Coverage Performance Analysis of OFDM-based
Spatial Multiplexing Systems
Li-Chun Wang, Cheng-Wei Chiu, Chu-Jung Yeh and Wern-Ho Sheen National Chiao Tung University, Taiwan
Email : lichun@cc.nctu.edu.tw, kimula.cm93g@nctu.edu.tw, teensky.cm93g@nctu.edu.tw, whsheen@mail.nctu.edu.tw
Abstract— 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 antennas 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 paper by means of order statistics andGlivenko-Cantelli theorem, we develop an analytical expressions for the link outage probability and cell coverage reliability of OFDM-based spatial multiplexing systems in a frequency selective fading channel, respectively.
Index Terms— MIMO, OFDM, cell coverage, link
outage, multiplexing
I. Introduction
Orthogonal Frequency Division Multiplexing (OFDM) has become a popular modulation technique for transmis-sion of broadband signals. OFDM can convert a frequency selective fading channel into a parallel collection of fre-quency flat fading sub-channels and thus can overcome inter-symbol interference (ISI) [1] [2]. In the meanwhile, multi-input multi-output (MIMO) antenna techniques can provide spatial multiplexing gain and diversity gain to in-crease spectrum efficiency and link reliability, respectively [3] [4] [5] [6]. 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 fre-quency diversity gain without the need for channel state information (CSI) at the transmitter. In the first category, the results in [7] proposed a transmit diver-sity scheme in a frequency selective fading channel. A space-time code across space and frequency (rather than time) was shown in [8] to yield spatial diversity. In [9], [10], a low-density parity-check (LDPC)-based The work was supported jointly by the National Science Council and the MOE program for promoting university excellence under the contracts NSC 94-2213-E-009 -030 -. and MOE ATU Program 95W803C
space time code was proposed to exploit both spatial diversity and selective fading diversity for MIMO-OFDM system in correlated channel. [11] presented the space-frequency code that can achieve full diver-sity in space and frequency for non-coherent MIMO-OFDM systems, where neither transmitter and re-ceiver has perfect CSI. [12] investigated the perfor-mance of space-frequency coded MIMO-OFDM as a function of Riciean K-factor, angle spread, antenna spacing and power delay profile. In [13], a code design framework for achieving full rate and full diversity in MIMO frequency-selected fading channels was pro-posed.
• The main goal of the second class of MIMO-OFDM techniques is to increase capacity by exploiting multi-plexing gain in the spatial domain, i.e., transmitting independent data streams across antennas and tones. The V-BLAST system suggested in [5] is a well-known layered approach to achieve spatial multiplex-ing gain in multi-antenna systems. [14] 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 [15] and a multiuser scheduling solution to address this issue in MIMO flat-fading channels was pro-posed in [16]. 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 coefficient matrix by singular value decomposition (SVD) and construct pre-filter and post-filters at the transmitter and the receiver to achieve the capacity [17]. This technique requires perfect CSI available at both the transmitter and receiver.
In this paper, we focus on the second type of MIMO-OFDM systems and aim to derive the closed form
ex-pressions for link outage and cell coverage cell coverage of the OFDM-based spatial multiplexing systems over frequency-selective fading channels. The rest of this paper is organized as follows. In Section II, we describe the system model. In Section III, we define the link outage for MIMO-OFDM systems. In Section IV, we derive the exact analytical expression form of link outage of MIMO-OFDM systems. In Section V, we provide an approximation an-alytical form of link outage of MIMO-OFDM systems and discuss the coverage performance. In Section VI, we show numerical results and give concluding remarks in Section VII.
II. System model
We consider a point-to-point MIMO system with M transmit antennas and M receive antennas. In the mean-while, 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 subchan-nel is assumed to be smaller than the chansubchan-nel coherent bandwidth. Figure 1 shows the considered structure of the OFDM-based spatial multiplexing systems, where M N independent data streams are multiplexed in M trans-mit antennas and N subchannels. The transtrans-mit 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 [19]. With xn and
yndenoting the M×1 transmit and receive signal vectors,
respectively, we can write
yn=√gnHnxn+nn , (1)
where n is the subchannel index and Hn represents the
M× M MIMO channel matrix of the nthsubchannel and
each entry of Hn is an i.i.d. circular-symmetric complex
Gaussian variable [12]. Represent nn the M× 1 spatially
white noise vector with E[nnn∗n] = σ2nI where (·)∗ is the
transpose conjugate operation. At last gndepicts 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 [18]
10 log10(gn) =−10µ log10(r) + g0 [dB] , (2) where µ is the path loss exponent and g0 is a constant subject to certain path loss models.
III. Definitions
A. 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}
[20]. The link outage for the spatial multiplexing MIMO system in a flat fading channel is defined as the event
Fig. 1. OFDM-based spatial multiplexing systems.
when the receive SNR of any substream is less than γth
[15] [16]. 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 OFDM-based spatial multiplexing system 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 channel is dominated by the weakest substream. Considering the average weakest eigen-mode over a series of N ’s MIMO flat-fading subchannels, we de-fine the link outage probability of the spatial-multiplexing-based MIMO OFDM system as follows:
Pout= Pr 1 N N n=1 γn,M ≤ γth , (3) where γn,M represents the receive SNR of the weakest substream in subchannel n for n = 1, . . . , N .
B. Cell Coverage Reliability
With Pout being the link outage probability, we define (1− Pout) to be the cell coverage reliability for its cor-responding 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 γthis no less than
(1− Pout).
IV. Link Outage Analysis
To begin with, we first analyze the received SNR of the weakest substream (denoted by γn,M) at the nth MIMO
flat-fading subchannel. With {λn,i}Mi=1 representing the
eigenvalues of the Wishart matrixHnH∗n, we can express
γn,M as
where ρnis the average receive SNR at the nthsubchannel and is equal to ρn= Ptgn N σn2 = Pt 10 (g0/10) N σn2 rµ . (5) Arrange{λn,i}Mi=1 in the decreasing order so that λn,1≥
λn,2 ≥ ... ≥ λn,M ≥ 0. According to [16] [21], the
proba-bility density function (PDF) of the minimum eigenvalue
λn,M is exponentially distributed with parameter M as
follows
fλn,M(λ) = M e−Mλ , λ≥ 0; (6)
and its cumulative distribution function (CDF) can be written as
Fλn,M(λ) =
λ
0 fλn,M(x)dx
= 1− e−Mλ, λ≥ 0. (7) By applying the singular value decomposition (SVD) method, it can be shown that the MIMO-OFDM channel (Hn) is equivalent to M N 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− M2γ
ρn , γ≥ 0. (8)
For the i.i.d. exponentially distributed random vari-ables {γn,M}Nn=1, the sum of exponentially distributed random variable Ω = N1 Nn=1γn,M becomes the Erlang distributed random variable. Thus, the PDF of Ω is
fΩ(x) = N ( M2 ρn) N(N x)N−1e−M2N ρn x Γ(N ) , x > 0 (9) and its CDF is FΩ(x) = x 0 fΩ(x)dx = 1− e−M2Nρn x N−1 j=0 (Mρ2nNx)j j! , x > 0 (10)
Thus, for a given threshold γth> 0, the link outage
prob-ability of the OFDM-based spatial multiplexing systems can be expressed as Pout = Pr 1 N N n=1 γn,M ≤ γth = Pr(Ω≤ γth) = FΩ(γth) = 1− e−M2Nρn γth N−1 j=0 M2N ρn γth j j! . (11)
By substituting (5) into (11), the link outage can be represented as Pout= 1− e−X N−1 j=0 Xj j! , (12) where X =M 2N2σ2 n γth rµ Pt 10(g0/10) . (13)
In (12), Pout is a function of given parameters
{M, N, Pt, σn2, µ, r, γth}. The cell radius r is defined as
the farthest distance at which the link quality suffices for maintaining 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 (12).
V. Cell Coverage Performance
In this section we first provide another simple approx-imation to closed-form expression of the link outage (12) to facilitate the derivation of the closed-form expression of the cell coverage r associated with link outage probabil-ity of spatial-multiplexing-based MIMO-OFDM systems. Then we present an method to calculate the cell coverage reliability of MIMO-OFDM systems.
A. Approximation of link outage probability
For brevity, we omit the index of M and use γn to replace γn,M. We also use the index F (·) instead of Fγn(·)
for the CDF of γn,M. 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 ithorder statistic. It is assumed that γ(ω) is the value most close to
1
N
N
n=1γn. Then the link outage can be rewritten as
Pout = Pr 1 N N n=1 γn≤ γth Pr(γ(ω)≤ γth) , (14) where w 0.63N Nω (see the details in Appendix),
and Nω is an approximation integer value of ω. By doing
so, link outage probability can be transformed to another form – the probability that at least Nω of the γnare less
than or equal to γth. By applying the theories of order
statistics, we obtain Pout Pr(γ(Nω) ≤ γth) = F(Nω)(γth) = N i=Nω N i Fi(γth) [1− F (γth)]N−i = IF (γth)(Nω , N− Nω+ 1) = I[1−e−X](Nω , N− Nω+ 1) (15)
where X is a function of (M, N, µ, r) defined in (13) and
Ip(a, b) = p 0ta−1(1− t)b−1dt 1 0ta−1(1− t)b−1dt For a > 0, b > 0 and 0≤ p ≤ 1 (16)
0 5 10 15 10−3 10−2 10−1 100 Transmit power P t (dBW) Link outage M=2 M=3 M=4 M=5
Approximately analytical result Analytical result
Simulation result
Fig. 2. Link outage probability v.s. transmit powerPtfor different values ofM when N = 128, noise power= −103dBm, µ = 3, r = 1km andγth= 2dB.
is the incomplete beta function. Now we have another closed-form approximation for the approximate link out-age probability. From (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.
B. Cell Coverage Reliability
To derive cell coverage r from (15), we first introduce the inverse incomplete Beta function which is shown as follows
z = Iw(a, b) ⇒ w = Iw−1(a, b) . (17)
By substituting (13) and (17) into (15), the cell coverage is given by r Pt N σ2n 10g010 M2γth · log 1 1− IP−1out(Nω , N− Nω+ 1) 1 µ . (18)
Note that (18 ) can be viewed as an analytical closed form approximation for the cell radius for OFDM-based spa-tial 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.
VI. Numerical Results
In this section, we present numerical examples to il-lustrate how the number of antennas 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 2 shows the simulative, analytical, and
the approximate link outage performances with various numbers of transmit and receive antennas for N = 128.
−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. Cell coverage radius v.s. transmit power Pt for different values ofM when N = 128, noise power= −103dBm, µ = 3, Pout= 0.1 and γth= 2dB.
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 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.
VII. Conclusions
In the paper, we have analyzed the link outage and cell coverage performance of the spatial multiplexing MIMO-OFDM systems over frequency-selective fading channels. We present an analytical formula that can evaluate the link outage probability for spatial multiplexing MIMO-OFDM system. We also provide another simplified approximation of the exact link outage probability that can applied to calculated the cell coverage associated a certain link outage probability. From our numerical results, we validate the accuracy of the analytical model and approximation method by simulation. We also present examples to illus-trate how and to what extend the number of antennas affect the link outage and the cell coverage for the spatial multiplexing MIMO-OFDM system. In the future, we will develop an optimal scheduling algorithm for multiuser OFDM-based spatial multiplexing systems, and see how to exploit the multiuser and frequency diversity to improve link quality of the diversity-deficient spatial multiplexing systems.
Appendix
In this appendix we discuss how to obtain the approxi-mate value ω. To this end, we introduce a function called
empirical distribution. The empirical distribution for an
i.i.d. sequence{γ1, ...γN} is a random variable defined as
FN(γ) = 1 N N n=1 I(−∞,γ](γn) , (19) where I(·] represents indicator function.
Fig. 4 shows the diagram of FN(γ). Because {γ(1) <
γ(2) < ... < γ(N)} and γ(ω) is the ωth order statistic for
the sequence{γ1, ...γN}, we can know that
FN(γ(ω)) = 1 N N n=1 I(−∞,γ(ω))(γn) = ω N . (20)
Furthermore, based on Glivenko-Cantelli Theorem, we know that the random variable
DN = sup
γ∈R|FN(γ)− F (γ)| (21)
converges to 0 with probability 1 when the value of N is large. Since IEEE 802.11a employs fast Fourier transform (FFT) with 64 carriers and IEEE 802.16 uses 256 carriers furthermore [22], we can assume that the value of N is very large. (20) can be written as
ω = N· FN(γ(ω)) N · F (γ(ω)) N · F ((γ)) (22) where (γ) is the expectation value of a function f (γ) in a variable γ, and (γ) = ∞ −∞γf (γ)dγ = ∞ −∞ γM 2 ρn e M2 ρnγdγ = ρn M2 . (23)
By substituting (23) into (22), we obtain
ω N · F ρn M2 = N· 1− e(−M2ρn M2ρn) = N· (1 − e−1) 0.63N Nω (24)
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