7. APPENDIX B
4.5 Fairness performance comparison for K = 5 and M = 3
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.
47
Approximately alalytical result by COSA Simulation result by FOSA
Approximately anaytical result by RR
Fig. 4.8: Cell coverage v.s. users for different µ while Pt = 1W, M = 3, N = 32, noise power= −103 dBm, Pout = 0.1 and γth= 2dB.
Fig. 4.7 shows that the cell coverage increases as the number of users increases.
As shown in the figure, it will increase more quickly when the number of antennas decreases. It indicates that the coverage area is hard to hold the link quality when we want to maintain M times of capacity.
Fig. 4.8 shows how the pass loss exponent could affect the cell coverage for different numbers of users. 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.
Concluding Remarks
The objective of this thesis is to efficiently assign the subcarriers in order to enhance the small coverage reliability of OFDM-based spatial multiplexing systems. This thesis includes the following research topics:
1. Define the link outage probability and provide an analytical closed form expression of it.
2. Provide another simple analytical closed form approximation of the link outage probability in order to further develop the cell coverage for the non-coherent OFDM-based spatial multiplexing systems.
3. Provide a low-complexity coverage-oriented subcarriers assignment algorithm (COSA) which can achieve larger cell coverage then fairness-oriented subcarriers assignment (FOSA) for non-coherent OFDM-based spatial multiplexing systems.
4. Develop an approximately analytical closed form expression for link outage and cell coverage by using COSA.
5. Demonstrate how the total transmit power, the number of antennas, the number of users and pass loss exponents can affect the cell coverage reliability under COSA algorithm over frequency-selective fading channels.
49
5.1 Link Outage and Coverage Analysis for
OFDM-based Spatial Multiplexing Systems
In Chapter 3 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 ap-proximation 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 simu-lation. We also present some results to illustrate how and to what extend the number of antennas, frequency selectivity order, and the pass loss exponent affect the link outage and the cell coverage for the spatial multiplexing MIMO-OFDM system.
5.2 Coverage Enhancement for
Spatial-Multiplexing-Based MIMO OFDM Systems by Joint Multiuser Scheduling and Subcarriers Assignment
In Chapter 4 we first extend the fairness-oriented subcarriers assignment algorithm (FOSA) from SISO OFDM to MIMO OFDM. Next we propose another low-complexity coverage-oriented subcarriers assignment algorithm (COSA). We demonstrate that COSA can achieve larger cell coverage of spatial-multiplexing based MIMO-OFDM systems than FOSA, while achieving almost the same fairness performance as FOSA.
With respect to COSA, we further derive an analytical expression form for link
out-age probability and cell coverout-age reliability by means of order statistics and Glivenko-Cantelli Theorem. Simulation results validate the accuracy of the analytical model and approximation method. We also present some numerical results to illustrate how the total transmit power, the number of antennas, the number of users, pass loss exponents, and various subcarriers assignment algorithms affect the cell coverage for the spatial multiplexing MIMO-OFDM system.
5.3 Suggestion for Future Work
For the future research of the thesis, we provide the following suggestions to extend our work:
• Compare the link quality and cell coverage for single-user OFDM-based spatial multiplexing systems with Space-Time-Frequency coded MIMO-OFDM system.
• Consider the uplink case and see how large the coverage reliability is for OFDM-based spatial multiplexing systems.
• To design a criterion to describe the tradeoff between cell coverage and fairness.
• Consider the time correlation between each time slot for each user, and see how the doppler effect could effect the cell coverage.
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Chapter 6 APPENDIX A
In Appendix A we discuss how to obtain the approximate 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
Furthermore, based on Glivenko-Cantelli Theorem, we know that the random variable DN = sup
γ∈R
|FN(γ) − F (γ)| (6.5)
converges to 0 with probability 1 when the value of N is large. In other words, P r
is a much stronger statement, and sup means the supremum upper bound. Since IEEE 802.11a employs fast Fourier transform (FFT) with 64 carriers and IEEE 802.16 uses 256 carriers furthermore [26], we can assume that the value of N is very large. (7.6) can be written as
ω = N · FN(γ(ω)) ' N · F (γ(ω))
' N · F (²(γ)) (6.7)
where ²(γ) is the expectation value of a function f (γ) in a variable γ, and
²(γ) =
By substituting (7.10) into (7.9), we obtain ω ' N · F³ ρn
where Nω is an approximation integer value of ω.
57
Fig. 6.1: Illustration of the empirical distribution for an i.i.d. sequence.
APPENDIX B
In Appendix B we discuss how to obtain the approximate 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 which is defined as
Furthermore, Glivenko-Cantelli Theorem says that the random variable DN = sup
γ∈R
|FN(γ) − Fk∗,n(γ)| (7.5) converges to 0 with probability 1 when the value of N is large. In other words,
P r
59 is a much stronger statement, and sup means the supremum upper bound. We assume that the value of N is large enough. (7.4) can be written as
ω = N · FN(γk∗,(ω)) ' N · Fk∗,n(γk∗,(ω))
' N · Fk∗,n(²(γk∗,n)), (7.7) where ²(γ) is the expectation value of a function f (γ) in a variable γ, and
fk∗,n(γ) = 1
where B(a, b) represents a Beta function. By substituting (7.9) into (7.7) w ' Nω(N, K) =
N · Ã
1 − exp
à 1
B(K, 1)
K−1X
i=0
CiK−1(−1)K−i 1 (K − i)2
!!K ,
(7.10) where Nω(N, K) is an approximation integer value of ω.
61
Vita
Cheng-Wei Chiu was born in Taiwan in 1981. He received the B.S. degree in Electrical Engineering from National Central University in 2004. From July 2004 to June 2006, he works his Master degree in the Wireless Network Lab of the Department of Communication Engineering at National Chiao Tung University. His research interests are in the field of radio resource management.