Performance of FHSS Multiple-Access Networks Using MFSK

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 44, NO. 11, NOVEMBER 1996

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erformance of Kyungwhoon Cheun, Member, IEEE, and Kwonhue Choi

Abstract-The main concern of this paper is to estimate the to be detrimental. It is only recently that researchers began symbol error probabilities of synchronous and asynchronous investigating the tightness of this bound [I]-[4]. Most recently frequency-hop spread-spectrummultiple-access (FHSS-MA) net- in [l], an approximation for the error probability was derived works through semi-analytic Monte Carlo simulations. We concentrate on systems transmitting one M-ary ( M 2) FSK that very accurately fits the simulated results for the case modulated symbol per hop with noncoherent demodulation. The when M = 2. Unfortunately, the search for an accurate usual practice when analyzing the performance of such networks and computationally efficient analytical approximation for the is to upper-bound the probability of symbol error when a hop is cases when M > 2 proved unfruitful. hit by I(' 2 1 interfering users with ( M - 1)/M or 1. Recent One observation made in [1] is that upper bounding the work [I] on the derivation of accurate approximations to this probability for the case when M = 2 has indicated that the symbol error probability whenever a hop is hit by one or more aforementioned bound not only gives excessively pessimistic re- interfering users by ( M - 1)/M (this will be referred to as sults but may also lead to wrong tradeoff decisions. In this paper, the ( M - 1)/M-bound) results in an excessively pessimistic using the simulated values for the error probabilities, we show that similar argument holds for the cases when M > 2 as well. estimate of the network performance. Another more interesting Also, by employing a normalized throughput measure that takes observation is that the bounding technique could give misinto account the bandwidth and time expansion associated with leading results when comparing the performance of systems the modulation order M , we find that there exists an optimum employing dfferent strategies (thus of different complexity). value of M that achieves the maximum possible throughput for the cases when binary and M-ary error correcting codes are One good example is that of comparing a system using employed. Throughput results are also given for the case when perfect side-information to erase all the symbols transmitted the signals from the active users in the network suffer from in the hops that are known to be hit with a system which independent Rayleigh fading. simply makes hard decisions. The analysis using the 1/2-

>

I. INTRODUCTION N this paper, estimates for the symbol error probabilities obtained via semi-analytic Monte Carlo simulations are used to accurately assess the performance of synchronous and asynchronous frequency-hop spread-spectrum multiple-access (FHSS-MA) networks. We concentrate on systems employing M-ary frequency-shift-keyed (MFSK) data modulation to transmit one M-ary symbol per hop at the transmitters along with noncoherent matched filter demodulation at the receivers. The lowest level of performance criteria on which all other performance criteria depend on in such networks is the probability of error of an M-ary symbol given that the hop in which the symbol is transmitted is hit by K' interfering users. Most of the previous analytical work on such networks are based on the simplifying bound that the probability of symbol error is ( M - l)/M or 1 whenever a hop is hit by one or more interfering users. In other words, hits by any number of interfering users in the network are considered

'

Paper approved by J. S. Lehnert, the Editor for Modulation and Signal Design of the IEEE Communications Society. Manuscript received June 9, 1993; revised December 15, 1994 and January 4, 1996. This work was supported by the Korea Science and Engineering Foundation (KOSEF) #9410900-059-2. This paper was presented in part at the 1993 IEEE Singapore Intemational Conference on Networksfinternational Conference on Information Engineering, Singapore 1993, and the 1994 Conference on Information Science and Systems, Princeton, NJ, 1994. The authors are with the Department of Electronic and Electrical Engineering, Pohang University of Science and Technology, Pohang 790-784, Korea. Publisher Item Identifier S 0090-6778(96)08596-0.

bound for M = 2 predicts that the system using perfect side-informationto erase all hit symbols performs much better than the one without perfect side-information and hence the additional complexity required to accurately detect the hits is justified. On the other hand, analysis based on the accurate approximations for the symbol error probability indicates that the system making simple hard decisions without perfect sideinformation performs significantly better than the system using perfect side-information to erase all hit symbols. In fact, assuming codes that achieve channel capacity, the system making simple hard decisions provides more than twice the maximum throughput. Another example is the comparison between synchronous and asynchronous hopping networks where synchronization of the hopping patterns of the users in the network may introduce additional complexity if at all possible. Again, under the 1/2-bound, it is predicted that synchronous hopping networks perform much better than

asynchronous hopping networks. But the results using the accurate values for the error probabilities indicate that the asynchronous hopping networks achieve better Performance. In summary, what these results indicate is that unless accurate values for the symbol error probabilities are employed, we may not only get an overly pessimistic estimate for the network performance but also be led to make incorrect design trade-off decisions. Unfortunately, expressions for the symbol error probabilities for the case when M > 2 that are both accurate and

0090-6778/96$05.00 0 1996 IEEE

CHEUN AND CHOI: PERFORMANCE OF FHSS MULTIPLE-ACCESS NETWORKS

numerically efficient are not yet known. The main purpose of this paper is to derive a description of the FHSS-MA network which lends itself to efficient Monte Carlo simulations for the symbol error probabilities. Using the simulation results, comparisons are made between systems employing different modulation orders (M) by defining a throughput measure that is normalized by the bandwidth and time expansion associated with the modulation order. It is found that there exists an optimum M that should be employed in order to achieve the largest possible maximum throughput. The case when the signals from the users suffer from independent Rayleigh fading is also considered. The structure of the paper is as follows. In Section 11, a brief description of the system and the channel models is given. Expressions for the receiver matched filter outputs used for the simulations are derived in Section Ill and in Section IV, an appropriately normalized throughput measure is defined. Finally, in Section V, simulation results for the symbol error probabilities are provided along with the normalized throughput results assuming codes that achieve channel capacity. 11. SYSTEM AND CHANNEL MODEL The system considered in this paper is a FWSS-MA network with K identical active users (transmitter-receiverpairs) identical to that described in [l] except for the fact that MFSK modulation with ( M > 2) is also employed. It is assumed that the users employ independent Markov hopping patterns [5]. The reason for assuming Markov hopping instead of the more popular memoryless hopping is that this simplifies the expressions for the receiver matched filter outputs and the simulation itself while providing results that are almost identical to those obtained assuming memoryless hopping [ 11. The transmitters send one M-ary symbol per hop in one of the q available frequency slots using MFSK modulation. At baseband, the MFSK frequencies allotted to the M symbols are

I

-Hz,

T

l=O,l,**.,(M-l)

(1)

where T is the hop (symbol) duration. This guarantees orthogonality of the MFSK signals under noncoherent demodulation using matched filters. Assuming that the hopping pattern of a receiver is perfectly synchronized with that of the corresponding transmitter, the complex baseband equivalent of the signal presented at a given receiver (say receiver number one) during a hop duration (say duration [0,TI) can be written as follows when the hop is hit by K' interfering users

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It is assumed that perfect synchronization is maintained between paired users giving = 0. For the special case of synchronous hopping, 71~ = 0, k = 2 , 3 , . . . , K ' +1; symbol transmitted by the kth user assumed to be independent between users and uniformly distributed on { O , l , . . . , M - l}; random phase of the kth user assumed to be independent between users and uniformly distributed on [O, 2n); complex additive white Gaussian noise (AWGN) process with E { z ( t ) z * ( r ) }= 2lv0S(t - r ) where No12 is the two-sided power spectral density of the AWGN at the receiver input. Also, S(.) is the Dirac delta function and x* denotes the complex conjugate of x;

{ ::t

E [O,Tl otherwise. After observing r ( t ) in the interval [0,TI, the receiver computes the M decision variables IUz I,1 = 0,1, . . . , M - 1 given below and chooses the index of the largest decision variable as the estimate 7iz of the transmitted symbol, Le., riZ = argmaxl IUll (Fig. 1)

111. THE DECISIONVARIABLES In this section, we derive expressions for the decision variables 1 UZI used in the Monte Carlo simulations. We first consider the asynchronous hopping case without fading and then the case when the signals from different transmitters suffer from independent Rayleigh fading. The results for the synchronous hopping case may be obtained from the asynchronous hopping case by setting T k = 0, k = 2,3, . . . , K'+l. A. No Fading By inserting (2) into (3) and noting that r1 = 0, it is straightforward to show that UZmay be written as

Uz =

iT

r(t)enp( - j y ) dt

K'+1

(4)

,.r

K'+1

k=l

+Yk)] +

(2)

(5)

received signal power of the kth user; delay of the kth user assumed to be uniformly distributed on [-T, T ) and independent between users.

The value of the first integral is T when ml = 1 and zero otherwise. The last integral denoted zl represents a complex Gaussian random variable (r.v.) with zero mean and E { z ~ z ; }= 2NoT&,, where S l , j = 1 when j = 1 and zero

x exp[j(2n?(t

- Tk)

Z(t)

where Sk

rk

IEEE TRANSACTIONS ON COMMUNICATIONS. VOL. 44, NO. 11, NOVEMBER 1996

1516

1

choose index of t h e largest

Fig. 1. The receiver.

TABLE I SIMULATED ERRORPROBABILTY GIVENm k FOR M = 8, K' = 1 AND 3 = 10 dB. ASYNCHRONOUS HOPPING. No FADING " 0

signal energy from the paired transmitter I

K'+l

r

7.24 x 107.58 x 107.70 x 107.73 x 10-

x [(l-

lpkI)Smk,l

-

IpkIsinc(;r(mk - i ) p k ) ( l - Smk,l)] (6)

where p k = and sinc(z) = sin(z)/z. After normalizing Uz by a T and defining

as the new decision variables, we arrive at the following expression for lQl I in terms of p k , m k , s k , and the signal-

B. Independent Ruykigh Fading When the signals arriving at the receiver suffer from Rayleigh fading with independent fade statistics, the decision variables are given as

x exp(j(pk - 8 ( 1 , m k , p k ) ) A ( 1 , m k , p k )

+ vZ .


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TABLE I1 SIMULATED AVERAGE ERRORPROBABILTY

FOR

= 2 , 4 , 8 , 1 6 , 3 2 , 6 4 AND

Here, r k are independent Rayleigh r.v.’s with probability density functions given by

% = 30 dB. ASYNCHRONOUS HOPPING, NO FADING

where H E A g1 and p1 Here, it is easily shown “1. . that HI, and ,Uk are zero mean independent and identically distributed (i.i.d.) Gaussian r.v.’s with E{HkH,*} = 2 and E { p ~ p ;= } where E, = 20,2E, is the average received signal energy from the paired transmitter. Also, s k = 2a:Sk is the average received signal power from the kth transmitter. Equations (8) and (15) are used in the Monte Carlo simulations to estimate the average M-ary symbol error probability given that a hop is hit by K’ interfering users.

&

Defining

Gk 4 r k e J P k

(13)

we know that Gk are independent zero mean Gaussian r.v.’s with E{GkG;} = E { r ; } = 2 ~ ; . After normalizing Rl by 01 and defining

IV. NORMALIZED THROUGHPUT The performance measure considered in this paper is an appropriately normalized network throughput which allows us to make fair comparisons between systems with different modulation orders. In order to keep the discussion general, we assume codes achieving channel capacity. Let W,, Hz be the total bandwidth allotted to the network

(%I

= Hl&n,,l

&

and Hz be the information bit rate. Since the duration of an MFSK signal is Tb log2 M seconds, we define the normalized throughput w ( K ) given K active users as follows:

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TABLE-I11 FOR M = 2 , 4 , 8 , 1 6 , 3 2 , 6 4 AND = 30 dB. ASYNCHRONOUS HOPPING, INDEPENDENT RAYLRIGH FADING SIMULATED AVERAGE ERRORPROBABILTY

%

K'\M 1 2 3 4 5 6 7 8

9 10

17 18 19 20 21 22 23 24 25 26 27 28 29 30

I

2 1.21 x 10-1 1.93 x 10-1 2.38 x 10-1 2.77 x 10-1 3.10 x 10-1 3.33 x 10-1 3.52 x 10-1 3.53 x 10-1 3.74 x 10-1 3.81 x 10-1

4.25 x 10-1 4.29 x lo-' 4.41 x 10-1

4.29 x 4.35 x 4.40 x 4.40 x 4.51 x 4.44 x 4.42 x 4.46 x 4.59 x 4.58 x 4.57 x

10-1 10-1 IO-' 10-1 1O-I 10-1

10-1 10-1

10-1 IO-' 10-1

I

4 8 16 32 1.66 x 10-1 1.90 x 10-1 1.97 x lo-' 2.07 x 10-1 2.64 x lo-' 3.05 x 10-1 3.26 x 10-1 3.37 x 10-1 3.47 x 10-1 3.86 x lo-' 3.99 x 10-1 4.14 x 10-1 3.92 x lo-' 4.41 x lo-' 4.70 x 10-1 4.86 x 10-1 4.33 x 10-1 4.86 x 10-1 5.20 x lo-' 5.25 x 10-1 4.62 x 10-1 5.25 x 10-1 5.48 x 10-1 5.63 x 10-1 4.95 x lo-' 5.57 x 10-1 5.81 x 10-1 5.93 x 10-1 5.24 x 10-1 I 5.75 x 10-1 I 6.04 x 10-1 I 6.26 x 10-1 5.34 x 10-1 6.03 x lo-' 6.32 x 10-I 6.52 x 10-1 5.47 x 10-1 6.18 x 10-1 6.46 x 10-1 6.70 x lo-'

6.10 x 6.15 x 6.29 x 6.37 x 6.37 x 6.38 x 6.51 x 6.43 x 6.51 x 6.51 x 6.72 x 6.60 x 6.60 x ,6.67 x

I

10-1 10-1 lo-' lo-'

lo-' 10-1

10-1 10-1 10-1 lo-' 10-1 10-1 lo-' 10-1

7.00 x 7.09 x 7.18 x 7.22 x 7.26 x 7.41 x 7.40 x 7.38 x 7.44 x 7.43 x 7.54 x 7.41 x 7.69 x 7.55 x

lo-' 10-1 10-1 10-1 10-1 10-1 10-1 lo-' 10-1 10-1 10-1 10-1 10-1 10-1

I

7.37 x 7.44 x 7.59 x 7.57 x 7.65 x 7.70 x 7.76 x 7.78 x 7.86 x 7.99 x 7.93 x 8.02 x 8.11 x 8.08 x

10-1 10-1

lo-' lo-' 10-1 10-1 10-1

10-1 10-1 10-1 10-1 10-1 10-1 lo-'

7.52 x 7.68 x 7.80 x 7.64 x 7.82 x 7.91 x 7.92 x 7.96 x 8.09 x 8.02 x 8.16 x 8.24 x 8.20 x 8.19 x

10-1 10-1 10-1

10-1 10-1 lo-' 10-1 10-I 10-1 10-1 10-1 10-1 10-1 10-1

I

I

64 2.10 x 10-1 3.30 x 10-1 4.19 x 10-1 4.69 x 10-1 5.22 x 10-1 5.67 x 10-1 6.07 x 10-1 6.21 x 10-1 6.46 x 10-1 6.69 x 10-1

7.60 x 7.65 x 7.76 x 7.76 x 7.93 x 7.91 x 8.05 x

10-1

10-1 10-1 10-1 10-I

10-1 10-1

8.11 x 10-1 10-1

8.13 x 8.14 x 8.26 x 8.29 x 8.29 x 8.23 x

10-1 10-1 10-1

10-1 10-1

where

where qBFSK = is the number of hopping slots available to a network employing BFSK modulation and C ( K ) is the channel capacity in [information bitdchannel use]. This gives the total average number of successfully transmitted information bits per second per Wz. When M-ary codes are employed, we model the coding channel as an M-ary symmetric discrete memoryless channel' with average symbol error probability P,(K) [6] for the case when the receiver simply makes hard decisions on each symbol without side-information. The channel capacity in this case can be shown to be [6]

'The channel is in fact not symmetric for M > 2 as discussed in Section V. The throughput may be improved upon by employing a nonuniform channel input distribution.

and P h = [5] denotes the probability of a hit when K = 2 and p ( K ' ) denotes the average probability of an M-ary symbol error given that a hop is hit by K' interfering users that are obtained through simulations for K' 2 1. For the case when perfect side-information is available to the receiver and the receiver uses this information to erase the symbols that were hit, we model the coding channel as an M ary symmetric discrete memoryless errors and erasure channel [6]. In this case, the probability of correct decision, error m d erasure denoted by P,( K ), P, ( K ) and P, ( K ), respectively, are given as

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CHEUN AND CHOI: PERFORMANCB OF FHSS MULTIPLE-ACCESS NETWORKS

2

1

3

5

4

a

7

6

10

9

K'

Fig. 2. Comparison of semi-analytic and waveform level simulations. No fading, asynchronous hopping, M = 8,

1

I

I

I

I

I

8= 30 dB.

I

0.9 - . . . . . . . .t . . . . . . . ..; . . . . . . . . .:. ........ .:, ........ ;. . . . . . . . . : . . . . . . . . . : .

.

j 30dE . . . . . . . . .:..

:. . . . . . . . . .: . . . . . . . . . ;

....................

I : - : ; /;

0.3 ........:............................................................. 0.2

........'.....

........................................

...I....

! / !/

.... . . . . . ' . . . . . . ~ . . . . .. . . I . .... . . .

0.1 "0

0.1

0.2

0.3

0.4

0.5

0.6

I

.........

0.7

j . . . . . . . . . . . . . . . . . .

0.8

0.9

1

Z

Fig. 3. Plots of f ( z ) for M

= 64,

% = 10,30 dB.

where p~ ( K )= (1- p h ) K - l is the probability that the symbol is not hit and p ( 0 ) is the probability of error due to background thermal noise only. The channel capacity for this channel can be shown to be [6]

CdK) = P H ( W log,(M(l - P ( 0 ) ) )

We may also consider binary coding in which case an M ary symbol is interpreted as being composed of log2 M binary

symbols. In this case, (18)-(23) may be employed to compute the channel capacity with M = 2 and p ( K ' ) replaced with 2 ( E 1 ) P(K').

v.

NUMERICALRESULTS

In this section, we present the simulation results for the symbol error probabilities when a hop is hit by K' interfering users and the normalized throughput computed from these


1520

0.3 . . . . . . . . . ..:. . . . . . . . . . .

‘“I

2 n I r.

. . . . . . . . .!. . . . . . . . . . ._:.. . . . . . . . ..;

. . . . . . . . . ..:. . . . . . . . . . . .

j”

01

, . \ ,

.....................

O

O

Y I I

OOL-

50

100

150

200

250

300

350

400

K Fig. 4. Normalized throughput for

$ = 30 dB.

No fading, M-ary coding, asynchronous hopping.

given above, we model the channel (no side-information) as an M-ary symmetric channel with error probability equal to the average error probability. Table I1 tabulates the simulated average symbol error problA. Symbol Error Probability abilities p(K’) for the above mentioned cases without fading Simulated estimates for the symbol error probabili- and Table I11 shows the simulated average symbol error ties for M = 2,4,8,16,32,64, K‘ = 1 , 2 , . . . , 3 0 and probabilities when the signals from different users are assumed Eb/No(Eb/No)= (E,/No)/log,M = 30 dB with and to suffer from independent Rayleigh fading for the case without Rayleigh fading are presented in this subsection. One of asynchronous hopping. Again 5000 errors were collected point to be noted before the simulation results are presented for each data point. Notice that the simulated symbol error is the fact that for M > 2 and asynchronous hopping, the probabilities are much smaller compared to the ( M - l ) / M symbol error probabilities depend on which of the M symbols bound. For example, when M = 32 without fading, using are actually transmitted. This is due to the fact that the the ( M - 1)/M-bound would mean that we use the value of amount of interference presented to each of the matched 0.96875 for p(K’) for all K’ 2 1 where the actual value filters in the receiver are different since the distribution of of p(K’) is 0.00901 for K’ = 1,0.0590 for K’ = 2 and the neighboring MFSK signal frequencies are different. For 0.513 for K’ = 10. We will see in the next subsection example, the statistic of the interference observed by the when normalized throughput is computed that this leads to matched filter matched to the MFSK signal located on the unrealistically pessimistic results. Finally, in order to verleft edge (corresponding to m k = 0) is different from that ify the semi-analytic simulation results, full waveform level observed by a matched filter matched to say m k = M/2. simulations were performed for the case when M = 8 and On top of this, given that a specific signal is transmitted, the & / N o = 30 dB without fading. The results are plotted in probability of error to each of the other possible signals are all Fig. 2 which shows good agreement with the semianalytic different. This results in a coding channel that is asymmetric. simulation results. To study the degree of the asymmetry, we present the In order to compute the normalized throughput, we also need simulated conditional error probability without fading given to evaluate the probability of symbol error when K’ = 0, Le., mk = 0 , 1 , 2 , 3 for M = 8, K’ = 1 and Eb/N0 = 10 dB in symbol error probability due to background thermal noise only. Table I where 5000 error events were collected for each data For the case without fading, we use the following formula for point. It is easily seen that due to symmetry, the average error the symbol error probability given in [7] probability when the transmitted symbol is M - 1 - m k is identical to the case when the transmitted data is m k . We note that the difference between the error probabilities for different m k is not excessively large. To get around the difficulty of having to deal with asymmetric channels and from the results results. We assume for simplicity that the received signal powers from all the users in the network are identical.


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Fig. 5. Comparison of normalized throughput for various systems for M = 4, when

where Q M ( . ,.) is the Marcum Q-function [ll]. The more simple closed form of [7, (2.14)] is not used since it involves alternating series computations and hence is numerically unstable for large M and high SNR's. For Rayleigh fading, the symbol error probability given by the following equation [8] is also inadequate for large M since it too involves numerical evaluation of an alternating series

Reference [8] suggests using numerical integration to evaluate the following expression instead of (25) for large M :

2 = 30 dB.

TABLE IV AVERAGEERROR PROBABILTY OF SYNCHRONOUS HOPPINGSYSTEM FOR M = 4 AND = 30 dB

SIMULATED

K' 11 N o fadine 1 Ravleinh fadinn

5.04 x 105.58 x 10-

6 7 8 9 10 11 12 13

But for high SNR's where the symbol error probability is small, the above p ( 0 ) = 1 - a form is again numerically unstable. To remedy this situation a change of variables may be used to rewrite (26) as (27) where (28) The numerical evaluation of (27) still results in convergence problems as can be seen in Fig. 3 where the integrand f ( x ) (which can easily be shown to be monotonically increasing) is plotted for M = 64 with E s / / N o as the parameter. The

No fading, M-ary coding.

14 15 16 17 18 19 20

5.96 x 106.18x lo-' 6.35x lo-' 6.48x lo-' 6.52x lo-' 6.53x lo-' 6.68~ lo-' 6.73 x lo-' 6.79x lo-' 6.79 x lo-' 6.83x lo-' 6.86x lo-' 6.90x lo-' 6.86x lo-' 6.97x lo-' 7.05x lo-'

6.06x 6.34x 6.29x 6.52x 6.65x 6.71x 6.75x 6.82 x 6.85x 6.97 x 6.89x 6.96x 6.92x 7.01 x 6.99x 7.05x

10-

lo-' lo-' lo-' lo-' lo-' 10-1 lo-'

lo-' lo-' lo-'

lo-' lo-' lo-' lo-' lo-'

resolution used in the integration is wasted in regions where the value of f ( z ) is nearly constant near zero and one thus, reducing the accuracy. One solution to this inefficiency is to evaluate the following approximation instead of (27):


1522

K

Fig. 6. Normalized throughput for

% = 30 dB.

Independent Rayleigh fading, A!-ary coding, asynchronous hopping.

0:

0

/

I

30.1 m

7

e

ir .: m

$ 0

00

K


where XI and A 2 are chosen to guarantee that the total error resulting from the truncation is less than 2 ~ Specifically .

$ = 30 dB. Independent Rayleigh fading, M-ary coding.

Rewriting (30) and (31) in terms of A1 and

A2,

we have


1523


2 = 30 dB. No fading, binary coding, asynchronous hopping.


x2

30 for the asynchronous system and K' > 20 for the synchronous system. In Fig. 4 is the plot of the normalized throughput for the asynchronous hopping case without fading with M-ary codes for &/No = 30 dB. This plot is actually a lower bound since the ( M - 1)/M-bound is employed for the symbol error probability when the number of interfering

1524



$ = 30 dB. Independent Rayleigh fading, binary coding, asynchronous hopping.

probabilities for the synchronous hopping system with M = 4 and &/No = 30 dB are tabulated in Table IV. Note that due to the fact that all hits are full hits unlike the asynchronous case, the error probabilities are very large even for small values of K’. The corresponding throughput is plotted in Fig. 5 (note 30. that p h = l / q for synchronous hopping.) We observe that Finally, Figs. 8-1 1 show the throughput curves when binary the decrease in the hit probability by employing synchronous codes are employed. We note that a practically manageable hopping is out-weighted by the increase in p ( K ’ ) and the value of M = 4 offers the maximum normalized throughput performance degrades compared to the asynchronous case as in all cases. with the case when M = 2 given in [l]. At this point, let us make a simple comparison with a narOne other observation made in [l] is that a system employrowband FDMA system. The maximum possible throughput ing perfect side-information to erase the symbols that were that may be obtained with a narrowband FDMA system using hit results in very poor performance compared to a systern MFSK modulation is bits/sec/Hz giving throughputs that simply makes hard decisions. In Fig. 5 , the normalized of 0.25 and 0.15625 for M = 16 and 32, respectively. throughput of a system using perfect side-information to On the other hand, from Fig. 5 , we see that the maximum erase all hit symbols for M = 4 and Eb/No = 30 dB is throughput for AFHSS-MA networks with M = 16 and plotted. We note that, as with the M = 2 case considered 32 are approximately 0.30 and 0.25 for Eb/No = 30 dB in [l], the system that simply makes hard decision on the when K = 150 and 120, respectively. This shows that it symbols without side-information far outperforms a systern is possible to obtain a larger throughput and thus, improved employing side-information to erase all the symbols that were bandwidth efficiency with asynchronous FHSS-MA compared hit. This is due to the fact that erasing all the hops that to narrowband FDMA using MFSK modulation for some are hit results in excessive erasures of symbols that would modulation orders.* This is mainly due to the fact that with have been demodulated correctly if they were not erased. large M , the average error probability is maintained at a low Also shown in Fig. 5 are the normalized throughput curves enough level to be handled by the channel code even when K computed using the ( M - l)/M-bound for comparison. These exceeds the number of hopping slots. plots compared to the one computed using the simulated Next, we compare the performance of synchronous and symbol error probabilities clearly show that the bound is overly asynchronous hopping systems. The simulated symbol error pessimistic, especially for the asynchronous case. Next, in Figs. 6-7, similar results are presented for the *Of course, this is assuming channel capacity codes and disregarding network delays Also, with narrowband FDMA, more bandwidth efficient case with independent Rayleigh fading. It is clear that fading increases the vulnerability due to background thermal noise modulation schemes may be employed

users exceed 30. From this plot, we find that there exists a value of M that results in the maximum achievable throughput. The abrupt decrease in the nqmalized throughput for large K for M = 32 and 64 are due to the ( M - 1)/M upper bounding of the symbol error probabilities for K’ >


1525

I

I

K


but it is unclear whether it should increase or decrease the degrading effects of the interfering users compared to the case without fading since not only the intended signal but also the interfering signals fade. The throughput results show that independent Rayleigh fading results in approximately a 30% decrease in maximum normalized throughput for M = 4 and E,/No = 30 dB. VI. CONCLUSION In this paper, we obtained simulated estimates for the symbol error probabilities of synchronous and asynchronous FHSS-MA networks using MFSK modulation with noncoherent detection for M = 2,4,8,16,32, and 64 with and without independent Rayleigh fading. An appropriately normalized throughput measure was defined in order to make a fair comparison between the performance of systems employing different modulation orders. Modeling the channel as an M ary symmetric memoryless channel, it was found that there exists an optimum value of M that should be used to obtain the largest possible throughput. It was also observed that the anomalies caused by the ( M - 1)/M-bound noticed in [l] for M = 2 are also present for the cases for M larger than two.

3 = 30 dB. Independent Rayleigh fading, binary coding.

with p k = F.In order to evaluate the above integral we need to evaluate the following integral:

First, let us consider the case when 0 5 r k 5 T . In this case the above integral is equal to (T - r k ) for the case when m k = 1. Otherwise, the integral may be evaluated to be

Next, when -T < Tk < 0, (35) is equal to ( T f r k ) when m k = 1. Otherwise the integral may be evaluated to be

Combining these results, it is a simple matter to show that (6) is valid.

APPENDIXI The derivation of (6) repeated below is presented in this appendix

REFERENCES [1] K. Cheun and W. E. Stark, “Probability of error in frequency-hop spreadspectrum multiple-access communication systems with noncoherent reception,” ZEEE Trans. Commun., vol. 39, pp. 1400-1410, Sept. 1991. [2] C. M. Keller, “An exact analysis of hits in frequency-hopped spreadspectrum multiple-access communications,” in Proc. Con$ Inform. Sci. Syst., Mar. 1988, pp. 981-986. [3] E. Geraniotis, “Multiple-access capability of frequency-hopped spreadspectrum revisited: An analysis of the effect of unequal power levels,” IEEE Trans. Commun., vol. 38, pp. 1066-1077, July 1990.

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[4] R T Short and C K. Rushforth, “Probability of error €or noncoherent frequency-hop spread-spectrum multiple-access communications,” Unisys Tech Rep. PX18829, May 1988 [5] E A. Geraniotis, “Coded FWSS communications in the presence of combined partial-band noise jamming, Rician nonselective fading, and multiuser interference,” IEEE J. Select Areas Commun , vol. SAC-5, no. 2, pp. 194-214, Feb 1987. 161 R. G. Gallager, Information Theov and Reliable Communications. New York Wiley, 1968 [7] M. K. Simon, J K. Omura, R A. Scholts, and B K. Levitt, Spread Spectrum Communications Rockville, MD Computer Science Press, vol I, 1985. [8] J. G Proakis, Digztal Communications New York McGraw-Hill, 1983 [9] S f i m and W. Stark, “Optimum rate Reed-Solomon codes for frequency-hop spread-spectrum multiple-access communication system,” IEEE Trans. Commun , vol 37, pp 138-144, Feh 1989. [IO] M B Pursley, “Frequency-hop transmission for satellite packet switching and terrestnal packet radio networks,” IEEE Trans. Inform Theory, vol 32, pp 652-667, Sept. 1986. [ll] J. I. Marcum, “Table of &-functions,” Rand Corporation Report RM339, Jan. 1950

Kyungwhoon Cheun (S’88-M’89) was born in Seoul, Korea, on December 16, 1962 He received the B A degree in electronic engineering from Seoul National University, Korea, 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 engineenng From 1987 to 1989, he was a Research Assistant at the Electronic Engineering Communication Systems Department at the University of Michigan, Ann Arbor and from 1989 to 1991 he was with the Electrical Engineering Department at the University of Delaware, Newark, as an Assistant Professor In 1991, he joined the Electronic and Electncal Engineering Department at the Pohang university of Science and Technology (POSTECH), Korea, where he is currently an Associate Professor. His current research interests include cellular and packet radio networks, algorithms, and VLSI design for US and European HDTV modems, military communication networks, synchronization, and equalization for radio systems, and free space laser optic communication systems. He has also served as an Engineering Consultant to various industries in the areas of modem, VLSI design, and cellular networks.

Kwonhue Choi was horn in Mokpo, Korea, in 1970 He received the B.S. and M.S. degrees in electronic and electrical engineenng from the Pohang University of Science and Technology (POSTECH), Korea, in 1994 and 1996, respectively. Since 1994, he has been a Research Assistant at the Department of Electronic and Electncal Engineenng, POSTECH where he is currently working toward the Ph.D. degree His research interests include spread spectrum communicaaons with emphasis on frequency hopped multiple access networks and synchronization for G A HDTV receivers.