. Next, we introduce two lemmas regarding the properties of gamma random variables, which will be used in Section 3.3.
Lemma 3.1 Let X1, X2,· · · , XK be independent gamma random variables with parameters pk and q, respectively. Let Y be the random variable given by Y = X1+· · · + XK. Then we have
Y ∼ G
K
k=1
pk, q
. (3.6)
Proof: Please refer to [80].
Lemma 3.2 Let X be a gamma random variable with parameters p and q. Let Y be the random variable given by Y = cX, c > 0. Then we have
Y ∼ G p, q
c
. (3.7)
Proof: The proof is completed by using a simple variable transformation fY(y) = fX(y/c)/c ∼ G
p, qc
.
3.2 System Capacity with Multiuser Scheduling
In this section, we derive the capacity expression for the multiuser scheduling system with only single antenna employed at the base station and all users. The basic operation of the multiuser scheduling system was described in Section 2.2. The base station is assumed to have the knowledge of {γk}Kk=1 by means of the correct and instantaneous feedback from all users. We also assume that the channel variation remains constant over one time slot, but independently varies between different time slots. Accordingly, the base station shall determine to service one target user in every time slot according to certain scheduling policies.
3.2.1 Scheduling Policy and Conditional Link Capacity
The scheduling policy considered in this chapter is to select a target user k∗ according to the following rule1
k∗ = arg max
k
γk
ρk , (3.8)
where ρk is introduced in the denominator to resolve the fairness issue due to the near-far effect. In this chapter, we assume ρk = ρ for all k to focus on the impact of the small-scale channel fading. The case of different ρk with large-scale path loss effect will be studied in the next chapter. Thus, the scheduling rule in (3.8) is reduced to
k∗ = arg max
k γk . (3.9)
Let fγmax(γ; p, q, K) be the PDF of max(γ1,· · · , γK), where {γk}Kk=1 denotes a set of K independent and identically distributed (i.i.d.) gamma variates with parameters p and q.
From [87], we can write
fγmax(γ; p, q, K) = Kfγ(γ) [Fγ(γ)]K−1 , (3.10) where fγ(γ) and Fγ(γ) are defined in (3.3) and (3.4). Let flinkk∗ (γ) be the conditional PDF of the received SNR given that the connection between the base station and user k∗ is established. Clearly, according to (3.9), we have
flinkk∗ (γ) = fγmax(γ; p, q, K) . (3.11) Once the target user is determined, adaptive modulation is applied to transmit as many information bits as possible. From [81], the link capacity between the base station and the selected target user k∗ then can be written as
Clinkk∗ =
∞
0
log(1 + γ)flinkk∗ (γ)dγ . (3.12) Note that the capacity expression in (3.12) is normalized to the bandwidth, thereby having the unit of nats/sec/Hz.
1The rule is slightly different from the proportional fair scheduling introduced in [9, 11]. If the data rate supported by the channel is proportional to SNR, both criteria are the same.
3.2.2 System Capacity Analysis
With link capacity, now we proceed to derive the system capacity. We define the system capacity as the sum of the link capacity delivered to each user on average. Let pk∗ denote the average probability of user k∗ receiving services from the base station. Then the system capacity can be expressed by
C = K k∗=1
Clinkk∗ pk∗ . (3.13)
We assume that the channel variations among multiple users are mutually independent so that {γk}Kk=1 are i.i.d. across k. Thus, each user has the same probability to receive services from the base station or pk∗ = 1/K. Furthermore, the system capacity in (3.13) can be written as Equation (3.14) implies that system capacity is equal to the conditional link capacity when a specific user is chosen from K users. In other words, the capacity gain is achieved by providing the system with multiuser diversity, i.e., more selections of independent channel variations experienced by multiple users. Finally, substituting (3.10) and (3.12) to (3.14) yields When the parameter p is an integer value, (3.15) can be derived as (see Appendix A)
C = K
where aki for 0≤ i ≤ k(p − 1) can be recursively calculated by and E1(·) is the exponential integral function of the first kind defined by [82]
E1(x) =
∞
x
e−t
t dt . (3.18)
As a result, for integer values of Nakagami fading parameter m, we have
C = Λ1
Some special cases of (3.19) can be further derived as follows:
• For the Rayleigh fading case (m = 1) with K-fold multiuser diversity, (3.19) is simpli-fied to
• For the single user case K = 1, multiuser diversity gain vanishes and (3.19) reduces to
C = which is identical to the link capacity of the Nakagami fading channel with optimal rate control [83, 84].
• For m = 1 and K = 1, (3.19) is simply reduced to the link capacity of the Rayleigh fading channel as shown in [85, 86]
C = e1/ρE1
1 ρ
. (3.22)
When m is not restricted to an integer, (3.15) can be also efficiently computed with the help of the orthogonal Laguerre polynomial as follows:
C K
where wi and zi are the weights and zeros of the Laguerre polynomial [82], and NL is the order of polynomial chosen to make the approximation error negligibly small. Consequently, for the general Nakagami-m fading environments, the capacity of the multiuser scheduling system with K-fold multiuser diversity can be expressed by
C = Λ2
m,m
ρ, K
. (3.24)
With the expression (3.19) or (3.24), we have established a closed-form capacity formula that includes the contribution of K-fold multiuser diversity and the impact of Nakagami-m channel fading. Next, we give a numerical example using (3.19).
3.2.3 Impact of Channel Fading
Figure 3.1 shows the impact of Nakagami fading on the capacity of the multiuser scheduling system. The mean SNR is set to one, i.e. ρ = 0 dB, in this example. The information capacity of the AWGN channel with the same mean SNR is also plotted for comparison.
From this figure, one can see that the capacity for K = 1 is always lower than that of the AWGN channel. Moreover, a more scattering environment with a smaller value of m yields lower capacity when K = 1. However, the story becomes totally different when more than one user reside in the system. One can find that, in the presence of multiuser diversity (K ≥ 2), the capacity of the multiuser scheduling system is the highest for the Rayleigh fading channel (m = 1) and decreases as m increases. When m→ ∞, the channel becomes the AWGN channel and no scheduling gain can be exploited any more.
As shown in Fig. 3.1, a more scattering fading environment is beneficial for the con-sidered multiuser scheduling system since the larger channel variations enable the scheduler to arrange transmissions at higher peaks of channel fading more likely. On the other hand, considering the possibly contradictory goal of stabilizing the fading link for most antenna diversity techniques, one should be careful in employing the fading mitigation based antenna schemes on top of the multiuser scheduling system. In the next section, we will examine the capacity achieved in the multiuser scheduling system with some typical fading mitigation
0 5 10 15 20 25 30 35 0.6
0.8 1 1.2 1.4 1.6 1.8
Number of users K
Nats/sec/Hz
m=1 (Rayleigh) m=2
m=4 m=8 AWGN
0.5 1 1.5 2
0.58 0.64 0.7
Figure 3.1: Impact of Nakagami-m channel fading on the capacity of the multiuser scheduling system.
based antenna schemes.