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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 4, APRIL 2011
Performance of Soft Decision Decoded Synchronous FHSS Multiple Access Networks Using MFSK Modulation under Rayleigh Fading Sungnam Hong, Changkyu Seol, and Kyungwhoon Cheun, Member, IEEE
Abstract—In this paper, we analyze the performance of soft decision decoded, synchronous, fast frequency-hopping spreadspectrum multiple-access networks using 𝑀 -ary frequency shift keying in Rayleigh fading channels under the assumption that the number of frequency hopping slots is sufficiently large. The joint probability density function (pdf) of the absolute values of the correlator outputs is derived to compute the channel capacity and evaluate the soft decision decoding performance of the 3GPP Turbo and convolutional codes. Results indicate that soft decision decoding based on the derived pdf drastically outperforms traditional decoders based on the Gaussian approximation of the effect of the multiple access interference (MAI). The sensitivity of the derived soft decision decoder to estimation errors in system parameters and the required computational complexities of the derived soft decision decoding metrics are also investigated. Index Terms—Frequency-hopping, frequency shift keying, multiple-access, soft decision decoding, Rayleigh fading.
I. I NTRODUCTION
C
ONSIDERABLE amount of research results are available in the literature dealing with the performance of fast frequency hopping1 spread spectrum multiple access (FHSSMA) networks using 𝑀 -ary frequency shift keying (MFSK) [1]–[17]. However, most of the theoretical results available in the literature focus on the uncoded error performance under hard decisions. In [1], accurate analytical expressions for the error probabilities of FHSS-MA networks using BFSK under additive white Gaussian noise (AWGN) channels were derived for both synchronous and asynchronous hopping networks. In [2], the results were extended to FHSS-MA networks using MFSK (𝑀 ≥ 2) under both AWGN and Rayleigh fading Paper approved by G. M. Vitetta, the Editor for Equalization and Fading Channels of the IEEE Communications Society. Manuscript received February 22, 2009; revised November 26, 2009 and August 12, 2010. This research was supported by the MKE (The Ministry of Knowledge Economy), Korea, under the ITRC (Information Technology Research Center) support program supervised by the NIPA (National IT Industry Promotion Agency) (NIPA-2010-(C1090-1011-0011)), and the WCU (World Class University) program through the Korea Science and Engineering Foundation funded by the Ministry of Education, Science, and Technology (Project No. R31-2008-000-10100-0). K. Cheun and S. Hong are with the Division of Electrical and Computer Engineering, Pohang University of Science and Technology (POSTECH), Pohang 790-784, Republic of Korea (e-mail:
[email protected];
[email protected]). C. Seol was with the Division of Electrical and Computer Engineering, POSTECH. He is now with the Semiconductor Business, Samsung Electronics Co., Ltd., Republic of Korea (e-mail:
[email protected]). Digital Object Identifier 10.1109/TCOMM.2011.012711.090109A 1 Here, we use the term fast frequency hopping to refer to the case when one coded symbol is transmitted per hop [5], [27]–[29].
channels via semi-analytic Monte Carlo methods. For the special case of synchronous hopping with MFSK, analytic expressions for the error probabilities under Rayleigh fading channels were derived in [3] assuming perfect power control. In [4], improved expressions for the error probabilities of synchronous FHSS-MA networks were derived without the restriction of perfect power control. While many studies have dealt with soft decision decoding in partial band Gaussian jamming channels and direct sequence code division multiple access (DS-CDMA) networks [20]–[23], only a few attempts have been made to apply soft decision decoding in FHSS-MA networks. In [5] and [6], a variety of ad-hoc, suboptimal soft metrics were proposed to be used with Turbo and convolutional coded FHSS-MA networks. Simulation results indicated the possibility that soft decision decoding with appropriately designed soft metrics can significantly outperform the hard decision decoding. These results motivate us to derive the optimum maximum likelihood (ML), or at least, near ML soft metrics to be used in fast FHSS-MA networks with MFSK. In this paper, we derive and analyze the ML soft metric in synchronous2 fast FHSS-MA networks with MFSK and Rayleigh fading under the assumption that the number of frequency hopping slots is sufficiently large. This assumption results in the absolute values of the correlator outputs being statistically independent conditioned on the number of active users in the network and numerical results indicate that this approximation is valid for all practical purposes. Both cases with and without the channel state information (CSI) at the receiver are considered where the CSI is defined as the magnitude response of the channel. The derived joint pdf of the absolute values of the correlator outputs is used to compute the channel capacity and the soft decision decoding performance of 3GPP Turbo and convolutional coded systems [34]. We also consider two Gaussian approximations to model the effect of the multiple access interference (MAI) on the correlator outputs. The first, legacy approximation [17]–[19] approximates the effect of the MAI conditioned on the number of active users in the network as following the complex Gaussian distribution, which we denote as GAI . The second, improved approximation [5] approximates the effect of the MAI conditioned on the number of hits in a particular 2 Examples of the synchronous FHSS-MA networks in commercial applications include WiMAX (diversity mode) [35], 3GPP LTE (virtual resource blocks of distributed type) [36] and Bluetooth [37].
c 2011 IEEE 0090-6778/11$25.00 ⃝
HONG et al.: PERFORMANCE OF SOFT DECISION DECODED SYNCHRONOUS FHSS MULTIPLE ACCESS NETWORKS USING MFSK MODULATION . . .
hop as following the complex Gaussian distribution, denoted as GAII . The decoders corresponding to the two Gaussian approximations are referred to as Gaussian decoders I and II and are denoted as GDI and GDII , respectively. Numerical results show that the ML soft decision decoder (MLD) drastically outperforms the legacy Gaussian decoder, i.e., GDI . For example, with 64–FSK and 3GPP Turbo coding, the MLD with the CSI supports 247 users at a frame error rate (FER) of 10−2 with 24 frequency hopping slots whereas the GDI only allows 19 users, a thirteen-fold increase (Table I). The GDII performs better than the GDI , especially for small modulation orders but still falls significantly short of the performance achievable with the MLD. Moreover, results show that the CSI used in conjunction with the MLD is much more valuable than with the GDI or the GDII . With 64–FSK and 3GPP Turbo coding, the number of active users that may be supported at an FER of 10−2 , increases by five-fold with the availability of the CSI with the MLD (Table I). However, for the GDI and the GDII , the corresponding increase is only approximately two-fold. We also find that there exists an optimum modulation order that should be used in order to maximize the network throughput for a given set of network parameters. Moreover, we observe that the derived MLD is quite insensitive to estimation errors in system parameters such as the number of active users in the network and the received signal-to-noise ratio (SNR) which are required to compute the ML soft metric at the receiver. Though the MLD understandably exhibits an increased sensitivity to channel estimation errors compared to the Gaussian decoders, it significantly outperforms the Gaussian decoders under all cases of practical interest. Finally, we investigate the required computational complexities of the soft metrics for the derived MLD, the legacy Gaussian decoder, GDI and a simplified version of the MLD. The numerical results show that the proposed simplified version of the MLD significantly reduces the complexity requirement of the MLD with performance indistinguishable from that of the MLD. The remainder of the paper is organized as follows. In Section II, a brief description of the system and channel models are given. The joint pdf of the absolute values of the correlator outputs is derived in Section III, and in Section IV, we derive the channel capacity and the corresponding normalized network throughput. In Section V, the soft decoding metrics are derived and numerical results are presented in Section VI. Finally, conclusions are drawn in Section VII. II. S YSTEM M ODEL The system considered is a synchronous FHSS-MA network with 𝐾, identical active users (transmitter-receiver pairs) in the network operating under independent Rayleigh fading channels, similar to that assumed in [2]. Each user transmits one MFSK modulated symbol per hop in one of the 𝑞 available frequency hopping slots and independently chooses a hopping slot for each symbol with equal probability. It is assumed that the hopping patterns of the users are synchronized on a hopby-hop basis, i.e., synchronous hopping. At baseband, the MFSK frequencies allotted to the 𝑀
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symbols are 𝑙 Hz, 𝑙 = 0, 1, . . . , 𝑀 − 1 (1) 𝑇 where 𝑇 is the hop (symbol) duration. Under the assumption that the hopping pattern of a receiver is perfectly synchronized with that of the corresponding transmitter, the complex baseband equivalent of the dehopped signal as seen by a given receiver (say receiver number one) during a hop duration (say duration [0, 𝑇 ]) can be written as follows: ) ( 𝐾 √ ∑ 𝑚𝑘 + 𝑀 (ℎ𝑘 − ℎ1 ) ¯ 𝑟1 (𝑡) = 𝑡 2𝑆𝑘 𝐻𝑘 𝑝𝑇 (𝑡) exp 𝑗2𝜋 𝑇 𝑘=1
+ 𝑧(𝑡),
0 ≤ 𝑡 ≤ 𝑇.
(2)
Here, 𝑝𝑇 (𝑡) = 1, for 𝑡 ∈ [0, 𝑇 ] and zero, otherwise, 𝑆¯𝑘 is the average received signal power from the 𝑘th transmitter, {𝐻1 , 𝐻2 , ⋅ ⋅ ⋅ , 𝐻𝐾 } are zero-mean, independent and identically distributed (i.i.d.), proper complex Gaussian RVs with 𝐸{𝐻𝑗 ∗ 𝐻𝑘 } = 𝛿𝑗,𝑘 where 𝛿𝑗,𝑘 = 1, if 𝑗 = 𝑘 and 𝛿𝑗,𝑘 = 0, otherwise and 𝑥∗ denotes the complex conjugate of 𝑥. Also, 𝑚𝑘 is the symbol transmitted by the 𝑘th user assumed to be independent between users and uniformly distributed on {0, 1, . . . , 𝑀 − 1} and ℎ𝑘 is the frequency hopping slot used by the 𝑘th user, assumed to be independent between users and uniformly distributed on {1, 2, . . . , 𝑞}. Finally, 𝑧(𝑡) is a zero-mean, complex, AWGN process with 𝐸{𝑧(𝑡)𝑧 ∗ (𝜏 )} = 2𝑁0 𝛿(𝑡 − 𝜏 ) where 𝑁0 /2 is the two-sided power spectral density, 𝛿(⋅) is the Dirac delta function [25]. After observing 𝑟1 (𝑡) in the interval [0, 𝑇 ], the receiver computes the 𝑀 , complex correlator outputs, 𝑈𝑙 given by ( ) ∫ 𝑇 2𝜋𝑙𝑡 𝑟1 (𝑡) exp −𝑗 𝑈𝑙 ≜ 𝑑𝑡, 𝑙 = 0, 1, . . . , 𝑀 − 1. (3) 𝑇 0 Assuming that the average received signal powers for the users are √ identical, i.e., 𝑆¯1 = 𝑆¯2 = ⋅ ⋅ ⋅ = 𝑆¯𝐾 [2] and normalizing by 2𝑆¯1 𝑇 , we have, 𝑈𝑙 Ω𝑙 ≜ √ = 𝐻1 𝛿𝑚1 ,𝑙 + 𝑍𝑙𝐾 , 𝑙 = 0, 1, . . . , 𝑀 − 1. (4) 2𝑆¯1 𝑇 Here, 𝑍𝑙𝐾 ≜
𝐾 ∑
𝐻𝑘 𝛿𝑚𝑘 +𝑀(ℎ𝑘 −ℎ1 ),𝑙 + 𝜇𝑙
(5)
𝑘=2
is the contribution of the MAI and the AWGN on Ω𝑙 , {𝜇0 , 𝜇1 , ⋅ ⋅ ⋅ , 𝜇𝑀−1 } are zero-mean, i.i.d., proper complex Gaussian RVs with 𝐸{𝜇𝑗 ∗ 𝜇𝑘 } = 𝛿𝑗,𝑘 /𝛾 where 𝛾 ≜ 𝐸¯𝑠 /𝑁0 and 𝐸¯𝑠 = 𝑆¯1 𝑇 is the average signal energy received from the paired transmitter. For the case when the frequency hopping slot occupied by the desired user is hit by 𝐾 ′ interfering users, the normalized correlator outputs, Ω𝑙 are given by ′
Ω𝑙 = 𝐻1 𝛿𝑚1 ,𝑙 + 𝑍𝑙𝐾 , 𝑙 = 0, 1, . . . , 𝑀 − 1 where ′ 𝑍𝑙𝐾
≜
′ 𝐾 +1 ∑
𝐻𝑘 𝛿𝑚𝑘 ,𝑙 + 𝜇𝑙
(6)
(7)
𝑘=2
is the combined contribution of the MAI and the AWGN on Ω𝑙 3 . 3 Here, without loss of generality, we have assumed that the first 𝐾 ′ users hit the hop under consideration.
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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 59, NO. 4, APRIL 2011
III. T HE J OINT PDF OF THE C ORRELATOR O UTPUTS In this section, we derive the joint pdf of the absolute values of the normalized correlator outputs, 𝑅𝑙 ≜ ∣Ω𝑙 ∣ for the cases with and without the CSI. Clearly, 𝑅𝑙 follows the Ricean ∑ distribution conditioned on the variables 𝐻1 , 𝑚1 and 𝐾 ′ +1 𝐵1,𝑙 ≜ 𝑘=2 𝛿𝑚𝑘 ,𝑙 where 𝐵1,𝑙 is the number of interfering users with 𝑚𝑘 = 𝑙 and ℎ𝑘 = ℎ1 . Thus, the pdf of 𝑅𝑙 conditioned on 𝐵1,𝑙 , 𝑚1 and ∣𝐻1 ∣ is given by [19]
where the summation is over k such that
∑𝑀−1 𝑙=0
𝑘𝑙 = 𝐾 ′ and
2𝑟𝑙 𝛿𝑚1 ,𝑙 + 𝐵1,𝑙 + 1/𝛾 ( ) 𝑟𝑙2 × exp − . 𝛿𝑚1 ,𝑙 + 𝐵1,𝑙 + 1/𝛾
𝑓𝑅𝑙 (𝑟𝑙 ∣𝐵1,𝑙 , 𝑚1 ) =
(13)
The required computational complexity for computing 𝑓R (r∣𝐾, 𝑚1 , ∣𝐻1 ∣) and 𝑓R (r∣𝐾, 𝑚1 ) is directly proportional to the number of possible combinations of the {𝐵1,0 , 𝐵1,1 , ⋅ ⋅ ⋅ , 𝐵1,𝑀−1 } for a given value of 𝐾 and 𝑀 𝑓𝑅𝑙 (𝑟𝑙 ∣𝐵1,𝑙 , 𝑚1 , ∣𝐻1 ∣) ( ) ( ) given by 2 2𝑟𝑙 ∣𝐻1 ∣ 𝛿𝑚1 ,𝑙 𝑟2 + ∣𝐻1 ∣ 𝛿𝑚1 ,𝑙 2𝑟𝑙 𝐾−1 ∑ (𝑘 + 𝑀 − 1) exp − 𝑙 𝐼0 = 𝐵1,𝑙 + 1/𝛾 𝐵1,𝑙 + 1/𝛾 𝐵1,𝑙 + 1/𝛾 . (14) 𝑘 𝑘=0 (8) Clearly, the complexity is huge even for modest values of 𝐾 where 𝐼0 (⋅) is the zeroth-order modified Bessel function and 𝑀 . For example, for 𝑀 = 4 and 𝐾 = 20, the number of of the first kind [25]. Since {𝑅0 , 𝑅1 , ⋅ ⋅ ⋅ , 𝑅𝑀−1 } possible combinations of the {𝐵1,0 , 𝐵1,1 , ⋅ ⋅ ⋅ , 𝐵1,𝑀−1 } that conditioned on the {𝐵1,0 , 𝐵1,1 , ⋅ ⋅ ⋅ , 𝐵1,𝑀−1 }, 𝑚1 need to be considered is on the order of 103 and for 𝑀 = 8 and ∣𝐻1 ∣ are statistically independent, the joint and 𝐾 = 50, this value is on the order of 109 which are pdf of R ≜ (𝑅0 𝑅1 ⋅ ⋅ ⋅ 𝑅𝑀−1 ) conditioned on clearly impractical. B1 ≜ (𝐵1,0 𝐵1,1 ⋅ ⋅ ⋅ 𝐵1,𝑀−1 ), 𝑚1 and ∣𝐻1 ∣ is given by ∏𝑀−1 In order to circumvent this problem, we assume that 𝑍𝑙𝐾 , 𝑓R (r∣B1 , 𝑚1 , ∣𝐻1 ∣) = 𝑙=0 𝑓𝑅𝑙 (𝑟𝑙 ∣𝐵1,𝑙 , 𝑚1 , ∣𝐻1 ∣) where 𝑙 = 0, 1, ⋅ ⋅ ⋅ , 𝑀 − 1 in (5), are i.i.d. conditioned on 𝐾. This r ≜ (𝑟0 𝑟1 ⋅ ⋅ ⋅ 𝑟𝑀−1 ). Then averaging 𝑓R (r∣B1 , 𝑚1 , ∣𝐻1 ∣) is equivalent to assuming that {𝐵 , 𝐵1,1 , ⋅ ⋅ ⋅ , 𝐵1,𝑀−1 } are 1,0) (𝐾−1 over B1 , we have, 𝐾−1−𝑑 = 𝑑) = i.i.d. with Pr(𝐵 𝑝𝑑ℎ,𝑀 (1 − 𝑝ℎ,𝑀 ) 1,𝑙 𝑑 ( ) ′ ∑ 1 𝐾 and 𝑝ℎ,𝑀 = 1/(𝑞𝑀 ) which leads to the absolute values of 𝑓R (r∣𝐾 ′ , 𝑚1 , ∣𝐻1 ∣) = ′ 𝐾 the correlator outputs being statistically independent. Under 𝑀 𝑘0 𝑘1 ⋅ ⋅ ⋅ 𝑘𝑀−1 k this assumption, for the case with the CSI, the joint pdf 𝑀−1 ∏ of the ( absolute values of)the correlator outputs denoted by × 𝑓𝑅𝑙 (𝑟𝑙 ∣𝐵1,𝑙 = 𝑘𝑙 , 𝑚1 , ∣𝐻1 ∣) (9) ˆ ˆ0 𝑅 ˆ1 ⋅ ⋅ ⋅ 𝑅 ˆ𝑀−1 conditioned on 𝐾, 𝑚1 and ∣𝐻1 ∣, R≜ 𝑅 𝑙=0 is given by ∑𝑀−1 where the summation is over k such that 𝑙=0 𝑘𝑙 = 𝐾 ′ . 𝑓R ˆ (r∣𝐾, 𝑚1 , ∣𝐻1 ∣) Since the number of available frequency hopping slots is 𝑞 and 𝑀−1 each user independently chooses the frequency hopping slots ∏ Ψ𝐵 (𝐾, 𝑝ℎ,𝑀 , 𝑓𝑅𝑙 (𝑟𝑙 ∣𝐵1,𝑙 , 𝑚1 , ∣𝐻1 ∣)) . (15) = with equal probability, the hit probability of a hop is given 𝑙=0 by 𝑝ℎ ≜ 1/𝑞. Thus, by averaging over 𝐾 ′ (9), we obtain the joint pdf of R conditioned on 𝐾, 𝑚1 and ∣𝐻1 ∣: ˆ conditioned on For the case without the CSI, the joint pdf of R 𝐾 and 𝑚1 is given by further averaging 𝑓R ˆ (r∣𝐾, 𝑚1 , ∣𝐻1 ∣) 𝑓R (r∣𝐾, 𝑚1 , ∣𝐻1 ∣) = Ψ𝐵 (𝐾, 𝑝ℎ , 𝑓R (r∣𝐾 ′ , 𝑚1 , ∣𝐻1 ∣)) over ∣𝐻 ∣ as 1 (10) where
𝑓R ˆ (r∣𝐾, 𝑚1 ) = ′
𝑀−1 ∏
Ψ𝐵 (𝐾, 𝑝ℎ,𝑀 , 𝑓𝑅𝑙 (𝑟𝑙 ∣𝐵1,𝑙 , 𝑚1 )) .
𝑙=0
Ψ𝐵 (𝐾, 𝑝ℎ , 𝑓R (r∣𝐾 , 𝑚1 , ∣𝐻1 ∣)) (16) 𝐾−1 ∑ (𝐾 − 1) 𝐾−1−𝑘 The following theorem establishes the fact that as the ≜ 𝑓R (r∣𝐾 ′ = 𝑘, 𝑚1 , ∣𝐻1 ∣) . 𝑝𝑘ℎ (1 − 𝑝ℎ ) 𝑘 number of frequency hopping slots, 𝑞 increases, the distri𝑘=0 (11) butions 𝑓R (r∣𝐾, 𝑚1 , ∣𝐻1 ∣) and 𝑓R ˆ (r∣𝐾, 𝑚1 , ∣𝐻1 ∣) and also (r∣𝐾, 𝑚 ) converge to identical distri𝑓R (r∣𝐾, 𝑚1 ) and 𝑓R 1 ˆ For the case without the CSI, the joint pdf of R condi- butions. 𝐾 −1 tioned on 𝐾 and 𝑚1 can be found by further averaging , Theorem 1: For a given 𝜆 ≜ 𝑓R (r∣𝐾, 𝑚1 , ∣𝐻1 ∣) over ∣𝐻1 ∣ and is given by 𝑞 𝐾−1 lim 𝑓R (r∣𝐾, 𝑚1 , ∣𝐻1 ∣) = lim 𝑓R ∑ (𝐾 − 1) ˆ (r∣𝐾, 𝑚1 , ∣𝐻1 ∣) 𝑞→∞ 𝑞→∞ 𝑓R (r∣𝐾, 𝑚1 ) = 𝑝𝑘ℎ (1 − 𝑝ℎ )𝐾−1−𝑘 𝑘 and 𝑘=0 ( ) ∑ 1 𝑘 × lim 𝑓R (r∣𝐾, 𝑚1 ) = lim 𝑓R ˆ (r∣𝐾, 𝑚1 ) . 𝑞→∞ 𝑞→∞ 𝑀 𝑘 𝑘0 𝑘1 ⋅ ⋅ ⋅ 𝑘𝑀−1 ×
k 𝑀−1 ∏ 𝑙=0
𝑓𝑅𝑙 (𝑟𝑙 ∣𝐵1,𝑙 = 𝑘𝑙 , 𝑚1 )
(12)
Proof : See Appendix. These results drastically simplify the design and analysis of the ML soft metric.
HONG et al.: PERFORMANCE OF SOFT DECISION DECODED SYNCHRONOUS FHSS MULTIPLE ACCESS NETWORKS USING MFSK MODULATION . . .
Average SER
10
10
10
Average SER
10
0
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IV. C HANNEL C APACITY AND N ORMALIZED T HROUGHPUT In this section, we compute the binary channel capacity and the corresponding normalized throughput for the synchronous FHSS-MA network under consideration. Since the 𝐾 {𝑍0𝐾 , 𝑍1𝐾 , ⋅ ⋅ ⋅ , 𝑍𝑀−1 } conditioned on 𝐾 are i.i.d. under the large 𝑞 assumption, the resulting channel is symmetric. Assuming that perfect interleaving of the binary code symbols is performed prior to MFSK modulation, the channel capacities given the number of active users, 𝐾 are given by [24]
-1
Simulation, q=2 Simulation, q=4 Simulation, q=8 Simulation, q=16 Analyses , Pe ( K )
-2
2
4
K
6
8
𝐶CSI (𝐾) 10
(a) BFSK modulation
0
[ [ { } = 1 − 𝐸∣𝐻1 ∣ 𝐸R∣ ∣𝐻1 ∣ log2 1 + Λ0 (𝑓R (R∣𝐾, 𝑚1 , ∣𝐻1 ∣)) ]] 𝐾, 𝑏01 = 0, ∣𝐻1 ∣ ∼ ˆCSI (𝐾) = 1 =𝐶 [ [ { ( ( ))} 0 ˆ − 𝐸∣𝐻1 ∣ 𝐸R∣ 𝑓R ˆ ∣𝐻1 ∣ log2 1 + Λ ˆ R∣𝐾, 𝑚1 , ∣𝐻1 ∣ ]] 0 𝐾, 𝑏1 = 0, ∣𝐻1 ∣ (18)
and 10
10
𝐶NCSI (𝐾)
-1
} ] [ { = 1 − 𝐸R log2 1 + Λ0 (𝑓R (R∣𝐾, 𝑚1 )) 𝐾, 𝑏01 = 0 ∼𝐶 ˆNCSI (𝐾) = ] [ { ( ( ))} 0 ˆ = 1 − 𝐸R 𝑓R 𝐾, 𝑏01 = 0 ˆ log2 1 + Λ ˆ R∣𝐾, 𝑚1 (19)
Simulation, q=2 Simulation, q=4 Simulation, q=8 Simulation, q=16 Analyses , Pe ( K )
-2
2
4
K
6
8
10
(b) 64–FSK modulation
𝑘∈𝐴𝜈
Fig. 1. Uncoded SER of synchronous FHSS-MA networks under independent ¯𝑏 /𝑁0 =20 [dB]. Rayleigh fading channels versus 𝐾. 𝐸
In order to evaluate the accuracy of the approximation for finite values of 𝑞, we compare the uncoded symbol error rate (SER) predicted by (16) to the simulated uncoded SER for several values of 𝑞. The uncoded SER, 𝑃𝑒 (𝐾) predicted by (16) is given by [26] ∫ 𝑃𝑒 (𝐾) = 1 −
0
∞
[𝑀−1 {∫ ∏ 𝑙=1
0
𝑦
}] 𝑓𝑅𝑙 (𝑥∣𝐾, 𝑚1 = 0) 𝑑𝑥
× 𝑓𝑅0 (𝑦∣𝐾, 𝑚1 = 0) 𝑑𝑦
for the cases with and without the CSI, respectively, where ∑ 𝑓R (r∣𝐾, 𝑚1 = 𝑘, ∣𝐻1 ∣)
(17)
where 𝑓𝑅𝑙 (𝑟𝑙 ∣𝐾, 𝑚1 ) = Ψ𝐵 (𝐾, 𝑝ℎ,𝑀 , 𝑓𝑅𝑙 (𝑟𝑙 ∣𝐵1,𝑙 , 𝑚1 )). In Figs. 1(a) and (b), the simulated uncoded SER curves along ¯𝑏 /𝑁0 = with 𝑃𝑒 (𝐾) are plotted for BFSK and 64–FSK when 𝐸 ¯ ¯ 20 dB where 𝐸𝑏 ≜ 𝐸𝑠 / log2 𝑀 is the average received signal energy per bit. The results indicate that the large 𝑞 assumption based on Theorem 1 is very accurate even for relatively small value of 𝑞. Even for 𝑞 = 2, the error resulting from the large 𝑞 assumption is practically negligible. Though not shown, simulations were carried out for a wide range of system parameters which consistently support this conclusion. Henceforth, this approximation is employed in all subsequent derivations except for the Gaussian approximations.
Λ𝜈 (𝑓R (r∣𝐾, 𝑚1 , ∣𝐻1 ∣)) ≜ ∑1
𝑘∈𝐴𝜈 0
𝑓R (r∣𝐾, 𝑚1 = 𝑘, ∣𝐻1 ∣)
.
(20)
Here, 𝑏𝜈1 , 𝜈 = 0, ⋅ ⋅ ⋅ , log2 𝑀 − 1 is the 𝜈th bit contained in the MFSK modulated symbol 𝑚1 and 𝐴𝜈𝑖 is the set of MFSK modulated symbols corresponding to 𝑏𝜈1 = 𝑖, 𝑖 ∈ {0, 1}. For both cases, the expectation operations may be evaluated using the Monte-Carlo integration technique [30]. The normalized network throughputs, defined as the average number of successfully transmitted information bits per unit time per unit bandwidth, assuming channel capacity achieving codes are defined as [2] 𝑊CSI (𝐾) ≜
ˆCSI (𝐾) 𝐾𝐶CSI (𝐾) ∼ ˆ 𝐾𝐶 = 𝑊CSI (𝐾) ≜ 2𝑞BFSK 2𝑞BFSK (21)
and 𝑊NCSI (𝐾) ≜
ˆNCSI (𝐾) 𝐾𝐶NCSI (𝐾) ∼ ˆ 𝐾𝐶 = 𝑊NCSI (𝐾) ≜ 2𝑞BFSK 2𝑞BFSK (22)
for the cases with and without the CSI, respectively. Here, 𝑞BFSK is the number of available frequency hopping slots assuming BFSK modulation, representing the normalized bandwidth.
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V. S OFT D ECISION D ECODING M ETRICS In this section, we derive the soft metrics for Turbo and convolutional decoding4. In Subsection V.A, the ML soft metrics based on the approximate joint pdfs of R are derived and in Subsections V.B and V.C, two soft metrics based on Gaussian approximations of the MAI are derived. A. ML Decoder (MLD) The log-likelihood ratios used as the soft metrics for ML soft decision decoding are given by 𝜈 𝜈 𝐿MLD CSI (𝑏1 ∣𝐾, r, ∣𝐻1 ∣) ≜ ln Λ (𝑓R (r∣𝐾, 𝑚1 , ∣𝐻1 ∣)) ( ) ∼ ˆ MLD (𝑏𝜈 ∣𝐾, r, ∣𝐻1 ∣) ≜ ln Λ𝜈 𝑓 ˆ (r∣𝐾, 𝑚1 , ∣𝐻1 ∣) =𝐿 CSI 1 R (23)
and 𝜈 𝜈 𝐿MLD NCSI (𝑏1 ∣𝐾, r) ≜ ln Λ (𝑓R (r∣𝐾, 𝑚1 )) ( ) ∼𝐿 ˆ MLD (𝑏𝜈 ∣𝐾, r) ≜ ln Λ𝜈 𝑓 ˆ (r∣𝐾, 𝑚1 ) = NCSI
1
R
(24)
for the cases with and without the CSI, respectively. B. Gaussian Decoder I (GDI ) Recall that the GDI is based on the GAI , i.e., 𝐾 {𝑍0𝐾 , 𝑍1𝐾 , ⋅ ⋅ ⋅ , 𝑍𝑀−1 } conditioned on 𝐾, are i.i.d., zeromean complex Gaussian RVs. Under this assumption, for the case with the CSI, the joint pdf of the abso˜ ≜ lute values of the )correlator outputs denoted by R ( ˜ ˜ ˜ 𝑅0 𝑅1 ⋅ ⋅ ⋅ 𝑅𝑀−1 conditioned on 𝐾, 𝑚1 and ∣𝐻1 ∣ is given ∏𝑀−1 by 𝑔R ˜ (r∣𝐾, 𝑚1 , ∣𝐻1 ∣) = ˜𝑙 (𝑟𝑙 ∣𝐾, 𝑚1 , ∣𝐻1 ∣) where 𝑙=0 𝑔𝑅 𝑔𝑅˜𝑙 (𝑟𝑙 ∣𝐾, 𝑚1 , ∣𝐻1 ∣) is given by replacing (𝐵1,𝑙 + 1/𝛾) in (8) with { } 2 𝑍𝑙𝐾 2 𝐾 = 𝐾 − 1 + 1 . (25) ≜ 𝐸 𝜎GA I 𝑞𝑀 𝛾 ˜ conditioned on For the case without the CSI, the joint pdf R ∏of 𝑀−1 𝐾 and 𝑚1 is given by 𝑔R ˜ (r∣𝐾, 𝑚1 ) = ˜𝑙 (𝑟𝑙 ∣𝐾, 𝑚1 ) 𝑙=0 𝑔𝑅 where 𝑔𝑅˜𝑙 (𝑟𝑙 ∣𝐾, 𝑚1 ) is given by replacing (𝐵1,𝑙 + 1/𝛾) in 2 . Thus, the soft metrics (13) with 𝜎GA I ( for the GDI are )given GDI 𝑚1 , ∣𝐻1 ∣) and by 𝐿CSI (𝑏𝜈1 ∣𝐾, r, ∣𝐻1 ∣) (≜ ln Λ𝜈 𝑔R ˜ (r∣𝐾, ) GDI 𝜈 𝜈 𝐿NCSI (𝑏1 ∣𝐾, r) ≜ ln Λ 𝑔R ˜ (r∣𝐾, 𝑚1 ) for the cases with and without the CSI, respectively [17]–[19]. C. Gaussian Decoder II (GDII ) Recall that the GDII is based on the GAII , i.e., ′ ′ 𝐾′ } conditioned on 𝐾 ′ , are i.i.d., zero{𝑍0𝐾 , 𝑍1𝐾 , ⋅ ⋅ ⋅ , 𝑍𝑀−1 mean complex Gaussian RVs [5]. Under this assumption, for the case with the CSI, the joint pdf of the abso¯ (lute values of the) correlator outputs denoted by R ≜ ¯ ¯ ¯ 𝑅0 𝑅1 ⋅ ⋅ ⋅ 𝑅𝑀−1 conditioned on 𝐾, 𝑚1 and ∣𝐻1 ∣ is given ′ by 𝑔R ¯ (r∣𝐾, 𝑚1 , ∣𝐻1 ∣) = Ψ𝐵 (𝐾, 𝑝ℎ , 𝑔R ∏𝑀−1 ¯ (r∣𝐾 ,′ 𝑚1 , ∣𝐻1 ∣)) ′ where 𝑔R ¯ (r∣𝐾 , 𝑚1 , ∣𝐻1 ∣) = ¯ 𝑙 (𝑟𝑙 ∣𝐾 , 𝑚1 , ∣𝐻1 ∣) 𝑙=0 𝑔𝑅 and 𝑔𝑅¯ 𝑙 (𝑟𝑙 ∣𝐾 ′ , 𝑚1 , ∣𝐻1 ∣) is given by replacing (𝐵1,𝑙 + 1/𝛾) in (8) with { } 𝐾′ 1 𝐾 ′ 2 ′ 2 𝜎GAII ≜ 𝐸 𝑍𝑙 𝐾 = + . (26) 𝑀 𝛾 4 The metrics are, in fact, applicable to all codes accepting soft inputs at the decoder.
Thus, the soft metric for the GDII with the CSI is given by 𝜈 𝜈 II 𝐿GD ¯ (r∣𝐾, 𝑚1 , ∣𝐻1 ∣)). CSI (𝑏1 ∣𝐾, r, ∣𝐻1 ∣) ≜ ln Λ (𝑔R ¯ For the case without the CSI, the joint pdf of R (r∣𝐾, 𝑚 , ∣𝐻 ∣) over ∣𝐻 can be found by averaging 𝑔R ¯ 1 1 1 ∣. ¯ conditioned on 𝐾 and 𝑚1 is Thus, the joint pdf of R ′ given by 𝑔R ¯ (r∣𝐾, 𝑚1 ) = Ψ𝐵 (𝐾, 𝑝ℎ , 𝑔R ¯ (r∣𝐾 , 𝑚1 )) where ∏𝑀−1 ′ ′ ′ 𝑔R ¯ (r∣𝐾 , 𝑚1 ) = ¯ 𝑙 (𝑟𝑙 ∣𝐾 , 𝑚1 ) and 𝑔𝑅 ¯ 𝑙 (𝑟𝑙 ∣𝐾 , 𝑚1 ) 𝑙=0 𝑔𝑅 2 . Thus, is given by replacing (𝐵1,𝑙 + 1/𝛾) in (13) with 𝜎GA II the soft metric for the GDII without the CSI is given by 𝜈 𝜈 II 𝐿GD ¯ (r∣𝐾, 𝑚1 )). NCSI (𝑏1 ∣𝐾, r) ≜ ln Λ (𝑔R VI. N UMERICAL R ESULTS In this section, we first present the normalized throughput curves of synchronous FHSS-MA networks using MFSK under Rayleigh fading channels as defined in (21) and (22). We present the FER performance of the derived soft metrics which are applied to rate-1/3 3GPP Turbo and convolutional codes [34] with a frame length of 5114 bits. We also employed rate-2/3 Turbo code which is derived from the rate-1/3 3GPP Turbo code via the puncturing pattern specified in [33]. For ¯𝑏 /𝑁0 is set to 20 dB, unless otherwise all numerical results, 𝐸 ¯ specified, where 𝐸𝑏 is the average received energy per infor¯𝑏 ≜ 𝐸 ¯𝑠 /(𝜌 log 𝑀 ) where 𝜌 denotes the mation bit with 𝐸 2 5 code rate . Also, the number of available frequency hopping slots assuming BFSK modulation (𝑞BFSK ) is assumed to be 128 and thus, the number of available frequency hopping slots 2𝑀 with MFSK modulation is 𝑞MFSK = 2 log 𝑞BFSK . 𝑀 A. Normalized Throughput In this subsection, we present the normalized throughput ˆ and the two Gausresults based on the derived joint pdf of R 𝐾 𝐾′ sian approximations for 𝑍𝑙 and 𝑍𝑙 . These results give us an idea of the validity (or the invalidity) employing Gaussian models of R from the viewpoint of system modelling. Figs. 2–5 compare the normalized throughput based on 𝑓R ˆ (r∣𝐾, 𝑚1 , ∣𝐻1 ∣), 𝑓R ˆ (r∣𝐾, 𝑚1 ) and two Gaussian models. In these figures, 𝑊 Exact depicts the normalized throughput results obtained from (21) and (22) and 𝑊 GAI and 𝑊 GAII denote the normalized throughput results under the two Gaussian approximations. The results for 𝑊 GAI are computed using (21) and (22), but with 𝑓R ˆ (r∣𝐾, 𝑚1 , ∣𝐻1 ∣) in (18) and 𝑓R ˆ (r∣𝐾, 𝑚1 ) in (19) replaced with 𝑔R ˜ (r∣𝐾, 𝑚1 , ∣𝐻1 ∣) and 𝑔R ˜ (r∣𝐾, 𝑚1 ), respectively. The results for 𝑊 GAII are also computed using (21) and (22), but with 𝑓R ˆ (r∣𝐾, 𝑚1 , ∣𝐻1 ∣) in (18) and 𝑓R ˆ (r∣𝐾, 𝑚1 ) in (19) replaced with 𝑔R ¯ (r∣𝐾, 𝑚1 , ∣𝐻1 ∣) and 𝑔R ¯ (r∣𝐾, 𝑚1 ), respectively6 . Figs. 2–5 indicate that WExact predicts that the maximum normalized throughputs for the cases with and without the CSI 5 For
the normalized throughput results, 𝜌 is the channel capacity. would like to emphasize at this point that 𝑊 GAI and 𝑊 GAII ′ represent the normalized throughput results when 𝑍𝑙𝐾 and 𝑍𝑙𝐾 are indeed i.i.d. complex Gaussian RVs possessing variance identical to the actual distributions. Hence, these results demonstrate the accuracy (or inaccuracy) of the Gaussian approximations in the system modelling and evaluation process. On the other hand, in the FER results presented in Tables I and II, the GDI and the GDII represent the results with the soft metrics assuming 𝑍𝑙𝐾 and ′ 𝑍𝑙𝐾 are complex Gaussian RVs when 𝑍𝑙𝐾 is actually generated according to (5). Hence, these results evaluate the performances of practical receivers based on the Gaussian approximations under real interference conditions. 6 We
HONG et al.: PERFORMANCE OF SOFT DECISION DECODED SYNCHRONOUS FHSS MULTIPLE ACCESS NETWORKS USING MFSK MODULATION . . .
1
Normalized Throughput
0.8 0.7 0.6
BFSK 4FSK 8FSK 16FSK 32FSK 64FSK
W Exact
W GA
0.5
I
Normalized Throughput
0.9
0.5 0.4 0.3 0.2
0.4
BFSK 4FSK 8FSK 16FSK 32FSK 64FSK
W GA
W Exact
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I
0.3 0.2 0.1
0.1 100
200
300
400
500
600
0
700
100
200
Fig. 2. Normalized throughput curves of synchronous FHSS-MA networks under independent Rayleigh fading channels versus 𝐾. 𝑊 Exact and 𝑊 GAI ¯𝑏 /𝑁0 =20 [dB]. The comparison of between with the CSI, 𝑞BFSK =128, 𝐸 𝑊 Exact and 𝑊 GAI demonstrates the accuracy (or inaccuracy) of the Gaussian approximation on MAI.
0.7 0.6
W
Exact
W
0.5
GA II
Normalized Throughput
Normalized Throughput
0.8
BFSK 4FSK 8FSK 16FSK 32FSK 64FSK
0.5 0.4 0.3
400
500
600
Fig. 4. Normalized throughput curves of synchronous FHSS-MA networks under independent Rayleigh fading channels versus 𝐾. 𝑊 Exact and 𝑊 GAI ¯𝑏 /𝑁0 =20 [dB]. The comparison of between without the CSI, 𝑞BFSK =128, 𝐸 𝑊 Exact and 𝑊 GAI demonstrates the accuracy (or inaccuracy) of the Gaussian approximation on MAI.
1 0.9
300 K
K
0.4
BFSK 4FSK 8FSK 16FSK 32FSK 64FSK
W Exact
W GA
II
0.3 0.2 0.1
0.2 0.1
0 100
200
300
400
500
600
700
K Fig. 3. Normalized throughput curves of synchronous FHSS-MA networks under independent Rayleigh fading channels versus 𝐾. 𝑊 Exact and 𝑊 GAII ¯𝑏 /𝑁0 =20 [dB]. The comparison of between with the CSI, 𝑞BFSK =128, 𝐸 𝑊 Exact and 𝑊 GAII demonstrates the accuracy (or inaccuracy) of the Gaussian approximation on MAI.
are achieved with 𝑀 = 8 and 𝑀 = 4, respectively. However, WGAI and WGAII erroneously predict that the maximum normalized throughputs are achieved with 𝑀 = 64 for both cases with and without the CSI. Also, the maximum normalized throughputs predicted by the Gaussian approximations are approximately 40% smaller compared to the actual maximum normalized throughput for the case with the CSI. For the case without the CSI, the maximum normalized throughputs predicted by the Gaussian approximations are approximately 30% larger than the correct value. Finally, note that 𝑊 Exact predict that we may expect approximately a 2-fold increase in the maximum normalized throughput with the CSI whereas the Gaussian approximations only predict a maximum of 10% gain with the CSI. From these results, we conclude that the use of accurate statistical models for the effect of the MAI is
100
200
300
400
500
600
K Fig. 5. Normalized throughput curves of synchronous FHSS-MA networks under independent Rayleigh fading channels versus 𝐾. 𝑊 Exact and 𝑊 GAII ¯𝑏 /𝑁0 =20 [dB]. The comparison of between without the CSI, 𝑞BFSK =128, 𝐸 𝑊 Exact and 𝑊 GAII demonstrates the accuracy (or inaccuracy) of the Gaussian approximation on MAI.
extremely important not only in predicting the performance of FHSS-MA networks but also in optimizing system parameters. B. Soft Decision Decoding Performance We now turn our attention to the actual performances of the derived soft metrics when applied to practical error correcting codes. The codes considered are rate-1/3 (𝜌 = 1/3), Turbo and convolutional codes as specified in the 3GPP standard [34] and rate-2/3 (𝜌 = 2/3), Turbo code which is derived from the rate1/3 3GPP Turbo code via the puncturing pattern specified in [33]. The number of information bits carried in a code frame is taken to be 5114 bits and the binary code symbols are uniformly interleaved prior to being mapped onto a stream of MFSK modulated symbols. The Turbo code is decoded via the iterative log-MAP decoder [32] with 8 decoder iterations per
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TABLE I ¯𝑏 /𝑁0 = 20 [dB], q T HE MAXIMUM NUMBER OF ACTIVE USERS SUPPORTED AT FER S OF 10−1 AND 10−2 . 𝐸 BFSK = 128, R ATE -1/3, 3GPP T URBO AND CONVOLUTIONAL CODES WITH FRAME LENGTH 5114.
𝑀
FER
2
10−1 10−2
4
10−1 10−2
8
10−1 10−2
16
10−1 10−2
32
10−1 10−2
64
10−1 10−2
CSI
Theoretical limit
O X O X O X O X O X O X O X O X O X O X O X O X
300 200 300 200 440 240 440 240 460 200 460 200 440 140 440 140 380 90 380 90 300 55 300 55
MLD 263 174 257 167 388 209 380 202 415 169 403 164 385 117 372 114 325 75 315 72 258 46 247 44
frame and the convolutional code is decoded using the Viterbi decoder [31]. Table I shows the maximum number of active users supported at FERs of 10−1 and 10−2 for various values of 𝑀 , with and without the CSI for the MLD, the GDI and the GDII ¯𝑏 /𝑁0 = 20 dB and 𝜌 = 1/3. The values for theoretical when 𝐸 limit on the maximum number of active users that may be supported by the network are obtained by finding the values of ˆNCSI (𝐾) = 𝜌 = 1/3 ˆCSI (𝐾) = 𝜌 = 1/3 and 𝐶 𝐾 satisfying 𝐶 for cases with and without the CSI, respectively. Note that the derived ML soft metric performs very close to theoretical limit
Turbo code GDI GDII 82 185 33 143 79 181 29 138 89 193 32 129 84 187 29 125 72 144 25 87 68 139 23 84 51 95 19 53 49 92 18 51 34 59 15 31 32 57 14 29 20 35 11 17 19 33 10 16
Convolutional MLD GDI 170 35 90 7 140 22 70 3 250 40 115 9 220 27 92 5 270 34 100 10 230 24 77 6 240 25 67 9 200 16 53 6 200 17 45 7 165 11 35 5 155 10 27 6 125 6 21 1
code GDII 115 75 95 57 130 75 105 60 100 54 83 40 67 32 55 23 42 17 33 13 23 9 17 6
for all modulation orders while the Gaussian approximations fall significantly short, especially for large modulation orders. Though the GDII gives relatively satisfactory performance for 𝑀 = 2, the performance of the legacy decoder, i.e., the GDI is especially disappointing. The result for the convolutional code shows trends similar to those of the Turbo code but offers performance only about 50% of that of the Turbo code. Table II shows the maximum number of active users supported at an FER of 10−2 for code rates 𝜌 = 1/3, 2/3 and ¯𝑏 /𝑁0 = 10 and 20 dB. First, note that at received SNRs of 𝐸 ¯ 𝐸𝑏 /𝑁0 =10 dB for 𝜌 = 1/3, i.e., when the channel is thermal
TABLE II ¯𝑏 /𝑁0 = 10 [dB], 𝜌 = 1/3 AND 𝜌 = 2/3. ¯𝑏 /𝑁0 = 20 dB, 𝐸 T HE MAXIMUM NUMBER OF ACTIVE USERS SUPPORTED AT AN FER OF 10−2 FOR 𝐸 𝑞BFSK = 128, 3GPP T URBO CODES WITH FRAME LENGTH 5114.
𝑀
2 4 8 16 32 64
CSI O X O X O X O X O X O X
𝜌 = 1/3 ¯𝑏 /𝑁0 = 10 [dB] ¯𝑏 /𝑁0 = 20 [dB] 𝐸 𝐸 MLD GDI GDII MLD GDI GDII 43 24 37 257 79 181 14 7 12 167 29 138 134 43 81 380 84 187 56 13 32 202 29 125 152 35 65 403 68 139 58 10 23 164 23 84 134 25 40 372 49 92 44 7 13 114 18 51 103 16 23 315 32 57 29 4 7 72 14 29 71 10 12 247 19 33 18 3 4 44 10 16
𝜌 = 2/3 ¯𝑏 /𝑁0 = 20 [dB] 𝐸 MLD GD I GDII 110 9 50 50 2 40 164 10 50 60 3 30 170 8 40 48 4 23 152 6 25 33 3 13 124 4 15 20 3 7 96 3 8 13 2 4
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TABLE III T HE MAXIMUM NUMBER OF ACTIVE USERS SUPPORTED AT AN FER OF 10−2 AS A FUNCTION OF THE ESTIMATION ERRORS FOR 𝐾 AND 𝛾 OF THE MLD. ˆ AND 𝛾 ¯𝑏 /𝑁0 = 20 [dB], q 𝐾 ˆ ARE THE ESTIMATES OF 𝐾 AND 𝛾 USED AT THE RECEIVER , RESPECTIVELY. 𝐸 BFSK = 128, R ATE -1/3, 3GPP T URBO CODES WITH FRAME LENGTH 5114.
ˆ − 𝐾)/𝐾} × 100 [%] {(𝐾 With CSI 10 log10
𝛾 𝛾 ˆ
[dB]
ˆ − 𝐾)/𝐾} × 100 [%] {(𝐾 Without CSI 10 log10
𝛾 𝛾 ˆ
[dB]
𝑀 =2 242 253 257 255 248 249 257 246 159 166 167 167 164 158 167 156
-80 -40 0 40 80 -3 0 3 -80 -40 0 40 80 -3 0 3
noise limited, though the MLD still significantly outperforms the Gaussian decoders (especially the legacy decoder GDI ), the achievable performance gain compared to the interference ¯𝑏 /𝑁0 = 20 decreases, as expected. On the limited case of 𝐸 other hand, when the code rate is increased to 𝜌 = 2/3, though the overall performance degrades due to the weakened error correcting capability, the performance gains of the MLD compared to the Gaussian decoders increase significantly. In Tables III and IV, we study the sensitivity of the ML soft metric to estimation error in 𝐾, 𝛾 and ∣𝐻1 ∣. First, Table III lists the maximum number of active users supported at an FER of 10−2 for various levels of estimation error in 𝐾 and 𝛾 when ¯𝑏 /𝑁0 = 20 dB and 𝜌 = 1/3. Note that the performance of 𝐸 the derived MLD is quite immune to estimation errors in SNR and the number of active users in the network requiring only a very crude estimates of these network parameters.
𝑀 =4 360 375 380 377 370 365 380 362 192 200 202 202 201 192 202 187
𝑀 =8 380 396 403 400 394 385 403 380 157 163 164 164 164 154 164 147
𝑀 =16 354 367 372 370 367 355 372 350 109 113 114 114 114 106 114 97
𝑀 =32 300 311 315 315 312 297 315 295 70 72 72 72 72 67 72 58
𝑀 =64 235 242 247 245 245 232 247 229 42 44 44 44 44 41 44 33
In order to study the effects of the CSI estimation error, i.e., the error in estimating ∣𝐻1 ∣ at the receiver, we need to incorporate a model for the estimation error. The actual model of the CSI estimation error depends on the specifics of the estimation algorithm and the employed pilot structure. In most scenarios, the CSI estimation error is assumed to be zero mean, i.i.d., complex Gaussian distributed [38]–[40], due either to the fact that the pilot structure does not allow hits or the effect of hits are assumed to be Gaussian. Here, we model the CSI estimate as being i.i.d. and is given by ∣𝐻1 + 𝑍𝑝 ∣ where 𝑍𝑝 is the contribution of the MAI and the AWGN. Since the hit probability of the pilot symbols will usually be designed to be less than that of the data symbols, we model the hit probability of the pilot symbols to be 𝛼𝑝ℎ , 0 ≤ 𝛼 ≤ 1 where 𝑝ℎ = 1/𝑞 is the hit probability of the data symbols. In this case, the pdf of 𝑍𝑝 conditioned on 𝐾 is given by 𝑓𝑍𝑝 (𝑧∣𝐾) =
TABLE IV T HE MAXIMUM NUMBER OF ACTIVE USERS SUPPORTED AT AN FER OF 10−2 AS A FUNCTION OF THE DESIGN PILOT STRUCTURE PARAMETER , 𝛼. ¯𝑏 /𝑁0 = 20 [dB], q 𝐸 BFSK = 128, R ATE -1/3, 3GPP T URBO CODES WITH FRAME LENGTH 5114.
MLD
GDI
GDII
M
Perfect CSI
2 4 8 16 32 64 2 4 8 16 32 64 2 4 8 16 32 64
257 380 403 372 315 247 79 84 68 49 32 19 181 187 139 92 57 33
0 236 341 347 305 246 184 77 83 67 48 32 19 172 181 136 90 56 32
0.25 194 246 210 153 100 62 71 77 63 45 30 18 147 153 116 77 48 28
𝛼 0.5 174 211 173 121 78 47 68 73 59 42 28 17 132 137 104 70 44 25
0.75 162 191 154 106 66 40 66 71 57 41 27 17 122 126 96 65 40 24
1.0 154 178 142 96 60 36 64 69 55 40 27 17 115 118 90 61 38 23
Without CSI 167 202 164 114 72 44 29 29 23 18 14 10 138 125 84 51 29 16
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TABLE V T HE REQUIRED COMPUTATIONAL COMPLEXITIES FOR MMLD , MGDI AND MAPP MLD PER RECEIVED MFSK SYMBOL WITH AND WITHOUT THE CSI.
MMLD MGDI MAPP MLD
CSI O X O X O X
+ 2𝐾𝑀 − 1 2𝐾𝑀 − 𝑀 − 1 𝑀 −1 𝑀 −1 2𝑄𝑀 − 1 2𝑄𝑀 − 𝑀 − 1
× 6𝐾𝑀 + 2𝑀 + 2 4𝐾𝑀 + 𝑀 𝑀 +1 2𝑀 6𝑄𝑀 + 2𝑀 + 2 4𝑄𝑀 + 𝑀
÷ 𝑀 𝑀 0 0 𝑀 𝑀
exp (⋅) 2𝐾𝑀 2𝐾𝑀 0 𝑀 2𝑄𝑀 2𝐾𝑀
𝐼0 (⋅) 𝐾𝑀 0 𝑀 0 𝑄𝑀 0
ln (⋅) 2 2 2 2 2 2
TABLE VI T HE MAXIMUM NUMBER OF ACTIVE USERS SUPPORTED AT AN FER OF 10−2 FOR MLD AND MLDAPP . qBFSK = 128, R ATE -1/3, 3GPP T URBO CODES WITH FRAME LENGTH 5114.
𝑀 2 4 8 16 32 64
CSI O X O X O X O X O X O X
MLD 257 167 380 202 403 164 372 114 315 72 247 44
¯𝑏 /𝑁0 = 20 [dB] 𝐸 MLDAPP 𝑄=1 𝑄=2 𝑄=3 91 245 256 36 157 157 97 363 378 49 195 195 70 388 400 40 162 162 45 363 371 34 114 114 28 310 314 22 72 72 16 244 246 13 44 44
𝑄=4 257 167 379 202 401 164 371 114 315 72 247 44
Ψ𝐵 (𝐾, 𝛼𝑝ℎ , 𝑓𝑅𝑙 (𝑟𝑙 ∣𝐵1,𝑙 , 𝑚1 ∕= 𝑙)). Table IV lists the maximum number of active users supported at an FER of 10−2 for various values of 𝛼 when ¯𝑏 /𝑁0 = 20 dB and 𝜌 = 1/3. Note that the loss due 𝐸 to estimation errors in the CSI is more severe compared to estimation errors in 𝐾 or 𝛾 and the loss increases with increasing modulation order. This is expected since optimum decoders, in general, are more sensitive to channel estimation errors. Also, as the hit probability for the pilot symbols increase beyond around 0.5𝑝ℎ , i.e., 𝛼 = 0.5, the performance of the MLD with the CSI falls below that of the MLD without the CSI. However, note that even the MLD without the CSI offers the performance in excess of twice that of the legacy Gaussian decoder, i.e., the GDI with the perfect CSI. These results suggest that in practice, the choice of the modulation order and whether or not to make use of the channel estimation information should be made depending on the channel condition. C. Computational Complexity of Soft Decision Decoding Metrics In this subsection, we investigate the required computational complexities of the soft metrics for the derived MLD, the legacy Gaussian decoder, GDI and a simplified MLD denoted by MLDAPP . The numerical results will demonstrate that the proposed simplified MLD significantly reduces the complexity requirement of the MLD with performance indistinguishable from that of the MLD. For the cases with and without the CSI, the soft metrics for the MLD are given by (27), as shown at the top of the next
MLD 44 15 134 56 153 57 134 44 103 30 72 18
¯𝑏 /𝑁0 = 10 [dB] 𝐸 MLDAPP 𝑄=1 𝑄=2 𝑄=3 22 43 43 7 14 14 44 132 134 13 56 56 37 152 152 11 57 57 26 134 134 7 44 44 19 103 103 5 29 29 10 72 72 3 18 18
𝑄=4 44 14 134 56 152 57 134 44 103 29 72 18
page, and 𝜈 ˆ MLD 𝐿 NCSI (𝑏1 ∣𝐾, r) ∑𝐾−1 MLD ∑ 𝛼MLD (𝑘,𝑝ℎ,𝑀 ,𝛾) exp(𝛽NCSI (𝑘,𝛾)𝑟𝑙2 ) ∑𝑘=0 𝐾−1 MLD 2 𝑙∈𝐴𝜈 1 𝑘=0 𝛼MLD (𝑘,𝑝ℎ,𝑀 ,𝛾) exp(𝛽CSI (𝑘,𝛾)𝑟𝑙 ) = ln ∑ , (28) ∑𝐾−1 MLD 2 𝑘=0 𝛼MLD (𝑘,𝑝ℎ,𝑀 ,𝛾) exp(𝛽NCSI (𝑘,𝛾)𝑟𝑙 ) ∑ 𝐾−1 MLD 2 𝑙∈𝐴𝜈 0 𝑘=0 𝛼MLD (𝑘,𝑝ℎ,𝑀 ,𝛾) exp(𝛽CSI (𝑘,𝛾)𝑟𝑙 ) (𝐾−1) 𝑘 1 𝑝ℎ,𝑀 respectively, where 𝛼MLD (𝑘, 𝑝ℎ,𝑀 , 𝛾) ≜ 𝑘+1/𝛾 𝑘 𝐾−1−𝑘
−1 MLD MLD (1 − 𝑝ℎ,𝑀 ) , 𝛽CSI (𝑘, 𝛾) ≜ 𝑘+1/𝛾 and 𝛽NCSI (𝑘, 𝛾) ≜ −1 . For values of 𝐾 and 𝑞 of interest, i.e., when 𝐾 is not 1+𝑘+1/𝛾 prohibitively larger than 𝑞, 𝛼MLD (𝑘, 𝑝ℎ,𝑀 , 𝛾) decrease very rapidly with increasing 𝑘. Hence, we may safely approximate ˆ MLD and 𝐿 ˆ MLD as follows: (29), as shown at the top of the 𝐿 CSI NCSI next page, and
ˆ MLDAPP (𝑏𝜈 ∣𝐾, r) 𝐿 1 NCSI ∑𝑄−1 MLD 2 ∑ 𝑘=0 𝛼MLD (𝑘,𝑝ℎ,𝑀 ,𝛾) exp(𝛽NCSI (𝑘,𝛾)𝑟𝑙 ) 𝜈 𝑙∈𝐴1 ∑𝑄−1 𝛼MLD (𝑘,𝑝ℎ,𝑀 ,𝛾) exp(𝛽 MLD (𝑘,𝛾)𝑟 2 ) CSI 𝑘=0 𝑙 = ln ∑ , ∑𝑄−1 MLD (𝑘,𝛾)𝑟 2 𝛼MLD (𝑘,𝑝ℎ,𝑀 ,𝛾) exp(𝛽NCSI 𝑘=0 𝑙) ∑ 𝜈 𝑄−1 MLD 2 𝑙∈𝐴0 𝑘=0 𝛼MLD (𝑘,𝑝ℎ,𝑀 ,𝛾) exp(𝛽CSI (𝑘,𝛾)𝑟𝑙 )
(30)
respectively, where 𝑄 is a positive integer less than equal to 𝐾. The soft metrics for the legacy Gaussian decoder, GDI are given by ) ( ( 2 ) ∑ GDI 𝐼 ∣𝐻 ∣ 𝛽 𝜎 𝑟 𝜈 𝑙 1 GAI 𝑙∈𝐴1 0 CSI 𝜈 I ( ) 𝐿GD ( 2 ) CSI (𝑏1 ∣𝐾, r, ∣𝐻1 ∣) = ∑ GDI 𝐼0 𝛽CSI 𝜎GAI 𝑟𝑙 ∣𝐻1 ∣ 𝑙∈𝐴𝜈 0 (31)
HONG et al.: PERFORMANCE OF SOFT DECISION DECODED SYNCHRONOUS FHSS MULTIPLE ACCESS NETWORKS USING MFSK MODULATION . . .
∑
∑𝐾−1
ˆ MLD (𝑏𝜈 ∣𝐾, r, ∣𝐻1 ∣) = ln 𝐿 CSI 1 ∑
∑𝐾−1
𝑙∈𝐴𝜈 1
𝑘=0
𝑙∈𝐴𝜈 0
∑ ˆ MLDAPP (𝑏𝜈 ∣𝐾, r, ∣𝐻1 ∣) = ln 𝐿 1 CSI ∑
and
∑
𝜈 I 𝐿GD NCSI (𝑏1 ∣𝐾, r) = ∑
𝑙∈𝐴𝜈 0
MLD (𝑘,𝛾) 𝑟 2 +∣𝐻 ∣2 MLD 𝛼MLD (𝑘,𝑝ℎ,𝑀 ,𝛾) exp(𝛽CSI (𝑙 1 ))𝐼0 (−2𝛽CSI (𝑘,𝛾)𝑟𝑙 ∣𝐻1 ∣) ∑𝐾−1 MLD 2 𝑘=0 𝛼MLD (𝑘,𝑝ℎ,𝑀 ,𝛾) exp(𝛽CSI (𝑘,𝛾)𝑟𝑙 )
∑𝑄−1 𝑙∈𝐴𝜈 1 𝑙∈𝐴𝜈 0
𝑘=0
∑𝑄−1 𝑘=0
( 2 ) 2) 𝜎GAI 𝑟𝑙 exp ( ( ) 2) GDI 2 exp 𝛽NCSI 𝜎GA 𝑟𝑙 I (
𝑙∈𝐴𝜈 1
𝑘=0
MLD MLD 𝛼MLD (𝑘,𝑝ℎ,𝑀 ,𝛾) exp(𝛽CSI (𝑘,𝛾)(𝑟𝑙2 +∣𝐻1 ∣2 ))𝐼0 (−2𝛽CSI (𝑘,𝛾)𝑟𝑙 ∣𝐻1 ∣) ∑𝐾−1 MLD 2 𝑘=0 𝛼MLD (𝑘,𝑝ℎ,𝑀 ,𝛾) exp(𝛽CSI (𝑘,𝛾)𝑟𝑙 )
GDI 𝛽NCSI
MLD MLD 𝛼MLD (𝑘,𝑝ℎ,𝑀 ,𝛾) exp(𝛽CSI (𝑘,𝛾)(𝑟𝑙2 +∣𝐻1 ∣2 ))𝐼0 (−2𝛽CSI (𝑘,𝛾)𝑟𝑙 ∣𝐻1 ∣) ∑𝑄−1 MLD (𝑘,𝛾)𝑟 2 𝛼 (𝑘,𝑝 ,𝛾) exp 𝛽 ( ) MLD ℎ,𝑀 CSI 𝑘=0 𝑙 MLD (𝑘,𝛾) 𝑟 2 +∣𝐻 ∣2 MLD 𝛼MLD (𝑘,𝑝ℎ,𝑀 ,𝛾) exp(𝛽CSI (𝑙 1 ))𝐼0 (−2𝛽CSI (𝑘,𝛾)𝑟𝑙 ∣𝐻1 ∣) ∑𝑄−1 MLD (𝑘,𝛾)𝑟 2 𝛼 (𝑘,𝑝 ,𝛾) exp 𝛽 ( ) MLD ℎ,𝑀 CSI 𝑘=0 𝑙
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(27)
(29)
A PPENDIX (32)
for the cases the CSI, respectively, ( 2with ) and without ( 2 ) GDI GDI 2 and 𝛽NCSI 𝜎 ≜ 2/𝜎 𝜎GAI ≜ where 𝛽 GAI )I ( 2 CSI 4 GA 1/ 𝜎GAI + 𝜎GAI . In the computation of the above MLD MLD soft metrics, (𝑘, 𝑝(ℎ,𝑀 , 𝛾), ( 2 ) 𝛼MLDGD ) 𝛽CSI (𝑘, 𝛾), 𝛽NCSI (𝑘, 𝛾), GDI 2 I 𝛽CSI 𝜎GAI and 𝛽NCSI 𝜎GAI are independent of 𝑟𝑙 and are not included in the complexity comparisons. The soft metrics for the MLD and the approximate MLD will collectively be referred to as MMLD and MAPP MLD , respectively, and those for the legacy Gaussian decoder will be referred to as MGDI . Table V lists the required computational complexities per received MFSK modulated symbol for MMLD , MAPP MLD and MGDI and Table VI shows the maximum number of active users that may be supported at an FER of 10−2 for various values of 𝑄. Note that the computational complexity requirements of MMLD is much larger than that of MGDI and may be prohibitively large for large value of 𝐾. However, as mentioned in Subsection VI.C, since the term 𝛼MLD (𝑘, 𝑝ℎ,𝑀 , 𝛾) very quickly decreases to zero with increasing 𝑘, we may safely approximate MMLD (27) and (28) as MAPP MLD (29) and (30), respectively. Table VI shows that the performance of the MLDAPP with very small values of 𝑄 (i.e. 𝑄 = 3, 4) results in performance nearly indistinguishable from those of the MLD. Hence, we conclude that though the required computational complexity of the MLDAPP is still significantly large compared to the legacy Gaussian decoder, the huge performance gains achievable with the MLDAPP more than justifies the increase in complexity which is no longer prohibitively large. VII. C ONCLUSION In this paper, we analyzed the performance of soft decision decoded synchronous FHSS-MA networks using Turbo and convolutional coded MFSK under Rayleigh fading channels. The joint pdf of the absolute values of the correlator outputs was derived to compute the channel capacity/normalized throughput and design the optimum soft decoding metric which in turn exhibited performance far exceeding that of the legacy Gaussian decoding metric. The results indicate that accurate statistical modelling of the effect of the MAI is extremely important not only in improving and predicting the performance of FHSS-MA networks but also in optimizing system parameters such as the modulation order.
P ROOF OF T HEOREM 1 In this Appendix, to prove Theorem 1, we first derive the characteristic functions of) Ω ≜ (Ω0 , Ω1 , ⋅ ⋅ ⋅ , Ω𝑀−1 ) ( ˆ 𝑙 is the normalized ˆ ˆ ˆ ˆ 𝑀−1 where Ω and Ω ≜ Ω0 , Ω1 , ⋅ ⋅ ⋅ , Ω correlator output under the assumption that the correlator outputs are statistically independent. Then, by using the charˆ we show that the characteristic acteristic functions of Ω and Ω, ˆ functions of R and R converge to the same function as the number of frequency hopping slots approaches infinity. Clearly, from (6), Ω𝑙 conditioned on 𝐵1,𝑙 , 𝑚1 and ∣𝐻1 ∣, is given by 𝐵1,𝑙
Ω𝑙 = ∣𝐻1 ∣𝑒
𝑗𝜃1
𝛿𝑚1 ,𝑙 +
∑
𝐻𝑘 + 𝜇 𝑙
(A1)
𝑘=2
where 𝜃1 is uniformly distributed on [−𝜋, 𝜋). Since 𝜃1 is uniformly distributed on [−𝜋, 𝜋), the characteristic function of 𝑒𝑗𝜃1 is 𝐽0 (𝑠) where 𝐽0 (⋅) is the zeroth-order Bessel function of the first kind [25], [41]. Also, since 𝐻𝑘 and 𝜇𝑙 are complex Gaussian RVs, the characteristic functions of 𝐻𝑘 and 𝜇𝑙 2 2 are 𝑒−𝑠 /4 and 𝑒−𝑠 /(4𝛾) , respectively [26]. Since the RVs 𝑗𝜃1 𝑒 , 𝐻𝑘 and 𝜇𝑙 are statistically independent, the conditional characteristic function of Ω𝑙 conditioned on 𝐵1,𝑙 , 𝑚1 and ∣𝐻1 ∣ is given by ( 2 ) 𝑠𝑙 (𝐵1,𝑙 + 1/𝛾) ΦΩ𝑙 (𝑠𝑙 ∣𝐵1,𝑙 , 𝑚1 , ∣𝐻1 ∣) = exp − 4 (A2) × 𝐽0 (𝑠𝑙 ∣𝐻1 ∣ 𝛿𝑚1 ,𝑙 ) . Also, since {Ω0 , Ω1 , ⋅ ⋅ ⋅ , Ω𝑀−1 } conditioned on the {𝐵1,0 , 𝐵1,1 , ⋅ ⋅ ⋅ , 𝐵1,𝑀−1 }, 𝑚1 and ∣𝐻1 ∣ are statistically independent, the characteristic function of Ω conditioned on B1 , 𝑚1 and ∣𝐻1 ∣ is given by ΦΩ (s∣B1 , 𝑚1 , ∣𝐻1 ∣) =
𝑀−1 ∏
ΦΩ𝑙 (𝑠𝑙 ∣𝐵1,𝑙 , 𝑚1 , ∣𝐻1 ∣)
(A3)
𝑙=0
where s ≜ (𝑠0 𝑠1 ⋅ ⋅ ⋅ 𝑠𝑀−1 ). Then averaging (A3) over B1 , we have, ΦΩ (s∣𝐾 ′ , 𝑚1 , ∣𝐻1 ∣) ) ∑ 1 ( 𝐾′ = 𝑀 𝐾 ′ 𝑘0 𝑘1 ⋅ ⋅ ⋅ 𝑘𝑀−1 k
×
𝑀−1 ∏ 𝑙=0
ΦΩ𝑙 (𝑠𝑙 ∣𝐵1,𝑙 = 𝑘𝑙 , 𝑚1 , ∣𝐻1 ∣)
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=
𝑀−1 ∏
𝐽0 (𝑠𝑙 ∣𝐻1 ∣ 𝛿𝑚1 ,𝑙 ) 𝑒
𝑠2 𝑙 − 4𝛾
(
𝑙=0
𝑀−1 1 ∑ − 𝑠2𝑛 𝑒 4 𝑀 𝑛=0
)𝐾 ′ . (A4)
∑𝑀−1 where the summation is over k such that 𝑙=0 𝑘𝑙 = 𝐾 ′ . Since the number of available frequency hopping slots is 𝑞 and each user independently chooses the frequency hopping slots with equal probability, the hit probability of a hop is given by 𝑝ℎ ≜ 1/𝑞. Averaging (A4) over 𝐾 ′ , the characteristic function of Ω conditioned on 𝐾, 𝑚1 and ∣𝐻1 ∣ is given by ΦΩ (s∣𝐾, 𝑚1 , ∣𝐻1 ∣) 𝐾−1 ∑ (𝐾 − 1) 𝐾−1−𝑘 = ΦΩ (s∣𝐾 ′ = 𝑘, 𝑚1 , ∣𝐻1 ∣) 𝑝𝑘ℎ (1 − 𝑝ℎ ) 𝑘 =
𝑘=0 𝑀−1 ∏
𝑠2 𝑙
(
×
𝑀−1 1 ∑ − 𝑠42𝑛 1 𝑒 +1− 𝑞𝑀 𝑛=0 𝑞
)𝐾−1 .
(A5)
On the other hand, under the assumption that {𝐵1,0 , 𝐵1,1 , ⋅ ⋅ ⋅ , 𝐵1,𝑀−1 } are i.i.d., the characteristic ˆ conditioned on 𝐵1,𝑙 , 𝑚1 and ∣𝐻1 ∣ is given by function of Ω ΦΩ ˆ (s∣𝐾, 𝑚1 , ∣𝐻1 ∣) )𝐾−1−𝑘𝑙 𝑀−1 ∏ 𝐾−1 ∑ (𝐾 − 1) ( 1 )𝑘𝑙 ( 1 = 1− 𝑘𝑙 𝑞𝑀 𝑞𝑀 𝑙=0 𝑘𝑙 =0
× ΦΩ𝑙 (𝑠𝑙 ∣𝐵1,𝑙 = 𝑘𝑙 , 𝑚1 , ∣𝐻1 ∣) =
𝑀−1 ∏
𝑠2 𝑙
𝐽0 (𝑠𝑙 ∣𝐻1 ∣ 𝛿𝑚1 ,𝑙 ) 𝑒− 4𝛾
𝑙=0 𝑀−1 ∏ (
×
𝑛=0
1 − 𝑠2𝑛 1 𝑒 4 +1− 𝑞𝑀 𝑞𝑀
)𝐾−1
.
(A6)
Now, as 𝑞 → ∞ for a fixed 𝜆 ≜ (𝐾 − 1)/𝑞, we have, lim ΦΩ (s∣𝐾, 𝑚1 , ∣𝐻1 ∣)
𝑞→∞
=
𝑀−1 ∏
𝑠2 𝑙
𝐽0 (𝑠𝑙 ∣𝐻1 ∣ 𝛿𝑚1 ,𝑙 ) 𝑒− 4𝛾
𝑙=0
}𝑞𝜆 ) 𝑀−1 ( 𝑠2 1 ∑ 𝑛 𝑒− 4 − 1 + 1 × lim 𝑞→∞ 𝑞𝑀 𝑛=0 { 𝑀−1 ( )} 𝑀−1 ∏ 𝑠2 𝑠2 𝜆 ∑ 𝑙 − 4𝛾 − 4𝑛 = 𝐽0 (𝑠𝑙 ∣𝐻1 ∣ 𝛿𝑚1 ,𝑙 ) 𝑒 exp −1 𝑒 𝑀 𝑛=0 𝑙=0 (A7) {
and lim ΦΩ ˆ (s∣𝐾, 𝑚1 , ∣𝐻1 ∣)
𝑞→∞
=
𝑀−1 ∏
𝑠2 𝑙
𝐽0 (𝑠𝑙 ∣𝐻1 ∣ 𝛿𝑚1 ,𝑙 ) 𝑒− 4𝛾
𝑙=0
× lim
𝑞→∞
𝑀−1 ∏ { 𝑛=0
1 𝑞𝑀
( 𝑒
−
𝑠2 𝑛 4
𝐽0 (𝑠𝑙 ∣𝐻1 ∣ 𝛿𝑚1 ,𝑙 ) 𝑒
𝑠2
𝑙 − 4𝛾
{ exp
𝑙=0
)} 𝑀−1 ( 𝑠2 𝜆 ∑ − 4𝑛 −1 , 𝑒 𝑀 𝑛=0 (A8)
resulting in lim ΦΩ (s∣𝐾, 𝑚1 , ∣𝐻1 ∣) = lim ΦΩ ˆ (s∣𝐾, 𝑚1 , ∣𝐻1 ∣) . (A9)
𝑞→∞
𝑞→∞
Hence, we have, lim ΦR (s∣𝐾, 𝑚1 , ∣𝐻1 ∣) = lim ΦR ˆ (s∣𝐾, 𝑚1 , ∣𝐻1 ∣) .
𝑞→∞
𝑞→∞
(A10)
For the case without the CSI, ΦΩ (s∣𝐾, 𝑚1 ) and ΦΩ ˆ (s∣𝐾, 𝑚1 ) are obtained by averaging (A5) and (A6) over ∣𝐻1 ∣, respectively. Thus, ΦΩ (s∣𝐾, 𝑚1 ) and ΦΩ ˆ (s∣𝐾, 𝑚1 ) 𝑠2 𝑙 𝛿𝑚1 ,𝑙
𝐽0 (𝑠𝑙 ∣𝐻1 ∣ 𝛿𝑚1 ,𝑙 ) 𝑒− 4𝛾
𝑙=0
=
𝑀−1 ∏
) −1 +1
}𝑞𝜆
are obtained by replacing 𝐽0 (𝑠𝑙 ∣𝐻1 ∣ 𝛿𝑚1 ,𝑙 ) with 𝑒− 4 in (A5) and (A6). Similarly, as 𝑞 → ∞ for a fixed 𝜆, ΦΩ (s∣𝐾, 𝑚1 ) and ΦΩ ˆ (s∣𝐾, 𝑚1 ) are also obtained by replacing 𝐽0 (𝑠𝑙 ∣𝐻1 ∣ 𝛿𝑚1 ,𝑙 ) with 𝑒− resulting in
𝑠2 𝑙 𝛿𝑚1 ,𝑙 4
in (A7) and (A8),
lim ΦΩ (s∣𝐾, 𝑚1 ) = lim ΦΩ ˆ (s∣𝐾, 𝑚1 ) .
𝑞→∞
𝑞→∞
(A11)
Hence, we have, lim ΦR (s∣𝐾, 𝑚1 ) = lim ΦR ˆ (s∣𝐾, 𝑚1 ) .
𝑞→∞
𝑞→∞
(A12)
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HONG et al.: PERFORMANCE OF SOFT DECISION DECODED SYNCHRONOUS FHSS MULTIPLE ACCESS NETWORKS USING MFSK MODULATION . . .
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Sungnam Hong was born in Seoul, Korea, on May 7, 1980. He received the B.S. degree in information and communication engineering from Sungkyunkwan University, Suwon, Korea, in 2006, and he is currently working toward the Ph.D. degree in electronic and electrical engineering at Pohang University of Science and Technology (POSTECH), Pohang, Korea. Since 2006, he has been a Research Assistant with the Division of Electrical and Computer Engineering, POSTECH. His current research interests include OFDM, MIMO systems, and synchronization algorithm for OFDM systems. Changkyu Seol was born in Seoul, Korea, on January 26, 1978. He received the B.S. degree in electronic and electrical engineering and chemical engineering from Pohang University of Science and Technology (POSTECH), Pohang, Korea, in 2003, where he is currently working toward the Ph.D. degree in electronic and electrical engineering. Since 2003, he has been a Research Assistant with the Division of Electrical and Computer Engineering, POSTECH. His current research interests include OFDM, MIMO systems, and synchronization algorithms for OFDM systems. Kyungwhoon Cheun (S’88-M’90) was born in Seoul, Korea, on December 16, 1962. He received his B.A. degree in electronics engineering from Seoul National University, in 1985, and the M.S. and Ph.D. degrees from the University of Michigan, Ann Arbor, in 1987 and 1989, respectively, both in electrical engineering. From 1989 to 1991, he was with the Electrical Engineering Department, University of Delaware, Newark, as an Assistant Professor. In 1991, he joined the Division of Electrical and Computer Engineering, Pohang University of Science and Technology (POSTECH), Pohang, Korea, where he is currently a Professor and the Director of the Center for Broad-band OFDM Multiple Access (BrOMA) supported by the Ministry of Knowledge Economy (MKE), Korea. He has also served as an engineering consultant to various industries in the areas of mobile communications and modem design, and currently serves as the CTO at Pulsus Technologies, a fabless SOC company specializing in full digital amplifiers. His current research interests include OFDM, turbo and turbo-like codes, space-time codes, MIMO systems, software-defined radio, and audio signal processing.
