A REDUCED COMPLEXITY DETECTOR FOR FAST ... - CiteSeerX

A REDUCED COMPLEXITY DETECTOR FOR FAST FREQUENCY HOPPING Helene Tayong Arlene Cole-Rhodes

A. Brinton Cooper III

Alexander Flaig Gonzalo Arce

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

Abstract

communications and resistance to hostile jamming include the commercially popular direct sequence code division multiple access (DS-CDMA) and frequency hopping multiple access (FHMA).

An improved version of the Fuzzy Rank Order Detector (FROD) is presented. It oers nearly the same performance and exibility as the original FROD while requiring only a fraction of the computational time and eort. We show performance improvements over the reduced rank sum (RRS) detector and the order statistics-normalized envelope detector (OS-NED) for fading, multiuser, fast frequency hopping communications. Keywords |frequency

A DS-CDMA receiver is vulnerable to excessive interference from nearby friendly users when trying to receive a weaker signal from a more distant user. One solution to this nearfar problem [1] requires the receiver to exert control over the transmitted power of all transmitters communicating with it and is, therefore, practical only in a cellular network architecture. The eective use of DS-CDMA in a peer-to-peer system would require multiple, simultaneous power control algorithms, one for each received signal. By contrast, interference from other users in FHMA systems is manifested only when two or more users hop to the same frequency.1 Even if such a collision completely destroys the transmission of one or both users on any hop, decoding and signal processing in the receiver can, in many cases, recover the information, thus obviating the neeed for power control. In addition, frequency hopping provides important levels of resistance to hostile jamming. Together, these features establish the importance of FHMA as a strong candidate for use in building wireless battle eld communications networks and clearly demonstrate the high value to the Army of the research reported herein.

hopping, non-coherent detection, fuzzy

ranks, order statistics

M

I. Introduction

OBILE, terrestrial military communications supporting fast-moving forces presents unique challenges not addressed by commercial architectures. Whereas civilian wireless cellular telephony provides wireless access to a wired communications infrastructure, mobile military wireless communications is the infrastructure. Hence, in the forward areas of the battlespace, wireless network architectures requiring centralized control and routing through \base stations" are impractical and vulnerable to hostile action. Local wireless networks in the forward areas must have relatively non-hierarchical architectures in which each node can act both as communications terminal and as network relay. Such a node is, therefore, likely to communicate with a varying number of its peers. Communications receivers in these peer-to-peer networks must oer robust performance when receiving signals from multiple users of widely disparate energies and must oer resistance to hostile jamming. They should provide increased spectral eciencies and oer a high capacity multiuser communciations capability since the competition for radio spectrum precludes new Army allocations.

Recently, the fuzzy rank order detector (FROD) was proposed [2] as a robust and ecient detector that oers competitive FHMA performance on fading channels. Advantages of the FROD accrue because it permits a receiver to capture information contained in the relative amplitudes (ranks) of the received samples of signals in a multiuser environment and in the statistical spread of those amplitudes. (Since power control algorithms of the DS-CDMA type are not presently feasible in peer-to-peer networks, the statistical spread of interference values may be signi cantly greater for FHMA.) The use of ranking in FHMA detectors follows from the fact that samples taken in the presence of a signal are stochastically larger than those originating purely from noise; i.e., FS +N (x)  FN (x); 8x so that samples containing signal tend to have larger ranks than those of noise only.

Spectrum spreading waveforms that oer ecient multiuser Dr. Cooper is also Adjunct Professor of Electrical and Computer Engineering at Morgan State University and at the University of Delaware. Prepared through collaborative participation in the Advanced Telecommunication and Information Distribution Research 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.

1 We ignore the case of a strong signal in an adjacent frequency bin which the receiver lter may not be able to attenuate suciently.

1

In this paper, we present the reduced FROD (R-FROD), a computationally less complex detector having performance comparable to that of the original FROD. It continues to provide both the robustness and the exibility of the FROD while requiring only a fraction of the number of computations of the original. We compare the performance of the RFROD with the original FROD and with competing FHMA detectors.

of the channel so that the tones fade independently. Thus, values in any row but the correct one are distributed according to H0 (below) and those of the correct row, according to H1, where H0 : H1 :

x  p1 e? x + (1 ? p)0e? x x  1 e? x : 1

0

1

The basic system model is presented in Section II. Section III reviews some important conventional detectors. Performance comparisons of the R-FROD and some conventional detectors are shown in Section IV. Section V draws some conclusions and discusses future work.

The foregoing represents a general framework used in the analyses of many fast frequency hopping (FFH) diversity combining methods in the literature including those which we now review.

II. System Model

III. Diversity Combining Methods

As in [2], we use the convenient system model rst introduced by Goodman et al [3]. Consider the kth user. During the mth symbol interval of length T seconds, the information source produces an M-ary data symbol xk (m) which is mapped to one of M orthogonal tones. The kth user is uniquely identi ed by an M-ary address ak = (ak1 ; : : :akL ) of length L symbols and duration T seconds, i.e., once per information symbol. The chip length is Tc = TL . Address symbols are mapped onto the same set of M orthogonal tones. The tone and the signature sequence are added modulo M to produce the transmitted signal yk (m) = [xk(m) + ak1 ; : : :; xk(m) + aki; : : :; xk(m) + akL ] (1) every T seconds.

Among the many FHMA detectors found in the literature, two are of interest because of their resistance to jamming and partial band interference, their robustness to channel fading, and their relative simplicity. The maximum rank sum receiver (MRSR) [4] is a nonparametric combiner that rank orders the set of received samples according to their energies, sums the rank orders (integers) of all the chips at each tone frequency, and selects the one producing the largest sum. It turns out that, asymptotically in L, the rank sums are jointly normal [5]. Although examples presented below do not approach the asymptotic state it has been reported [4] that previous simulations for moderate but non-asymptotic cases agree well with analytic, asymptotic results.

For a network of size K, a receiver sees the sum of K waveforms weighted by unknown amplitudes Ak ; k = 1; : : :; K, plus noise n(t). Assume synchronous operation. The kth receiver takes the Fourier transform of the received ensemble (equivalent to passing it through a bank of chip detectors) to estimate non-coherently the energy in each frequency/time bin, producing the M  L matrix Y = [Yi;j ]; i = 1; : : :; M; j = 1; : : :; L. Then the kth signature sequence is subtracted from the row index of Y to give the decoded matrix h i Xk = Xi;jk  = Yi?a ;j : (2) Thus, the correct row of Xk contains values of the transmitted signal plus noise, and the others contain only noise or noise plus interfering signals from other FHMA users that are not despread in the foregoing manner, so do not contribute solely to one row of the matrix. Examples of Xk matrices can be found in [2], [4], and others.

The reduced rank sum receiver (RRR) [4] is a variant of the MRSR and aords essentially the same performance at a substantial reduction in complexity [4] for channels where fading and interference are independent on successive hops. Diversity combining based on order statistics (OS) for single user M-ary frequency shift keying (MFSK) systems [6] sums all but the largest order statistic on each hop, rejecting the maximum variate in order to prevent outliers from aecting the computations. The tone having the largest OS sum is selected. OS was shown by simulation (analysis may be intractable) to sustain low error probabilities in a single user multitone jamming environment [6], so is clearly an attractive candidate for FHMA with multiuser interference.

k j

One of the earliest detectors, normalized envelope detection (NED) [7], divides each detected envelope value by the sum of all the envelope values occurring during the same hop interval. Thus, it adapts to changes in interference and is useful for mitigating the eects of very large interfering signals.

For M tones, and L chips, the probability that one out of K users will experience collision with at least one of the others (assuming random hopping) is [4]

A form of near-far [1] interference exists in FHMA systems whenever energy from another nearby FHMA signal causes signi cant interference values to appear in the spurious (not correct) rows of the decoded matrix. Naturally, an interfering user's signal coded with a signature sequence somewhat


1 ? M ?1 K ?1

?

p =1? (3) We use a simpli ed Rayleigh fading channel model and assume that the tone spacing exceeds the coherence bandwidth 2

orthogonal to ak would not appear entirely in one spurious row of Xk , but a sucient number of such interfering tones from a large number of such users can so populate the received data matrix as to confound the decision for the correct tone. It has been found [8] that combining the useful features of OS diversity combining and NED in order to reduce the eects of these large spurious products can be more eective than either technique alone. Called OS-NED, the technique rst uses OS diversity to sort the received values at each frequency from smallest to largest, then calls upon NED to reduce the in uence of the strongest interferers on the sums. Since values in the correct row are stochastically larger than those in any other row, FS +N (x)  FN (x), normalization by NED has a smaller eect on the sum of the correct samples than on the interfering values.

1. The ML energy values in this vector are rst sorted and ordered from smallest value to the largest. The latter may represent signal plus noise and/or multiple access interference from other users. In this way we obtain a new rank ordered vector xL given by xL = (x(1); x(2); : : :; x(N ) ); x(1)  x(2)  : : :  x(N ) : (4) Now de ne a spread function  which will provide a measure of how closely related two sample values are as a function of the absolute value of the dierence between them. The spread function maps this dierence onto the interval [0; 1]. A value of one indicates that the two sample values are identical (zero dierence) and the larger dierences are mapped to values closer to zero, i.e. 0  (x; y) = f(jx ? yj)  1: (5) The continuum of values can be thought of as the degree to which they are similar. 2. The spread function is centered on each of the N sample values in the vector x and evaluated for each of the ordered values in the vector xL . These are stored in the form of an N  N time-rank matrix T. The function  has a Gaussian shape, the width of which is controlled by the parameter . We call the spread (or fuzziness) parameter. The time-rank matrix T is de ned as   T( ) = i(j ) (6) where i(j ) = e?(x ?x ) =2 : (7) We note that as ! 0,  approaches an impulse function and will have a nonzero value only for argument pairs having values which are very close, while as ! 1  is atter in shape and tends to assign the same value to all dierences. 3. The vector < of real-valued ranks for energy values in the vector x is calculated as follows. First, normalize the matrix T along its rows so that all its row sums are equal to unity. Call the matrix of normalized rows ?: Multiply ? by the vector of time location values, t = [1; 2; : : :; N]T to obtain < as < = ?
t (8)

IV. The Reduced FROD In this section, we brie y review the FROD, which was proposed and discussed in [2], and we introduce a computationally more ecient version of this detector. In prior work, the performance of the FRO detector was compared with that of two other conventional detectors, the MRSR (above) and the hard decision majority vote [9] (HDMV) detector.2 The FROD was found to outperform both of these detectors on a frequency selective Rayleigh fading channel corrupted with additive Gaussian noise, especially for low SNR. In some sense, the FROD is a generalization of the MRSR since its algorithm is similar except that the FROD assigns real-valued (as opposed to integer-valued) ranks to the entries of the received matrix, thus exploiting more fully the distribution of energy among the received samples.

i

A. Theory of Operation

Consider a FFH network which uses the spread spectrum modulation scheme known as frequency hopped multilevel FSK [3] on a Rayeigh fading channel with additive Gaussian noise. If there are K users in the network transmitting sequences of L tones in any time frame, the received matrix is decoded using the address ak of the preferred (kth ) user. For a set of M possible transmission frequencies, the decoded matrix at the transmitter will be of dimension M  L. We proceed to assign non-integer ranks to the entries of this decoded matrix, using the scheme outlined below.

(j)

2

A.2 Implementation in the Fuzzy Rank Order Detector The ML entries in the decoded matrix are the replaced by their corresponding fuzzy ranks from the vector