On the Capacity of Cellular Networks with MIMO Links - University of

On the Capacity of Cellular Networks with MIMO Links Muhammad Naeem Bacha, Jamie S. Evans and Stephen V. Hanly ARC Special Research Centre for Ultra-Broadband Information Networks (CUBIN), Department of Electrical and Electronic Engineering, University of Melbourne, Vic. 3010, Australia, {m.bacha,jse,hanly}@ee.unimelb.edu.au

Abstract— We provide scaling results for the sum capacity of the multi-access, uplink channel in a flat fading environment, when there is interference from other cells. We consider a scaling regime where the number of antennas per user remains fixed but the number of antennas at the base station and the number of users in each cell grow large together. We characterize the asymptotic behaviour of the spectral efficiencies in each cell, in three scenarios: 1) single cell processing with full frequency reuse 2) single cell processing, with frequency re-use partitioning of adjacent cells and 3) base station cooperative decoding (macrodiversity). It is shown that base station cooperation provides very significant gains in spectral efficiency over single cell processing.

I. I NTRODUCTION It is now well known that the use of multi-antennas at each end in a wireless link provides a multiplexing gain in capacity proportional to the minimum of the number of antennas of the two ends of the link in a flat-fading channel [1], [2]. In the multi-cell context, there are various possible cellular architectures that can be considered. The traditional approach is to use single cell processing, same-cell users are provided with orthogonal channels, and frequency reuse partitioning is used to limit the co-channel interference. If this architecture is considered, and mobiles and base stations are equipped with antenna arrays, then each mobile to base station link can be considered as a point to point MIMO channel, as above. However, with the use of MIMO, many other options are available. Multiple antennas at the base station can be used to suppress both intra and inter-cell interference, so it is no longer necessary to use frequency re-use partitioning, nor to separate same-cell mobiles into orthogonal channels. Furthermore, if all (or many) mobiles in the network are multiplexed onto the same channel, it is worthwhile to consider the possible gains from macrodiversity: allowing joint processing (cooperation) amongst all the base stations in the network. In recent work [3], [4], we considered the uplink of a cellular network with orthogonal channels, and full frequency reuse between cells. In the present paper, we extend the model to allow multiple users in each cell to share the same channel, so there is intra-cell, as well as inter-cell interference. We compare and contrast the above architectural choices. As in [3], we take an asymptotic approach to our analysis, so as to obtain insightful capacity formulas, which can be

evaluated numerically for different choices of parameters, and used in optimization of those parameters. We use recent results on the spectral properties of large dimensional random matrices [5] to obtain asymptotic spectral efficiencies when the number of antennas per user remain fixed and the number of users in each cell and the number of antennas at the base station grow large. This approach enables us to consider the impact of various key parameters, such as the traffic loading in each cell, and the average level of interference from a mobile in an adjacent cell. It also allows us to efficiently calculate the asymptotic capacity of the network for all the architectures considered above. A. Related work In [6] the idea of cooperative, joint decoding by the base stations (macrodiversity) was introduced. In this context, it was shown that reuse partitioning was not able to achieve the capacity of a general cellular network. With only a single antenna at each mobile and base station, and without multipath fading in the model, it was shown that reuse partitioning only achieves capacity when the average “power profile” [6] of each group of users being partitioned, is the same. This is a very special case, so re-use partitioning is in general suboptimal. In the present paper, we revisit the issue of reuse partitioning of the spectrum amongst the cells, when each mobile and base station is equipped with an antenna array, under flat Rayleigh fading assumptions. We revisit also the related issue of whether or not to orthogonalize the users within a cell. In [7], Wyner introduced a very tractable model for the characterization of the information theoretic limits of cellular networks. He considered both an infinite linear array of cells, and also an infinite hexagonal 2D cellular array model. The received signal at a base station consists of the active users (transmitting on the uplink) of that cell plus a gain α ∈ (0, 1] times the signals of the active users from adjacent cells. Thus the interference is considered to be caused by adjacent cells’ users only, and the parameter α provides a simple way to model the level of inter-cell interference. In [7], this model was used to analyze the achievable rates for single antenna systems without fading. Multi-path fading was later considered in [8], [9] and in the present paper, we provide an extension of the model to the MIMO setting.

1337 1-4244-0355-3/06/$20.00 (c) 2006 IEEE This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2006 proceedings. Authorized licensed use limited to: UNIVERSITY OF MELBOURNE. Downloaded on December 4, 2008 at 01:44 from IEEE Xplore. Restrictions apply.

4.5

N=5 N=10 N=20 N=100 First Order

4

3.5

nats/symbol/ N log N

More recent work on MIMO in cellular networks study the problem from the point of view of multiuser detection [10], and single user detection [11]–[14]. The case of no base station cooperation has been treated in [12] but for a single user per cell. [14] has characterized these results for more general cases of correlation in the channel and coloured noise. The characterization of MIMO capacity for the multiaccess channel appears in [15], [16]. The mathematics used in this paper has been reviewed in the survey paper [5] with derivations published in [17]. In [17] these results have been used to characterize the effect of antenna correlation on the capacity of a single link MIMO fading channel. However, the questions asked in the present paper, and the physical model itself, are quite different from those considered in [17]. Very recently the sum rate characterization of cooperating base stations has been considered in [18] when there is a single antenna at each end.

3

2.5

2

1.5

1

0.5

0

0

0.2

0.4

0.6

0.8

1

β

1.2

1.4

1.6

1.8

2

Fig. 1. Single cell users: Spectral efficiency per N log N (N is the number of BS antennas) against β

II. S INGLE CELL ANALYSIS We first consider a multi-access scheme for the uplink such that K active users are transmitting simultaneously to a base station. Each user has t antennas and the base station (BS) has N antennas. The BS is able to decode all of the users simultaneously. We consider only Rayleigh fading channels such that each path from a users’ antenna to a BS antenna fades independently. The BS is assumed to have channel state information (CSI) whereas the transmitter does not. The base band received signal y at the BS is given as: y

=

K


(N )   Csum − min(1, β) log N = min(1, β) C(β, P ) (2) N →∞ N where

gi hi xi + n

lim

i=1

= Hx + n where x = [xT1 ....xTK ]T is the complex transmitted signal vector (concatenation of individual transmit vectors of all the users). hi is a N × t complex matrix representing the fast fading channel from user i to the BS and gi represents the average (slow) gain from user i to the BS where i = 1, ..., K. We fix the slow gains to be equal to unity for all users however the results can easily be extended to unequal gains for all users. Thus, H = [h1 .....hK ] is the N × Kt complex matrix representing the overall multi-access channel from the users to the BS with entries which are i.i.d. circularly symmetric complex Gaussian with zero mean and unit variance. n is the N × 1 vector representing the noise with i.i.d. complex Gaussian entries of unit variance. All the users are transmitting independent signals to the BSs, with the same average transmit power, and so the spatial covariance matrix of the transmitted signal of the K users is thus E[xx† ] = Pt I(Kt) , where I(Kt) is Kt × Kt identity matrix. It is well known that when the channel state information (CSI) is only available at the BS, the sum capacity Csum of the system is given by [2], [16] P HH† )]. (1) t We consider the regime in which the number of users and the number of antennas at the BS grow large, with their ratio Csum = E[log det (IN +

β fixed, i.e. N → ∞, K → ∞ and K N → t . The number of antennas at each user, t, is fixed. Our main result for this case is the explicit expression for the sum capacity in this asymptotic regime using results from Random Matrix Theory. Of particular interest is how the spectral efficiency varies with the parameter β. β Theorem 1: As K and N grow large such that K N → t, the sum capacity per user grows logarithmically in the limit in addition to a constant factor:

C(β, P ) =

max(1, β)P 1−γ 1 log ( ) − 1 + log ( ) γ 1−γ t

(3)

such that γ = min(β, β1 ). Proof: This theorem falls out on straightforward application of the random matrix results contained in [19] and [20] concerning the almost sure weak convergence of F (N ) to F , where F (N ) , is the empirical distribution function of ¯H ¯ † ≡ 1 HH† , and where F the eigenvalues of the matrix H N is the Marcenko-Pastur distribution [19]. For the details of our proof, see [21]. Notice that the expression for the sum capacity involves a term growing to infinity at a logarithmic rate, so it is tempting to consider a normalization by this factor to obtain first order asymptotic spectral efficiency expressions. Thus, using (2), we obtain (N ) Csum = min(1, β). (4) lim N →∞ N log N If one applies this notion of spectral efficiency to a system with a large, but fixed number of BS antennas, N , then by varying β, one can gauge the effect of varying the number of users in the cell. It is interesting to note that the right hand side of (4) is increasing in β for β < 1, but it saturates at 1 nat/symbol/N log N when β > 1. However, examination

1338 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2006 proceedings. Authorized licensed use limited to: UNIVERSITY OF MELBOURNE. Downloaded on December 4, 2008 at 01:44 from IEEE Xplore. Restrictions apply.

of Figure 1 shows that this first order asymptotic result is not accurate for realistic system sizes. On the other hand, including the constant term in (3) as an important second-order effect does provide great accuracy. In Figure 1 the dot signs (plus, circle and cross) are for Monte-carlo simulations of a finite system with 500 independent realizations of the channel and the dashed lines correspond to the asymptotic results. (This is also valid for the remaining plots in the paper). Thus, (2), (3) provide a useful definition of asymptotic spectral efficiency. In [3], we considered a cellular network in which mobiles in the same cell were given orthogonal channels. However, Figure 1 suggests that this is a suboptimal approach, since spectral efficiency increases with the traffic loading, β. It is indeed straightforward to verify that orthogonalizing the mobiles in the cell is suboptimal. As in [6], we use the strict concavity of log(det(.)) to show that SS Csum = E log det(IN +

>

K P
hi h†i ) t i=1

K KP 1
Orth E log det(IN + hi h†i ) = Csum K i=1 t

SS where Csum is the sum rate when the mobiles share the Orth spectrum, and Csum is the sum rate when the mobiles are given orthogonal channels (e.g. separate time slots). To quantitatively measure the loss incurred when we orthogonalize the users, we consider the asymptotic regime as KP † Orth = E log det(It + h h1 ) N grows large. Note that Csum t 1 1 † h1 h1 → It , it follows that and since lim N →∞ N

1 Orth log N Csum ∼ , N N

1 SS Csum ∼ log N. N

For this reason, in the following sections, we will assume that all mobiles in each cell share the same channel. III. W YNER C IRCULAR C ELLULAR ARRAY MODEL An interesting question is to investigate whether the logarithmic growth in spectral efficiency occurs in multiple cell systems. Wyner [7] provided a simple and elegant model to capture the effect of intercell interference such that the linear model consists of an infinite number of cells arranged on a line. A BS in one cell receives signals on the uplink from users in its own cell and from the two adjacent cells. In Wyner’s model, it was efficient to orthogonalize mobiles within the cell, but this is clearly not the case here, since each BS is equipped with an antenna array. We adapt this model to the MIMO setting of the present paper. However, it is worth noting at this stage that the results can be extended to two dimensional network. We consider a finite circular network of M cells (circular in order to avoid edge effects) such that each BS in a cell can hear signals from the users of exactly two adjacent cells. We consider the asymptotics in which the antenna arrays at the BSs, and the numbers of users in each cell, grow large together. We have K active users in each cell with t antennas per user and N antennas at each BS. The received signal at the BS in cell m is then given by:

Fig. 2.

(A slice of) Circular Array with cells m − 1, m, m + 1

ym = αHm(m−1) x(m−1) + Hmm xm + αHm(m+1) x(m+1) + nm

(5)

where Hmm represents the channel gain from users in cell m. The channel gain from intercell users in m − 1 and m + 1 is the average gain α times the fast fading gain matrices Hm(m−1) and Hm(m+1) respectively. Each fast fading channel gain matrix Hml (l can have values of m − 1, m + 1 and m) is N × Kt matrix. The entries of each of these matrices form an i.i.d Gaussian collection with zero mean, independent real and imaginary parts, each with variance 1/2. xl ∈ CKt×1 is transmit vector from users in cell l, ym ∈ CN ×1 is the received signal vector and nm ∈ CN ×1 is noise vector both at the BS in cell m and has independent, circularly symmetric complex Gaussian entries; thus, nm ∼ N (0, IN ). The spatial covariance matrix of the transmitted signals of the users with in each cell is Ql = Pt I(Kt) , [16]. A. Single cell processing (SCP) 1) with co-channel interference: This is the more traditional approach where a BS in a cell is only aware of the codebooks of intracell users. The received signals of intercell users cause co-channel interference and the decoder has to decode the intracell users in spite of the inter-cell interference. The mutual information at the BS in cell m in this case is given as (For consistency we use the term sum capacity although in this case the mutual information is just an achievable rate.) I(xm ; (ym , Hm(m−1) , Hmm , Hm(m+1) )) = C1 − C2

where

  ¯ 1Q ¯ 1H ¯ †) C1 = E log det(IN + H 1   ¯ 2Q ¯ 2H ¯ †) C2 = E log det(IN + H 2

(6)

(7) (8)

and ¯ 1 = [Hm(m−1) Hmm Hm(m+1) ] H   ¯ 1 = P diag α2 I(Kt) , I(Kt) , α2 I(Kt) Q t 2 ¯ 2 = [Hm(m−1) Hm(m+1) ] Q ¯ 2 = α P I(2Kt) H t

We consider the asymptotic regime, the number of users in each cell and the number of antennas at the BS go to infinity keeping their ratio fixed i.e., Kt N → β. Using Theorem 2.45 in [5], we can characterize each of the terms C1 and C2 in equation (6). Thus as K, N → ∞ we have for C1 

lim

N →∞

 1 C1 − min(1, 3β) log(N ) = C1 N

1339 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2006 proceedings. Authorized licensed use limited to: UNIVERSITY OF MELBOURNE. Downloaded on December 4, 2008 at 01:44 from IEEE Xplore. Restrictions apply.

(9)

2.5

 α2 P 1  + γ Pt ) + log( γe )  2β log(1 + γ t ) + β log(1 1 α2 P P + φ ) + 2β log(φ te ) + β log(φ te ) C1 = log(1

   1 2 log( α2 P ) + log( P ) 3

te

te

Spectral Efficiency / N log N (nats/s/Hz)

where

β > 1/3 β < 1/3 β = 1/3 (10)

and where γ and φ satisfy the following equations 

 2 2

1 + γ α tP

1 + 1 + γ Pt

1 =3− , β

1 − 3β φ= 3β

(11)

respectively. Similarly for C2 

lim

N →∞

 1 C2 − min(1, 2β) log(N ) = C2 N

Half Reuse Full Reuse α=0.1 Full Reuse α=0.5 Full Reuse α=1 2

1.5

1

0.5

0

0

Fig. 3.

where  α2 P (2β−1) 2β  ) β > 1/2 2β log( 2β−1 ) + log( te (1−2β)α2 P 1 C2 = log( 1−2β ) + 2β log( 2βte ) β < 1/2   2 β = 1/2 log( αteP )

(13)

We now write down our main result for the spectral efficiency of SCP when there is co-channel interference: Result 1: Denote the sum capacity for cell m in this “in(N ) terference” model by Cint . As K and N get large such that Kt N → β, we have C1 − C2 = lim

   

N →∞  



(N )

Cint N (N ) Cint N (N ) Cint N

− β log N

β < 1/3

− (1 − 2β) log N

1/3 < β < 1/2

1

2

3

4

5

6

7

8

9

10

β

(12)

Single Cell Site Processing (SCP) schemes comparison

2) with Reuse partitioning: We consider a traditional reuse architecture with a reuse factor of one-half in our circular array of cells. Thus, we divide the available time slots equally between two adjacent cells, and every user will transmit in half of the available time slots. All users in the same cell transmit in the same time slots, and when cell m users are transmitting, BS m will receive no interference from cell m − 1 nor m + 1. The mutual information in this case for cell m is provided by: I(xm ; ym , Hmm ) = E[log det(IN + Hmm Qm H†mm )]

β > 1/2

where C1 and C2 are the constants appearing in equations (10) and (13) respectively. We note that in this case, the capacity per user in the cell of interest grows logarithmically with the number of antennas at the BS provided that there are more antennas at BS m than transmit antennas in the interfering cells (m − 1 and m + 1), which is the condition that β < 1/2. The logarithmic rate of growth depends on whether the system as a whole, viewed as a single cell, is saturated or not (saturated when β > 1/3). When β > 1/2, there is too much interference for BS m to null it out completely, and the limiting spectral efficiency is finite. The main message of this section is that with SCP and full frequency re-use in each cell, the traffic loading in each cell must be tightly controlled. The asymptotic scaling with log N does not depend on the inter-cell interference factor, α, but only on the traffic loading parameter, β. However, the important constant term, characterized in equations (10) and (13) does depend on α. Again, the asymptotics are extremely accurate provided the constant term is included. Since the intercell interference factor enters via the constant term, it is of interest to examine the re-use architecture where half the bandwidth is given to each cell, to remove the intercell interference effect.

Since each cells’ users are sharing the available time slots, the power spectral density of each user will be twice as compared 2P I(Kt) . The sum to without reuse architecture, i.e., Qm = t capacity in this case can be characterized in a similar fashion to the single cell analysis of section II (equation (2) and (3)) except that the transmit power of every user will be doubled and (the sum capacity) will be normalized by the reuse factor 2. We now provide some numerical results to compare the asymptotic spectral efficiency of single cell processing (SCP) schemes. We are interested in how the asymptotic spectral efficiency varies as the number of users changes, keeping the number of antennas at the BS fixed. Let there be 10 antennas at each BS and just a single antenna at each user. However, we vary the number of users per cell from 1 to 100. All users are considered to be transmitting at equal power of 20dB above noise level (we consider a normalized noise power level of 0dB). In case of single cell processing, both full frequency reuse and a reuse factor of one-half are considered. We plot the asymptotic spectral efficiency against the ratio β in Figure 3 for different values of α. The spectral efficiency of SCP with full reuse is a function of average gain α due to the presence of adjacent cell interference. The results for reuse partitioning are independent of α due to the absence of co-channel interference. Full reuse performs well in the presence of weak cochannel

1340 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE ICC 2006 proceedings. Authorized licensed use limited to: UNIVERSITY OF MELBOURNE. Downloaded on December 4, 2008 at 01:44 from IEEE Xplore. Restrictions apply.

interference from the adjacent cells. This observation even extends to the case of relatively strong interference (such as α = 0.5) as long as the BS receiver has enough degrees of freedom to null out the interference. Thus when β < 1/3, the receive antennas at the BS outnumber the transmit antennas of intracell and intercell users, and the BS can efficiently suppress the interference and perform well. When β > 1/3, there are not enough degrees of freedom at the BS to null out the interference, so SCP only does well in this case when α is very small. The figure also shows that for larger values of β (e.g. β > 1.5), the re-use partitioning scheme does best of those schemes considered. B. Macrodiversity We now consider the model in which the BSs are cooperating and sharing data to jointly decode all the users in the cellular array. Thus, there exists a hyper-receiver that has knowledge of the code books of all the users and has delayless access to data from all the BSs. We consider the received signals at all M BSs in our circular array and from equation (5) can write it in block form as ˜x+n ˜ = H˜ ˜ y (14) T T ˜ = [y1T , ..., yM ] is the concatenation of the received where y ˜ = [xT1 , ..., xTM ]T is signal vectors at all the BSs. Similarly x a vector which is a concatenation of the transmitted signals ˜ is the overall channel matrix of all the users in the array. H from the users to the BSs in the system and is given as



H11

  αH21   ˜ H= 0   .  . . αHM 1

αH12

0

H22

αH23

αH32 .. . 0

H33 .. . 0

··· .. . .. . .. . ···

αH1M 0 0 .. .

        

(15)

HM M

[nT1 , .., nTM ]T

˜= and n is the noise vector. Using our earlier result in [3] we can thus write ˜ = E[log det(I(M N ) + H ˜Q ˜H ˜ † )] ˜ , H) I(˜ x; y

(16)

where the capacity is achieved with circularly symmetric ˜ with zero mean complex Gaussian transmit signal vector x and block diagonal covariance matrix [16] 

˜ = diag Q

P P I(Kt) , ..., I(Kt) t t





N →∞

where Cmac

 1 Cmac − min(1, β) log(N ) = Cmac MN

 β 2 P  log((1 + 2α )(β − 1) t e ) + β log( β−1 ) 2 P 1 − β)(1 + 2α ) t e ) + log( 1−β ) = β log((1

2  log (1+2α )P te

IV. C ONCLUSIONS

(17)

The relevant large-system result is Theorem 2.54 in [5]. ˜Q ˜H ˜ † , we obtain the main Applying this result to the matrix H result of this section: Result 2: Assume that M ≥ 3. The asymptotic sum capacity Cmac normalized by the number of antennas at the BS in case of cooperating BSs to jointly decode all the users in the limiting regime is given as lim

and β = Kt N . An interesting conclusion is that the sum capacity per cell is independent of the total number of cells M in the system (unless M < 3). This might be related to the fact that in the cellular model considered here, each BS can only receive data from two adjacent cells in addition to its own cell users. However, this is not the whole story: that was also true in the model considered in [7], but in that scenario the capacity was sensitive to the network size. In our case, the MIMO asymptotics wash out the effect of a finite cellular network, and provide an expression that is insensitive to the network size (provided M ≥ 3). We now use the above theory to numerically evaluate the asymptotic spectral efficiency using full macrodivesity, and we compare it with the previous schemes. As in Section III-A, each BS has 10 antennas, and every user has a single antenna which is transmitting at 20dB above noise level. All the BSs in the network are cooperating to share the received data and use a hyper receiver to jointly decode all the users in the system. We plot the sum capacity against β for varying number of users in each cell for different values of α in Figure 4. As α increases the sum capacity increases. This is because increasing α increases the receiver diversity, and the amount of received power collected from each mobile. Figures 4 (a) and (b) indicate that cooperative joint processing, using macrodiversity, provides significantly higher spectral efficiency than does single cell processing, even for moderately small values of α, such as 0.1. Comparing macrodiversity with SCP and re-use partitioning, we see that at high β, the gain from macrodiversity is by more than the re-use factor. With re-use partitioning, we lose not only the bandwidth (by the re-use factor) but also pay the price of single cell processing, both in terms of diversity and received power. An important issue is how to implement the hyper-receiver, taking into account practical issues such as synchronization, decentralization, delay etc. Although many issues remain to be resolved, we note here recent work [22], [23] which shows that message passing algorithms, in which messages are passed between adjacent BSs, can distribute the computations involved in joint, cooperative decoding.

β>1 β