2017 IEEE International Symposium on Information Theory (ISIT)

Multi-Cell Aware Opportunistic Random Access Huifa Lin† and Won-Yong Shin‡ † Communications ‡ Computer

& Networking Laboratory, Dankook University, Yongin 16890, Republic of Korea Science and Engineering, Dankook University, Yongin 16890, Republic of Korea Email: [email protected]; [email protected]

Abstract—We propose a decentralized multi-cell aware opportunistic random access (MA-ORA) protocol that almost achieves the optimal throughput scaling in a K-cell random access network with one access point (AP) and N users in each cell. Under our MA-ORA protocol, users opportunistically transmit with a predefined physical layer data rate in a decentralized manner if the desired signal power to the serving AP is sufficiently large and the generating interference power to other APs is sufficiently small. It is proved that the aggregate throughput scales as K (1 − ) log(SNR log N ) in a high signal-to-noise e K−1

ratio (SNR) regime if N scales faster than SNR 1−δ for small constants , δ > 0. Our analytical result is validated by computer simulations.

I. I NTRODUCTION Random access in wireless communications has received growing attention to support massive users owing to a relatively low protocol overhead with high spectral efficiency. In random access networks, due to the explosive growth of users and their generated data packets, there is an intention to densely deploy access points (APs) to achieve higher system throughput along with unchanged frequency bandwidth. Thus, it is crucial to fully understand the nature of random access networks that consist of multiple cells sharing the same frequency band, so called multi-cell random access networks.1 On the other hand, there have been extensive studies on the usefulness of fading in single-cell broadcast channels by exploiting the multiuser diversity gain as the number of users is sufficiently large (e.g., opportunistic beamforming [1] and random beamforming [2]). Moreover, opportunism can be utilized in multi-cell downlink networks through a simple extension of [2]. More recently, for the interfering multiple access channel (IMAC), which is one type of multi-cell uplink networks, the optimal throughput scaling was analyzed by showing that the full multiuser diversity gain can be achieved by a distributed user scheduling strategy in each cell, provided that scheduling criteria are properly determined and the number of users in each cell is larger than a certain level [3]. The benefits of opportunistic transmission can also be exploited in random access networks. The idea of single-cell aware opportunistic random access (SA-ORA) (also termed channel-aware slotted This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2017R1A1A2A10000835) and by the Ministry of Science, ICT & Future Planning (MSIP) (2015R1A2A1A15054248). 1 Here, we use the term “cell” to denote the domain of an AP and the associated users.

978-1-5090-4096-4/17/$31.00 ©2017 IEEE

ALOHA in the literature) was proposed for slotted ALOHA networks with a single AP [4]. By assuming that channel state information (CSI) can be acquired at the transmitters, the SA-ORA protocol in [4] was shown to achieve the multiuser diversity gain without any centralized scheduling. This idea was further extended to carrier sense multiple access networks [5] and multichannel wireless networks [6]. Nevertheless, all the protocols in [4]–[6] deal only with the single-AP problem, and thus cannot be straightforwardly applied to multi-cell random access networks where inter-cell interference exists. In this paper, we consider a K-cell slotted ALOHA random access network, consisting of one AP and N users in each cell. We then introduce a decentralized multi-cell aware opportunistic random access (MA-ORA) protocol that almost achieves the optimal throughput scaling in the network model. Unlike opportunistic scheduling in [3] for cellular multiple access environments where base stations select users via feedback information, in our MA-ORA protocol designed for random access, both the intra-cell collision and inter-cell interference need to be mitigated solely by users’ opportunistic transmission in a decentralized manner. To do so, we assume that uplink channel gains to multiple APs are available at the transmitters by exploiting the uplink/downlink reciprocity in time-division duplex (TDD) mode. We utilize this local CSI at the transmitter (CSIT) to design our protocol. In the initialization phase, two thresholds and a physical layer (PHY) data rate are computed offline as system parameters. Thereafter, each user first estimates its uplink channel gains via the downlink channel in each time slot. Then, each user determines whether both 1) the channel gain to the serving AP is higher than one threshold and 2) the total inter-cell interference generated by this user to the other APs is lower than another threshold. Users opportunistically transmit with the PHY data rate if the above two conditions are fulfilled. By virtue of such opportunistic transmission, when N is large, it is possible to successfully decode the desired packets sent from multiple users of different cells with high probability, yielding the optimal throughput scaling in the K-cell slotted ALOHA random access network. In our network model, it is shown that the aggregate throughput achieved by the proposed MA-ORA protocol scales as K e (1 − ) log(SNR log N ) in a high signalto-noise ratio (SNR) regime, provided that N scales faster than K−1 SNR 1−δ for an arbitrarily small constant  > 0 and a constant 0 < δ < 1. This reveals that even without any centralized

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2017 IEEE International Symposium on Information Theory (ISIT)

scheduling from the APs, the proposed protocol achieves the multiuser diversity gain, while obtaining the multiplexing gain of 1e in a cell, which is the best we can hope for under slotted ALOHA-type protocols [7]. Our analytical result is validated by computer simulations. II. S YSTEM AND C HANNEL M ODELS We consider a multi-cell random access network consisting of K ≥ 1 APs that share the same frequency band, where N users are served in each cell. Each AP attempts to decode received packets from its belonging users independently. All the users and APs are equipped with a single antenna. A slotted ALOHA-type protocol is adopted and we assume the perfect slot-level synchronization not only between the users and their serving AP but also among the APs. We assume fully-loaded traffic such that each user has a non-empty queue of packets to transmit. For each user, a head-of-line packet in the queue is transmitted with a probability of p at random, regardless of the number of retransmissions, i.e., each packet is assumed to be the same for all retransmission states. We adopt a modified signal-to-interference-plus-noise ratio (SINR) capture model, where an AP is able to decode the received packet if its SINR exceeds a given decoding threshold, while treating the intercell interference (the interfering signals from the other-cell users) as noise. Concurrent intra-cell transmission such that two or more users in the same cell simultaneously transmit causes collision, and thus the receiver (AP) fails to decode any packet. In this paper, we adopt single-user detection at each AP rather than any sophisticated multiuser detection, since using multiuser detection schemes cannot fundamentally increase degrees of freedom under our random access system model. Let hij→k denote the channel coefficient from the i-th user in the j-th cell to the k-th AP for i ∈ {1, · · · , N } and j, k ∈ {1, · · · , K}, which is modeled by an independent and identically distributed (i.i.d.) complex Gaussian random variable. We consider a quasi-static fading model, i.e., the channel coefficients are constant during one time slot and vary independently in the next time slot. We assume the local CSIT such that the users can acquire the uplink channel gains to multiple APs. For instance, the channel gain from the i-th user i in the j-th cell to the k-th AP, denoted by gj→k = |hij→k |2 , is available at the i-th user. The CSIT can be acquired via the downlink channel by exploiting the uplink/downlink reciprocity in TDD mode. Practical CSIT acquisition methods have been introduced for various multi-cell multi-antenna systems [8] and wireless local area networks (LANs) [9]. In the previous seminal literature on SA-ORA with one AP deployment [4], the local CSIT was simply assumed to be available. Recently, the CSIT acquisition process was described in more detail for the multichannel SA-ORA [6]. Inspired by [4], [6], a channel gain acquisition process for our model deploying multiple APs is described as follows. Under the slotted ALOHA protocol, each AP broadcasts (0, 1, e) feedback to inform the belonging users of the reception status after each time slot via the downlink channel, where 0 means

that no packet is received (idle); 1 means that only one packet is received (successful transmission); and e indicates that two or more packets are transmitted simultaneously (collision). In our multi-cell random access scenario, each AP broadcasts the feedback message in an orthogonal mini-time slot, which requires a small amount of coordination among the APs. By exploiting the uplink/downlink channel reciprocity, each user is capable of estimating the channel gains to multiple APs through the received feedback messages. That is, it is possible for each user to perform multi-cell aware channel gain estimation. It is worthwhile noting that since only the amplitude information of the CSIT (but not the phase information) is required in our protocol, the length of feedback messages can be greatly shortened by using quantization. The received signal yk ∈ C at the k-th AP is given by yk =

nk X

π(u ) π(uk )

hj→kk xk

nj XX

π(u ) π(uj )

hj→kj xj

+zk ,

j6=k uj =1

uk =1

|

+

{z

}

desired signal

|

{z

inter-cell interference

}

π(u )

where xk k is the transmitted signal from the π(uk )-th user in the k-th cell and the binomial random integer nk ∼ B(N, p) is the number of transmitting users in the k-th cell. The received signal is corrupted by the i.i.d. complex additive white Gaussian noise (AWGN) zk ∈ C with zero-mean and the variance N0 . For each transmission, there is an average   π(uk ) 2 transmit power constraint E xk ≤ PTX . III. M ULTI -C ELL AWARE O PPORTUNISTIC R ANDOM ACCESS (MA-ORA) We first describe the overall procedure of the proposed MAORA protocol. In the initialization phase, the APs broadcast two thresholds ΦG and ΦI as well as the PHY data rate R for opportunistic transmission in our multi-cell random access network. During the data communication phase, it is examined that the maximum medium access control layer (MAC) throughput of the conventional slotted ALOHA protocol deploying one AP is achieved at the transmission probability p = N1 for large N .2 Similarly, in our MA-ORA protocol, the transmission probability p is also set to N1 to avoid excessive intra-cell collisions or idle time slots, enabling us to find a relationship between ΦG and ΦI (to be discussed in Section III-A). In each time slot, each user first estimates the uplink channel gains by using the feedback messages sent from the APs. Then, for all the users, the i-th user in the j-th cell compares the channel gains with the given two thresholds to examine whether the following two inequalities are fulfilled: i gj→j ≥ ΦG

and

K X

i gj→k ≤ ΦI ,

(1)

(2)

k=1 k6=j

2 The MAC throughput here is defined as the average number of successful transmissions per time slot, where only one user transmits in a cell, and thus is given by N p(1 − p)N −1 .

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2017 IEEE International Symposium on Information Theory (ISIT)

where (1) indicates a “good” channel condition to the serving AP that leads to a large desired signal power and (2) means that the inter-cell interference generated by this user is well confined due to a “weak” channel condition to the other APs. In each cell, users satisfying both (1) and (2) transmit with the PHY data rate R (to be selected in Section III-B), while the other users keep idle. Each AP receives and decodes the desired packet by treating all the interference as noise.

where the binomial random variable nk ∼ B(N, p) is the number of simultaneously transmitting users in the k-th cell and π(u) denotes the index of transmitting users in each cell. (j) The throughput at the j-th AP for j ∈ {1, · · · , K}, Rsum , is then given by

A. The Selection of the Two Thresholds

where N p(1 − p)N −1 is the MAC throughput and R is the target PHY data rate. From the fact that p = N1 , the aggregate throughput of the K-cell random access network is given by

In this subsection, we show how to select the thresholds ΦG and ΦI . First, according to the opportunistic transmission conditions (1) and (2), the probability that each user accesses the channel is expressed as   K X  i  i p = Pr gj→j ≥ ΦG , gj→k ≤ ΦI 

(j) Rsum = N p(1 − p)N −1 ·R · ps , | {z } MAC throughput

Rsum =


 i = Pr gj→j ≥ ΦG Pr 

K X

N −1 1 =K 1− · R · ps N  N −1   1 ΦG R ≥K 1− · R · Pr >2 −1 , N ρ−1 + n ˜ ΦI (4) 

  i gj→k ≤ ΦI  ,

k=1 k6=j

where the second equality holds since the channel gains to different APs are independent of each other. From the fact that p is set to N1 , we have   K X
1   i i gj→k ≤ ΦI  = . Pr gj→j ≥ ΦG Pr  N k=1 k6=j

Then, the relationship between ΦG and ΦI is given by
ΦG = FG−1 1 − (FI (ΦI )p−1 )−1
= FG−1 1 − (FI (ΦI )N )−1 , where FG and FI denote distribution functions PKthe cumulative i i (CDFs) of gj→j and , respectively. k=1 g j→k k6=j In our MA-ORA protocol, the threshold ΦI is set to ρ−1 to achieve the optimal throughput scaling as ρ increases (to be proven in Section IV), where ρ , PNTX0 denotes the average SNR. In consequence, the two thresholds are given by ( ΦI = ρ−1
ΦG = FG−1 1 − (FI (ρ−1 )N )−1 . B. The Selection of the PHY Data Rate In this subsection, we show how to select the PHY data rate R. We start with computing the resulting successful decoding probability. Even if an AP receives only one packet from one of the belonging users, this received packet is still corrupted by the noise and the inter-cell interference. Thus, it is required that the received SINR of the desired packet exceeds a certain decoding threshold given by 2R − 1. The successful decoding probability, ps , is then expressed as   i PTX gj→j ps = Pr  > 2R − 1 , (3) PK Pnk π(u) N0 + k=1 u=0 PTX gk→j k6=j

(j) Rsum

j=1

k=1 k6=j



K X

where n ˜ ∼ B((K − 1)N, p) is a binomial random variable, representing the total number of signals from the Pinterfering K other cells, and is given by n ˜ , k=1 nk . Here, the inequality k6=j comes from (1) and (2). Now, we focus on computing a lower bound on the successful decoding probability, p˜s , shown below:   ΦG R p˜s = Pr ≥2 −1 ρ−1 + n ˜ ΦI (5)   (K−1)N X ΦG R Pr = ≥ 2 − 1 Pr(˜ n = i). ρ−1 + iΦI i=0 Let us consider an integer ν ∈ {0, 1, · · · , (K − 1)N }. If R is ΦG G set to a value such that ρ−1 +(ν+1)Φ < 2R − 1 ≤ ρ−1Φ+νΦ , I I   ΦG R then the probability Pr ρ−1 +iΦI ≥ 2 − 1 is given by 1 and 0 for i ∈ {0, · · · , ν} and i ∈ {ν + 1, · · · , (K − 1)N }, respectively. Based on this observation, the entire feasible range of 2R − 1 (i.e., the decoding threshold) can be divided into the following (K − 1)N + 1 sub-ranges:   ΦG 0, −1 ,··· , ρ + (K − 1)N ΦI     ΦG ΦG ΦG , ,··· , ,∞ . ρ−1 + (ν + 1)ΦI ρ−1 + νΦI ρ−1   G , ∞ , we have p˜s = 0, which In particular, for R ∈ ρΦ−1 is thus  neglected in our  work. Using the fact that the term R G Pr ρ−1Φ+iΦ ≥ 2 − 1 in (5) is an indicator function of R, I we set R to the maximum value under the condition i that 2R − ΦG ΦG 1 lies in each sub-range ρ−1 +(ν+1)ΦI , ρ−1 +νΦI , which is given by   ΦG R = log2 1 + −1 (6) ρ + νΦI for ν ∈ {0, 1, · · · , (K − 1)N }.

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2017 IEEE International Symposium on Information Theory (ISIT)

IV. AGGREGATE THROUGHPUT S CALING L AW We first introduce two important lemmas to present our main result. To obtain the first lemma, we start from revisiting the lower bound on the aggregate throughput in (4). To simplify this lower bound, we derive an explicit expression of ΦG as i a function of ΦI . Since the channel gain gj→j follows the exponential distribution whose CDF is given by FG (x) = K P i 1 − e−x , the sum of channel gains gj→k follows the chik=1 k6=j

square distribution and the corresponding CDF is given by with 2(K −1) degrees of freedom, where FI (x) = γ(K−1,x/2) Γ(K−1) Rx γ(a, x) = 0 ta−1 e−t dt is the lower incomplete Gamma function and Γ(x) = (x − 1)! is the Gamma function. Thus, we can establish the following relationship between ΦG and ΦI :
ΦG = FG−1 1 − (FI (ΦI )p−1 )−1 = ln (FI (ΦI )N ) . (7) By using (7) in (4), we have  N −1   ln (FI (ΦI )N ) 1 R >2 −1 . ·R·Pr Rsum ≥ K 1 − N ρ−1 + n ˜ ΦI {z } | {z } | p˜s

MAC throughput

(8) The following lemma is presented since the resulting form of p˜s is still not analytically tractable. Lemma 1. When the PHY data rate R is selected from the discrete set in (6), we have p˜s = I1− N1 ((K − 1)N − ν, ν + 1) ,

(9)

where Ip (·, ·) is the regularized incomplete beta function and p = 1 − N1 . Proof: See Appendix A. From this lemma, we are able to equivalently transform the expression of p˜s in (8) to the well-known regularized incomplete beta function. In Fig. 1, the function I1− N1 ((K − 1)N − ν, ν + 1) versus ν is plotted for various K = {2, 3, 4}. This function tends to get steeply increased for small ν and then to gradually approach one. That is, it is monotonically increasing with ν. For example, for K = 2, it is numerically found that the pairs of (˜ ps , ν) are given by (0.9, 3), (0.99, 5), and (0.999, 7). Based on this observation, it is possible to select a proper finite value of ν that makes p˜s approach almost one. Using (6), (8), and (9), the aggregate throughput is now lower-bounded by  N −1   1 ln (FI (ΦI )N ) Rsum ≥K 1 − · log2 1 + −1 N ρ + νΦI | {z } (10) PHY data rate (R)

· I1− N1 ((K − 1)N − ν, ν + 1) . However, it is still not easy to find the closed form expression of (10) due to the complicated form of FI (ΦI ). This motivates us to introduce the following lemma.

Fig. 1: The function I1− N1 ((K − 1)N − ν, ν + 1) versus ν for K = {2, 3, 4}. Lemma 2. For any 0 ≤ x < 2, FI (x) is lower-bounded by FI (x) ≥ c1 x(K−1) , where c1 = SNR.

e−1 2−(K−1) (K−1)Γ(K−1)

is a constant independent of N and

Proof: We refer to [3, Lemma 2] for the proof. Using this lemma leads to a more tractable lower bound on the aggregate throughput. We are now ready to establish the following theorem, which presents the aggregate throughput scaling law achieved by the proposed MA-ORA protocol. Theorem 1. Consider the MA-ORA protocol in the K-cell random access network. Suppose that ΦI = ρ−1 . Then, the MA-ORA protocol achieves an aggregate throughput scaling of   K Θ (1 − ) log ρ(log N ) (11) e with high probability in the high SNR regime if  K−1  N = Ω ρ 1−δ ,

(12)

where ρ is the average SNR,  > 0 is an arbitrarily small constant, and 0 < δ < 1 is a certain constant.3 Proof: A brief sketch of the proof is only provided here due to page limit. We select finite ν ∗ (K, ) that makes I1− N1 ((K − 1)N − ν, ν + 1) = 1 − , where  > 0 is an
N −1 arbitrarily small constant. Then, since 1 − N1 ≥ 1e for −1 N > 1 and ΦI = ρ , the aggregate throughput Rsum is lower-bounded by   ρ · ln(FI (ρ−1 )N ) K · log2 1 + · (1 − ) Rsum ≥ e ν ∗ (K, ) + 1 !
K−1 ρ · ln(c1 ρ−1 N) K ≥ · log2 1 + · (1 − ), e ν ∗ (K, ) + 1 3 We use the following notation: i) f (x) = O(g(x)) means that there exist constant C and c such that f (x) ≤ Cg(x) for all x > c, ii) f (x) = Ω(f (x)) if g(x) = O(f (x)), and iii) f (x) = Θ(g(x)) if f (x) = O(g(x)) and g(x) = O(f (x)).

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2017 IEEE International Symposium on Information Theory (ISIT)

where the second inequality holds owing to Lemma 2. In order to achieve the logarithmic gain (i.e., the power gain with
K−1 increasing N ), it is required that c1 ρ−1 N ≥ N δ for  K−1  0 < δ < 1, which finally yields N = Ω ρ 1−δ . Under this condition, Rsum can be lower-bounded by   K δ · ρ ln(N ) Rsum ≥ (1 − ) · log2 1 + ∗ , e ν (K, ) + 1 ∗ which scales as K e (1 − ) log ρ(log N ) since δ and ν (K, ) are some constants independent of N .

The following discussions are provided along with our analytical result in Theorem 1. Remark 1: An aggregate throughput scaling of K e log ρ(log N ) can be obtained by a genie-aided removal of all the inter-cell interference, which corresponds to a system of K interference-free parallel SA-ORA networks. This is an upper bound on the aggregate throughput for our K-cell slotted ALOHA random access, which matches the achievable throughput scaling in (11) to within a factor of  > 0. Remark 2: The aggregate throughput scaling of K log ρ(log N ) is achieved by the opportunistic scheduling protocol for the IMAC, where collisions can be avoided by adopting user selection at each base station [3]. In comparison with our MA-ORA protocol, we observe some interesting consistency as follows: 1) the user scaling conditions required by the two protocols are exactly the same; and 2) after removing the intra-cell contention loss factor (given by 1e ), these two protocols achieve the same throughput scaling to within an arbitrary small  > 0 gap. V. N UMERICAL E VALUATION We validate our analytical result by evaluating performance of the proposed MA-ORA protocol. The parameter N is set to K−1 a scalable value according to ρ, i.e., N = ρ 1−δ in (12). We assume that  = 0.01 and δ = 0.1, which can be set to different values according to other parameter settings. Then, the value of ν is calculated according to the relationship between  and ν. We set ΦI = ρ−1 and ΦG = ln(FI (ΦI )N ) (see (7)). Based on these parameters, the PHY data rate R can be computed from (6). In Fig. 2, we evaluate the aggregate throughput [bps/Hz] achieved by the MA-ORA protocol versus SNR in dB scale for K = 3. In the figure, the solid line is also plotted from the theoretical result in Theorem 1 with a proper bias to check the slope of K e (1 − ) log ρ(log N ). One can see that the slope of the simulated curve coincides with the theoretical one in the high SNR regime. A PPENDIX A P ROOF OF L EMMA 1 As shown in (6), the PHY data rate R is a function of the integer  ν ∈ {0, 1, · · · ,(K − 1)N }. We recall that the term G Pr ρ−1Φ+iΦ ≥ 2R − 1 in (5) is given by 1 and 0 for i ∈ I {0, · · · , ν} and i ∈ {ν + 1 · · · , (K − 1)N }, respectively, if R ΦG G is set to a value such that ρ−1 +(ν+1)Φ < 2R − 1 ≤ ρ−1Φ+νΦ . I I

Fig. 2: The aggregate throughput of the MA-ORA protocol versus SNR, where K = 3 and N scales according to the user scaling condition.

Thus, by dividing the whole range of i into two sub-ranges, (5) can be expressed as p˜s = =

ν X i=0 ν X

(K−1)N

1 · Pr(˜ n = i) +

X

0 · Pr(˜ n = i)

i=ν+1

Pr(˜ n = i) = Pr(˜ n ≤ ν),

i=0

which is the CDF of the binomial random variable n ˜ ∼ B((K − 1)N, N1 ). It is known that this CDF can be expressed as the regularized incomplete beta function [10], which is given by p˜s = I1− N1 ((K − 1)N − ν, ν + 1). R EFERENCES [1] P. Viswanath, D. N. C. Tse, and R. Laroia, “Opportunistic beamforming using dumb antennas,” IEEE Trans. Inf. Theory, vol. 48, no. 6, pp. 1277– 1294, Aug. 2002. [2] M. Sharif and B. Hassibi, “On the capacity of MIMO broadcast channels with partial side information,” IEEE Trans. Inf. Theory, vol. 51, no. 2, pp. 506–522, Jan. 2005. [3] W.-Y. Shin, D. Park, and B. C. Jung, “Can one achieve multiuser diversity in uplink multi-cell networks?” IEEE Trans. Commun., vol. 60, no. 12, pp. 3535–3540, Dec. 2012. [4] X. Qin and R. A. Berry, “Distributed approaches for exploiting multiuser diversity in wireless networks,” IEEE Trans. Inf. Theory, vol. 52, no. 2, pp. 392–413, Jan. 2006. [5] G. Miao, Y. Li, and A. Swami, “Channel-aware distributed medium access control,” IEEE/ACM Trans. Netw., vol. 20, no. 4, pp. 1290–1303, Dec. 2012. [6] Z. Khanian, M. Rasti, F. Salek, and E. Hossain, “A distributed opportunistic MAC protocol for multichannel wireless networks,” IEEE Trans. Wireless Commun., vol. 15, no. 6, pp. 1–1, Mar. 2016. [7] D. Bertsekas and R. Gallager, Data networks (2nd Ed.). New Jersey, USA: Prentice-Hall, Inc., 1992. [8] P. Komulainen, A. T¨olli, and M. Juntti, “Effective CSI signaling and decentralized beam coordination in TDD multi-cell MIMO systems,” IEEE Trans. Sig. Process., vol. 61, no. 9, pp. 2204–2218, May 2013. [9] R. Liao, B. Bellalta, M. Oliver, and Z. Niu, “MU-MIMO MAC protocols for wireless local area networks: A survey,” IEEE Commun. Surveys Tuts., vol. 18, no. 1, pp. 162–183, 2014. [10] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York, NY, USA: Dover, 1966.

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