nc → ∞), in the next step the first message of MT 2 can be decoded at BS 2 without interference of the first message of MT 1. This procedure continues until the last message is decoded with rate rπ¯ ,L , where π ¯ ∈ M \ π. Due to the Gaussian nature of all involved random variables, the rate of MT k follows from a Shannon type expression [6] rk (p) = 



    H   pk,l (hk,k ) hk,k   log2 1+ L , L
  P P H H l=1   0 h h +Φ p h (h ) + p 0 ¯ ¯ ¯ nn k,l k,k k,k k,l k, k k, k    l0 =L−l

L X

l0 =

L−l+1,π = k ¯ L−l, π = k

(3) ¯ where k ∈ K \ k. In order to compare the achievable performance for a certain power assignment and decoding order, a performance point is defined as the tuple of achievable data rates r1 , r2 and the required backhaul rate rBh gIDIS (p, π) = (r1 (p, π), r2 (p, π), rBh (p, π)) ,

(4)

rBh (p, π) = r1 (p, π) + r2 (p, π) − rπ¯ ,L (pπ¯ ,L , π).

(5)

where

For the case of L = 1, the outlined cooperation scheme corresponds to distributed interference subtraction (DIS), which is presented in, for example, [2], [3], [7]. The corresponding performance points are denoted by gDIS . As discussed in [3], a further reduction of the required backhaul rate is possible by exploiting side-information at the receiving BS by employing Slepian-Wolf coding. IV. C OMPARISON OF DIS AND IDIS This section compares the performances that are achievable when IDIS is employed instead of DIS. The performance of both BSC schemes is compared for the setup that is depicted in Figure 1. To concentrate on the interference mitigating ability of the proposed cooperation schemes, single antenna BSs (Nbs = 1) are assumed in the following. The number of parameters that are required for the channel model is reduced by assuming an upper layer power control mechanism that

regulates the power gain of each MT to its assigned BS to a fixed value (λ1,1 = λ2,2 = 1). Furthermore, as the achievable rates in (3) only depend on |hm,k |2 = λm,k , a parameter reduced channel model can be derived as p   λ1,2 1 p . (6) Hred = λ2,1 1 A. IDIS with Unlimited Iterations Before proceeding with the actual comparison, the maximum sum rate

achievable rate of MT 2 is given by  r2∞= lim L→∞

L  Pmax 1 X    L ln1 + . P P ln(2)  (L − l) Lmax + (L − l) Lmax λ + σn2 l=1 | {z } SINRl

Because SINRl → 0 for L → ∞, we can substitute limL→∞ ln(1 + SINRl ) = SINRl and get L

L l=1

r1 (p, π) P max p1,l , L l=1 p2,l ≤Pmax ,∀π∈M

+ r2 (p, π) (7)

of IDIS is established for the case that the number of iterations — and therefore the number of transmitted superimposed messages — is unlimited. In this section no backhaul constraint is assumed (RBh → ∞). Due to the nonconvexity of the maximum sum rate problem, the determination of this upper bound is a complex problem. Moreover, for L → ∞, the discrete power assignment becomes a density which is difficult to optimize. However, it is straightforward to show that an iterative exchange of information is particularly beneficial for symmetric channel realizations, where λ2,1 = λ1,2 = λ. The solution for this special case is stated in the following lemma. Lemma 1: Assuming capacity-achieving codes, the sum rate achievable with an IDIS cooperation scheme in a symmetric two-cell scenario (λ < 1) is maximized if both MTs split their transmissions into an infinite number of messages that are assigned infinitesimally small power each, i.e. L → ∞. Proof: The proof, which is based on recursion, is only sketched here. Assume that each terminal transmits one message with maximum power assignment p1 . If these messages are each split into two messages with power p21 (i.e. adding another iteration to the IDIS process), we can calculate the difference of the sum of the rates with and without the additional iteration as ( p21 +

p1 4 (1 − λ)λ p1 2 σn )( 2 λ + σn2 )( p21 λ

| = lim

L→∞

1 ln(2)

(L − l) PLmax + (L − l) PLmax λ + σn2 {z } SINRl

L X

1+λ+

l=1

In the limit L → ∞, becomes an integral

1 L

2 σn Pmax

+ σn2 )

(8)

To further increase the sum rate, all messages are split up again. The positive increase that corresponds to these further iterations, follows from a similar expression than (8). We just need to substitute p1 with pL1 , the power of the messages at the considered iteration, and we need to account for the additional interference by adding an extra term to σn2 . However, each further division of the transmit power on more and more messages increases the achievable sum rate, which concludes the proof. In particular, Lemma 1 is independent of the actual power distribution. Therefore, in the case L → ∞, an assignment of equal power to all messages is optimal. Assuming that the first message is decoded at BS 1, in this case the maximum

−

(9)

!

1 L l L (1

= dx as well as

l L

. + λ) = x, and the sum

Z1 1 dx = 2 σ n ln(2) − x(1 + λ) 1 + λ + Pmax 0   1  2 σn log2 1 + λ + Pmax − x(1 + λ) (10)  = −(1 + λ) 0   log2 1 + Pσmax (1 + λ) 2 n . = 1+λ For a very large number of iterations, the decoding order does not have an influence on the rate of the MTs; thus, r2∞ = r1∞ , and the sum rate is   log2 1 + Pσmax 2 (1 + λ) n ∞ . (11) rsum = r1∞ + r2∞ = 2 1+λ Since all but the last message are exchanged, for L → ∞ the rate of this message becomes infinitesimally small; hence, the required backhaul rate equals the sum rate: r2∞

∞ rBh = rsum .

> 0 , 0 < λ < 1.

!

Pmax L

1 X lim L→∞ ln(2) l=1

rsum = P



(12)

Figure 2 shows the achievable sum rates for four different BSC schemes as a function of the interference power gain of a symmetric channel. In this plot, no backhaul constraint is considered. The optimum power assignments for the IDIS schemes with L = 2 and L = 3 iterations were determined by an exhaustive search over the parameter space. From this result, it can be deduced that the major share of the gain that is possible by employing IDIS instead of DIS is already achievable for L = 2 iterations. Hence, only this scheme is considered in further examinations. All results that are presented in the following are found by an exhaustive search over the parameter space. If a very asymmetric Z-channel is observed, where one interference power gain (either λ2,1 or λ1,2 ) is zero, an iterative exchange of information always has a detrimental effect, and,

Maximum sum rate [bit/channel use]

7

6.5

L→∞

L=3 6

L=2 5.5

L = 1 (DIS)

5

(a) PDIS

4.5

Fig. 3: Performance regions for DIS and IDIS for a symmetric channel (λ − 5 dB, Nbs = 1, SNR = 10 dB, Pmax = 1)

4 -15

-10

λ[dB]

-5

(b) PIDIS (L = 2)

0

Fig. 2: Comparison of the maximum sum rate of DIS and IDIS (L = 2, L = 3, L → ∞) for symmetric channels (SNR = 10 dB, RBh → ∞, Nbs = 1, Pmax = 1)

1.08

0

1.07 1.06 −5

B. Performance Regions In this section, the performance of DIS and IDIS is compared for a symmetric example channel by means of the performance region, which is defined as the convex hull around all achievable performance points:   [ P = coh  g(p, π) . (13) PL

l=1

P p1,l , L l=1 p2,l ≤Pmax ,∀π∈M

The achievability of all points on this convex hull can be proven by a time-sharing argument. Figure 3 shows the performance regions of DIS and IDIS for the setup with two MTs. Since DIS schemes are a subset of IDIS schemes, the performance region of IDIS is always greater or equal than the corresponding DIS region: PDIS ⊆ PIDIS .

(14)

The DIS region is bounded by the performance points, gDIS (pmax , π = 1) and gDIS (pmax , π = 2) that correspond to the two possible decoding orders: either BS 1 (π = 1) or BS 2 (π = 2) decodes the first message and forwards it to the other BS. In order to achieve any of these points, both messages are

1.05

λλ2,1 [dB] [dB]

1.04

B a

in the proposed model, zero power would be assigned to all but one message per terminal. First, the message that can be decoded without interference is decoded. Subsequently, this message is exchanged over the backhaul in order to decode the message of the other MT without interference as well. This approach corresponds to DIS. However, note that a strategy that is based on superposition coding and partial interference subtraction as proposed in [8] achieves higher rates in low backhaul regimes. The preceding discussion characterizes the gains that are achievable by employing IDIS instead of DIS. In summary, highest gains are achievable for symmetric channels; no gains are achievable for asymmetric Z-channels. As will be explored in more detail in Section IV-D, the gains for the wide range of other channel realizations lie in between these limits.

1.03

−10

1.02 1.01 −15 −15

−10

λ [dB] [dB] λ1,2

−5

0

1

A b

Fig. 4: Ratio of the max. sum rate of IDIS and DIS as a function of the interference gains λ2,1 and λ1,2 . (SNR = 10 dB, bit Nbs = 1, RBh = 4 channel use , L = 2, Pmax = 1) transmitted with full power, and to achieve any performance point on the straight line in between, time-sharing can be used. If IDIS is employed, Figure 3 shows that by allocating the available power appropriately and by investing the required amount of backhaul capacity, the performance region can be increased beyond the straight time-sharing line. C. Evaluation of IDIS Gains In this section, the achievable gains of IDIS are evaluated for a broader class of channel realizations. Figure 4 shows the ratio of the maximum sum rates of IDIS and DIS for an SNR of 10 dB. It can be seen that for a backhaul capacity of bit 4 channel use , the gain of IDIS over DIS is at most ≈ 8%. As predicted, the gain of IDIS is greatest for symmetric channels and very poor for highly asymmetric ones. Certainly, achievable gains further depend on the available backhaul. More concisely, it follows from the discussion in the last section that possible gains increase with the backhaul rate, however, ∞ the sum rate is always bounded by rsum . For the case of a symmetric channel with very strong interference (λ = 0 dB), decoding of both messages at the same BS achieves the same rate as any DIS scheme. Therefore, IDIS does not provide any further gains. Another measure for the comparison of DIS and IDIS is the ratio of the maximum sum rates of both schemes for any channel realization. Figure 5 shows this ratio for the case that

6 Sum capacity [bit/channel use]

Max. gain of IDIS over DIS [%]

25 20 (SNR) ΨSW 4

15 10 Ψ4 (SNR)

5 0

0

5

10

15 20 SNR [dB]

25

Slepian-Wolf source coding is employed [3]: max

{Hred |λ2,1 ,λ1,2 ≤1}

SW rsum,IDIS (RBh , Hred )

(15)

rsum,IDIS (RBh , Hred ) . rsum,DIS (RBh , Hred )

(16)

or that Slepian-Wolf coding is not permitted: ΨRBh (SNR) =

max

{Hred |λ2,1 ,λ1,2 ≤1}

IDIS (L = 2)

4.5 4

DIS

3.5 3 0

1

2 3 SNR-Gap Γ[dB]

4

5

Fig. 6: Comparison of the maximum sum rate of DIS and IDIS (L = 2) as a function of the SNR-Gap Γ for symmetric channels (SNR = 10 dB, λ = −5 dB, RBh → ∞, Nbs = 1) V. C ONCLUSIONS

,

SW rsum,DIS (RBh , Hred )

5

2.5

30

Fig. 5: Maximum achievable gain of IDIS when compared to bit DIS as a function of the SNR (Nbs = 1, RBh = 4 channel use )

ΨSW RBh (SNR) =

5.5

Interestingly, the combined employment of IDIS and Slepian-Wolf coding provides stronger gains than IDIS without Slepian-Wolf coding. The use of Slepian-Wolf coding is particularly beneficial in the low backhaul and high SNR regime. When the available backhaul is very large, SlepianWolf coding is not beneficial at all, which is obvious since Slepian-Wolf coding only reduces the rate that is required for lossless exchange of decoded messages. In the low backhaul regime and for channels with weak interference, IDIS provides no gains compared to DIS. In higher backhaul regimes, the relative gain is much more equally distributed over the complete set of observed channels. The gain of an employment of IDIS instead of DIS is maximum when no backhaul constraint is considered (RBh → ∞), because the iterative exchange of information requires additional backhaul compared to DIS (see Section IV-A). D. Employment of IDIS in Practical Systems It is well known that communication schemes that are based on superposition coding are exposed to additional practical challenges. A main obstacle is that the performance of practical codes is always worse than the theoretical limit; they show a certain gap to capacity. For a variety of uncoded and coded transmissions over an AWGN channel, this gap can be approximated by a scaling factor Γ applied to the SNR when the probability of error is fixed [9]. In the presented framework, the SNR gap is applied to each transmitted message, which results in a penalization of superposition coding. In Figure 6 maximum sum rates of DIS and IDIS with two iterations are compared for a channel with moderate SNR = 10 dB. The plot shows that the gain of IDIS almost completely vanishes for a gap greater than 3 dB, which is still challenging for practical codes, especially in a fading environment that theoretically requires a certain codebook for each channel realization.

The introduced scheme iterative distributed interference subtraction (IDIS) is based on the iterative exchange of decoded messages which are transmitted at the mobile terminals by using superposition coding. The scheme shows explicit gains for a wide range of channel realizations when compared to its non-iterative counterpart distributed interference subtraction (DIS). IDIS is particularly beneficial for symmetric channel realizations. To evaluate achievable performance gains, the maximum sum rate limit of IDIS has been determined for the case that an infinite number of iterations is allowed. The results show that two iterations (and therefore the transmission of two superimposed messages per terminal) are sufficient to achieve the biggest share of the gain that IDIS provides compared to DIS. By applying the SNR gap theory, it could be shown that a practical implementation of IDIS is only beneficial as long as codes are available that perform close to the Shannon limit which makes the practical benefit of the proposed BSC scheme questionable. However, the presented information theoretic analysis states upper bounds on data rates that are achievable for a class of base station cooperation schemes that is based on an iterative exchange of reliability information on decoded messages. R EFERENCES [1] A. del Coso and S. Simoens, “Distributed compression for the uplink channel of a coordinated cellular network with a backhaul constraint,” in SPAWC, 2008. [2] A. Sanderovich, O. Somekh, and S. Shamai, “Uplink macro diversity with limited backhaul capacity,” ISIT 2007, pp. 11–15, June 2007. [3] P. Marsch and G. Fettweis, “On Uplink Network MIMO under a Constrained Backhaul and Imperfect Channel Knowledge,” ICC, 2009. [4] T. Mayer, H. Jenkac, and J. Hagenauer, “Turbo base-station cooperation for intercell interference cancellation,” Communications, 2006. ICC ’06. IEEE International Conference on, vol. 11, pp. 4977–4982, June 2006. [5] S. Khattak and G. Fettweis, “Distributed iterative detection in an interference limited cellular network,” IEEE VTC2007, pp. 2349–2353, 2007. [6] T. M. Cover and J. A. Thomas, Elements of Information Theory 2nd Edition. Wiley-Interscience, July 2006. [7] P. Marsch and G. Fettweis, “On backhaul-constrained multi-cell cooperative detection based on superposition coding,” PIMRC, 2008. [8] W. Yu and L. Zhou, “Gaussian z-interference channel with a relay link: Achievability region and asymptotic sum capacity,” submitted for pub. [9] J. Forney, G.D. and G. Ungerboeck, “Modulation and coding for linear gaussian channels,” Inform. Theory, IEEE Trans. on, vol. 44, no. 6, 1998.