3. Analysis for Coverage Performance for OFDM-based Spatial Multiplexing Sys-
3.6 Numerical Results
In this section, we present numerical results to illustrate how the number of antennas, number of subchannels (the order of frequency selectivity), and the pass loss exponent could affect the link outage and cell coverage in OFDM-based spatial multiplexing systems. We first assume a predetermined value γth = 2dB, noise power = −103dBm, g0 = −32, and r = 1km. Figure 3.4 shows the simulative, analytical, and the approx-imate link outage with various order of frequency selectivity in the case of M = 3.
The performance metric shown in the x−axis is the transmit power Ptin dB domain.
We can see that the link outage reduces as Ptincreases. Furthermore, the link outage probability will reduce more quickly when the number of subchannels increases. We can also see that the analytical value is very close to the simulation result, and the
0 5 10 15 10−3
10−2 10−1 100
Transmit power P
t (dBW)
Link outage M=5 M=4 M=3 M=2
Approximately analytical result Analytical result
Simulation result
Fig. 3.5: Link outage probability v.s. transmit power Pt for different values of M when N = 128, noise power= −103dBm, µ = 3, r = 1km and γth = 2dB.
approximately value especially when the number of subchannels is large.
Figure 3.5 shows the link outage performances with various numbers of transmit and receive antennas for N = 128. One can see that when for the case M = 2, Pt increases, the link outage would first reduce. It indicates that the link outage probability become higher as the number of antennas increases. Thus it is hard to maintain M times of capacity for a large number of M.
Figure 3.6 shows the link outage performances with different pass loss exponents.
The link outage probability is high for a large pass loss exponent .
Figure 3.7 shows the cell coverage with different numbers of antennas in the case of N = 128 and µ = 2. We can see that the cell coverage increases as Pt increases, and it will increase more quickly with fewer antennas. That is, it indicates that the coverage area is easier to maintain M times of capacity when M is small.
27
0 5 10 15
10−3 10−2 10−1 100
Transmit power P
t (dBW)
Link outage µ=3.2 µ=3.1 µ=3 µ=2.9
Approximately analytical result Analytical result
Simulation result
Fig. 3.6: Link outage probability v.s. transmit power Pt for different values of µ when M = 3, N = 128, noise power= −103dBm, r = 1km and γth= 2dB.
Figure 3.8 shows the cell coverage with various frequency selectivity orders when M = 3 and µ = 2. It shows that the cell coverage is slightly larger in the high-frequency selectivity than in the low high-frequency-selectivity. Figure 3.9 shows the cell coverage with various pass loss exponents in the case of M = 3 and N = 128. Notice that µ = 2 is for free space, and µ = 3.5 ∼ 4 is for two-path model of an urban radio channel. We can see that the cell coverage would increase more quickly for µ = 2.
−100 −5 0 5 10 500
1000 1500 2000 2500 3000 3500
Transmit power P
t (dBW)
Cell coverage (m)
M=1 M=2
M=4
M=3
Simulation result
Approximately analytical result
Fig. 3.7: Cell coverage radius v.s. transmit power Pt for different values of M when N = 128, noise power= −103dBm, µ = 3, Pout = 0.1 and γth= 2dB.
29
−10 −5 0 5 10
200 400 600 800 1000 1200 1400 1600
Transmit power P
t (dBW)
Cell coverage (m)
N=128
N=32 N=8
Simulation result
Approximately analytical result
Fig. 3.8: Cell coverage radius v.s. transmit power Pt for different N when M = 3, noise power= −103dBm, µ = 3, Pout= 0.1 and γth= 2dB.
−10 −5 0 5 10 102
103 104
Transmit power Pt (dBW)
Cell coverage (m)
µ=2.5
µ=3
µ=3.5
Simulation result
Approximately analytical result
Fig. 3.9: Cell coverage power v.s. transmit powerPt for different values of µ while M =3, N = 128, noise power= −103dBm, Pout = 0.1 and γth = 2dB.
Chapter 4
Coverage Enhancement for
Spatial-Multiplexing-Based MIMO OFDM Systems by Joint Multiuser Scheduling and Subcarriers Assignment
Thanks to the orthogonal frequency multiplexing (OFDM) modulation to overcome the inter-symbol-interference in the frequency-selective fading channel and the multi-input multi-output (MIMO) antenna techniques to deliver high multiplexing capacity gain, MIMO-OFDM has become an attraction option for the next-generation high-speed wireless systems. Due to the fact that the multiplexing gain and diversity gain in the spatial domain are difficult to be maximized simultaneously and that the to-tal transmit power is split over the multiple antenna, the spatial-multiplexing-based MIMO system faces a coverage reliability issue. To improve the coverage reliability of the diversity-deficient spatial-multiplexing-based MIMO system, we suggest exploit-ing the multiuser diversity from schedulexploit-ing techniques and frequency diversity from OFDM subcarrier assignment in the frequency selective fading simultaneously. For this purpose, an efficient and low-complexity sub-carriers assignment scheme com-bined with multi-user scheduling and is proposed in this chapter. By means of the analytical techniques of the order statistics and Glivenko- Cantelli theorem, we de-velop an analytical expression for the link outage probability and cell coverage relia-bility. Our results show that the proposed joint multi-user scheduling and sub-carriers assignment scheme can significantly improve the reliability of the spatial multiplexing
gain of MIMO-OFDM systems. Moreover, we show that the proposed scheme not only can improve the cell coverage performance of the spatial-multiplexing based MIMO-OFDM systems, but successfully preserve the multiusers’ fairness performance.
4.1 Introduction
Orthogonal Frequency Division Multiplexing (OFDM) can convert a frequency se-lective fading channel into a parallel collection of frequency flat fading sub-channels and thus can overcome inter-symbol interference (ISI) [7] [8]. In the meanwhile, multi-input multi-output (MIMO) antenna techniques can provide spatial multiplex-ing gain and diversity gain to increase spectrum efficiency and link reliability, respec-tively [9] [10] [11] [12]. Combining MIMO with OFDM (MIMO-OFDM) becomes an attractive air-interface solution for the next generation high speed wireless systems.
The MIMO-OFDM system combines the advantages of both techniques in providing simultaneously increased data rate and elimination of the effects of delay spread.
Spatial-multiplexing based OFDM is one of the categories of MIMO-OFDM techniques. The main goal of spatial-multiplexing based MIMO-MIMO-OFDM tech-nique is to increase capacity by exploiting multiplexing gain in the spatial domain, i.e., transmitting independent data streams across antennas and tones. However, increasing the number of transmit antennas results in a smaller signal-to-noise ra-tio (SNR) per degree of freedom because the total available transmit power is split uniformly across transmit antennas. This leads to link outage or coverage issue of the spatial multiplexing MIMO system. This issue has been investigated originally in [6] and a multiuser scheduling solution to address this issue in MIMO flat-fading channels was proposed in [3].
In a multiuser MIMO system over flat fading channels, multiuser diversity can be
33 exploited to improve downlink capacity. [3] proposed a fair scheduling scheme called strongest-weakest-normalized-subchannel-first (SWNSF) which requires only limited amount of feedback and can significantly increase the coverage of the multiuser MIMO system while further improving system capacity. In a multiuser orthogonal frequency division multiplexing (OFDM) system over frequency selective fading channels, [2]
described an optimal subcarriers allocation algorithm and proposes a low-complexity suboptimal adaptive subcarriers allocation algorithm which performs almost as well as the optimal solution. In a multiuser MIMO OFDM system over frequency selec-tive fading channels, [4] derived the optimal subcarrier allocation criterion and the optimal power loading criterion for downlink MIMO OFDM systems. [25] applied an optimization algorithm to obtain a joint subcarrier and power allocation scheme based on MIMO OFDM combined with dirty paper coding (DPC) when fully instantaneous channel state information (CSI) is available.
In this chapter we focus on the assignment of each resource dimension to only one user to avoid the complexity and the assignment requires only limited amount of feedback. Besides, we assume frequency selective quasistatic channels where chan-nels do not vary within a block of transmission. We first extend the suboptimal subcarriers assigning algorithm proposed in [2] from SISO OFDM to MIMO OFDM systems, and we call it fairness-oriented subcarriers assignment (FOSA) in this chap-ter. Besides, we propose a low-complexity coverage-oriented subcarriers assignment algorithm (COSA) which can achieve larger cell coverage than FOSA for spatial-multiplexing based MIMO-OFDM systems. Furthermore, we derive an approximately analytical closed form expression for link outage and cell coverage by using COSA, and show how the total transmit power, number of antennas, number of users and the pass loss exponent could affect the cell coverage under this subcarriers assignment algorithm over frequency-selective fading channels.
Fig. 4.1: Architecture for scheduling in multiplexing-based MIMO-OFDM systems. Assume that N = 2, NT = 4, and M = 2
4.2 System model
We consider K users in a MIMO system with M transmit antennas in base station, and M receive antennas at each mobile. We adopt OFDM modulation with total NT subcarriers and let a group of adjacent NT/N subcarriers form a subchannel.
The total bandwidth of each subchannel is assumed to be smaller than the coherent bandwidth of the channel. Fig. 4.1 shows the considered structure of the spatial-multiplexing based MIMO-OFDM systems. The resource scheduling algorithms are carried out at the base station. In order to keep the scheduling complexity low, we divide the NT subcarriers into N groups which are made up of NT/N neighboring subcarriers and are the minimum resource units to be allocated. Fig. 4.2 shows an illustration for MIMO-OFDM subchannel assignment where H(k,n) represents the flat and independent fading M×M channel matrix corresponding to the kthuser at the nth
35
Fig. 4.2: Illustration for MIMO OFDM subcarriers assignment. Assume that N = 4, K = 2
subchannel. In the spatial-multiplexing based MIMO-OFDM systems, M transmit antennas and N OFDM sub-channels construct the transmitted symbol vector by multiplexing MN independent data streams. The transmit power is uniformly split to M transmit antennas. It is assumed that the length of the cyclic prefix (CP) in the OFDM system is greater than the length of the discrete-time baseband channel impulse response so that the frequency-selective fading channel indeed decouples into a set of parallel frequency-flat fading channels [22]. With xk,n and yk,n denoting the M × 1 transmit and receive signal vectors for the user k, respectively, we can write
yk,n =√
gk· H(k,n)· xk,n+ nk,n, (4.1)
where n is the sub-channel index while index k represents a specific user. H(k,n) represents the flat and independent fading M × M channel matrix corresponding to the kth user of the nth subchannel and each entry of H(k,n) is an i.i.d. circular-symmetric complex Gaussian variable [4] [18]. Represent nk,n the M × 1 spatially white noise vector with E[nk,n· n∗k,n] = σ2k,nI, where (·)∗ is the transpose conjugate operation. While H(k,n) captures the channel fading characteristics, gk depicts the large-scale behavior of the channel gain. For a user k at a distance of rk from the base station, gk can be written as [23].
10 log10(gk) = −10µ log10(rk) + g0, [dB] (4.2)
where µ is the path loss exponent and g0 is a constant subject to certain path loss models.
4.3 Definitions
4.3.1 Link Outage Probability
Define the link outage probability reflect to what extend a system can reliably support the corresponding link quality. For a single-input single-output (SISO) system in flat fading channel, link outage is usually defined as the probability that the received SNR is less than a predetermined value γth, i.e. Pout = Pr{γ < γth} [5]. The link outage for the spatial multiplexing MIMO system in a flat fading channel is defined as the event when the receive SNR of any substream is less than γth [6] [3]. When all the degrees of freedom in the spatial domain of a MIMO system are used for the transmission of parallel and independent data streams to exploit the spatial multiplexing gain, the data stream with the lowest SNR in the MIMO system will dominate the link reliability performance especially when the link reliability likely of high percentile, such as 90% or even higher, is concerned.
The spatial-multiplexing based MIMO-OFDM systems in a frequency selective fading channel can be viewed as the sum of flat fading MIMO channels. As discussed before, the high-percentile link reliability performance of each MIMO flat-fading chan-nel is dominated by the weakest substream. Considering the average weakest eigen-mode over a series of N’s MIMO flat-fading subchannels, we define the link outage probability of the spatial-multiplexing-based MIMO OFDM system for user k as fol-lows:
where γk,n,M represents the receive SNRs of the weakest substream corresponding to
37 the kth user in subchannel n for n = 1, . . . , N .
4.3.2 Cell Coverage Reliability
With Poutk being the link outage probability for user k, we define the cell coverage for all the users in a cell is the farthest distance at which the link quality suffices for maintaining a required receive SNR γth with cell coverage reliability (1 − Poutk ). What we focus on is the farthest user in the boundary of the cell coverage. In other words, if the farthest user maintains the link quality, the other (K − 1) users will maintains it too.
4.4 Subcarriers assignment algorithms
If the channel is static and is perfectly known to the transmitter and the receiver, joint power and subcarrier allocation is known to be optimal for MIMO OFDM systems [4].
However, solving the optimization problems could be very complex to implementation.
In [2], the authors show that the total system performance will be close to optimum as long as the energy is poured only into subchannels with good channel gains. They also bring up a concept that a flat transmit power spectral density (PSD) would hardly reduce the system performance if each subchannel is assigned to a user whose channel gain is good for it. Therefore, the authors propose a suboptimal subcarriers assigning algorithm for single-input single-output (SISO) OFDM systems and its complexity is almost negligible compared with the complexity of finding optimal allocation by solving a joint power and subcarrier optimization equation.
In this section, we present two suboptimal subcarriers assignment algorithms for spatial-multiplexing based MIMO-OFDM systems. At first, we show an extended suboptimal subcarriers assignment algorithm – FOSA based on the work in [2]. Then
Fig. 4.3: Fairness-oriented subcarriers assignment (FOSA)
we propose a low-complexity but performance-enhanced COSA algorithm.
In the following subcarriers assignment algorithms, equal amount of power is allocated to each subchannel. Define A = {1, 2, . . . , N}, B = {1, 2, . . . , K}, and Rk
as the subchannel index set and the allocated weakest subchannel metric for kth user in the following two flow charts, respectively. Notice that λk,n = λmin(H(k,n)· H(k,n)∗) represent the weakest substream in H(k,n).
39
Fig. 4.4: Coverage-oriented subcarriers assignment (COSA)
4.4.1 Fairness-Oriented Subcarriers Assignment (FOSA)
The authors in [2] proposed a suboptimal subcarriers assignment for SISO OFDM systems. Based on the designed principles, we extend the algorithm to a MIMO OFDM system and denote as FOSA in the chapter. The main advantage of FOSA is the negligible complexity compared with the optimal allocation. Furthermore, FOSA can ensure all the mobile users to receive at almost the same times of subchannels, and it can achieve good fairness performance at the same time. Figure 4.3 shows the flow chart of the FOSA algorithm.
1 2 3 4 5 6 7 8 9 10
Fig. 4.5: Fairness performance comparison for K = 5 and M = 3
4.4.2 Coverage-Oriented Subcarriers Assignment (COSA)
We provide an low-complexity coverage-oriented subcarriers assignment algorithm (COSA) shown in Fig. 4.4. In the following sections, we will show COSA achieve larger cell coverage than FOSA for spatial-multiplexing based MIMO-OFDM systems by applying multiuser diversity. Furthermore, we will show that COSA can achieve almost the same fairness performance as FOSA when the number of subchannels is large.
41
4.5 Fairness Performance
We first define a fairness index F in the multiuser systems as follows
F =
where Ti is the number of times the subchannels allocated to ith user. For F = 1, it is the fairest condition between users, and it is not fair as F ¿ 1. Fig. 4.5 shows the fairness performance comparison for FOSA and COSA, and we can see that COSA achieves almost the same fairness performance as FOSA when the number of subchan-nels is large. Since COSA can achieve almost the same fairness performance, and it can further achieve larger cell coverage than FOSA (we’ll show it in numerical results), we focus on COSA and derive an approximately analytical closed form expression for link outage and cell coverage by using this subcarriers assignment algorithm.
4.6 Link Outage by COSA
To begin with, we first analyze the received SNR of the weakest substream (denoted by γk,n,M) at the nth subchannel for user k. With {λk,n,i}Mi=1representing the eigenvalues of the Wishart matrix H(k,n)H(k,n)∗, we can express γk,n,M as
γk,n,i= ρk,nλk,n,i/M, (4.5)
where ρk,n is the average receive SNR at the nth subchannel and is equal to ρk,n = Ptgk
Nσ2k,n = Pt 10(g0/10)
N2 σk,n2 rkµ (4.6)
Arrange {λk,n,i}Mi=1 in the decreasing order so that λk,n,1 ≥ λk,n,2 ≥ ... ≥ λk,n,M ≥ 0.
According to [3] [24], the marginal probability density function (PDF) of the minimum eigenvalue λk,n,M is exponentially distributed with parameter M.
fλk,n,M(λ) = Me−M λ , λ ≥ 0. (4.7)
The marginal cumulative distribution function (CDF) of it can be written as Fλk,n,M(λ) =
Z λ
0
fλk,n,M(x)dx
= 1 − e−M λ, λ ≥ 0. (4.8)
By applying the singular value decomposition (SVD) to H(k,n), we know that an open-loop MIMO-OFDM channel is enhanced by forming MN parallel subchannels, each of which has effective output SNR γk,n,i = ρk,nλk,n,i/M at the receive antenna. Notice that γk,n,M is also an exponentially distributed random variable with the following marginal CDF
Fγk,n,M(γ) = 1 − e−M 2γρk,n , γ ≥ 0. (4.9)
For brevity, we omit the index of M and use γk,nto replace γk,n,M. In (4.9), {γk,1, ...γk,N} are identically and independently exponential distributed random variables with marginal CDF Fγk,n(γ) = 1 − e−M 2γρk,n. By using COSA, any user k competes for services with other (K − 1) users at each time slot for each subchannel n. Assume that the select user at each time slot for each subchannel n is k∗. The CDF of γk∗,n is given by ith order statistic. First we assume that γk∗,(ω) is the value which is most close to
43 We’ll show the details in Appendix B, and Nω(N, K) is an approximation integer value of ω. In (4.12), link outage is transformed to another form of probability which means at least Nω(N, K) of the γk∗,n are less than or equal to γth. By applying the theories of order statistics, we obtain
Pout ' Pr(γk∗,(Nω(N,K))≤ γth)
where is the incomplete beta function. Now we have analytical closed form approximation for the link outage. We will show how we can derive the cell coverage from this approximate link outage formula in the next section.
4.7 Cell Coverage by COSA
We can define that the cell coverage for all the users in a single cell is the farthest distance at which the link quality suffices for maintaining a required receive SNR γth with the probability no less than (1 − Poutk ). We focus on the farthest user in the boundary of the cell coverage. In other words, if the farthest user maintains the link quality, the other (K − 1) users will maintains it too. In order to derive r (or rk for the farthest user) from (4.15), we first introduce the inverse incomplete Beta function which is shown as follow
z = Iw(a, b) ⇒ w = Iw−1(a, b). (4.16) By substituting (4.14) and (4.16) into (4.13), we can obtain the cell coverage as
r '
45
Approximately alalytical result by COSA Simulation result by FOSA
Approximately anaytical result by RR
Fig. 4.6: Cell coverage v.s. users for different Pt while N = 32, M = 3, noise power= −103 dBm, µ = 3, Pout= 0.1 and γth= 2dB
Now we have an analytical closed form approximation of the cell coverage for spatial-multiplexing based MIMO-OFDM systems over frequency selective fading channel, and it is a function composed of given parameters {M, N, K, Pt, σ2n, µ, γth} and required Pout (usually 0.1).
4.8 Numerical Results
In this section, we present some numerical results to illustrate how the total transmit power, the number of antennas, the number of users and pass loss exponents can affect the cell coverage in spatial-multiplexing based MIMO-OFDM systems. We
2 4 6 8 10 12 14 16 18 20
Approximately alalytical result by COSA Simulation result by FOSA
Approximately anaytical result by RR
Fig. 4.7: Cell coverage v.s. users for different M while Pt = 1W, N = 32, noise power=
−103 dBm, µ = 3, Pout= 0.1 and γth= 2dB.
first assume a predetermined value γth = 2dB, noise power = −103dBm, g0 = −32, and r = 1km.
From Figs. 4.6 ∼ 4.8, we validate the accuracy of the analytical cell coverage of COSA by simulation. For comparison, the coverage performance of FOSA and the round-robin (RR) scheduling algorithm is also presented. Notice that the RR algorithm achieves the fairness among users, but takes no multiuser diversity. Clearly, the coverage performance by COSA is better than those by FOSA and RR from Figs.
4.6 to 4.8. It indicates that by using COSA the multiuser diversity can be more effectively exploited to improve the cell coverage.
Fig. 4.6 shows how the total transmit power Pt can affect the cell coverage with different numbers of mobile users. When there are two times of total transmit power, the cell coverage is about to increase a quarter.