Optimal Frame Splitting for Downlink MIMO Channels ... - IEEE Xplore

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IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 57, NO. 6, NOVEMBER 2008

case, the algorithm results in the following optimum transmission powers for the five stations: {70.00%, 0.00%, 60.00%, 80.00%, 60.00%}. Notice that the second station can now be removed since it no longer contributes to the coverage.

Optimal Frame Splitting for Downlink MIMO Channels With Distributed Antenna Arrays Chen Sun, Member, IEEE, Thomas Hunziker, Member, IEEE, Jun Cheng, Member, IEEE, Makoto Taromaru, Member, IEEE, and Takashi Ohira, Fellow, IEEE

VIII. C ONCLUSION In this paper, we have proposed a novel approach for the placement of wireless BSs. The proposed approach computes the number of BSs, their locations, and the transmission powers that satisfy the power coverage requirements. The proposed approach provides a flexible means for choosing arbitrary power propagation and demand patterns, making it potentially suitable for real applications. The proposed approach uses the 2-D convolution as a core process. This results in substantial reduction in complexity by utilizing available fast methods for computing convolution. Simulations of the new algorithm show its efficiency and flexibility in solving wireless placement problems. ACKNOWLEDGMENT The author would like to acknowledge Prof. S. Selim for his valuable reviews and comments on this paper.

Abstract—A frame-splitting (FS) scheme is proposed to exploit spatial diversity in the downlink wireless transmission from a base station (BS) to a mobile station (MS) that has multiple receive antennas. The BS has multiple geographically distributed arrays, each consisting of multiple transmit antennas. The scenario comprises a number of downlink multiple-input–multiple-output (MIMO) channels from different BS arrays to an MS with mutually independent Rayleigh-fading processes. A data frame from the BS for the MS is split into portions, which are consecutively transmitted from multiple BS arrays. For the FS transmission scheme, the distribution of information capacity is formulated on the basis of the FS fractional lengths of the portions. Analytical evaluation of the outage probability reveals the optimal setting of FS fractional lengths for the maximum diversity advantage based on knowledge of the long-term average signal-to-noise ratios (SNRs) of the downlink MIMO channels. Index Terms—Distributed antennas, diversity, multiple-input multipleoutput (MIMO), outage capacity, Rayleigh fading, Wishart matrices.

I. I NTRODUCTION R EFERENCES [1] H. D. Sherali, C. M. Pendyala, and T. S. Rappaport, “Optimal location of transmitters for micro-cellular radio communication system design,” IEEE J. Sel. Areas Commun., vol. 14, no. 4, pp. 662–673, May 1996. [2] Q. Hao et al., “A low-cost cellular mobile communication system: A hierarchical optimization network resource planning approach,” IEEE J. Sel. Areas Commun., vol. 15, no. 7, pp. 1315–1326, Sep. 1997. [3] P. Calegari et al., “Genetic approach to radio network optimization for mobile systems,” in Proc. IEEE Veh. Technol. Conf., May 1997, vol. 2, pp. 755–759. [4] M. H. Wright, “Optimization methods for base station placement in wireless applications,” in Proc. IEEE Veh. Technol. Conf., May 1998, vol. 1, pp. 387–391. [5] J. K. Han, B. S. Park, and Y. S. Choi, “Genetic approach with a new representation for base station placement in mobile communications,” in Proc. IEEE VTC, Oct. 7–11, 2001, vol. 4, pp. 2703–2707. [6] R. C. Santiago and V. Lyandres, “A sequential algorithm for optimal base stations location in a mobile radio network,” in Proc. IEEE Int. Symp. Pers., Indoor, Mobile Radio Commun., Sep. 5–8, 2004, vol. 4, pp. 2895–2899. [7] J. K. L. Wong et al., “Base station placement in indoor wireless systems using binary integer programming,” Proc. Inst. Elect. Eng.—Commun., vol. 153, no. 5, pp. 771–778, Oct. 2006. [8] B. Park, J. Yook, and H. Park, “The determination of base-station placement and transmit power in an inhomogeneous traffic distribution for radio network planning,” in Proc. IEEE Veh. Technol. Conf., Sep. 2002, vol. 4, pp. 2051–2055. [9] A. Elnaggar, H. M. Alnuweiri, and M. R. Ito, “A new recursive algorithm for multidimensional convolution,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 46, no. 5, pp. 652–654, May 1999. [10] K. Berberidis, “An efficient partitioning-based scheme for 2-D convolution and application to image registration,” in Proc. Int. Conf. Electron., Circuits Syst., Sep. 15–18, 2002, vol. 3, pp. 843–846. [11] I. Chiang and W. C. Chew, “Fast real-time convolution algorithm for microwave multiport networks with nonlinear terminations,” IEEE Trans. Circuits Syst. II, Exp. Briefs, vol. 52, no. 7, pp. 370–375, Jul. 2005. [12] R. Janaswamy, Radiowave Propagation and Smart Antennas for Wireless Communications. Norwell, MA: Kluwer, 2000.

Deploying multiple antennas is an effective means to improve the performance of wireless communications. Multiple transmit and multiple receive antennas are installed to construct a multiple-input multiple-output (MIMO) wireless channel. Analysis of the information capacity distribution of a MIMO channel in [1] and [2] suggested a great increase in spectrum efficiency. Transmitting independent data streams in parallel through multiple antennas (for example, the Bell Laboratories layered space–time architecture (BLAST) [3]) exploits the high spectrum efficiency of the MIMO channel. This effect is known as spatial multiplexing [4]. The distributed antenna system (DAS) was proposed in [5] and [6]. Instead of being colocated at a wireless base station (BS), multiple antennas are deployed at geographically dispersed locations within a wireless service area and are connected with a central BS by fiber/coaxial cables. From the viewpoint of wireless system architecture, the DAS brings many benefits, such as a reduction in transmission power, tolerance to large-scale fading, and improvement in link quality and coverage [5]–[7]. To gain the benefits from both the MIMO channel and the DAS wireless architecture, the antennas that are deployed at dispersed locaManuscript received October 7, 2006; revised July 10, 2007, December 19, 2007, and December 21, 2007. First published February 2, 2008; current version published November 12, 2008. This work was supported by the Ministry of Internal Affairs and Communications under the grant “Research and development of fundamental technologies for advanced radio frequency spectrum sharing in mobile communication systems.” The review of this paper was coordinated by Prof. M. Juntti. C. Sun was with the ATR Wave Engineering Laboratories, Kyoto 619-0288 Japan. He is now with the Ubiquitous Mobile Communications Group, National Institute of Information and Communications Technology (NICT), Yokosuka 239-0847 Japan (e-mail: [email protected]). T. Hunziker is with the University of Kassel, D-34121 Kassel, Germany (e-mail: [email protected]). J. Cheng is with the Doshisha University, Tatara, Kyotanabe, Kyoto 6100321 Japan (e-mail: [email protected]). M. Taromaru is with the ATR Wave Engineering Laboratories, Kyoto 6190288 Japan (e-mail: [email protected]). T. Ohira is with the Toyohashi University of Technology, Toyohashi 4418580 Japan (email: [email protected]). Digital Object Identifier 10.1109/TVT.2008.917254

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Fig. 1. Schematic diagram of FS transmission from a BS that has NB = 3 distributed BS arrays to an MS with multiple receive antennas. Each frame of K data bits from the BS is split into M = 3 portions with K = K1 + K2 + K3 bits, which are transmitted through NB = 3 distributed BS arrays in TDMA channels.

tions in a DAS are replaced with multiple arrays in [8]. However, the success of this system requires very strict conditions. The signals from those distributed BS arrays must arrive at the MS at the same time [9]. Perfect symbol and frame synchronization among such a large number of antennas poses implementation difficulties due to practical limitations. To circumvent the difficulty of synchronization, the BS can select one of the NB BS arrays that provides the maximum downlink channel capacity [10]. The system requires the feedback of the fading status of all the NB downlink MIMO channels at each fading block. We call this scheme selective transmission. In a block fading environment, as in [11], the selective transmission scheme requests the feedback of the status of each fading block. However, continuously tracking the short-term fading status and the heavy feedback pose a practical difficulty for implementation. In this paper, we study the downlink with distributed antenna arrays to an MS reported in [8]. There are totally NB antenna arrays that are connected by fiber cables with the central BS. These arrays are distributed to NB distinct locations within a service area. Each array, (i) which is henceforth referred to as a BS array, consists of Nt , i = 1, 2, . . . , NB transmit antennas. A mobile station (MS) having Nr receive antennas observes a downlink MIMO channel consisting of NB (i) Nt transmit antennas and Nr receive antennas. To avoid the i=1 complicated synchronization of the transmission system in [8], we consider that these NB distributed BS arrays do not simultaneously transmit to the MS. Rather, we assume that these NB distributed BS arrays collaborate in transmission. At any time instance, only one of the NB distributed BS arrays is involved in the communication (i) link. Therefore, at any time instance, the MS observes an Nr × Nt dimensional downlink MIMO channel, where i ∈ {1, 2, . . . , NB }. We consider a block fading environment [11] and assume that the channel fading status is fixed during the transmission period of each data frame but changes over multiple data frames. The data frame from the central BS is split into M (M ≤ NB ) portions, which are called subframes. Each subframe is transmitted through a different one of the NB distributed BS arrays to the MS in time-division multiple access (TDMA) channels. We call this scheme frame-splitting (FS) transmission. An example of an FS transmission of M = 3 from a BS with NB = 3 distributed BS arrays to an MS that has Nr receive antennas is illustrated in Fig. 1.

Given the aforementioned model, the problem that arises is how to partition a data frame to achieve the maximum diversity advantage. To address this, we formulate the distribution of information capacity for FS transmission on the basis of the fractional lengths of the subframes. The optimization of the FS fractional lengths for the maximum diversity benefit is built on knowledge of the average signal-to-noise ratio (SNR) over a few fading blocks for each of the NB downlink MIMO channels. Tracking this long-term statistical property requires much less feedback than the selective transmission scheme. Furthermore, FS transmission avoids the complicated synchronization of [8] among a large number of antennas. In Section II, we review the random capacity of a single MIMO channel. In Section III, we obtain an analytical model of the distribution of information capacity for the FS scheme. In Section IV, we use this analytical model to investigate the outage probability and reveal the optimal splitting of data frames based on knowledge of the long-term average SNRs. Finally, in Section V, we summarize our conclusion.

II. P RELIMINARIES To obtain an analytical distribution of information capacity for FS transmission, in this section, we review the random capacity of a single MIMO channel. In our model, only one of the NB distributed BS arrays communicates with the MS at any instance. The information capacity for the FS transmission through multiple downlink MIMO channels will be examined in Section III. The following notations are used throughout the paper: Let |A| be the determinant of matrix A having the lkth element {A}l,k . In addition, [al,k ]l,k=1,2,...,L is an L × L matrix with the lkth element al,k . The superscript (·)H denotes conjugate transpose. Let E[·] denote expectation, and Ev [·] is the expectation with respect to a particular random variable (RV) v.

A. MIMO Channel Let s be an Nt × 1 vector of transmitted symbols from Nt transmit antennas. The received Nr × 1 symbol vector y at the Nr receive

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III. D ISTRIBUTION OF I NFORMATION C APACITY OF FS T RANSMISSION

antennas is y = Hs + n

(1)

where n denotes noise modeled as an Nr × 1 vector of independent circular symmetric complex Gaussian RVs with zero mean and variance σ 2 . The Nr × Nt Rayleigh-fading MIMO channel H is given by 1/2

H = Σ1/2 r Hw Σt

(2)

where Σt and Σr are Hermitian matrices that represent fading correlation at transmitter and receiver ends, respectively. The elements of matrix Hw are modeled as independent circular symmetric complex Gaussian RVs with zero mean and unit variance [12], [13]. We assume that Nr ≤ Nt and that the fading correlation exists only at the receive antenna array. According to the exponential correlation model [14], [15], matrix Σr = [r|l−k| ]l,k=1,2,...,Nr with a fading correlation coefficient r ∈ (0, 1]. It is assumed that there is no fading correlation at the BS arrays due to sufficient space for installing multiple antennas. Thus, Σt is an Nt × Nt identity matrix.

As illustrated in Fig. 1, multiple subframes are transmitted through M out of NB distributed BS arrays in a sequential manner. Based on the analytical c.f. of the random capacity of a MIMO channel given in Section II, here, we derive the analytical c.f. of the random information capacity for the FS transmission scheme as a function of the FS fractional lengths. Let Hm be the MIMO channel from the mth distributed BS array (m) consisting of Nt transmit antennas to the MS having Nr receive an(m) tennas. Denote by Γm the long-term average SNR of this Nr × Nt downlink MIMO channel Hm . In addition, denote by Cm the information capacity of the channel Hm . As described in Section I, we consider a block fading environment. Every transmitted data frame from the BS consists of K information bits. The data frame is split into M subframes, each having Km information bits satisfying M

Km = K.

(8)

m=1

B. Random MIMO Channel Information Capacity When H in (1) is perfectly known only by the receiver and not by the transmitter, the channel capacity is given by [2] C=

Nr l=1

log2

Pt G 1 + 2 λl σ Nt

Here, we introduce an FS fractional length parameter τm of each subframe, which is defined as Δ

τm = (3)

where Pt is the total transmitter power, irrespective of Nt . In addition, G denotes the distance-related path loss. The long-term average SNR Δ of a MIMO channel is defined as Γ = Pt G/σ 2 per receive antenna. Due to poor scattering conditions, signals at antennas are correlated. Here, λ1 , λ2 , . . . , λNr are the ordered nonzero eigenvalues of HHH , which is a full-rank random complex Wishart matrix [16]. Owing to the “random” nature of fading channels, the channel capacity C is modeled as an RV, which differs from Shannon channel capacity. The characteristic function (c.f.) of C can be written in the following compact form [17], [18]: φC (ω) = KΣr |G(ω)|

(4)

Km , K

m = 1, 2, . . . , M.

As stated in Section I, the mth subframe (Km bits) is transmitted from the mth distributed BS arrays to the MS. The transmission of the entire data frame is complete after all the subframes are received at the MS. These K bits can equivalently be considered as being transmitted in parallel through M distinct MIMO channels. Denote by CFS the information capacity for the entire frame transmission being normalized by the total number of channel usages. Under the assumption of the block fading environment, we can write the information capacity of this K-dimensional channel as KC FS = K1 C1 + K2 C2 + · · · + KM CM .

CFS = xNt −Nr +k−1 e−x/αl ϕ(x, ω, Γ, Nt )dx

{G(ω)}l,k =

(5)

Here, ϕ(x, ω, Γ, Nt ) = (1 + (Γ/Nt )x)jω/ ln 2 , and KΣr is a normalizing constant given by KΣr =

π Nr (Nr −1) |Σr |−Nr

Nr (Nt )Γ Nr (Nr ) |V(α)| Γ

·

Nr

(k − 1)!

(6)

k=1

(n − l)!.

(7)

M

ECm [exp{jωτm Cm }]

m=1

Δ

l=1

(11)

φCFS (T , ω) = ECFS [exp{jωCFS }] =

eigenvalues of Σr are denoted by vector α = [α1 , α2 , . . . , αNr ] with α1 ≥ α2 ≥ · · · ≥ αNr ≥ 0. The complex multivariate gamma funcNr (n) in (6) is given by [16] tion Γ Nr

τm Cm .

Assuming that the MIMO channels H1 , H2 , . . . , HM are statistically independent, we model the channel capacities C1 , C2 , . . . , CM Δ as independent RVs. Letting vector T = [τ1 , τ2 , . . . , τM ], we can write the c.f. of CFS as

with {V(α)}l,k = (−αk )1−l , for l, k = 1, 2, . . . , Nr . The ordered

Nr (n) = π Nr (Nr −1)/2 Γ

M m=1

0

l, k = 1, 2, . . . , Nr .

(10)

Therefore, we have

with

∞

(9)

=

M

φCm (ωτm )

(12)

m=1

where φCm (ωτm ) is the c.f. of the mth MIMO channel capacity that (m) can be obtained by calculating (4) for arguments ωτm , Γm and Nt .


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Thus, we have φCFS (T , ω) = (KΣr )M ·

M

|Gm (ωτm )|

(13)

m=1

where we have used the fact that KΣr in (6) is common to all MIMO channels. This is because the distributed BS arrays communicate with the common MS antenna array. In (13), Gm (ωτm ) is defined in (5) for the mth MIMO channel. Therefore, by applying the Gil-Pelaez inversion formula [19] to (13), we can obtain the pdf fCFS (T , x) and the cumulative distribution function (cdf) FCFS (T , x) of CFS as

x FCFS (T , x) =

fCFS (T , v)dv −∞

1 = − 2

∞

φCFS (T , ω) −jωx e dω. j2πω

(14)

−∞

In (11), we have expressed the information capacity CFS for the FS transmission as a function of the FS fractional lengths τ1 , τ2 , . . . , τM . The function is in the form of a weighted sum of RVs C1 , C2 , . . . , CM , where the fractional lengths are the weights. Thus, using (13) and (14), we can analytically evaluate the distribution of CFS for a given value of vector T . In the following section, we will use the analytical cdf of CFS to study the outage performance of the FS transmission scheme. Furthermore, we will give the optimal fractional lengths for the lowest outage probability and, thus, the maximum diversity benefit. Note that the integral in (14) can be efficiently implemented by the fast Fourier transform [20]. Remark: When all MIMO channels have the same dimensions and equal average SNRs, the mean information capacity of the FS transmission is fixed when M increases. This is because E[CFS ] =

M Km m=1

= E[C1 ]

K

Fig. 2. CDF of CFS from an analytical approach and Monte Carlo simulations at different values of M . The average SNRs Γ1 = Γ2 = Γ3 , Nt = Nr = 3, and the fading correlation coefficient r = 0.5. Each frame is evenly split into M subframes.

investigating the influence of the long-term average SNRs on the FS scheme. The analytical model we obtained in Section III applies to the FS transmission through multiple MIMO channels, where the fading correlation at the receiver side of each MIMO channel is described by the exponential correlation model with an arbitrary fading correlation coefficient r ∈ (0, 1]. As an example, we hereafter set the fading correlation coefficient r = 0.5 because the capacity loss of a MIMO channel is negligible, even with the fading correlation coefficient r as large as 0.5 [17], [21], [22].

A. Diversity Advantage of FS Transmission E[Cm ]

M Km m=1

K

= E[C1 ]

(15)

where we have used the fact that H1 , H2 , . . . , HM follow identical and independent distributions and that the capacity of each MIMO channel is independent of the FS fractional lengths and the number of subframes. Therefore, the FS scheme does not directly increase the mean information capacity in this situation. However, it will be shown in the following section that the FS scheme significantly reduces the outage probability. IV. A NALYSIS AND D ISCUSSION We have obtained the cdf of CFS as a function of τm ’s. In this section, we study the influence of different values of τm on the diversity performance. First, we will look at a special case where the SNRs of all MIMO channels are equal. Then, we examine a general situation where the SNRs of MIMO channels are different. For both cases, we will obtain the optimal FS fractional lengths that give the lowest outage probability of the FS transmission. The number of antennas at the M distributed BS arrays is assumed (1) (2) (M ) = Nt . This is convenient for equal, i.e., Nt = Nt = · · · = Nt

First, we examine the diversity advantage of the FS transmission in a special case where the average SNRs of all the MIMO channels are the same. Because the MIMO channels have the same statistical properties, we intuitively split the data frame into M equal-length subframes, i.e., τ1 = τ2 = · · · = τM = 1/M . In Section IV-C, we will show that evenly splitting the data frame in this special case is optimal. Each subframe is transmitted from a distinct distributed BS array to the MS in a sequential manner. To obtain the cdf of CFS , we calculate its c.f. using (13) and numerically calculate the integral in (14). In Fig. 2, we plot the analytical cdfs on a logarithmic scale at SNRs of 5, 10, and 15 dB. For comparison, we also obtain the empirical cdfs of CFS on the basis of 105 realizations of each downlink MIMO channel. The close match between the analytical and empirical cdfs verifies our analytical formulation on the distribution of CFS for equal SNRs. For unequal SNRs, we also verify the formulation, but this is omitted here due to limited space. For simplicity, we use the cdf of CFS obtained from the analytical approach to study the outage performance of FS transmission in the following sections. The curves in Fig. 2 show that in the same SNR situation, transmitting portions of a data frame through multiple MIMO channels reduces the outage probability, although the mean information capacity does not increase, as mentioned in Section III. For example, when the SNR is 5 dB, the transmission through M = 3 BS arrays achieves a decrease of outage rate at 3 bits/s/Hz by two orders of magnitude as compared with the transmission without FS (M = 1). This indicates a significant improvement in channel stability.

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Fig. 3. Achieved information capacities (in bits per second per hertz) of an FS transmission scheme and a selective transmission scheme at 0.01 outage rate. Fading correlation coefficient r = 0.5 for the case Nt = Nr = 3.

B. Outage Capacity of FS Transmission Here, we examine the outage capacity of the FS transmission. Fig. 3 depicts the achieved capacity at a 1% outage rate for the FS transmission of M = 2 and τ1 = τ2 = 0.5 with different numbers of transmit and receive antennas. In a scalar Rayleigh-fading channel (Nt = Nr = 1), the FS scheme provides a 7.5-dB gain at 2 bits/s/Hz. As expected, the spatial multiplexing advantage of a MIMO channel (Nt = Nr = 3) increases the outage capacity. In addition, the FS scheme achieves approximately a 1.5-dB diversity gain at 10 bits/s/Hz to further improve performance. The results show that the spatial diversity gain achieved through FS improves the transmission stability, thus increasing the outage capacity. Without tracking the instantaneous channel fading status, the FS transmission scheme splits the data frame from the BS into portions that are transmitted through distributed BS arrays at different locations in a sequential manner. If the tracking of the short-term fading status of multiple downlink MIMO channels from the distributed BS arrays to the MS is available, the optimal solution for the maximum diversity is to select the BS array that provides the highest capacity to transmit the entire data frame. This selective transmission scheme is statistically equivalent to that in [10]. As stated in Section I, the scenario is possibly difficult to achieve in practice due to practical limitations (such as heavy feedback and tracking of short-term fading status), but we use it here as an upper bound on the performance obtained with the practical FS scheme. It is worth noting in Fig. 3 that the FS scheme evades the task of tracking the short-term fading status of all MIMO channels and approaches the capacity bound by exploiting the spatial diversity with only the feedback of the long-term SNRs. C. Outage Probability Over FS Fractional Lengths In the previous sections, we have evaluated the distribution of CFS of different SNR situations, including equal SNRs as a special case. Here, we investigate the performance of the FS transmission with different FS fractional lengths in general cases of downlink transmission with distributed BS arrays. That is, the average SNRs of received signals at the MS from distributed BS arrays can be different. The difference reflects distance-related path losses from the BS arrays that are distributed at different locations from the MS. We will obtain the optimal FS fractional lengths that give the lowest outage probability of the

Fig. 4. Outage probability at 8 bits/s/Hz over FS fractional length. M = 2, τ1 = 1 − τ2 , Nt = Nr = 3, and fading correlation coefficient r = 0.5.

FS transmission and examine the SNR condition under which the FS brings a diversity advantage. First, we examine the influence of the relative difference between SNRs. Fig. 4 depicts the outage probability of information capacity at 5 bits/s/Hz with different settings of FS fractional lengths, where Γ1 = 10 dB. The frame is split into two subframes. As shown in the figure, when the average SNRs are different, the lowest outage rate is achieved by sending a large portion of the frame over the channel with a relatively high average SNR. For example, when the relative difference in decibels between the values of Γ1 and Γ2 is Γ1 − Γ2 = 2 dB, it is advantageous to transmit approximately 60% of the frame over the channel of Γ1 and to transmit the remaining 40% over the channel of Γ2 . This shows that the optimal FS fractional lengths are τ1 = 0.6 and τ2 = 0.4 in this case. As can be seen in the figure, given the dimensions of the MIMO channels and the fading correlation property, the choice of optimal FS fractional lengths is affected by the relative difference of the SNRs of these channels. Using the analytical model, we can determine the optimal FS fractional lengths of unequal SNRs by plotting the outage rate over the fractional lengths shown in Fig. 4. Note that when Γ1 = Γ2 = 10 dB, the outage curve is symmetric and achieves the minimum at τ1 = τ2 = 0.5. This is a trivial solution for equal-SNR cases, as we predicted in Section IV-A. In addition, as shown in Fig. 4, when Γ1 = 10 dB, the FS transmission can bring a lower outage probability, compared with that of no FS if Γ2 > 5 dB. Furthermore, the lowest outage probability can be achieved by properly setting FS fractional lengths. When the SNR Γ2 of the second downlink MIMO channel drops to 5 dB, the lowest outage probability of the FS transmission is achieved at τ1 = 1. In other words, the optimal FS fractional lengths in this situation are τ1 = 1 and τ2 = 0. In this situation, the whole data frame should be transmitted through the first distributed BS array. This situation of unequal SNRs happens when the MS is located close to the first distributed BS array but relatively far away from the second distributed BS array. To examine the influence of the values of SNRs in addition to their relative differences, we fix the relative difference at Γ1 − Γ2 = 3 dB and plot the normalized outage rate at 5 bits/s/Hz when Γ1 changes from 5 to 15 dB in Fig. 5. The normalized outage rate is defined as the outage rate with FS as divided by the outage rate without FS, i.e., τ2 = 0. As shown in the figure, the FS transmission gains improvement in the outage performance when Γ1 ≥ 9 dB. Furthermore, these curves indicate the optimal fractional lengths at different SNRs. For


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In the previous sections, we have obtained the optimal FS fractional lengths for different situations of unequal SNRs using Figs. 4 and 5. Based on these results, we build a table of optimal FS fractional lengths for different SNR situations. Knowing that the FS fractional lengths are dependent on SNRs, we implement a lookup-table-based FS transmission scheme with adjustable fractional lengths for a case of unequal SNRs, where Γ1 − Γ2 = 3 dB. Here, we use these optimal FS fractional lengths for the FS transmission. Accordingly, when Γ1 ≥ 9 dB, the FS transmission is employed with the optimal fractional lengths. The achieved outage rate is shown in Fig. 6 as a star-marked line. However, when Γ1 < 9 dB, the transmission is fulfilled without FS, i.e., τ2 = 0. As can be seen, the lowest outage probability, i.e., the maximum diversity benefit, of the transmission is achieved by changing the FS fractional lengths in accordance with the long-term average SNRs. V. C ONCLUSION Fig. 5. Outage probability at 8 bits/s/Hz over FS fractional length as Γ1 changes from 7.5 to 22.5 dB. M = 2, τ1 = 1 − τ2 , Nt = Nr = 3, and fading correlation coefficient r = 0.5.

Fig. 6. Outage probability at 8 bits/s/Hz over SNR Γ1 . M = 2, τ1 = 1 − τ2 , Nt = Nr = 3, and fading correlation coefficient r = 0.5.

We have proposed an FS scheme for downlink transmission with distributed antenna arrays. The scheme splits a data frame into portions, with each being sent from a distinct distributed BS array to the MS. To analyze the performance of the FS scheme, we gave a theoretical framework to the information capacity of consecutively transmitting different portions of a frame through multiple MIMO channels. We performed numerical evaluation of the outage probability using the analytical model, given long-term SNRs of MIMO channels, and intuitively illustrated the optimal FS fractional lengths from the curves of outage probabilities. These optimal FS fractional lengths give the lowest outage probability at a given target transmission data rate for the FS transmission. In the situation of equal SNRs, the data frame is evenly split to achieve the lowest outage probability. In the presence of unequal SNRs, a lookup table of optimal FS fractional lengths is helpful for FS transmission. The FS scheme avoids complicated synchronization among a large number of antennas and heavy feedback of the short-term channel fading status. Through FS transmission, the spatial uniqueness of multiple BS arrays that are distributed at different locations is exploited to improve the transmission stability in a multipath fading environment. The investigation of the outage probability for the FS scheme, as an information-theoretic measure, is helpful for future research on practical (e.g., coding) implementation of the scheme to achieve the diversity advantage. R EFERENCES

example, when Γ1 = 13 dB, splitting the frame with τ1 = 0.6 and τ2 = 0.4 provides the lowest outage probability. In addition, when Γ1 = 15 dB, the optimal FS fractional lengths are τ1 = 0.55 and τ2 = 0.45, which give the lowest outage probability. Therefore, a lookup table can be built to give the optimal FS fractional lengths at different situations of unequal SNRs. As also shown in Fig. 5, the transmission should be fulfilled without FS when Γ1 < 9 dB. In Fig. 6, we plot the outage probability at 5 bits/s/Hz for different SNR situations. When Γ1 − Γ2 = 3 dB, the outage curves for the transmission without FS and that with FS of τ1 = 0.55 cross at approximately Γ1 = 10 dB. This shows that when the SNR Γ1 is larger than 10 dB (moderate-to-high SNR situation), it is advantageous to split the data frame and transmit over two channels, even if they experience a 3-dB relative difference in SNRs. However, when the SNR is lower than 10 dB (relatively low SNR situation), the transmission should be fulfilled without FS. In addition, the figure confirms that splitting a data frame evenly provides a diversity advantage over that of no FS when Γ1 = Γ2 .

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An Improved Closed-Form Approximation to the Sum of Arbitrary Nakagami-m Variates Daniel B. da Costa, Student Member, IEEE, Michel D. Yacoub, Member, IEEE, and J. C. S. Santos Filho

Abstract—The aim of this paper is threefold: 1) to propose a simple accurate closed-form approximation to the probability density function of the sum of arbitrarily distributed Nakagami-m random variables; 2) to propose a simple accurate closed-form approximation to the level crossing rate for the sum of Nakagami-m random processes; and 3) to show some possible applications for the proposed formulations. With such an aim, we choose the α–μ distribution for which the parameters are estimated from the sum of the Nakagami-m envelopes. As shall be shown from sample representative examples, the proposed approximations are simple, versatile, and highly accurate. The approach used here can easily be extended to other applications such as bit error rate and channel capacity calculations among others. Index Terms—Approximation methods, moment-based estimators, Nakagami-m sums, α–μ distribution.

I. I NTRODUCTION The sum of fading envelopes occurs in several wireless communications applications, such as diversity combining techniques [e.g., equalgain combining (EGC) and maximal-ratio combining (MRC)], signal detection, linear equalization, etc. The exact evaluation of some of the statistics [e.g., probability density function (pdf) and level crossing rate (LCR)] of the resulting envelope for the general application can be rather cumbersome, for it may require a multidimensional convolution or integral of products of moment-generating functions (MGFs). These approaches become computationally impracticable as the number of signals increases. Hence, simple approximate solutions are of interest. Among the various fading scenarios, Nakagami-m [1] has received special attention due to its wide range of applicability and to the ease of manipulation of the formulations involved. In [2], the pdf of the signalto-noise ratio (SNR) at the output of multibranch EGC receivers for independent Nakagami-m fading signals was derived. The computation gets ahold of an infinite series [3], whose accuracy is controlled both by a sampling parameter and by the number of terms in the series. A closed-form expression for the pdf of the sum of Nakagami-m random phase vectors was derived in [4], which is a solution restricted to integer fading parameters. Recently [5], the distribution of the sum of independent identically distributed (i.i.d.) Nakagami-m variates has been derived that involves nested infinite summations of gamma and hypergeometric functions, which, in the authors’ own words, “are hard to evaluate in general.” An approximation to the sum of Nakagami-m Manuscript received September 24, 2007; revised January 1, 2008 and January 30, 2008. First published February 15, 2008; current version published November 12, 2008. This work was supported in part by the Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) under Grant 05/59259-7. The review of this paper was coordinated by Prof. J. Wu. D. B. da Costa was with the Wireless Technology Laboratory (WissTek), Department of Communications, School of Electrical and Computer Engineering, State University of Campinas, 13083-852 Campinas, SP, Brazil. He is now with INRS-EMT, University of Quebec, Montreal, QC, Canada (e-mail: [email protected]). M. D. Yacoub and J. C. S. Santos Filho are with the Wireless Technology Laboratory (WissTek), Department of Communications, School of Electrical and Computer Engineering, State University of Campinas, 13083-852 Campinas, SP, Brazil (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TVT.2008.918725

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