IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 55, NO. 5, SEPTEMBER 2006
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Optimum Parameters for Maximum Throughput of FHMA System With Multilevel FSK Kwonhue Choi, Member, IEEE, and Kyungwhoon Cheun, Member, IEEE
Abstract—The authors optimize system parameters including modulation order M , diversity order L, and traffic density λ to maximize the throughput of frequency-hopping-multiple-access networks with multilevel frequency-shift keying. It is shown that optimal λ is close to ln 2 with the exception that optimal λ = 1 for the erasures coding channel with M = 2. For the erasures coding channel, optimum L is also derived, and for the hard-decision coding channel, optimum L is numerically searched. For the region of M < 64, the erasures coding channel achieves higher throughput than the hard-decision coding channel, and for the region of M > 64, the hard-decision coding channel achieves slightly higher throughput than the erasures coding channel. While the maximum throughput for the hard-decision coding channel monotonously increases as M increases, the maximum throughput for the erasures coding channel shows a local increase at M = 2, and it is significantly higher than that for the hard-decision coding channel with M = 2. Tradeoffs between system parameters to achieve the maximum throughput under a fixed bandwidth constraint are also presented. It is shown that the optimal set for (M, L, N (= number of available frequencies)) is approximately (G, log2 G, G) for both coding channels, where G is the total system bandwidth expansion normalized by bit rate. Index Terms—Capacity, frequency-hop communication, frequency-shift keying (FSK), multiaccess communication, multilevel FSK, throughput.
I. I NTRODUCTION
I
N THIS PAPER, we investigate the performance of a multilevel frequency-shift-keying (FSK) frequency-hoppingmultiple-access (FHMA) network in relation with system parameters and derive the optimum set of the system parameters achieving maximum throughput. This system was first proposed by Viterbi [1] and primarily intended for use over low-data-rate satellite links. Viterbi’s proposal has received a great deal of attention from researchers including Goodman and co-workers [2]–[4]. They examined the system for possible use in digital mobile radiotelephony and evaluated the maximum allowable users for appropriately designed system parameters. Recently, in [4], information-theoretic capacities of this system were calculated and compared with those of the FHMA system Manuscript received April 7, 2003; revised February 22, 2004, April 24, 2005, and November 13, 2005. This work was supported by Yeungnam University, Korea, and the Ministry of Information and Communication (MIC), Korea, through the ITRC programs supervised by the Institute of Information Technology Assessment (IITA) under IITA-2005-C1090-0501-0018 and IITA-2005-C1090-0502-0008. The review of this paper was coordinated by Prof. H. Leib. K. Choi is with the School of Electrical Engineering and Computer Science, Yeungnam University, Kyongsan, Korea (e-mail:
[email protected]). K. Cheun is with the Division of Electronic and Computer Engineering, Pohang University of Science and Technology (POSTECH), Pohang 790-784, Korea (e-mail:
[email protected]). Digital Object Identifier 10.1109/TVT.2006.878604
with the conventional MFSK modulation [5], [6], [8]–[10]. Another essential problem regarding the system design is how to select the system parameters in order to achieve the maximum throughput under various conditions. However, analytic throughput evaluation for this system with respect to the system parameters and optimization of the system parameters has not yet been made. This paper is concerned with optimization for system parameters, including modulation order, diversity order, and traffic density to maximize the network throughput for FHMA systems with multilevel FSK. Tradeoffs between system parameters to achieve a maximum throughput under fixed bandwidth constraint are also presented. II. S YSTEM M ODEL The system and channel model considered in this paper is similar to the frequency-hopping-spread-spectrum multiaccess (FHSS-MA) network patterns described in [1], [2], and [4], where the number of available tone positions within a system band N is equal to a data symbol alphabet that is the multilevel FSK signal M . Here, we extend to a more generalized case when N ≥ M . There are K users, each transmitting M -ary data with MFSK over N frequencies. In order to achieve diversity against a multiaccess interference, each M -ary data is repetitively transmitted L times using a random hopping pattern. In order to maintain orthogonality for noncoherent detection, a tone separation is set to L/TM , where TM denotes M -ary symbol duration equal to Tb log2 M . The block diagram of the considered system is shown in Fig. 1 along with the frequency-time matrix indicating the corresponding operation at each stage in the transmitter and receiver. Frequency index fl for L chips belonging to one M -ary is hopped over GF(N ) as follows: fl = ml ⊕ hl ,
1≤l≤L
(1)
where ml denotes L times repeated M -ary data and is uniformly distributed over [0, 1, 2, . . . , M − 1], and hl denotes the frequency-hopping sequence and is assumed to be independent and uniformly distributed on [0, 1, 2, . . . , N − 1]. Thus, fl is i.i.d. and uniformly distributed over [0, 1, 2, . . . , N − 1]. This model includes fast FHMA systems employing conventional MFSK. In fast FHMA systems employing a conventional MFSK modulation, hopping frequency hl is restricted to have discrete values of 0, M, 2M, . . . among GF(N ). However, it still holds that fl is uniformly distributed over [0, 1, 2, . . . , N − 1] and independent for l like the considered system. Consequently, the statistics of multiaccess interference
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Fig. 1. Block diagram of the considered system and signal processing operations with N = 8, M = 4, and L = 5.
of the synchronous fast FH system employing the conventional MFSK is identical to that of the considered system. Thus, we conjecture that the same remark regarding the optimum parameters and maximum throughput as obtained in this paper will be possible for the fast synchronous FHMA system employing the conventional MFSK with hard limited diversity combining. To recover transmitted data, each corresponding receiver synchronously dehops according to a hopping pattern and combines the decision statistics of L chips. Here, we focus on the case when the multiaccess interference is dominant over a background noise, and a threshold detection is performed for the energies of M tone positions prior to the diversity-combining FSK demodulator. After dehopping and threshold test, an M by L detection matrix shown in Fig. 1 is generated. Each row corresponds to a possible frequency index of transmitted M -ary data, and the elements of each column denote the detection results of the corresponding chip. Neglecting the background noise, with an arbitrary small detection threshold, the detection result can be perfectly set to one if there exists a signal transmitted in the corresponding cell and 0 otherwise. The row corresponding to the transmitted data is always complete, where the term complete means that the row is full of 1’s. Therefore, if there is only one complete row in a detection matrix, the received symbol is correctly demodulated. Recall that fl is i.i.d. and uniformly distributed over [0, 1, 2, . . . , N − 1]. Under this condition, the probability of insertion that a signal is detected at a certain cell of a detection matrix due to interference is calculated as follows [2]: K−1 1 PI = 1 − 1 − . N
(2)
We note that (2) does not depend on the individual signal energies in each cell of the time-frequency matrix but only depends on the system parameters N and K. This comes from the assumption that the background noise is negligible and that the detection is perfect. We can safely use (2), although there is fading or shadowing if they are not so severe that we can guarantee that the signal energy is still higher than the background-noise level.1 By defining λ to be traffic density given as K/N , (2) is written as follows: λN −1 1 . (3) PI = 1 − 1 − N With a well-known formula, limN →∞ (1 − 1/N )xN = exp(−x), (3) can be approximated for the sufficiently large N , which is practical as follows: PI ≈ 1 − exp(−λ).
(4)
Even though (4) does not give the explicit expression with the signal-to-interference ratio (SIR), other user interference is inherently considered in formulating (4), since the insertion occurs due to interference. Rather than by using the SIR, we express the insertion probability by using traffic density. By the nature of the frequency hopping, other user interference is not static hop by hop, and thus, it cannot be modeled as an equivalent Gaussian noise or other continuous random variable. 1 For the case of severe fading or shadowing, the detection error is not negligible, and (2) will deviate from the correct value. Accurate analysis for the case of severe fading or shadowing will be meaningful but is beyond the scope of this paper. We leave them for a further work.
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Two types of coding channel model are considered according to how to demodulate a symbol if there is more than one complete row. First, we consider the erasures coding channel, where the receiver erases the corresponding symbol with more than one complete row. This coding channel is characterized with the two-channel transition problem, symbol correct probability PC , and symbol erasure probability PX . Neglecting the background noise, the probability that a certain row other than the row corresponding to transmitted data is not complete is equal to 1 − PIL , and thus, the symbol erasure probability PX , i.e., the probability, in which there is more than one complete row, is calculated as follows: M −1 PX = 1 − 1 − PIL M −1 = 1 − 1 − (1 − exp(−λ))L (5) and the symbol correct probability PC is given as follows: PC = 1 − PX M −1 = 1 − (1 − exp(−λ))L .
Fig. 2. Union bound and exact symbol error probability when M = 64, L = 4, 8, 12, and 16.
(6)
Next, we consider the hard-decision coding channel, where the receiver makes an arbitrary decision among possible symbol values with the equal probability for the case of multiple complete rows. The probability that a certain low is complete and the receiver makes a wrong decision between this row and the correct row is given as follows: 2 complete rows = PM
1 L P . 2 I
2 complete rows PM = (M − 1) × PM
M −1 (1 − exp(−λ))L . 2
(8)
III. P ERFORMANCE A NALYSIS The performance measure considered here is an appropriately normalized network throughput given as follows: (9)
where C denotes the channel capacity, B denotes total system bandwidth, and TM denotes M -ary symbol duration. Since there are N frequency position with a spacing of 1/(TM /L), a total bandwidth B is given as follows: B=
LN . TM
Cλ CK = . LN L
(11)
From (11), we note that W is irrespective of N if the traffic density λ(= K/N ) is determined to be constant.
In this case, the considered coding channel is modeled as an M -ary memoryless erasures coding (MMEC) channel. Channel capacity for MMEC channel is given as [7] CX (PX ) = (1 − PX ) log2 M.
Fig. 2 shows the union bound along with the exact error probability for M = 64, L = 4, 8, 12, and 16. We note that the union bounds nicely fit the exact values in the region of PM < 0.1, which is the realistic region for the system with the nominal system parameters.
CK W = BTM
W =
A. Erasures Coding Channel (7)
Then, a simple union bound for symbol error probability is given as follows [5]:
=
With the substitution of (10) into (9), W is written as follows:
(10)
(12)
With the substitution of (12) and (5) into (11), W in itself takes the form of a complicated function of system parameters (M, L, λ), and thus, it is difficult to find optimum values of these parameters that achieve maximum W . From (5), we can express L as a function of PX as follows: 1
L = log1−exp(−λ) 1 − (1 − PX ) M −1 1 log 1 − (1 − PX ) M −1 . = log (1 − exp(−λ))
(13)
We multiply −1 to both the denominator and the numerator in (13) in order to make them positive; then, we get log L=
=
1
1
1−(1−PX ) M −1
log log
exp(λ) exp(λ)−1 1
1−(1−PX
1 ) M −1
λ − log (exp(λ) − 1)
.
(14)
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Substitution of (14) into (9) changes W into the form of the function of (M, PX , λ) as follows: WX (M, PX , λ) =
CX (PX )λ (λ − log (exp(λ) − 1)) 1 log 1 1−(1−PX ) M −1
=
(1 − PX ) log2 M 1 log 1
Note that (21) is less than one for M = 2. Since the practical diversity order should not be less than one, we take the optimum L for M = 2 as one. Also, optimum λ for M = 2 is also different from (18), which is obtained without a constraint on L. Therefore, we have to separately consider the optimization for M = 2. From (5), PX for M = 2 and L = 1 is written as follows: PX (M = 2) = 1 − exp(−λ).
1−(1−PX ) M −1
× λ (λ − log (exp(λ) − 1))
(15)
where PX is treated as just a parameter instead of L, and thus, optimization of (PX , λ) is equivalent to optimization of L. In other words, optimum L is determined from optimum (PX , λ) by (14), which is another expression of (5). From (15), we note that the throughput is factorized into two parts. The former is a function of M and PX , and the latter is a function of λ only. Therefore, throughput is maximized when two parts are maximized, respectively. Since two parts are the convex functions of PX and λ, respectively, the corresponding optimum λ and optimum PX are given as the roots of the following equations: d {λ (λ − log (exp(λ) − 1))} =0 dλ (1−P X ) log2 M 1 d log 1 1−(1−PX ) M −1 = 0. dPX
(17)
(19) where w−1 (x) is the first branch of the Lambert w function [11] and is defined to satisfy (20)
Substituting (18) and (19) into (14), optimum L is given as follows: − exp( M−1 −1 ) w−1 + M1−1 M −1 . (21) Lopt (M ) = − ln 2 We can check that a substitution of (18) and (21) into (5) gives (19). With (18) and (21), maximum throughput Wmax is given as follows: ln(2M ) (1 − exp(ζM ))M −1 −ζM
(24)
It can be easily shown that λ maximizing (24) is derived as follows: λopt (M = 2) = 1.
(25)
With the substitution of (25) into (24), maximum throughput is obtained as follows: 1 . e
(26)
B. Hard-Decision Coding Channel
(18) M −1 −1 − exp M −1 1 + PX,opt (M ) = 1− 1−exp w−1 M −1 M −1
WX,max (M ) =
WX (M = 2) = exp(−λ)λ.
(16)
w−1 (x) exp (w−1 (x)) = x, w−1 (x) ≤ −1.
With the substitution of (23) and (12) into (11), throughput is written as follows:
WX,max (M = 2) =
From (16) and (17), optimum PX and optimum λ are calculated as follows: λopt = ln 2
(23)
(22)
where ζM = w−1 (− exp(−1/M − 1)/(M − 1))+ 1/(M − 1).
In this case, the considered coding channel is modeled as an M -ary memoryless symmetric coding (MMSC) channel. The channel capacity for MMSC channel is given as [7] PM CHD(PM )= log2 M +PM log2 +(1−PM)log2 (1−PM). M −1 (27) By performing similar calculations in (13) and (14), we can express L as a function of PM from (8) as follows: −1 log M 2PM . (28) L= λ − log (exp(−λ) − 1) Again, substitution of (28) into (11) changes W into the form of the function of (M, PM , λ) as follows: WHD (M, PM , λ) =
CHD (PM )λ (λ − log (exp(λ) − 1)) −1 log M 2PM
=
CHD (PM ) × λ (λ − log (exp(λ) − 1)) . −1 log M 2PM
(29)
Since the latter factors of (29) are identical to that of (15), λopt is given as ln 2 as the erasures coding channel. In this case, closed-form PM,opt (M ), which maximizes the former factor of (29), does not exist. We numerically find PM,opt (M ) and, by using this, we calculate Lopt (M ) and WHD,max (M ) from (28) and (29).
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TABLE I Lopt (M ), λopt (M ), AND Wmax (M ) FOR BOTH CODING CHANNEL, (∗): OBTAINED VIA COMPUTER SEARCH
(10), total system bandwidth expansion with respect to bit rate Rb = 1/Tb , which is denoted by G, is given as follows: G=
B LN LN . = = Rb TM /Tb log2 M
(30)
For a given G, N is determined as a function of M , L, and G as follows: N=
Fig. 3.
Maximum throughput versus log2 M for each coding channel.
IV. P ARAMETER T RADEOFFS U NDER F IXED B ANDWIDTH C ONSTRAINT In this section, we investigate the tradeoffs between N , M , and L in maximizing W under a given total bandwidth. From
(31)
From (31), we note that there exists a tradeoff between L and N . That is, increasing of L in order to achieve more diversity has to pay the cost of an increase of PI in (3) due to a decrease of N . From (3) and (5), PX and PM are given as follows:
C. Numerical Results Table I shows the maximum throughput for several values of M along with λopt and Lopt for the erasures coding channel and the hard-decision coding channel. There is a difference between analytical results and the results via computer search. This is due to the fact that in computer search, L is optimized in the integer field for practical reasons. However, we note that Lopt via computer search is identical to the nearest integer to analytical results and that the maximum throughput shows good accordance with analytical results. In Fig. 3, the maximum throughput is plotted versus log2 M for each channel model. For the region of M < 64, the erasures coding channel achieves higher throughput than the hard-decision coding channel, and for the region of M > 64, the hard-decision coding channel achieves slightly higher throughput than the erasures coding channel. While the maximum throughput for the hard-decision coding channel monotonously increases as M increases, the maximum throughput for the erasures coding channel shows a local increase at log2 M = 1, i.e., M = 2, and it is significantly higher than that for the hard-decision coding channel.
G log2 M . L
PX = 1 − PM =
1 − 1 − exp
M −1 2
1 − exp
−KL G log2 M
−KL G log2 M
L M −1 (32)
L .
(33)
With a given G and M , L that minimizes PX and PM achieves a maximum throughput for each coding channel model. Since (32) and (33) have a common term (1 − exp(−KL/G log2 M ))L and they are the increasing functions of this term, L that minimizes this term also minimizes them. Thus, optimum L is obtained by solving the following equation: d (1 − exp(−T L))L dL
=0
(34)
where T replaces the term K/(G log2 M ). From (34), it is easily shown that optimum L for both coding channels is given as follows: ln 2 T G ln M = K
Lopt =
(35)
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Fig. 5. Maximum throughput when G = 256 and (L, N ) is optimally selected for M and K: erasures coding channel.
Fig. 4. Optimum L versus K for various M with G = 256.
and from the relation in (31), optimum N is given as follows: Nopt = =
G log2 M Lopt K . ln 2
(36)
Equations (35) and (36) are obtained without constraints on the region of (L, N ). However, in the practical systems, M cannot be larger than N , and L cannot be less than one, i.e., there exist practical constraints on (L, N ) as follows: M
