PERFORMANCE COMPARISON OF FREQUENCY

PERFORMANCE COMPARISON OF FREQUENCY-HOP SPREAD-SPECTRUM MULTIPLE-ACCESS NETWORKS USING DIFFERENT MODULATION ORDERS KYUNGWHOON CHEUN Department of Electrical and Electronics Engineering Pohang Institute of Science and Technology (POSTECH) P.O.Box 125 Pohang 790-600 KOREA

Abstract

The main concern of this paper is to estimate the symbol error probability of asynchronous FHSS multiple-access networks through Monte Carlo simulations and use this estimate to more accurately compare the performance of FHSS multiple-access networks with different modulation orders M . We concentrate on systems transmitting one M-ary ( M 2 2) orthogonal FSK modulated symbol per hop with noncoherent demodulation. The usual practice is to upper bound the probability of symbol error when a hop is hit by K’ 2 1 other users by ( M - 1)/M (or 1.) Recent work [l] on the derivation of accurate approximations to this probability for the case when M = 2 has indicated that the aforementioned bound not only gives excessively pessimistic results but may also lead us to make wrong trade-off decisions. In this paper, using the simulated values for the error probability we show that the same is true for the case when M > 2. Also by employing a normalized throughput measure that takes into account the bandwidth expansion associated with the modulation order M > 2, we show that there exists an optimum value of M for a given bandwidth that should be used to obtain the largest possible network throughput for a given bandwidth. Simulations and throughput results are also given for the case when the signals from the active users in the network suffer from independent Rayleigh fading.

I. INTRODUCTION In this paper, estimates for the symbol error probabilities obtained via Monte-Carlo simulations are used to accurately assess the performance of Asynchronous Frequency-Hop Spread-Spectrum MultipleAccess (AFHSS-MA) networks. We concentrate on systems employing orthogonal MFSK modulation to transmit one M-ary symbol per hop at the transmitters along with noncoherent demodulation at the receivers. The lowest level of performance criteria on which all other performance criteria depend for 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 works on such networks are based on the simplifying bound that the probability of symbol error is ( M 1)/M (or 1) whenever a hop i s h i t by one or more interfering users. In other words, hits by any number of other interfering users in the network is considered to be detrimental. It is only recently that researchers began investigating the validity and the tightness of this bound [I, 2, 3, 41. In [l],an approximation for the error probability was derived that very accurately fits the simulated results for the case when M = 2. The search for an accurate and computationally tractable analytical approximation for the cases when M > 2 proved unfruitful.

-

One observation made in 111 is that upper bounding symbol error probability whenever a hop is hit by more than one interfering user by ( M - 1)/M (this will be referred to as the ( M - 1)/M-bound) results in an excessively pessimistic estimate of the network performance. Another more interesting observation is that the bounding technique could give misleading results when comparing the perfor-

mance of systems employing different strategies (thus different complexity.) One good example is that of comparing a system using perfect side-informatioio to erase the hops that are known to be hit with a system that simply makes hard decision on all the received symbols. The analysis using the l/2-bound ( M = 2) predicta that the system with perfect, side-information performs much better than the one without perfect side-information and hence the additional complexity required t o accurately detect the hits is justified. On the other hand, analysis biised on the accurate approximations fcir the symbol error probability indicate that the system without perfect side-information performs significantly better than the one with perfect side-information. In fact, assuming codes that achieve channel capacity are employed, the system without perfect side-information provides more than twice the maximum throughput and more graceful degradation as the number of users increase. Another exaimple is the comparison between synchronous and asynchronous hopping networks where synchronizing the hopping patterns of the users in the network would introduce additional complexity. Again under the 1/2-bound, it is predicted that the synchronous hopping networks perform far better than the asynchronous hopping networks. But the results using the accurate values for the error probabilities indlicate 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 wrong design decisions. Unfortunately, expressions for the symbol error probabilities for M > 2 that are both accurate and numerically tractable is not yet known. The main purpose of this paper is to derive a description of the AFHSS-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 ) for a given bandwidth by defining a normalized throughput measure that is normalized by the bandwidth expansion caused by using a larger M for a given data rate. It is found that there exists an optimum M that should be employed in order to achieve the largest possible maximum throughput for a given bandwidth. We also consider the case when the signals from the users suffer from independent Rayleigh fading.

11. SYSTEM AND CHANNEL MODEL The system considered in this paper is that of an AFHSS-MA network with K identical active users (transmitter-receiver pairs.) 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 outputs of the receiver matched filters and the simulation itself while providing results that are almost identical to those obtained assuming memoryless hopping patterns. The transmitters send one M-ary symbol per hop in one of the q available frequency slots using noncoherently orthogonal MFSK

modulation. At baseband, the MFSK frequencies allotted to the A4 symbols are

lQl=A where T is the duration of one symbol (hop.) 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, T ] ) can be written as follows when the hop is hit by K’ interfering users.

l&Tll -

1 = 0,1,... , M - 1

as the new decision variables, we arrive at the following expression

for

I%l,

A ( [ ,mkt P k )

’

where pk =

WI

and E, = S I T . Also,

A ( [ , m , p ) = ( l - bkl)6rnk,l - lbkIsinc(r(mk - I ) P k ) ( l e([, m , p ) = - X p k ( m k 1 )

+

received signal power of the k-th user. delay of the k-th user assumed to be uniformly distributed on [-T, T ) and independent between users ( T I = 0.) symbol transmitted by the k-th user assumed to be independent between users and uniformly distributed on { O , l , .. ., M - 1 ) . random phase of the k-th user assumed to

-

b k , r )

(9)

and vi are zero mean complex Gaussian random variables with E{qvi+) &,j /(E,/”o).

B. Independent Rayleigh fading Similar procedures may be followed to obtain the following expression for the decision statsitics when the signals arriving at the receiver suffer from Rayleigh fading with independent fading statistics

be independent between users and uniformly distributed on [O, 27r). complex White Gaussian Noise (WGN) process with E { z ( t ) z * ( r ) }= 2 N 4 t - r ) where N0/2 is the twesided power spectral density of the AWGN at the receiver input. Also, 6(.) is the Dirac delta function and t* denotes the complex conjugate of z.

[o>a

5.

where Hj Here, it is easily shown that Hk and pk are zero mean i.i.d. Gaussian randomvariables with E { H k H ; } = 2 and E { / l k P i } =

A where 3,is the average received signal energy from the paired EsINo transmitter. Also, 31: is the average received signal power from the k-th transmitter.

1 t E 0 otherwise.

N.NORMALIZED THROUGHPUT After observing r ( t ) in the interval [ O , T l , the receiver computes the M decision variables lull, 1 = 0,1, . . . ,M - 1 given below and chooses the index of the largest decision variable as the estimate (Fig. 1 . )

=

/

/lT

r ( t )exp (-jF dt ) 1 = 0,1, . . . , M - 1.

(3)

The performance measure considered here is an appropriately normalized network throughput which allows us to compare systems with different modulation orders. We define a normalized throughput was W =

rKLPC(K)log2M QqM

(11)

where

111. THE DECISION VARIABLES In this section, we derive expressions for the decision variables used for the Monte Carlo simulations. We first consider the case when there is no fading and then the case when the signals from different transmitters suffer from independent Rayleigh fading.

A. No fadinn By inserting ( 2 ) into ( 3 ) and noting that forward to show that U1 may be written as

TI

= 0, it is straight-

r

=

rate of the error correction code used (1‘4 in [information symbols/channel symbol] probability of a packet being received error free( 13) (14) number of frequency hopping slots available number of M-ary code symbols per packet. (15)

Also, q~ is the relative bandwidth required for M > 2 per hopping slot relative to the case when M = 2 for a given data rate. If we approximate the bandwidth of a binary orthogonal FSK modulated signal operating at bit rate 1by and the bandwidth of an M-ary T4 orthogonal FSK modulated signal operating at the same bit rate by M Tbrog,M 9 V h f is given by

&

qM

+

210g2M.

(16)

dt.

Hence the factor Q ~ in M the denominator of the definition of the normalized throughput denotes the total normalized bandwidth used by a system and penalizes the throughput accordingly. The normalized throughput defined as above gives the average number of successfully transmitted information bits per packet over the network per unit bandwidth given that there are K active users.

T and defining

Here, in order to concentrate on the effects on the system peformance of using different modulation orders M , we restrict ourselver

iT (-jF) r(t)exp

After normalizing Ut by a

M

___

to codes achieving channel capacity. The normalized throughput in this case can be written as,

sible signals are all different. This results in a coding channel that is asymmetric.

To study the degree of asymmetry we present the simulated probability of error without fading given mk = 0 , l for M = 4, K" = 1 and = 10 dB in Table 1 where 5000 error events were collected for ea& data point. It is easily seen that due to symmetry, the error probability when the transmitted data is M - 1 - mk is identical to the case when the trancimitted data is mk. We note that the dlifference between the error probabilities for different mk is not excessively large. In Table 2 we present similar results for the case when M = 8, K' = 1 and = 10 dB. To get around the difficulty of having to deal with asymmetric channels and from the results given above we will model the channel (no side-information) as an M-ary symmetric channel with error probability equal to the average of the probabilities given above.

3

where C ( K ) is the channel capacity in [information bits/symbol] given that there me K active users in the network. This gives the average number of information bits that are successfully communicated over the network per code symbol per unit bandwidth. Modelling the coding channel as an M-ary symmetric discrete memoryless channel' with symbol error probability P e ( K )for the case when the receiver simply makes hard decisions on each symbol without sideinformation, the channel capacity can be shown to be [6]

where

a

and Ph = [5] denotes the probability of a hit when K = 2 and p(K') denotes the average probability of error given that a hop is hit by K' interfering users that are obtained through simulations for K' 2 1.

2

Tables 3-10 tabulate the simulated average symbol error probai bilities p(K') for the above mentioned cases without fading. Again 5000 errors were collected for each data point. Notice that the simulated symbol error probabilities are much smaller compared to the ( M - l)/M-bound. For example, using the ( M - 1)/M-bound would mean that we use the value of 0.96875 for p(K') for all K' >_ 1where the actual value of p(K') is 0.058 for K' = 1, 0.12 for K' = 2 and 0.54 for K' = 10. We will see in the next section when normalized throughput is computed that this leads to unrealistically pessimistic results.

B. Normalized throurrh~ut

For the case when perfect side-information is available to the receivers and the receivers use this information to erase the symbols that were hit, we model the coding channel as an M-ary discrete memoryless symmetric errors and erasure channel with correct, error and erasure probabilities given by

In this subsection, results for the normalized throughput defined in Section IV are presented. Recall that the normalization is such that the bandwidth penalty for employing a larger modulation order M is taken into account For the following numerical results, we take q to be 100.

p c ( K ) = (1- P(O))PH(K)

In Fig. 2 is the plots of the normalized throughput without fading for Eb/No=lO and 30 dB. Next in Fig. 3, the maximum attainable normalized throughput ILV,, is plotted as a function of the modulation order M . These plots are actually lower bounds in that we used the ( M - l)/M-bound for the symbol error probability when the number of interfering users exceed 10. From these plots we find that there exists a value of M that results in the maximum achievable throughput. Here M = 16 gives the best throughput performance giving not only the largiest maximum throughput but also throughput close to the maximum (among all possible M ) for all values of K . We also note that the value of K that achieves the maximum normalized throughput increases as M increase.

Pz(K) = 1 -PH(K)

(22)

where ~ H ( K= ) ( l - ~ h ) ~ - land p(0) is the probability of error due to background thermal noise only. The channel capacity for this channel can be shown to be,

C s ( K ) = PHKlogZ(M(1 -P(o)))

(23)

At this point, let

V. NUMERICAL RESULTS 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 results. We assume that the received signal power from all the users in the network are identical. A. Symbol error probability

Simulated estimates for the symbol error probabilities for M=2,4,8, K'=1,2, ...,10 and Eb/No=lO and 30 dB. One point to be noted before the simulation results are presented is the fact that for M > 2, the symbol error probabilities depend on which of the M symbols are actually transmitted. This is because the amount of interference presented to each of the matched filters in the receiver are different since the distribution of the neighboring MFSK signal frequencies are different. For example, the statistics of the interference observed by the matched filter matched to the MFSK signal on the left edge (corresponding to mk = 0) is different from that observed by a matched filter matched to say (A4 - 1)/2. O n top of this, given that a specific signal is transmitted, the probability of error to each of the other PO* Ithe channel is in fact not symmetric for M > 2 as discussed in Section v.

uti

make a simple comparison with a narrow-

band system, i.e. when q = 1. For a narrowband system with M = 16, the largest possible number of successful information bits thait can be transmitted per symbol per unit bandwidth is 4/r]ls = 4/2 = 2. On the other hand, the maximum normalized throughput achieved with the AFHSS-MA system with M = 16, q = 100 and &,/No = 10 dB is approximately 2.2. This implies that the AFHSS-MA system achieves a larger maximum throughput than the narrowband SYStem. The reason for this is that the hits in an asynchronous AFITSSMA network are mostly partial hits due to a small number of interfering users and the error probabilities induced by these hitci are small. If we were to employ hopping patterns that are synchronized among the users (synchronous hopping), it is not possible to obtain throughput larger than a narrowband system. For a more amurate comparison of synchronous and asynchronous hopping systems, we simulated the symbol error probability for a synchronous hopping system with M = 16 and Eb/N0=50 dB and the corresponding throughput is plotted in Fig. 4 (note that Ph = l / q for synchroioous hopping.) We note that as with M = 2 case[l], the asynchronous hopping system shows superior performance in terms of maximum achievable throughput, contradicting the predictions made using the ( M - l)/M-bound [9, 101. Hence at least for the system modell b o

ing considered in this paper, the additional complexity that must be introduced to synchronize the hopping patterns of the users in the network is not justified. Another interesting observation is that the degradation of the normalized throughput for the synchronous hopping network is extraordinarily graceful as the number of active users in the network increases. It achieves higher throughput than the asynchronous network for K 2 555 and maintains usable throughput beyond K = 1000. The reason for the asynchronous hopping network having superior performance over the synchronous hopping network for the case when K is not excessively large (< 555) is due to the fact that although the asynchronous network is usually hit by a larger number of interfering users compared to the synchronous network, the hits are partial hits due to a small (< 10) number of interfering users which are far less severe than the full hits suffered by the synchronous case. Hence in this region of K , the fact that the asynchronous system has a smaller symbol error probability for a given K’ overrides the fact that it is hit by a larger number of interfering users on the average. On the other hand, when K is very large the number of interfering users in a hop is usually quite large (most above 10 for K = 800) for the asynchronous case. Hence the resulting symbol error probabilities will be quite large ( ( M - 1)/M-bound is used for K’ > 10) for both the asynchronous and the synchronous systems and thus in this region, the synchronous system with a smaller Ph will be the one with a larger throughput. One other observation made in [l] is that a system employing perfect side-information to erase the hops that were hit gives very poor performance compared to a system that simply makes hard decisions. In Fig. 4, the normalized throughput of a system with perfect side-information for M = 16 and Ea/No=50 dB is plotted. We note that as with the M = 2 case considered in [l],the system that simply makes hard decision on the symbols without side-information far outperforms a system employing side-information to erase all the hops that were hit. This is due to the fact that erasing all the hops that are hit results in excessive erasures of symbols that would have been demodulated correctly if it were not erased. Also shown in Fig. 4 is the normalized throughput curve computed using the ( M - l)/M-bound for comparison. This plot compared to the one computed using the simulated symbol error probabilities clearly shows that the bound is overly pessimistic.

References

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[8] J.G. Proakis, Digital Communications, McGraw-Hill, 1983. [9] S. Kim and W. Stark, “Optimum rate Reed-Solomon codes for frequency-hop spread-spectrum multiple-access communication system,” IEEE Trans. Commun., Vol. COM-37, pp.138-144, Feb. 1989.

[lo] M.B. Pursley, “Frequency-hop transmission for satellite packet switching and terrestrial packet radio networks,” IEEE Trans. Inform. Theory, Vol. IT-32, pp.652-667, Sept. 1986. [ll] J.I. Marcum, “Table of &-functions,” Rand Corporation Report RM-339, Jan. 1950. Acknow1edegment:This work was supported in part by the Korea Telecommunications Research Center and the Korea Science and Engineering Foundation symbol error probabilty 8.52 x 109.20 x i n -

Table 1: Simulated error probabilty given mk for M = 4, K‘ = 1 and = 10 dB.

3

mk I symbol error probabilty 0 1 7.24 x 1 7.58 x 2 7.70 x lod2 3 7.73 x in-2

Table 2: Simulated error probabilty given mk for M = 8, K’ = 1 and $ = 10 dB.

9.59 x 1.77 x 2.35 x 2.66 x

10101010-

1.60 x 1.34 x 2.04 x 2.46 x

10101010-

[l] K. Cheun, W.E. Stark, “Probability of error in frequency-hop spread-spectrum multiple-access communication systems with noncoherent reception,” IEEE Trans. Commun., Vol. COM-39, pp. 1400-1410, Sept. 1991. [2] C.M. Keller, “An exact analysis of hits in frequency-hopped spread-spectrum multiple-access communications,” Proc. Conf. Inform. Sci. Syst., pp. 981-986, Mar. 1988. [3] E. Geraniotis, “Multiple-access capability of frequency-hopped spread-spectrum revisited: an analysis of the effect of unequal power levels,” IEEE Trans. Commun., Vol. COM-38, pp. 10661077, Jul. 1990.

Table 3: Simulated average error probabilty for M = 2 and and 30 dB.

%= 10

Table 4: Simulated average error probabilty for M = 8 and and 30 dB.

%= 10

[4] R.T. Short and C.K. Rushforth, “Probability of error for nonce herent frequency-hop spread-spectrum multiple-access communications,” Unisys technical report No. PX18829, May 1988. [5] E.A. Geraniotis, “Coded FH/SS 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.194214, Feb. 1987. [6] R.G. Gallager, Information Theory and Reliable Communications, John Wiley & Sons, Inc. 1968. [7] M.K. Simon, J.K. Omura, R.A. Scholts and B.K. Levitt, Spread spedrum communicaiions Volume I, Computer Science Press, 1985.

M

K'

8 8

1 2

P(K')IE~/N,, = 10 dB 7.62 x lo-'

8 1 4 1 8 1 5 1

10

1.75 x 2.68 x 3.50 x 4.09 x 4.54 x 5.00 x 5.30 x 5.64 x 5.89 x

P(K')IE~/N,, = 30 dB

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

I

I

1

101010-

1.40 x 1.25 x 2.27 x 3.16 x 3.80 x 4.33 x 4.76 x 5.12 x 5.46 x 5.72 x

10-1 lo-' 10-1 1010101010-

Table 5: Simulated average error probabilty for M = 8 and and 30 dB.

L

M 16 16 * 16 16 16 16 16 161

K' 1 2 3 4 5 6 7 8

I

~ ( K ' ) ~ E ~=/ 10 N ,dB , 6.67 x lo-' 1.50 x 10-1 2.30 x lo-' 3.06 x 10-1 3.74 x 10-1 4.30 x IO-1 4.79 x i n - 1 5.18 x lo-'

H

10-1

%= 10

P(K')IE~/N,, = 30 dB 1.14 x lo-" 9.01 x 10-2 1.74 x 10-1 2.53 x IO-' 3.30 x 10-1 3.89 x 10-1 4.m x 10-1 4.97 x 10-1

Table 9: Simulated average error probabilty for M = 128 and 10 and 30 dB.

L

2.27 x lo-'

3.72 x 10-1

Table 6: Simulated average error probabilty for M = 16 and and 30 dB.

321 321 32 32 32

6 7 8 9 10

I

I

3.82 x 4.24 x 4.70 x 5.13 x 5.40 x

lo-' 10-1

lo-' lo-' 10-1

I .

3.25 x 3.79 x 4.26 x 4.77 x 5.13 x

%= 10

Table 10: Simulated average error probabilty for M = 256 and 10 and 30 dB.

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

4I Table 7: Simulated average error probabilty for M = 32 and and 30 dB.

M

%= 10

1

1 K' I P(K')IE~,N,,= 10 dB I ~(K')IE,/N,,= 30 dB

641 64 64 64

1 II 2 3 4

5.38 x 1.09 x 10-1 1.66 x lo-' 2.19 x 10-1

I

1

7.32 x lo-" 3.83 x lo-' 8.12 x 1.33 x 10-1

Figure 1: The receiver. Table 8: Simulated average error probabilty for M = 64 and and 30 dB.

-&

$=

10

lud

n $=

1

I 2 5

2 5 1

2

s

A

A

Asynchronous hopping No ride-miormailon Simulated

B.

s

c.

s nchronoun hopplng d side-iniormatlon (M-l)/M-bound

D.

AS

E:

Arychro?our hopping No side-iniormation (M-l)/M-bound

1.5

nrhronous hopping d side-miormstlon simulated

nshronovr hopping S,dYe-infO"alion

1

0.5

0 O

100

200

300

400

500

600

100

800

900

IO00

R

Figure 2: Normalized throughput for

800

IO00

2 = 10 dB. No fading. Figure 4: Comparison of normalized throughput for various systems for M = 16, when = 50 dB. No fading.

2.5

2

m lOab

1.5

3Odb

5Qdb

1

0.5

0 10

"

Figure 3: wmor versus M . ho fading

100