PARAMETRIC DIVERSITY COMBINING IN FAST ... - CiteSeerX

PARAMETRIC DIVERSITY COMBINING IN FAST FREQUENCY HOPPING Helene Tayong Antoinette Beasley Arlene Cole-Rhodes

A. Brinton Cooper III

Gonzalo Arce

Department of Electrical & Computer Engineering Morgan State University Baltimore, MD 21251 ftayong, beasley, [email protected]

Computational & Information Sciences Directorate US Army Research Laboratory APG, MD 21005 [email protected]

Department of Electrical & Computer Engineering University of Delaware Newark, DE 19716 [email protected]

Abstract

nel. Much previous work on FFH/MFSK diversity combiners [1] [3],[5{9]uses the \unconventional" [10] model of FFH/MFSK in which the number of hopping carriers equals the number of MFSK modulation tones. This model was introduced by Goodman [11] who studied the dependence of FFH/MFSK error probability on system bandwidth, transmission rate/channel, number of simultaneous users, and noise and propagation. In previous work [1{3], we examined the performance of the fuzzy rank order detector (FROD) on the unconventional system model, observing that it performs at lower bit error rates thatn do some conventional threshold [5], rank-sum [6], and related [7{9] techniques. In the sequel, FFH/MFSK for which the number of hopping carriers exceeds the number of modulation tones will be called conventional FFH/MFSK

The Fuzzy Rank Order Detector (FROD) is extended to fast frequency hopping systems with M ?ary frequency shift keying modulation in which the number Q of hopping carriers is much greater than the number M of modulation tones. Comparisons with other diversity combiners show that the FROD continues to oer improved bit error probability at a cost of bandwidth eciency. Comments on the utility of this nding to battle eld communications are included. Keywords |frequency hopping, non-coherent MFSK, fuzzy rank order detector, bandwidth eciency

F

I. Introduction

REQUENCY hopping waveforms present a class of attractive choices for mobile, multiuser, battle eld communications waveform [1{3] designs. Previous work makes a compelling case for the use of frequency hopping to provide timely, secure, and high quality battle eld communications to the Objective Force in highly portable and mobile packages. The low multiple access interference (MAI) and lack of a need for complexities such as closed loop power control assure delivery of information with a quality of service that supports the war ghter's needs in a simple, robust system.

Although the unconventional MFSK/FFH system model seems to be heavily endowed with multiple access interference (MAI), it is easy to show that it oers maximizes bandwidth eciency for xed bit error probability. In military applications, however, the need to resist jamming calls for a very large set of hopping carriers. With MFSK modulation, this can be accomplished in two ways: by unconventional FFH/MFSK with a very large number of modulation tones and by conventional FFH/MFSK with a conventionally small number of modulation tones. The purpose of this paper is to study how the fuzzy rank order detector (FROD) aects the bandwidth eciency and the probability of bit error performance for conventional and unconventional FFH/MFSK

In this work, we focus on fast frequency hopping (FFH) in order to exploit its frequency diversity on the frequency selective fading channel. While coherent modulation provides somewhat superior performance to the noncoherent case [4], coherent reception may be dicult to achieve when using frequency hopping in the dicult environment posed by mobileto-mobile communications in the battlespace. Hence, we study noncoherent MFSK in an attempt to achieve good bandwidth eciency without insisting on a coherent chan-

We present the system and signal models in Section 2 and derive comparisons of the bandwidth utilization for the conventional and unconventional FFH/MFSK systems in Section 3. In Section 4, we compare experimental results in conventional FFH architectures for the FROD [1{3] and for other diversity combiners. Conclusions and near-term future plans are presented in Section 5. This continues to be work in progress.

Dr. Cooper is also Adjunct Professor of Electrical and Computer Engineering at Morgan State University and at the University of Delaware. Paper was prepared through collaborative participation in the Advanced Telecommunication and Information DistributionResearch Program (ATIRP) sponsored by the U.S. Army Research Laboratory under the Federated Laboratory Program, Cooperative Agreement DAAL029602-0002. The U.S. government is authorized to reproduce and distribute preprints for Government purpose, notwithstanding any copyright notation thereon.

II. System Models Let 1

0.002

(log (x)/log (2))/(5*1024)

1 M-ary FSK

M

Bandwidth Efficiency Per User

S/P

Q-ary FH

Fig 1. FH Transmitter

0.001

    

K = number of users Q = number of hop frequencies M = number of modulation tones L = number of hops per information symbol T = duration of an M -ary information symbol

100

800

is maximized when Q = M .

900

1000

2

Since there is no coordination among users, hits are assumed to be independent of one another and, hence, the number of hits on a hop is binomially distributed with a mean of  = Ph (KQ ? 1); H (3) where Ph is given by (4).

De nition: Bandwidth eciency is the information rate achieved per unit bandwidth at a speci ed bit error probability.

For illustration, x L and M and choose Q and K so that  = KM ? 1, where KM is the number of users in the unH conventional system. Then the average number of collisions per hop in the conventional system equals the actual number of collisions per hop in the unconventional system. This provides a comparison of the worst case condition in which a symbol in the conventional system experiences a hit on every hop.

Let 1=Th = L=T . This is the instantaneous bandwidth of the FFH/MFSK signal and the hopping carriers must be spaced by at least this amount. Each user transmits one M ?ary signal every T seconds at an information rate of log2 M=T b=s. The total system bandwidth occupied by the FFH signals is

By Lemma 1, the bit error probability will be smaller for the conventional than for the unconventional FFH system, provided that the conditional bit error probability given a hit is larger than the conditional bit error probability given no hit (which is a reasonable expectation). The data in the next section con rm this nding.

= Q
(1=Th ) = LQ=T and the bandwidth eciency per user is  = log2 M=T Wss

=

400 500 600 700 Number M of Modulation Tones

Lemma 2: In FFH/MFSK, the system bandwidth eciency

III. Bandwidth Eciency

LQ=T log2 M LQ

300

Fig 3. Per User Bandwidth Eciency vs Size of MFSK Constellation for Fixed System Bandwidth

Every T seconds, a binary source emits k independent symbols which are mapped into one M ?ary symbol. The value of this symbol determines the frequency of the output of an MFSK modulator. The MFSK signal over T seconds is hopped over a sequence of L carrier frequencies, as selected by a Q?ary sequence generator. Thus, the modulated signal hops to a new frequency every Th = T =L seconds.

u

200

IV. Experimental Results

(1)

In conventional FFH, the probability of a given user experiencing a hit from any of the other Kc ? 1 users is  1 Kc ?1 : (4) Ph = 1:0 ? 1 ? Q

(1) is plotted in Figure 3. The system bandwidth eciency easily follows as K
log2 M = : (2) LQ

Since K is a function of the output bit error rate, Pe ,  can be speci ed for a xed Pe in order to make fair system comparisons.

The average number of hops experiencing a hit in the conventional system is H c = Ph (Kc ? 1) where Kc is the number of users in the conventional system. Setting H c = Ku , the 2

FROD/MRSR vs Wideband FROD/MRSR

−1

0.1

10

"FROD_u.dat" "FROD_c.dat"

Probability of Bit Error

0.01

−2

Prob. of Error

10

Unconventional FFH 0.001

Conventional FFH

Q = 1024 L=5 M = 32

0.0001 −3

10

FROD Wideband FROD MRSR Wideband MRSR

1e-05 0

0.05

0.1

0.15 0.2 System Bandwidth Efficiency

0.25

0.3

0.35

−4

10

0

20

40

60 Number of Users

80

100

120

Fig 5. Probability of Bit Error vs System Bandwidth Eciency for FROD.

Fig 4. Probability of Bit Error vs User Population Size for Conventional and Unconventional FFH using FROD and MRSR Combiners. L = 5; Q = 1024; M = 32.

cious appetite for channel bandwidth. However, the picture is changed by jamming considerations. In unconventional FFH/MFSK, received energy at any hop frequency will appear in the receiver. Hence, any in-band jammer will provide interference in addition to the MAI with which the receiver must deal. In conventional FFH/MFSK, the probability that a given jammed frequency places energy in the diversity combiner is given by (4). So, for Q = 1000 and K ? c = 100, the probability that a jammed hop frequency places interference in the receiver is 0.09, and the conventional system oers an advantage in jam resistance that cannot be ignored. Finally, we have shown that the advantage of the FROD over competing diversity combiners continues in the conventional as well as the unconventional FFH/MFSK system.

number of users in the unconventional system creates a similar (but not identical) MAI environment in the conventional receiver for symbols aected by typical numbers of interferrers. We simulated both systems using these values of user population size in order to achieve the same range of bit error probability. The results are in Figures 4 and 6 (WE HOPE! ****). Figure 4 shows simulation results comparing the error probabilities of the FROD and the maximum rank sum receiver (MRSR) for unconventional FFH/MFSK [1] and new results for conventional FFH/MFSK for Q = 1000, L = 5, and M = 32: Using (2) with Figure 4 gives the relationship between error probability and system bandwidth eciency  for both conventional and unconventional FFH shown in Figure 5.

Near-term future plans include:  investigation of the increase in bandwidth eciency af-

forded by channel coding;  quanti cation of the performance for equal spread bandwidths;

V. Conclusions and Future Plans Only a few authors have examined the spectral eciency of FFH. Fiebig [10] studied FFH/MFSK systems with channel coding and obtained values as large as 0.40 for the Gaussian channel without fading. When frequency selective fading was included [12],  dropped to 0.36 for the chip synchronous case and 0.19 for the chip asynchronous case, both for bit error probability of 10?3 or lower. While our results are similar we have not yet introduced channel coding, the principal contributor to the results of [12].

REFERENCES [1] A. Flaig, G. R. Arce, A. B. Cooper, III, H. Tayong, and A. ColeRhodes, \Fuzzy rank-order detectors for frequency hopping networks," in Proceedings, 1999 ARL Telecommunications Federated Laboratory Symposium, (College Park, MD), Feb. 1999. [2] H. Tayong, A. Cole-Rhodes, A. B. Cooper III, A. Flaig, and G. Arce, \Exploiting the energy distribution in non-coherent frequency hopping diversity combining," in Proceedings, 34-th Conference on Information Sciences and Systems, Mar. 2000. [3] H. Tayong, A. Cole-Rhodes, A. B. Cooper III, A. Flaig, and G. Arce, \A reduced complexity receiver for fast frequency hopping," in Proceedings, Army Research Laboratory Federated Laboratory Symposium, Mar. 2000. [4] J. G. Proakis, Digital Communications. New York: McGraw-Hill, 3-rd ed., 1995. [5] R. Viswanathan and S. Gupta, \Performance comparison of likelihood, hard-limited, and linear combining receiver," IEEE Trans. Commun., vol. 31, pp. 670{677, May 1983.

Its superior bandwidth eciency explains the persistent interest in the unconventional FFH/MFSK model that is displayed in the literature. It is also an attractive candidate for tactical communications since the emerging concept of network-centric warfare is expected to have a vora3

[6] R. Viswanathan and S. Gupta, \Nonparametric receiver for FHMFSK mobile radio," IEEE Trans. Commun., vol. COM-33, Feb. 1985. [7] T. A. Gulliver, \Order statistics diversity combining in worst case noise and multitone jamming," in Conference Record, 1991 IEEE Military Communications Conference, (McLean, VA), Mar. 1991. [8] T. A. Gulliver, E. Felstead, R. Ezers, and J. Wight, \A uni ed approach to time diversity combining for fast frequency hopped NCMFSK," in Proceedings, Third Annual IEEE International Symposium on Spread Spectrum Techniques and Applications, pp. 303{308, July 1994. [9] T. A. Gulliver, R. Ezers, E. Felstead, and J. Wight, \The performance of diversity combining for fast frequency hopped NCMFSK in Rayleigh fading," European Transactions on Telecommunications, vol. 6, no. 1, 1995. [10] U.-C. Fiebig, \On the eciency of fast frequency hoppingmultipleaccess systems," in Proc. International Conference on Communications, (Chicago), 1992. [11] D. J. Goodman, P.S.Henry, and V. Prabhu, \Frequency-hopped multilevel FSK for mobile radio," Bell Syst. Tech. J., vol. 59, no. 10, 1980. [12] U.-C. Fiebig, \The eciency of FFH/CDMA systems in a mobile radio environment," in Proc. International Conference on Communications, ICC'94, 1994. The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the ocial policies, either expressed or implied, of the Army Research Laboratory or the U.S. Government.

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