2. Background
2.5 Some Definitions of Performance Metrics
2.5.3 Cell Coverage Reliability
, (2.2)
where γk,n,M represents the receive SNRs of the weakest substream corresponding to the kth user in subchannel n for n = 1, . . . , N .
2.5.3 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 ). Our focus is 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.
Analysis for Coverage Performance for OFDM-based Spatial Multiplexing
Systems
Combining multi-input multi-output (MIMO) antenna techniques with orthogonal frequency division multiplexing (OFDM) modulation (MIMO-OFDM) becomes an attractive air-interface solution for the next generation high speed wireless systems.
Nevertheless, because the total available transmit power is split uniformly across transmit antennas in MIMO-OFDM systems, increasing the number of transmit an-tennas leads to a smaller signal-to-noise ratio (SNR) per degree of freedom. Thus the coverage performance of this kind of MIMO-OFDM system becomes an essential issue. In this chapter by means of order statistics and Glivenko- Cantelli theorem, we develop an analytical expressions for the link outage probability and cell coverage re-liability of OFDM-based spatial multiplexing systems in a frequency selective fading channel, respectively.
3.1 Introduction
Orthogonal frequency division multiplexing (OFDM) modulation has become a pop-ular modulation technique for transmission of broadband signals. OFDM can convert a frequency selective fading channel into a parallel collection of frequency flat
fad-15 ing 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 multiplexing gain and diversity gain to increase spectrum efficiency and link reliabil-ity, respectively [9] [10] [11] [12]. Combining MIMO with OFDM (MIMO-OFDM) becomes an attractive air-interface solution for the next generation high speed wireless systems.
Generally, there are three categories of MIMO-OFDM techniques.
• The first aims to realize spatial diversity and frequency diversity gain without the need for channel state information (CSI) at the transmitter. In the first category, the results in [13] proposed a transmit diversity scheme in a frequency selective fading channel. A space-time code across space and frequency (rather than time) was shown in [14] to yield spatial diversity. In [15, 16], a low-density parity-check (LDPC)-based space time code was proposed to exploit both spatial diversity and selective fading diversity for MIMO-OFDM system in correlated. [17] presented the space-frequency code that can achieve full diversity in space and frequency for MIMO-OFDM systems, where neither transmitter and receiver has perfect CSI. [18] investigated the performance of space-frequency coded MIMO OFDM as a function of Riciean K-factor, angle spread, antenna spacing and power delay profile. In [19], a code design framework for achieving full rate and full diversity in MIMO frequency-selected fading channels was proposed.
• The main goal of the second class of MIMO-OFDM techniques is to increase ca-pacity by exploiting multiplexing gain in the spatial domain, i.e., transmitting independent data streams across antennas and tones. The V-BLAST system sug-gested in [11] is a well-known layered approach to achieve spatial multiplexing gain in multi-antenna systems. [20] showed that a MIMO delay spread channel can provide both higher diversity gain and multiplexing gain than MIMO
flat-fading channels. However, increasing the number of transmit antennas results in a smaller signal-to-noise ratio (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 ad-dress this issue in MIMO flat-fading channels was proposed in [3]. Nevertheless, the coverage performance for spatial-multiplexing-based MIMO-OFDM systems in frequency-selective fading channels has not been widely discussed so far.
• The third type of MIMO-OFDM technique is to decompose the channel coeffi-cient matrix by singular value decomposition (SVD) and construct pre-filter and post-filters at the transmitter and the receiver to achieve the capacity [21]. This technique requires perfect CSI available at both the transmitter and receiver.
In this chapter, we focus on the second type of MIMO-OFDM systems and aim to derive the closed form expressions for link outage and cell coverage cell cover-age of the OFDM-based spatial multiplexing systems over frequency-selective fading channels. The rest of this chapter is organized as follows. In Section II, we describe the system model. In Section III, we define the link outage for MIMO-OFDM sys-tems. In Section IV, we derive the exact analytical expression form of link outage of MIMO-OFDM systems, and provide an approximation analytical form of link out-age of MIMO-OFDM systems in Section V. In Section VI, we discuss the coverout-age performance of MIMO system. In Section VII, we show numerical results and give concluding remarks in Section VIII.
17
3.2 System model
We consider a point-to-point MIMO system with M transmit antennas and M receive antennas. In the meanwhile, we adopt OFDM modulation with total NT sub-carriers 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 channel coherent bandwidth.
Figure 3.1 shows the considered structure of the OFDM-based spatial multiplexing systems, where MN independent data streams are multiplexed in M transmit an-tennas and N subchannels. The transmit power is uniformly split to M transmit antennas. It is assumed that the length of the cyclic prefix (CP) in the OFDM sys-tem 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 xnand yn denoting the M × 1 transmit and receive signal vectors, respectively, we can write
yn =√
gnHnxn+ nn , (3.1)
where n is the sub-channel index and Hnrepresents the M ×M MIMO channel matrix of the nth subchannel and each entry of Hn is an i.i.d. circular-symmetric complex Gaussian variable [18]. Represent nn the M × 1 spatially white noise vector with E[nnn∗n] = σ2nI where (·)∗ is the transpose conjugate operation. At last gn depicts the large-scale behavior of the channel gain. For a user at a distance of r from the base station, gn can be written as [23]
10 log10(gn) = −10µ log10(r) + g0 [dB] , (3.2) where µ is the path loss exponent and g0 is a constant subject to certain path loss models.
Fig. 3.1: OFDM-based Spatial Multiplexing Systems.
3.3 Definitions
3.3.1 Link Outage Probability
To begin with, we first define the link outage probability which reflects how reliable a system can 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.
19
Fig. 3.2: The Eigenvalues in Each Subchannel.
The OFDM-based spatial multiplexing system in a frequency selective fading channel can be viewed as the sum of flat fading MIMO channels as shown in Fig. 3.2.
As discussed before, the high-percentile link reliability performance of each MIMO flat-fading channel is dominated by the weakest substream. Referring to Fig. 3.3 and 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 as follows:
Pout = Pr Ã
1 N
XN n=1
γn,M ≤ γth
!
, (3.3)
where γn,M represents the receive SNR of the weakest substream in subchannel n for n = 1, . . . , N.
Fig. 3.3: The Eigenvalues in Selective Fading Channel.
3.3.2 Cell Coverage Reliability
With Pout being the link outage probability, we define (1−Pout) to be the cell coverage reliability for its corresponding cell radius associated with the required SNR. That is, for a user at the cell radius with cell coverage reliability (1 − Pout), the probability of the received SNR being higher than the required threshold γth is no less than (1 − Pout).
3.4 Link Outage Analysis
To begin with, we first analyze the received SNR of the weakest substream (denoted by γn,M) at the n-th MIMO flat-fading subchannel. With {λn,i}Mi=1 representing the eigenvalues of the Wishart matrix HnH∗n, we can express γn,M as
γn,i = ρnλn,i/M , (3.4)
21 where ρn is the average receive SNR at the n-th subchannel and is equal to
ρn = Ptgn
Nσn2 = Pt 10(g0/10)
N2 σn2 rµ . (3.5)
Arrange {λn,i}Mi=1 in the decreasing order so that λn,1 ≥ λn,2 ≥ ... ≥ λn,M ≥ 0. Ac-cording to [3] [24], the probability density function (PDF) of the minimum eigenvalue λn,M is exponentially distributed with parameter M as follows
fλn,M(λ) = Me−M λ , λ ≥ 0; (3.6) and its cumulative distribution function (CDF) can be written as
Fλn,M(λ) = Z λ
0
fλn,M(x)dx
= 1 − e−M λ, λ ≥ 0. (3.7)
By applying the singular value decomposition (SVD) method, it can be shown that the MIMO-OFDM channel (Hn) is equivalent to MN parallel substreams, each of which has effective output SNR γn,i= ρnλn,i/M at the receive antenna. Notice that γn,M is also an exponentially distributed random variable of which CDF is written
Fγn,M(γ) = 1 − e−M 2γρn , γ ≥ 0. (3.8) For the i.i.d. exponentially distributed random variables {γn,M}Nn=1, the sum of exponentially distributed random variable Ω = N1 PN
n=1γn,M becomes the Erlang distributed random variable. Thus, the PDF of Ω is
fΩ(x) = N(Mρ2
Thus, for a given threshold γth > 0, the link outage probability of the OFDM-based spatial multiplexing systems can be expressed as
Pout = Pr
By substituting (4.5) into (4.11), the link outage can be represented as Pout = 1 − e−X radius r is defined as the farthest distance at which the link quality suffices for main-taining a required receive SNR γth with the probability no less than (1 − Pout). The objective is to derive an analytical closed-form expression for the cell radius r to be a function composed of given parameters {M, N, Pt, σn2, µ, γth} and the required Pout (usually 0.1). Because of complexity, it is not easy to derive an analytical closed-form expression for the cell radius r directly from (4.12).
3.5 Cell Coverage Performance
In this section we first provide another simple approximation to closed-form expression of the link outage (4.12) to facilitate the derivation of the closed-form expression of the
23 cell coverage r associated with link outage probability of spatial-multiplexing-based MIMO-OFDM systems. Then we present an method to calculate the cell coverage reliability of MIMO-OFDM systems.
3.5.1 Approximation of link outage probability
Considering the order statistics of a N random variables {γn}Nn=1, we reorder them and obtain {γ(1) < γ(2) < ... < γ(N )}. Then γ(i) is called the ith order statistic. It is integer value of ω. By doing so, link outage probability can be transformed to another form – the probability that at least Nω of the γn are less than or equal to γth. By applying the theories of order statistics, we obtain
Pout ' Pr(γ(Nω) ≤ γth)
where X is a function of (M, N, µ, r) defined in (4.13) and is the incomplete beta function. Now we have another closed-form approximation for the approximate link outage probability. From (4.15), we will derive the closed-form expression for the cell radius associated with link outage probability of MIMO-OFDM system, which will be discussed in the next section.
3.5.2 Cell Coverage Reliability
To derive cell coverage r from (4.15), we first introduce the inverse incomplete Beta function which is shown as follows
z = Iw(a, b) ⇒ w = Iw−1(a, b) . (3.17) By substituting (4.13) and (7.1) into (4.15), the cell coverage is given by
r '
Note that (7.2 ) can be viewed as an analytical closed form approximation for the cell radius for OFDM-based spatial multiplexing systems over frequency selective fading channels. It is a function composed of given parameters {M, N, Pt, σn2, µ, γth} and the required Pout.
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Fig. 3.4: Link outage probability v.s. transmit power Pt for different values of N when M = 3, noise power= −103dBm, µ = 3, r = 1km and γth= 2dB.
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
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