Impact of Transmit Array Geometry on Downlink System-Level Performance of MIMO Systems Afif Osseiran, Kambiz Zangi, and Dennis Hui Ericsson Research {Afif.Osseiran, Kambiz.Zangi, Dennis.Hui}@ericsson.com
Abstract—For a cellular system with a fixed number of transmit antennas at each base station, we investigate how the system-level performance varies as a function of the geometry of the transmit array used at each base station. Our results show that the system-level performance of a cellular system can be significantly improved by using transmit antenna geometries other than the uniform linear arrays that have been investigated extensively in the literature so far. For example with 4 transmit antennas at each base station, we show that the spectral efficiency for achieving a 5 percentile user data rate of 2 Mbps (in a 5 MHz bandwidth) is improved by 58% when a non-uniform, linear array is used instead of a traditional uniform linear array. Key Words: Array Gain, Array Geometry, OFDM, SIC, Spatial multiplexing, System Performance.
I. I NTRODUCTION Multiple-input, multiple-output transmission with four or more transmit antennas at each base station (BS) is being proposed as one of the key radio access technologies for IMTAdvanced. In [15], the authors proposed and demonstrated a real-time 4x4 MIMO system with peak data rate of 1 Gbps operating over a 100 MHz of bandwidth. The peak data rate was further extended in [16] to 5 Gbps using a real-time 12x12 MIMO system operating over the same bandwidth. In a MIMO system, the peak data rate sustainable over an extended time period depends crucially on the statistical properties of the MIMO channel, which are in turn affected by the relative positions of the transmit antennas. Recently, it has been shown [14], [13] that for a cellular system with a fixed number of transmit antennas at each BS, a fixed number of receive antennas at each user terminal (UT), and a fixed receive SNR, the achievable average data rate on the link between each BS and each UT varies considerably as a function of the geometry of the transmit antenna array. In this paper, we investigate how such variation in link performance due to the antenna geometry affects the overall system-level performance. To the best of our knowledge, the system-level performance of a cellular system with MIMO has only been investigated for one transmit antenna array geometry at the BS, namely the uniform linear array (ULA). The literature in this area can be divided in two groups: (a) uniform linear phased arrays, and (b) uniform linear diversity arrays (ULDA). In (a), the spacing between consecutive antenna elements are chosen small enough relative to λ (the wavelength at the carrier frequency) to ensure that all antennas are highly correlated, and typically just one stream of data is transmitted to each UT. In (b) the Ntx transmit antennas are positioned far
apart relative to λ such that every pair of transmit antennas are essentially independent, and typically Ntx independent streams are transmitted to each UT. It is worth mentioning that with a ULA transmit array geometry, the correlation matrix of the MIMO channel is often modeled as the Kronecker product of the correlation matrices at the transmitter and receiver sides [12], and these correlation matrices can be selected as Hermitian Toeplitz and parameterized by a single correlation factor ρ. In [11], system-level performance of a MIMO OFDM system with Ntx = 4 and ULA transmit array geometry was considered for beyond 3G systems (both phased array and diversity arrays were covered). In [10], the system-level performance of a MIMO OFDM system with Ntx = 2 and ULA transmit antenna geometry (diversity array only) was considered. The authors in [9] and [7], evaluated the system-level performance of a WiMax, MIMO, OFDM system with Ntx = 2 and ULA transmit array geometry using the correlation-based stochastic MIMO modeling approach of [12]. The same correlation-based model was used by the authors in [6] to evaluate the system-level performance of MIMO HSPA/WCDMA with ULA transmit array geometry and Ntx = 2. In [8], the system-level performance of a WiMax, MIMO, OFDM system with 2 independent transmit antennas at each BS was evaluated. The authors in [5] compared the system-level performance of WCDMA system with 2 independent transmit antennas to the performance of an evolved UTRA with 2 independent transmit antennas. The main contribution of this paper is to show that the system-level performance of a cellular system having multiple transmit antennas at each BS can be improved significantly by using transmit antenna geometries other than the uniform linear arrays that have been investigated extensively in the literature so far. The second contribution of this paper is to derive analytically the structure of the optimal precoder for the 2-clustered, non-uniform, linear array that was shown recently [14] to be nearly optimal from the link-level performance perspective. We believe that due to their cost and size constraints, the vast majority of future UTs will be limited to 2 receive antennas, similar to what is required in the long-term evolution of WCDMA standard [10]. Although there will certainly be UTs or other devices (e.g. laptops) with more than 2 receive antennas, it is unlikely that they will be the most commonly used devices. Hence in this paper, we focus on determining the impact of the geometry of BS’s transmit antennas on those
UTs that have 2 receive antennas. From a single-link perspective, [14] and [13] found that a particular non-uniform linear transmit array geometry (the socalled 2-clustered transmit array geometry whose details are presented in Section II) outperforms most other transmit array geometries over a wide range of SNRs. Motivated by these results, for Ntx = 4 and 8, we compare the downlink systemlevel performance of a cellular system using the 2-clustered transmit antenna geometry to the performance of a cellular system using the traditional ULA with diversity transmit antennas. Due to space constraints in this paper, we will not discuss ULA with phased arrays (i.e. pure beamforming with single stream transmission). The rest of this paper is organized as follows. The two studied transmit antenna geometries are presented in Section (II) along with a novel derivation of the optimal slow linear precoder for each geometry. In Section (III), our network deployment model is explained. Two criteria for evaluating the system-level performance are presented in Section (IV) followed by system-level simulation results. Lastly, Section (V) concludes the paper with some general guidelines for choosing the geometry of transmit antenna arrays for IMTAdvanced systems. II. T RANSMIT A NTENNA C ONFIGURATION X1 (f )
G(f )
S1 (f ) S2 (f )
Linear
X2 (f )
Y1 (f )
Precoding Y2 (f )
W SM (f )
XNtx (f )
Base Station
Fig. 1.
User Terminal
Generic model of a MIMO system on the downlink
Our system model is depicted in Fig. 1, where we consider an OFDM system with the following, base-band, discrete-time model: y[f ] = G[f ]x[f ] + n[f ]
f = 1, .., Nf ,
(1)
where Nf denotes the number of sub-carriers, x[f ] is an Ntx -dimensional vector transmitted from the BS, y[f ] is a 2-dimensional received vector, G[f ] is the (2, Ntx ) channel matrix, and n[f ] is an 2-dimensional complex noise vector. Throughout this paper, the transmitter is assumed to be only aware of the statistics of G[f ] ; however, the transmitter is not aware of the the particular realization of G[f ] (a reasonable assumption for IMT-Advanced systems with bandwidths exceeding 100 MHz), and we assume that the receiver has perfect
knowledge of the particular realization of G[f ]. We further assume that the transmit antennas are preceded by a linear precoding matrix W of size (Ntx , M ), i.e. x[f ] = Ws[f ], where s[f ] is a vector of length M, the number of parallel transmitted streams, and E s[f ]s† [f ] = I; hence, the covariance matrix of x[f ] is given by Φxx [f ] = W[f ]W† [f ].
(2)
The total transmit power from each BS is P , i.e. Nf
tr(W[f ]W† [f ]) = P.
(3)
f =1
The uniform linear diversity array with 10 λ inter-antenna spacing is the first geometry we will consider. Given that any pair of transmit antennas are independent with this geometry, we observe that E{gi [f ]gj [f ]† } = 0Ntx ×Ntx for all i = j, where gi [f ] denotes the i-th column of G[f ]. Combining this observation with the results in [17], [18], [14], we see that the optimal W for the uniform linear diversity array is NtxPNf I. This implies that with diversity transmit arrays, we should always transmit Ntx independent data streams, one stream from each transmit antenna. The 2-clustered non-uniform linear array is the second geometry we will consider. The 2-clustered geometry with Ntx = 8 transmit antennas is depicted in Fig. 2. The Ntx transmit antennas are grouped in 2 clusters where each cluster is configured as a ULA with λ2 spacing between consecutive antenna elements within each cluster. The two clusters are placed such that the intra-cluster spacing is 10 λ. Next we will determine the optimal W for the 2-clustered geometry. Referring to Fig. 2, let G1 denoted the submatrix of G formed by taking the first N2tx columns of G, and let G2 denoted the submatrix of G formed by taking the last N2tx columns of G. Note that G1 is the channel between the first cluster and the UT, and G2 is the channel between the second cluster and the UT. Given that the two clusters are placed far apart, and given that that the two clusters are symmetric, we observe that these two submatricies are independent and identically distributed. According to [18], [14], the transmit covariance matrix (denoted by RG = E G† [f ]G[f ] ) plays a crucial role in determining the optimal precoder; hence, we start by showing the special structure of this matrix when the 2-clustered geometry is used at the transmitter: † G1 G1 G†1 G2 RG = E (4) G† G1 G† G1 ⎡ 2 1 ⎤ 0 E G†1 G1
⎦ = ⎣ (5) 0 E G†2 G2 RG1 0 = , (6) 0 RG1 where we in going from Eq. (4) to Eq. (5), we have used the independence of G1 and G2 ; in going from Eq. (5) to Eq. (6), we have used the fact that G1 and G2 are identically
distributed; and we have defined RG1 = E G†1 [f ]G1 [f ] .
Since RG in Eq. (6) is block diagonal with 2 identical blocks on its diagonal, it is easy to see that the eigen vectors of RG must come in pairs of the form: vi 0 Ntx ×1 2 ; , (7) 0 Ntx ×1 vi 2
λ 2
S1 (f )
vmax
where vi ’s are the eigen vectors of RG1 , and 0 Ntx ×1 is 2 the N2tx -dimensional zero vector. This special structure of the eigen vectors of RG combined with the results from [18], [14] can then be used to easily show that the optimal precoder for the 2-clustered geometry is a block diagonal matrix, i.e. W = diag(Φ, Φ), where diag(A, B) is a block diagonal matrix having A and B as main diagonal blocks matrices, and where the off-diagonal blocks are zero matrices. Φ is an ( N2tx , N2tx ) matrix that can be expressed as: Φ = U diag(α1 , α2 , ..., α Ntx ), 2
10λ
λ 2
(8)
where U is a unitary matrix whose
columns are the eigenvectors of RG1 = E G†1 [f ]G1 [f ] , and an iterative procedure for computing the positive constants αi ’s can be found in [18], with (αi )2 = 2 PNf . This implies that the optimal transmitter for the 2-clustered geometry transmits two independent and identically distributed sets of data streams, each containing a maximum of Ntx /2 data streams transmitted over one cluster. The special geometry of antennas within each cluster allows us to derive a very simple approximation to Φ. More specifically, the antennas in each cluster form a uniform, linear, phased array; since, the antennas within each cluster are highly correlated due to the λ2 spacing between consecutive antennas in each cluster. It is well known in the beamforming literature [19] that with highly correlated
phased arrays † antennas lead to E G1 [f ]G1 [f ] having just one dominant eigen vector.
the unit-norm dominant eigen Let us denote † vector of E G1 [f ]G1 [f ] by vmax . This allows us to replace Φ with vmax with very little loss in performance. Hence, the optimal linear precoder forthe 2-clustered geometry can P be approximated by W = 2 Nf diag(vmax , vmax ), as depicted in Fig. 2 for the case with 8 transmit antennas. With this approximation, only two independent streams will be transmitted to each UT, one stream from each cluster. All the system-level results presented in the remainder of this paper for the 2-clustered geometry are based on this approximation. III. N ETWORK DEPLOYMENT MODEL A network deployment with seven sites where each site comprises three sectors is considered. The number of BS antennas per sector is four or eight. BS antennas are placed above rooftop. The network is assumed to operate at a carrier frequency of 3.5 GHz and OFDM with 128 sub-carriers is used within the 5 MHz transmission bandwidth. Table I provides a summary of the assumed system parameters.
S2 (f )
vmax
W
Fig. 2.
Transmitter with 2-clustered geometry and Ntx = 8
Parameter Number of sites Inter-site distance [m] Number of sectors per site Number of BS antennas per sector Sector output power BS receiver noise figure Number of UT transmit antennas UT output power UT receiver noise figure Carrier frequency Transmission bandwidth Sub-carrier bandwidth Number of sub-carriers Cyclic prefix length
Value 7 1000 m 3 4 or 8 36.5 dBm 5 dB 2 24 dBm 7 dB 3.5 GHz 5 MHz 39.0 kHz 128 3.2 µs
TABLE I S YSTEM AND S IMULATIONS PARAMETERS .
a scenario with macro BS installation above rooftops and UTs located outdoors on street level. None Line Of Sight propagation is assumed between the BS antennas and the UTs. Shadow fading is modeled as a log-normally distributed random variable with a standard deviation of 8 dB. The raybased channel model is an extension to the 3GPP spatial channel model (SCM) [3] with correlated shadow fading, delay spread and angular spread.
B. Receiver Structure A. Radio Channel Model The C2 metropolitan area pathloss and channel model from [2] is used in the evaluations. The model is applicable to
UTs are equipped with 2 antenna elements separated half a wavelength. A dual antenna MMSE receiver with successive stream cancellation is employed at the UTs.
C. Radio Network Algorithms
12 5 % User bitrate[Mbps]
UTs connect to the sector with the lowest path-loss, shadowing included, and the downlink beamforming gain is considered in the cell selection procedure. Signals are transmitted using a fixed output power and the modulation order and channel code rate is selected to maximize the data rate. Turbo coding with rates from 1/10 to 8/9 are used in combination with QPSK, 16QAM, or 64QAM to find an appropriate transmission format. Round-robin transmission scheduling is employed. Further one user per sector is scheduled for transmission.
14
10 8 6 4 2 0 1
D. Link-to-System Interface
IV. S YSTEM -L EVEL P ERFORMANCE R ESULTS A. Performance Criteria The spectral efficiency per sector is defined as the number of correctly received bits divided by the product of the number of sectors, the simulation time, and total bandwidth. Two performance criteria are used: the 5 percentile and 95 percentile user data rate. The 5 (resp. 95) percentile user data rate is defined as the 5 (resp. 95) percentile of the cumulative probability distribution of the average data rate delivered to each user. While the 5 percentile criterion can be seen as a measure of a minimum desired data rate for most users (including those on the cell edge), the 95 percentile criterion on the other hand measures the highest peak rate that can achieved. B. Coverage Results In this section we focus on the impact of transmit array geometry on the 5 percentile user data rate. Figures 3 and 4 plot the 5 percentile user data rate versus the spectral efficiency with 4 transmit (TX) antennas and 8 transmit (TX) antennas at each BS respectively. Fig. 3 shows that with 4 TX antennas, the 2-clustered transmit array geometry results in approximately 58% higher spectral efficiency than the ULDA at the 5 percentile data rate of 2 Mbps. Fig. 4 shows that with 8 TX antennas, the 2-clustered transmit array geometry results in approximately 76% higher spectral efficiency than the ULDA at the 5 percentile data rate of 2 Mbps.
1.5 2 Spectral Efficiency [b/s/Hz/sector]
2.5
Fig. 3. The 5 percentile user throughput versus the sector spectral efficiency for 4x2 ULDA and 2-clustered configuration.
14 12 5 % User bitrate[Mbps]
To estimate the packet decoding error probability of a channel coded block transmitted over a multi-state channel, a mutual information (MI) based link-to-system interface is used [4]. The model uses the post-receiver SINRs of the symbols in the channel coded block to calculate the average MI for bitinterleaved coded modulation. The average MI is then used to estimate the packet error probability.
2−clustered ULDA
2−clustered ULDA
10 8 6 4 2 0 1
1.5 2 2.5 Spectral Efficiency [b/s/Hz/sector]
3
Fig. 4. The 5 percentile user throughput versus the sector spectral efficiency for 8x2 ULDA and 2-clustered configuration.
efficiency with 4 transmit antennas and 8 transmit antennas at each BS respectively. Fig. 5 shows that with 4 TX antennas, the 2-clustered transmit array geometry results in approximately 47% higher spectral efficiency than the uniform linear diversity array (ULDA) at the 95 percentile data rate of 50 Mbps. Fig. 6 shows that with 8 TX antennas, the 2-clustered transmit array geometry results in approximately 76% higher spectral efficiency than the ULDA at the 95 percentile data rate of 50 Mbps. The relative gains in spectral efficiency of the 2-clustered geometry compared to the ULDA for the peak and coverage criterion (with 4 and 8 TX antennas) are summarized in Table II. It is worthwhile mentioning that the peak rates mentioned above are obtained assuming 5 MHz bandwidth. For an FDD, OFDM system with 100 MHz bandwidth, one can safely scale the peak rates obtained here by a factor of 20, resulting in peak rates of 1 Gbps (approximately the same as reported in [15]).
C. Peak Rate Results In this section, we focus on the impact of transmit array geometry on the 95 percentile user data rate. Figures 5 and 6 plot the 95 percentile user data rate versus the spectral
V. C ONCLUSION Using system-level simulations, in this paper we showed that the system-level performance of a cellular system hav-
Criterion 5% at 2 Mbps 95% at 50 Mbps
4 TX
8 TX
58% 47%
76% 76%
array transmit antenna geometry.
TABLE II S YSTEM - LEVEL PERFORMANCE GAINS WHEN USING 2- CLUSTERED GEOMETRY INSTEAD OF ULDA.
80
Part of this work has been performed in the framework of the IST project IST-4-027756 WINNER II, which is partly funded by the European Union. The authors would like to acknowledge the contributions of their colleagues in WINNER II, although the views expressed are those of the authors and do not necessarily represent the project.
60
R EFERENCES
120
95 % User bitrate[Mbps]
100
2−clustered ULDA
40 20 0 1
1.5 2 Spectral Efficiency [b/s/Hz/sector]
2.5
Fig. 5. The 95 percentile user throughput versus the sector spectral efficiency for 4x2 ULDA and 2-clustered configuration.
ing multiple transmit antennas at each base station can be improved significantly by using transmit antenna geometries other than the uniform linear arrays that have been investigated extensively in the literature so far. With a total of 4 (or 8) transmit antennas at each BS, we showed that the spectral efficiency resulting in a 5 percentile user data of 2 Mbps can be improved by than 58% (or 76%) if the 2-clustered transmit array geometry is used instead of the uniform linear array geometry. Furthermore, we analytically derived the structure of the optimal precoder for the 2-clustered transmit array geometry, and we offered a very simple approximation for implementing the precoding part of the optimal transmitter for the 2-clustered geometry. The results obtained in this paper suggest that in macro BS installation for IMT-Advanced system having 4 or more transmit antennas, the 2-clustered transmit array geometry should be considered as an alternative to the traditional uniform linear
120 100 95 % User bitrate[Mbps]
ACKNOWLEDGMENT
2−clustered ULDA
80 60 40 20 0 1
1.5 2 2.5 Spectral Efficiency [b/s/Hz/sector]
3
Fig. 6. The 95 percentile user throughput versus the sector spectral efficiency for 8x2 ULDA and 2-clustered configuration.
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