On Antijamming in General CDMA Systems-Part II: Antijamming Performance of Coded Multicarrier Frequency-Hopping Spread Spectrum Systems

©2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

888

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 3, MARCH 2008

On Antijamming in General CDMA Systems–Part II: Antijamming Performance of Coded Multicarrier Frequency-Hopping Spread Spectrum Systems Reza Nikjah, Student Member, IEEE, and Norman C. Beaulieu, Fellow, IEEE Abstract— In the first part of the paper, the capacity of a general multiuser code division multiple access (CDMA) jamming channel was analyzed for noncooperative and cooperative users in uplink and downlink static and Nakagami fading channels when the receiver has or lacks jammer state information. The results were applied to a unified channel model encompassing a variety of multiuser spread spectrum systems contaminated by jamming. It was found that the jammer should spread its energy evenly over all degrees of freedom in order to minimize the average capacity. In the second part of the paper, using a standard orthogonal frequency division multiplexing (OFDM) model, the performance of a coded version of a multicarrier frequencyhopping (MC-FH) CDMA system in static and Rayleigh fading jamming uplink channels is analyzed. The MC-FH system under study is a hybrid of the OFDM and frequency hopping concepts, and is also a practical example of the model developed in the first part. It is demonstrated that in the cases where the receiver knows the jammer state and the receiver lacks the jammer state, spreading and contracting the jamming power over the system bandwidth, respectively, will give rise to the worst performance for the communicators. Optimal decorrelator weights for the receiver soft outputs in different channels, valuable for practical system design, are also obtained. Index Terms— Code division multiple access, diversity, fading channels, multicarrier frequency-hopping, single-user detection.

I. I NTRODUCTION REQUENCY hopping (FH) is one of the well known spread spectrum techniques providing both multiaccess capability and potential resilience against hostile interference [1], [2, Part 1, Ch. 1]. Moreover, multicarrier modulation (MCM) is a recognized way of producing immunity to channel frequency selectivity, and has an efficient implementation via fast Fourier transform (FFT) devices [3]. Multicarrier frequency-hopping (MC-FH) systems have received great attention because they take advantage of both MCM and the FH concept, and because they can be implemented coherently at the receiver when appropriately and specifically designed [4]. It is generally known that considerable performance gain is achieved if a FH system is implemented coherently, rather than noncoherently, in jamming environments [5]–[9]. Nevertheless, realizing a coherent frequency synthesizer at the

F

Manuscript received August 24, 2006; revised September 4, 2007; accepted December 5, 2007. The associate editor coordinating the review of this paper and approving it for publication was V. K. N. Lau. The authors are with the Department of Electrical and Computer Engineering, The University of Alberta, 2nd Floor, ECERF, University of Alberta, Edmonton, Alberta, Canada, T6G2V4 (e-mail: {nikjah, beaulieu}@ece.ualberta.ca). Digital Object Identifier 10.1109/TWC.2008.060627.

receiver in a FH system poses much challenge in practice as the hopping carrier frequency at the transmitter makes carrier recovery difficult, if not impossible in cases, for the receiver [5]. This problem has been extensively addressed in the literature, and some solutions have been provided [6], [10]–[13], which include the use of a digital delay-lock loop for joint phase/timing tracking following coarse acquisition, or the transmission of reference symbols of known phase during each hop to assist the receiver in estimating and tracking the carrier phase. The solutions basically rely on the principle of maintaining a continuous phase for the hopping carrier at the transmitter to facilitate the carrier recovery at the receiver. However, the phase continuity cannot be sustained if the hopping span is large [4], [6], [12]. This issue is avoided in the MC-FH scheme proposed in [4] by keeping sufficiently small the bandwidth inside which a subcarrier is hopping, thereby making coherent implementation possible. In this paper, we are not concerned with phase tracking errors or their effect in degrading the performance; rather, we consider the MCFH scheme in [4] assuming it to be perfectly coherently implemented. There is a substantial body of literature introducing other novel MC-FH structures, each of some practical importance, and evaluating their performances [14]–[21]. Chen et al. introduced a modified multicarrier direct-sequence code division multiple access (DS-CDMA) scheme allowing for data substreams hopping over subchannels, and investigated the problem of optimizing the hopping pattern for the best bit error rate (BER) in a multiuser uplink channel [14]. A similar, but more general, framework for applying the FH concept to multicarrier DS-CDMA schemes was proposed in [15], and compared to conventional single-carrier and multicarrier DSCDMA systems in a Nakagami multipath fading multiuser environment with RAKE receiver structures. The idea was to select a constant number of carriers out of a number of available carriers according to a FH pattern, which was considered to be either random or uniform in the paper. The authors demonstrated the compatibility of their scheme with existing code division multiple access (CDMA) systems, its possible implementation by software-defined radios, and its capability in handling multirate services [16]. In [17], a multicarrier Mary frequency shift keying (MFSK)/FH-CDMA scheme was proposed which utilized FH patterns with cross correlation no greater than one, aimed at reducing mutual interference. The system BER was analyzed in multiuser nonfading and fading channels with noncoherent reception, and shown to be

c 2008 IEEE 1536-1276/08$25.00 

©2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

NIKJAH AND BEAULIEU: ON ANTIJAMMING IN GENERAL CDMA SYSTEMS—PART II

a decreasing function of the number of subcarriers. In [18], taking advantage of a bandwidth-efficient multicarrier on-off keying (MC-OOK) modulation, the authors introduced a MCFH CDMA system with a larger number of available hopping frequency slots as compared to a corresponding system with MFSK modulation. The larger number of the frequency slots results in a greater immunity to multiple access interference (MAI). The BER performance of the MC-OOK/FH multiaccess (FHMA) scheme with noncoherent receptions was compared to a commensurate MFSK/FH scheme with slow FH in [18], and with fast FH (FFH) in [19]. A multicarrier directsequence (DS)/slow FH CDMA system was proposed in [20], which was similar to a conventional multicarrier DS-CDMA system, except that the main frequency subbands were divided into a number of hopping frequency dwells. A similar FH technique was applied to a conventional multicarrier CDMA (MC-CDMA) system in [21], allowing for the narrowband frequency subcarriers of a user to hop within some groups of frequency slots. The multicarrier scheme in [21] is also similar to the MC-FH in [4], with a major difference that in [21] the hopping pattern for a user is the same in all frequency groups, while in [4], each user has independent FH patterns for distinct frequency subbands, giving more flexibility in system design. Elkashlan and Leung [21] evaluated their scheme in an uncoded multiaccess environment using a Gaussian assumption for the MAI. Among all these, we consider the MC-FH system in [4] in this paper as a representative example of the general model developed in the first part, because of its simple, yet flexible, structure, easy implementation via FFT techniques, and suitable design for coherent realization. Note that all MC-FH frameworks described in [14]–[21], plus the model of [4] considered in this paper, rely on a hybrid of the orthogonal frequency division multiplexing (OFDM) and FH concepts in having compact orthogonal frequency subbands [22, Ch. 2], and having subcarriers hop over these subbands. Other than the FH concept which is not employed in a standard OFDM system, the MC-FH system considered here, or the multiaccess model in the first part of the paper, differs from the OFDM system in that in the former, coded data substreams modulate the subcarriers while in the latter, independent parallel data substreams are sent over the subcarriers. Nevertheless, as we practice in this paper, the basic OFDM model can still be directly utilized to describe and analyze the MC-FH scheme. The MC-FH system introduced in [4] has been analyzed in different nonjamming channels in [8], [23], [24], and in uncoded form in jamming environments in [4], [25]. The multiuser performance of uncoded binary frequency shift keying coherent and noncoherent MC-FH systems in static and fading downlink channels using Gaussian approximation for the decision variable was investigated in [8]. The performances of uncoded noncoherent single-user MC-FH and FFH systems in frequency-selective Rayleigh fading channels were analyzed and compared in [23]. Ebrahimi and Nasiri-Kenari performed a multiuser performance analysis of uncoded and coded coherent MC-FH systems in static and fading channels; however, their analysis excluded hostile interference [24]. Also, simulation results for the performance of a coded MC-FH scheme with a multiuser iterative detector equipped with an add-on jamming estimator were presented in [26]. The current part of

889

our paper gives a detailed performance evaluation of multiuser coded MC-FH systems in static and Rayleigh fading uplink channels in the presence of deliberate interference, with new results and practical guidelines provided for the design of such schemes. Particularly, the focus is on having or lacking jammer state information (JSI) at the receiver, optimal weighting for the decorrelator soft outputs, and worst-case jamming. Some other relevant studies undertaken in the literature on coded, coherent, or FHMA systems operating in jamming environments include [9], [27], [28]. In [9], taking advantage of the turbo processing principle, the authors developed an iterative scheme for joint decoding and channel estimation in convolutionally coded and turbo-coded FH networks in the presence of additive thermal noise, additive partial-band noise jamming (PBNJ), additive MAI, and multiplicative flat Rayleigh fading. Unintentional or intentional interference was identically modeled as Gaussian processes, with their variance being estimated at the receiver. An iterative symbol-aided estimation algorithm was also proposed. In our work, we do not take into account estimation processes, and assume perfect coherent receivers. Also, the PBNJ and MAI in the system are treated and modeled separately, and the effects of the jamming contamination fraction and interfering users on the system performance are investigated. A link throughput analysis of a slotted ALOHA FHMA packet radio network under PBNJ or partial-band tone jamming (PBTJ), with noncoherent MFSK modulation and hard decisions at the receiver, was presented in [27]. The authors utilized the concept of the cutoff rate, as a practically maximum achievable rate in a coded system, to bring into play forward error correction coding for their scheme. In contrast, in our work, coherent detection with softinput receivers is considered, general block or convolutional coding is applied, and frequency diversity is achieved owing to multicarrier modulation. In [28], FHMA networks with noncoherent MFSK modulation in the presence of PBTJ and Rayleigh fading were analyzed. The system was uncoded, and a Gaussian approximation with a correction coefficient for the variance was considered for MAI. In our paper, we treat MAI exactly and derive analytical results without Gaussian approximation. Also, we only consider PBNJ and exclude the case of PBTJ in our analysis. The paper is organized as follows. The MC-FH system model is described in Section II. Performance analyses in static and fading channels are carried out in Section III. Section IV presents some numerical results. Finally, conclusions are drawn in Section V. II. MC-FH S YSTEM In this section, we describe the MC-FH system model as a practical and important example for the general framework developed in Section II of the first part of the paper. In the subsequent sections, we examine the performance of coded MC-FH systems in static and fading jamming channels. We consider a noncooperative CDMA model as described in Section I of the first part of the paper. In other words, the receiver employs a single-user detector (SUD) not attempting to detect and cancel MAI. The available bandwidth W , which contains N subcarriers in total, is divided into L frequency subbands. Each frequency subband has exactly h = N/L frequency subcarriers with

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 3, MARCH 2008

Information bits

Encoder

Interleaver BPSK modulator

Subband 1

Serial to parallel converter

Frequency hopping inside subband 1 BPSK modulator

Subband 2 Frequency hopping inside subband 2 Frequency hopping inside subband L

BPSK modulator

Subband L

(a) Schematic block diagram (after [4, Fig. 1])

dl

dL Demux L

Demux l

h

N = Lh

BPSK −1, +1 mapping

RF oscillator

d2 Demux 2

h

0, 1

Cyclic prefix & parallel to serial converter

d1 Demux 1

Interleaver

Digital to analog converter

Encoder

Inverse discrete Fourier transform

Information bits

Serial to parallel converter

©2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

890

h 1

0

νl − 1 νl νl + 1

0 dl 0

h

0

(b) Structural block diagram Fig. 1.

The transmitter block diagram for the MC-FH system.

the same spacing as that in a commensurate OFDM system. Each user in any signaling interval chooses L subcarriers, one from each frequency subband, based on the user’s hopping signature code. Selected subcarriers are transmitted simultaneously after phase modulation by the user’s encoded bits. Fig. 1(a) represents a schematic block diagram of the transmitter underlining the main functions carried out. The frequency hoppers operate at the symbol rate, and implement the users’ signature code. Fig. 1(b) shows a practical OFDM

implementation of the transmitter. In fact, the transmitter and receiver of MC-FH schemes can be implemented via FFT techniques as in OFDM systems. Note that in Fig. 1(b), the demux or demultiplexer modules realize the FH operation in Fig. 1(a), and are controlled by the user’s hopping signature code denoted by ν1 , · · ·, νL in the figure. Now, the received signal for User m can be written as s(m) (t) =

L  

(m) (m)

2Ssub gl,i dl,i

i

l=1     (m) × exp j2π fl + νl,i /Ts (t − iTs ) pTs (t − iTs ) (1)

where Ssub is the average received power per subcarrier (m) assumed to be the same for all users, gl,i is the complex flat channel coefficient influencing the lth transmitted subcarrier in (m) the ith signaling interval, dl,i is the lth coded bit in the ith interval assumed to be −1 or +1, and {f1 , f2 , · · · , fL } are (m) the lower limits of the frequency subbands. Parameter νl,i th th is the l chip of User m’s signature in the i interval that determines which subcarrier is selected from the lth frequency subband. Variable Ts is the signaling interval and pTs (t) is a rectangular pulse of duration Ts . The processing gain in this scheme is N = L h, i.e. the total number of subcarriers. (m) The νl,i ’s are considered to be independent random variables (RVs) each with discrete uniform distribution in the interval (m) [0, h − 1]. In static channels, gl,i = 1. However, in fading (m) uplink channels the gl,i ’s are assumed to be independent zero-mean circularlysymmetric complex Gaussian (CSCG) (m) 2 RVs with variance E |gl,i | = 1. Assuming that the transmissions are quasi-synchronous, and taking advantage of cyclic prefixes in multicarrier systems, we can consider the system to be fully synchronous. Also, we assume that the system is affected by a PBNJ1 . In this paper, the ambient noise is ignored relative to the jamming noise for clarity of exposition. Each available subcarrier in the frequency spectrum is considered to be either not hit or fully hit by the jammer. Partially hit subcarriers are subsumed under fully hit subcarriers for a conservative analysis. The total received baseband signal can then be written as r(t) =

K 

s(m) (t) + n(t)

(2)

m=1

where K is the number of users and n(t) is the complex Gaussian jamming noise. The jammer has a total power J and contaminates a fraction ρ of the total bandwidth W . Therefore, with NJ = J/W , n(t) has power spectral density NJ /ρ wherever in the spectrum that the jammer is present, and 0 elsewhere. It is assumed that 1/Ts ≤ ρW ≤ W or 1/N ≤ ρ ≤ 1. III. P ERFORMANCE E VALUATION Let User 1 be the desired user. Considering a coherent receiver, the decorrelator output for the lth transmitted subcarrier 1 PBNJ constitutes an important type of jamming signal [2, Part 1, Ch. 3], and has been widely considered in previous research [5]–[7], [9], [11], [12], [25]–[27], [29], [30].

©2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

NIKJAH AND BEAULIEU: ON ANTIJAMMING IN GENERAL CDMA SYSTEMS—PART II

in the ith interval is given by

Ts 1 (1) r(t + iTs ) rl,i = √ Ts 2Ssub 0     (1) × exp −j2π fl + νl,i /Ts t dt .

ˆ is defined as in which D(λ)     

ˆ ˆ (8c) D(λ) = E exp λ m+1 (r) − m−1 (r) d = −1 (3)

Using (1)–(3) and eliminating subscripts l and i and superscript (1) for simplicity, we obtain r = gd+

K 

g

(m) (m) (m)

d

η

+θn ˆ

891

(4)

m=2

where θ represents the presence or absence of the jammer and is a Bernoulli RV with parameter ρ; i.e.,  1 − ρ, x = 0 . (5) pθ (x) = ρ, x=1  2 The RV n ˆ is a zero-mean CSCG RV with variance E |ˆ n| = 1/(ργsub ) where Ssub Ts γsub = (6) NJ is the signal-to-jamming-noise ratio (SJR) per subcarrier per symbol. The RV η (m) is 1 when there is a hit by User m and (m) 0 otherwise. From the statistics of the νl,i ’s, it is concluded that the η (m) ’s are independent Bernoulli RVs with parameter 1/h. In other words, we can write  α, x = 0 (7) pη(m) (x) = β, x = 1 where α  1 − h1 and β  1 − α. Assume that the channel is either memoryless by itself or considered to be memoryless by the use of interleaving. If a received signal vector after symbol decorrelation to be mapped onto a codeword is (r1 , r2 , · · · , rs ) for some s, and the candidate codewords are (ci1 , ci2 , · · · , cis ) for different i’s, then the operation of any decoder, be it singleuser or not, can be expressed by using a metric function th mc (r). The sdecoder chooses the i0 codeword where i0 = arg maxi q=1 mciq (rq ). Now assume that the encoder is a binary block or convolutional code of rate R that accepts blocks of k input bits at a time and has a weight distribution polynomial T (δ, ). Polynomial T (δ, ) is the sum of all terms Au,v δ u v , where Au,v is the total number of output sequences of Hamming weight u generated by input sequences of Hamming weight v. Recall that block codes have finite-length weight distributions, whereas convolutional codes have infinite-length weight polynomials [31, Ch. 3 and Ch. 11]. If Pb is the BER of the system exploiting a soft decoder, and m−1 (r) and m+1 (r) are the metrics that the decoder assigns to the encoded bits −1 and +1, respectively, where r is the soft output given in (4), it has been shown that [2, Part 1, Ch. 4], [31, Ch. 12], [32], [33, Ch. 5]2 1 ∂T (D, )

Pb ≤ (8a)

2k ∂ =1 where ˆ (8b) D = min D(λ) ˆ λ≥0

2 The formulation given here is a structured version of those expressed in [2], [31]–[33].

where d is the same as in (4). If maximum-likelihood (ML) decoding is adopted, we have m−1 (r) = ln p−1 (r)

(9a)

m+1 (r) = ln p+1 (r)

(9b)

where p−1 (r) and p+1 (r) are the probability density functions of the soft output r when the transmitted bit, d, is −1 and +1, respectively. In fact, under ML decoding (8c) is rewritten as     ˆ = E exp λ ˆ Λ(r)

d = −1 D(λ) (10a) where Λ(r) is the log-likelihood ratio (LLR) function defined as p+1 (r) . (10b) Λ(r) = ln p−1 (r) Now, a SUD treats the MAI as Gaussian noise, and from (4) presumes that3   r ∼ CSCG g d, β(K − 1) + θ/(ργsub ) (11) where we have used the fact that the variance of the MAI term in (4) is given by β(K − 1). This variance can be derived using the statistics of the η (m) ’s given in (7). The suboptimal decoder utilizes the ML metrics given in (9), with the approximate distribution of r in (11). Consequently, (8a), (8b), (10), and (11) can be used to yield an upper bound on Pb for a SUD. Combining (4), (10b), and (11), and after some manipulations, we obtain Λ(r) =

4 Re {r g ∗ } . β(K − 1) + θ/(ργsub )

(12)

It is concluded from (12) that the LLR computed by the SUD is of the form wθ Re {r g ∗ }, where w0 (for θ = 0) is the weight that the receiver gives to the decorrelator soft output not hit by the jammer, and w1 (for θ = 1) is the weight given to the output hit by the jammer. Although one can obtain an expression for wθ from (12), we prefer to consider the LLR with general weightings, w0 and w1 , to be optimized for the best BER performance, thus obtaining a “general linear diversity combining”. Note that for a singleuser application, ML combining at the receiver with known JSI is a linear diversity combining with different weights given to different branches based on the JSI [7]. However, for a multiuser scenario which is considered here, no matter whether the receiver knows or lacks JSI, the weighting in (12) is not optimal since a SUD is suboptimal per se. Also, note that when the receiver lacks JSI, it cannot use (12), and instead, has to employ wθ Re {r g ∗ } for computing the LLR, with some value w0 = w1 . In such a case, the receiver amounts to the 3 Note that the use of a SUD in a multiuser environment is standard in the literature as can be found in [14], [15], [17]–[21], [24] without jamming, and in [8], [9], [25], [27], [28] in the presence of jamming. Although the SUD views the MAI as noise and detects the desired signal accordingly, in our analysis of the SUD performance we model and treat the MAI exactly. Among the references above, in [17]–[19], [24], [27] MAI is precisely analyzed, and in the rest, Gaussian approximation for the MAI, which may not apply to a small number of users, is utilized.

©2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

892

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 3, MARCH 2008

“linear combining” receiver described in [7]. The concept of arbitrarily weighted decorrelator outputs has been similarly used in [32]. Some nonlinear diversity combining techniques for FFH systems were also proposed in [7], [29], [30], such as hard decision majority vote, reduced rank sum, and clipping combining; these schemes are not considered in this paper. Now for a general linear diversity combining SUD, from (4), (8a), (8b), (10a), setting Λ(r) = wθ Re {r g ∗ }, and making ˆ we obtain the change of variable λ = w1 λ, Pb ≤

1 ∂T (D, )

2k ∂ =1

Now, from (13c)–(13e) we have



(13a)

where D = min D(λ)

(13b)

λ≥0

  E {exp(sX)} = exp σ 2 s2 /2





wθ λy w1

 = (1 − ρ) E exp(w λ y) θ = 0

 + ρ E exp(λ y) θ = 1

(16)

where we have used the distribution of η (m) in (7). Also, we have used the moment generating function of a real Gaussian RV X ∼ Normal(0 , σ 2 ) given as [34, Ch. 2]

in which 



  K (m) (m) D(λ) = (1 − ρ) e−w λ E ew λ m=2 η d    K (m) (m) E eλ n + ρ e−λ E eλ m=2 η d K−1  (2) (2) = (1 − ρ) e−w λ E ew λ η d  (2) (2) K−1 2 + ρ e−λ+λ /(4ργsub ) E eλ η d  K−1 = (1 − ρ) e−w λ α + β cosh(w λ)  K−1 2 + ρ e−λ+λ /(4ργsub ) α + β cosh(λ)

(17)

D(λ) = E exp

(13c)

for any complex s. We write the result in (16) as D(λ) = F (w λ) + G(λ) for compactness where  K−1 F (x)  (1 − ρ) exp(−x) α + β cosh(x)

where w0 w w1

and  G(x)  ρ exp −x +

and  y = Re {r g ∗ } = −|g|2 + Re

K 

(18a)

(13d)

 η (m) d(m) g (m) g ∗

+ θn.

m=2

(13e) In (13e), n = Re {ˆ n g ∗ } and therefore, the RV n is a conditionally zero-mean real Gaussian RV with variance  E n2 | g = |g|2 /(2ργsub )

  K−1 x2 α + β cosh(x) . (18b) 4ργsub

(14)

where γsub has been given in (6). The reason for the change ˆ to λ is that in subsequent analyses when of variable from λ using (13), we will handle only the ratio w0 /w1 , denoted by w in (13d) and called the weight factor, rather than w0 and w1 separately. In the next two subsections, we specialize the general analyses performed above to the cases of static and fading channels. Note that an extreme case of MC-FH schemes is obtained when h = 1, i.e. when there is no hopping and the system is equivalent to a MC-CDMA system. Users should then be distinguished by a DS signature instead of a hopping signature. In other words, L = N encoded bits are modulated by the N bits of the signature. All the analyses also hold for the case of MC-CDMA systems if we set h equal to 1.

A. Static Channels

The variable D (13b) is obtained by minimizing D(λ) over nonnegative λ. If the receiver has JSI, it can set w, the weight factor, to an optimal value, denoted by wopt , in order to minimize D(λ). Otherwise, w can be determined empirically; however, choosing w = 1 seems intuitive when the receiver lacks information. Table I presents some detailed information about the values of λopt (the minimizer of D(λ)), wopt , and D for different choices of K and w for static channels. These results have been obtained by scrutinizing the derivative of D(λ) in (16) for different cases. In this table,

x(s) opt = ln

 ⎧ ⎫ 2 ⎨α + β(K − 1) + α − β ⎬ ⎩

β(K − 2)

⎭

is the point at which F (x) reaches its minimum on x ≥ 0 for K > 2. The parameter x(f) opt and function Sq (w, λ) related to the fading channel case in Table I will be defined in the next subsection. A theoretical value of infinity for wopt in Table I means that the receiver sets w1 to zero; i.e., it ignores the subcarriers hit by the jammer. Note that λopt for the case of optimal weighting and K > 1 can be found as the minimizer of G(x) on x ≥ 0.

(m)

In static channels, gl,i = 1 for any l, i, and m. Therefore, (13e) reduces to y = −1 +

K  m=2

η (m) d(m) + θ n .

(15)

B. Rayleigh Fading Channels Parallel to the case of static (13c)–(13e)

 channels, from we need first to calculate E exp(w λ y) θ = 0 and

TABLE I O PTIMAL W EIGHTS AND E XPRESSIONS FOR D IN D IFFERENT C ASES

wopt = ∞ λopt = 2ργsub

K=1 Static channels

wopt = ∞ λopt ∈ (0, 2ργsub )

K=2

D = (1 − ρ)β/2 + G(λopt ) wopt = x(s) opt /λopt λopt ∈ (0, 2ργsub )

K>2

D = F (x(s) opt ) + G(λopt ) wopt = ∞ Fading channels

Optimal weight factor: w = wopt

D = ρ exp (−ργsub )

Static channels Fading channels

λopt = 2ργsub

K=1

D = ρ/(ργsub + 1) wopt = x(f) opt /λopt

K>1

K=1

Arbitrary weight factor w

©2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

NIKJAH AND BEAULIEU: ON ANTIJAMMING IN GENERAL CDMA SYSTEMS—PART II

 2 2 √ λopt ∈ K+1/(ργ , sub )−1 K+1/(ργsub )−1 K−1 K−1 K−1−q q D = q=0 α β Sq (w, λopt ) q

λopt ∈ [2ργsub , 2ργsub + 2(1 − ρ)wγsub ] D = F (wλopt ) + G(λopt ) λopt ∈ (0, 2γsub )

K=2

D = F (wλopt ) + G(λopt )   λopt ∈ 0, x(s) opt

K>2

K=1

K>1

893

η (m) , and d(m)2 = 1, and then applied (17). Also, in the forth equality in (19) we utilized the distribution of η (m) given in (7) and the fact that the η (m) ’s are independent. Applying the binomial expansion to (19) yields  K − 1

 K−1 αK−1−q β q E exp(w λ y) θ = 0 = q q=0     q w 2 λ2 |g|2 × E exp −w λ + 4   K−1−q q K−1  K−1 α β q = (20) qw2 2 q=0 − 4 λ + wλ + 1 provided that all summands are positive. The last equality in (20) results from the facts that |g|2 has exponential distribution with mean 1, and that if X is exponentially distributed with mean μ, then E {exp(sX)} = (1 − μs)−1 for any complex s satisfying Re{1 − μs} > 0 [34, Ch. 2]. Taking similar steps to those  in (19) and (20), we next calculate E exp(λ y) θ = 1 . From (13e) we have

 E exp(λ y) θ = 1   

(m) (m)

,d = E E exp(λ y) θ = 1 , g , η    2   K    λ

|g|2 = E exp −λ |g|2 E exp η (m) g 4 m=2   × E exp(λ n) | g 

λopt ∈ 2ργsub , √

2 1+1/(ργsub )−1



D = S0 (w, λopt )   1 1 λopt ∈ 2 min (K−1)w , K+1/(ργ , sub )−1   1 2 min √ 1 ,√ (

D=

K−1)w

K−1 K−1 q=0

q

α

K+1/(ργsub )−1

K−1−q

β q Sq (w, λopt )

 E exp(λ y) θ = 1 . From (13e) we can write

 E exp(w λ y) θ = 0   

= E E exp(w λ y) θ = 0 , g , η (m) , d(m)  2 2   K    w λ 2 2 (m) |g| η = E exp −w λ |g| exp 4 m=2    2 2   K    w λ 2 2 (m) |g| η = E exp −w λ |g| E exp

g 4 m=2    2 2 K−1    w λ |g|2 = E exp −w λ |g|2 α + β exp 4 (19) where in the second equality we have used the facts that g (m) ∼ CSCG(0, 1), the g (m) ’s are independent, η (m)2 =



  λ2 2 = E exp −λ + |g| 4ργsub   2 K−1  λ 2 |g| × α + β exp 4

D = F (wλopt ) + G(λopt )

where in the last equality we have used the fact that the RV n conditioned on g is a zero-mean Gaussian RV with variance given in (14), and then invoked (17). Now similar to (20) by using the binomial expansion and the moment generating function of an exponential RV, we obtain K−1 K−1−q q  α β  K−1 q E exp(λ y) | θ = 1 = 2 − [q + 1/(ργsub )] λ /4 + λ + 1 q=0 (21) provided that all summands are positive. Combining (13c), (20), and (21) we conclude that K−1  K − 1 αK−1−q β q Sq (w, λ) D(λ) = (22a) q q=0 where 1−ρ −q w2 λ2 /4 + wλ + 1 ρ . (22b) + − [q + 1/(ργsub )] λ2 /4 + λ + 1 Now D (13b) is obtained by minimizing D(λ) on all nonnegative values of λ for which all summands in the Sq (w, λ)’s are positive. At the same time, the jammer, if smart, will try to maximize D by selecting a suitable ρ, and the receiver, if aware of the JSI, will counter the jamming by choosing a Sq (w, λ) 

©2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

894

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 3, MARCH 2008

suitable w to minimize D. Note that for the special case of K = 1, from (22) we have ρ 1−ρ + . D(λ) = 2 wλ + 1 −λ /(4ργsub ) + λ + 1 In this case, for an arbitrary w it can be shown that λopt , the nonnegative minimizer of D(λ) yielding positive summands in (22), satisfies 2 2ργsub ≤ λopt <  . 1 + 1/(ργsub ) − 1 On the other hand, for K = 1 if the receiver knows JSI, it opts for wopt = ∞ to minimize D(λ), which will result in ρ D(λ) = 2 −λ /(4ργsub ) + λ + 1 and, thus, λopt = 2ργsub . Consequently, D = ρ/(ργsub + 1). A smart jammer in this case chooses to have ρ = 1 in order to increase D to its maximum value, 1/(γsub + 1). Now, for K > 1 and an arbitrary w, by investigating (22) it can be shown that λopt satisfies   1 1 , 2 min (K − 1)w K + 1/(ργsub ) − 1   1 1 ≤ λopt < 2 min √ , . ( K − 1)w K + 1/(ργsub ) − 1 On the other hand, in the case of K > 1 and w = wopt , from (22) it can be shown that λopt , which is the minimizing λ in (22), is also the minimizing λ in (21) and satisfies 2 2 ≤ λopt <  . K + 1/(ργsub ) − 1 K + 1/(ργsub ) − 1 Also, let x(f) opt be the positive minimizing x for   K−1−q q K−1  K−1 α β q q=0

−q x2 /4 + x + 1

under the condition that all summands are positive. Then, we (f) 2 2 ≤ x(f) have K−1 opt < √K−1 and wopt = xopt /λopt . Table I shows a summary of all the results obtained. IV. N UMERICAL R ESULTS To illustrate the performance of MC-FH systems in the presence of jamming, we consider variable-rate so-called superorthogonal codes [33, Ch. 5] from the class of convolutional codes, for these systems [24]. Superorthogonal codes exhibit near-optimal performance for coherent CDMA systems with large processing gains [33]. In these codes, there is k = 1 bit per input block and the rate is R = 22−c , where c ≥ 3 is the constraint length. The weight distribution of superorthogonal codes, needed in our analysis in (13), has been calculated and is given by [33] T (δ, ) =

 ζ c+2 (1 − ζ) 1 − ζ [1 +  (1 + ζ c−3 − 2ζ c−2 )]

where ζ = δ 1/(2R) . In all following numerical examples, it is assumed that the rate of the superorthogonal code used is R = 1/L, such that every input bit is encoded to L output bits, which modulate the L selected subcarriers in a symbol; therefore, the rate of transmission is one bit per symbol. As

R = 22−c , in the examples that follow, L is required to be an integral power of 2. It is also possible to send less or more than one bit per symbol by reducing or increasing the code rate, which poses a trade-off between throughput and BER performance. Such trade-offs are not investigated here. In all of the example graphs in this section, SJR denotes the (average) signal-to-jamming-noise ratio per symbol, which equals Lγsub (see (1) and (6)). Also, N is the total number of subcarriers, L is the number of frequency subbands, K is the number of users, R is the code rate, and ρ is the jamming fraction. Moreover, the labels “UB”, “GS”, and “SIM” respectively signify the precise analytical upper bound on the BER performance, the BER analytical upper bound using Gaussian approximation of the MAI, and Monte-Carlo simulation of the real BER performance. In the Gaussian approximation, the MAI is approximated with a zero-mean normal RV of the same variance. Fig. 2 depicts the BER versus the SJR in a coded MC-FH scheme for different values of N , L, and K, in a static channel with an omnipresent jammer, i.e. a jammer with ρ = 1. Note that when ρ = 1, the performance is independent of the weight factor w, because the jammer contaminates all subcarriers evenly. First, note that although the exact upper bound is not tight, it has the same trend as that of the simulation curve, and falls within almost 1 BER decade of the simulation result. A comparable result for a single-user uncoded MFSK system was observed in [2, p. 211] with a slightly better accuracy for a similar BER upper bound. Also, up to 2 BER decades gap between a similarly derived upper bound and a lower bound on the BER performance of a nonjamming multiuser coded MC-FH system was reported in [24]. Second, the Gaussian curve does not always match well with the upper-bound curve, and its deviation increases when the SJR is increased. In fact, Gaussian approximation of the MAI can be acceptable according to a central limit theorem if the MAI consists of a sum of a sufficiently large number of independent RVs. However, referring to (15) reveals that the average number of users interfering with a given subcarrier is not K − 1, but (K − 1)β = L(K − 1)/N . Therefore, for the Gaussian curve to be a good approximation of the upper-bound curve, it is necessary for L(K − 1)/N to be large enough. This happens for the case (N, L, K) = (128, 16, 22) in Fig. 2, which exhibits the best match between the Gaussian and exact upper-bound curves. Moreover, increasing the SJR translates into a decrease in the Gaussian jamming noise power, and equivalently, into more deviation of the MAI-plus-noise term from a normal density. Gaussian assumption for the MAI considerably simplifies the results, but is not always useful in the application considered in this work. The following examples confirm this assertion. In Fig. 3, the BER versus SJR performance of a MC-FH system in a fading channel is depicted for different values of ρ, with known JSI and optimal weighting at the receivers. The Gaussian approximation for the upper bound is obviously not acceptable for the case considered in this figure as L(K − 1)/N = 0.25 is not large enough. Also, note that the analytical upper-bound curve is not more than one BER decade greater than the simulation curve. Furthermore, as the SJR is increased an error floor in the BER is experienced due to implementing single-user detection in a multiuser environment; the MAI

100

N N N N N N N N N

ρ=1

10−1 10−2

= 64, L = 4, K = 10, UB = 64, L = 4, K = 10, GS = 64, L = 4, K = 10, SIM = 128, L = 16, K = 22, UB = 128, L = 16, K = 22, GS = 128, L = 16, K = 22, SIM = 256, L = 8, K = 30, UB = 256, L = 8, K = 30, GS = 256, L = 8, K = 30, SIM

895

100

10−5

10−10 BER

BER

10−3 10−4

L = 16

10−15

R=

1 16

SJR = 10 dB ρ = 0.5

10−5

12 14 11 6 10 8 13 7 9 Signal-to-jamming-noise ratio per symbol, SJR (dB)

15

Fig. 2. The BER versus the SJR in a MC-FH system with a broadband jammer for different values of N , L, and K for static channels. 10−1 10−2 10−3 10−4

1 4

N = 64

R=

L=4

K=5

ρ = 0.1, UB ρ = 0.1, GS ρ = 0.1, SIM ρ = 0.3, UB ρ = 0.3, GS ρ = 0.3, SIM ρ = 1, UB ρ = 1, GS ρ = 1, SIM

10−5 10−6

4

6

8

10 12 14 16 18 20 22 24 26 28 30 Number of users, K

Fig. 4. The BER versus the number of users in a MC-FH system with jamming fraction ρ = 0.5, and with JSI and optimal weighting at the receiver, for a fading channel. 10−2 10−3 10−5 10−7

N = 128 L=8

10−9

R=

1 8

SJR = 9 dB, w = 1 SJR = 9 dB, w = wopt SJR = 15 dB, w = 1 SJR = 15 dB, w = wopt

10−13 5

= 64, UB = 64, GS = 128, UB = 128, GS = 256, UB = 256, GS

10−11

10−7 10−8

10−25 2

BER upper bound

10−7 5

N N N N N N

10−20

10−6

BER

©2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

NIKJAH AND BEAULIEU: ON ANTIJAMMING IN GENERAL CDMA SYSTEMS—PART II

11 12 14 15 6 13 8 10 7 9 Average signal-to-jamming-noise ratio per symbol, SJR (dB)

10−15

2

4

6

14 12 10 8 Number of users, K

16

18

20

Fig. 3. The BER versus the SJR in a MC-FH system for different values of the jamming fraction, ρ, with JSI and optimal weighting at the receiver, for a fading channel.

Fig. 5. The BER upper bound versus the number of users in a MC-FH system with and without JSI, and with worst-case jamming, for a fading channel.

remains the limiting factor of the performance at large SJR. This MAI bottoming effect can also be observed in [27]. Fig. 4 gives the BER versus the number of users in a fading channel for different processing gains. This figure provides another illustration confirming the unsuitability of the Gaussian assumption for the MAI for MC-FH systems. The deviation of the Gaussian curve from the exact upperbound curve is mitigated only when L(K − 1)/N is raised, by increasing K or decreasing N . In the rest of this section, owing to its simplicity and efficiency, only the precise analytical performance overbound is utilized providing a conservative design. Fig. 5 provides another BER-versus-K graph in fading channels with two values of the SJR when the weight factor w is either 1 (unknown JSI) or optimal. The gain obtained from knowing JSI decreases as the number of users or the required BER increases. In fact, for a larger number of users, stronger MAI cancels out some of the gain achieved by the receiver when it knows JSI. Also, the gap between the performances under unknown JSI and optimal weighting decreases as the

SJR increases, which is due to the fact that an increase in the SJR, being equivalent to a decrease in the jammer strength, vitiates the weighting. The rapid deterioration of the performance with increasing K in Figs. 4 and 5, caused by the use of a SUD instead of a multiuser detector, is highly analogous to that in [9], [24], [28]. Figs. 6 and 7 show the BER versus ρ for different values of the SJR in the two cases of unknown and known JSI for static channels, respectively. The performance improvement achieved when increasing the SJR for a fixed value of ρ is almost linear. Also, the amount of the improvement when JSI is unknown appears to be independent of ρ. Interestingly, from these figures it is observed that the optimal value of ρ for the jammer when w = 1 tends to be the smallest possible value. In other words, a smart jammer contracts its bandwidth in the unknown-JSI scenario. A similar result has been reported for single-user coded DS binary phase shift keying and singleuser coded noncoherent FH/MFSK systems in [2, Part1, Ch. 4], and for a single-user uncoded DS/FH (or pure FH) coherent

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 3, MARCH 2008

100 N = 128 R =

10−1

L = 16

45

SJR = 4 dB SJR = 6 dB SJR = 8 dB SJR = 10 dB SJR = 12 dB

1 16

K = 10

10−3 10−4 10−5 10−6

35 30

15

10−8

5 0.3

0.5 0.6 0.7 0.4 Jamming fraction, ρ

0.8

0.9

1.0

Fig. 6. The BER upper bound versus the jamming fraction, ρ, in a MC-FH system for different values of SJR with unknown JSI for a static channel. 10−2

10−4

SJR = 4 dB SJR = 6 dB SJR = 8 dB SJR = 10 dB SJR = 12 dB

N = 128 R = L = 16

K = 10

10−8

10−10

0.2

0.3

0.5 0.6 0.7 0.4 Jamming fraction, ρ

0 0.1

0.2

0.3

0.5 0.6 0.7 0.4 Jamming fraction, ρ

0.8

0.9

1.0

Fig. 8. The optimal weight factor in dB versus the jamming fraction, ρ, in a MC-FH receiver, for different values of SJR, K, and L for a fading channel.

subcarriers are hit by the jammer which now has a weaker power spectral density. Therefore, it would be to the receiver’s advantage to lessen the weight factor appreciably in such cases. Also, note that wopt for ρ = 1 ranges from 1 dB to 8.5 dB for SJR = 14 dB, and from 11.8 dB to 24.4 dB for SJR = 5 dB. The optimal weight factor wopt typically takes on even greater values for smaller values of ρ. This shows that in many cases, the optimal weight is much greater than unity. Therefore, the receiver must avoid setting w = 1 if it has JSI but cannot compute wopt .

1 16

10−6

10−12 0.1

= 5 dB, K = 4, L = 64 = 5 dB, K = 4, L = 32 = 5 dB, K = 17, L = 64 = 5 dB, K = 17, L = 32 = 14 dB, K = 4, L = 64 = 14 dB, K = 4, L = 32 = 14 dB, K = 17, L = 64 = 14 dB, K = 17, L = 32

20

10

0.2

SJR SJR SJR SJR SJR SJR SJR SJR

25

10−7

10−9 0.1

N = 256

40

20 log10 wopt (dB)

BER upper bound

10−2

BER upper bound

©2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

896

0.8

0.9

1.0

Fig. 7. The BER upper bound versus the jamming fraction, ρ, in a MC-FH system for different values of SJR with known JSI for a static channel.

scheme in [6]. On the contrary, when w = wopt , ρ tends to be the largest possible value, i.e. 1. The tendency of a smart jammer to be wideband has been demonstrated under some conditions for a single-user uncoded MC-FH system in [4]. Furthermore, there is an error floor observed at large values of ρ in Fig. 6, i.e. where the JSI is unknown and w = 1. The error floor or relative flatness of the BER for large values of ρ can be explained as follows. The power spectral density of the jammer is NJ /ρ, which, as a function of ρ, changes more slowly at larger values of ρ. Now, as w in the case of unknown JSI is set to unity independently of ρ, the BER performance of such scheme tends to change more slowly at larger values of ρ. The BER bottoms out when ρ = 1, i.e. when the jammer is broadband. Note that this explanation for justifying the relative flatness does not carry over to Fig. 7 since in Fig. 7, w is optimal and adaptively changes with ρ. Fig. 8 presents the optimal weight factor in dB versus ρ for different values of the SJR, K, and L, when N = 256, in fading channels. When the jammer chooses to cover a greater fraction of the entire bandwidth, a greater number of

V. C ONCLUSION The performance of coded MC-FH multiple access schemes in the presence of jamming in static and fading channels was analyzed. It was shown that Gaussian approximation for the MAI is unacceptable for most cases. It was found that optimal weights for the receiver soft outputs in countering smart jamming will generally result in ρ = 1. On the other hand, equal weights cause a smart jammer to minimize its value of ρ. Stronger MAI or greater SJR reduces the gain attained by the optimal weighting. If the receiver has JSI but cannot compute the optimal weights, it should choose a weight factor greater than unity based on an adaptive method. R EFERENCES [1] R. L. Pickholtz, D. L. Schilling, and L. B. Milstein, “Theory of spreadspectrum communications—A tutorial,” IEEE Trans. Commun., vol. 30, no. 5, pp. 855–884, May 1982. [2] M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt, Spread Spectrum Communications Handbook, 2nd ed. New York: McGrawHill, 2002. [3] J. A. C. Bingham, “Multicarrier modulation for data transmission: An idea whose time has come,” IEEE Commun. Mag., vol. 28, no. 5, pp. 5–14, May 1990. [4] E. Lance and G. K. Kaleh, “A diversity scheme for a phase-coherent frequency-hopping spread-spectrum system,” IEEE Trans. Commun., vol. 45, no. 9, pp. 1123–1129, Sep. 1997. [5] M. K. Simon and A. Polydoros, “Coherent detection of frequencyhopped quadrature modulations in the presence of jamming—Part I: QPSK and QASK modulations,” IEEE Trans. Commun., vol. 29, no. 11, pp. 1644–1660, Nov. 1981. [6] G. Cherubini and L. B. Milstein, “Performance analysis of both hybrid and frequency-hopped phase-coherent spread-spectrum systems part I & part II,” IEEE Trans. Commun., vol. 37, no. 6, pp. 600–622, Jun. 1989.

©2008 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE.

NIKJAH AND BEAULIEU: ON ANTIJAMMING IN GENERAL CDMA SYSTEMS—PART II

[7] J. J. Kang and K. C. Teh, “Performance of coherent fast frequencyhopped spread-spectrum receivers with partial-band noise jamming and AWGN,” IEE Proc. Commun., vol. 152, no. 5, pp. 679–685, Oct. 2005. [8] Z. Z. Yazdi and M. Nasiri-Kenari, “Performance comparison of coherent and non-coherent multicarrier frequency-hopping code division multiple-access systems,” in Proc. IEEE Int. Symp. Personal, Indoor and Mobile Radio Commun. (PIMRC), Sep. 2004, vol. 1, pp. 165–169. [9] H. El Gamal and E. Geraniotis, “Iterative channel estimation and decoding for convolutionally coded anti-jam FH signals,” IEEE Trans. Commun., vol. 50, no. 2, pp. 321–331, Feb. 2002. [10] C.-M. Su and L. B. Milstein, “Comparison of joint phase/timing tracking loops for a coherent frequency hopped spread spectrum receiver,” in Proc. IEEE Global Telecommun. Conf. (GLOBECOM), Nov.–Dec. 1988, vol. 1, pp. 546–550. [11] ——, “Channel equalization and performance analysis of a coherent frequency hopped spread spectrum system,” IEEE J. Select. Areas Commun., vol. 7, no. 4, pp. 548–560, May 1989. [12] ——, “Analysis of a coherent frequency-hopped spread-spectrum receiver in the presence of jamming,” IEEE Trans. Commun., vol. 38, no. 5, pp. 715–726, May 1990. [13] J. S. Stadler, “A performance analysis of coherently demodulated PSK using a digital phase estimate in frequency hopped systems,” in Proc. IEEE Military Commun. Conf. (MILCOM), Nov. 1995, pp. 107–112. [14] Q. Chen, E. S. Sousa, and S. Pasupathy, “Multicarrier CDMA with adaptive frequency hopping for mobile radio systems,” IEEE J. Select. Areas Commun., vol. 14, no. 9, pp. 1852–1858, Dec. 1996. [15] L.-L. Yang and L. Hanzo, “Slow frequency-hopping multicarrier DSCDMA for transmission over Nakagami multipath fading channels,” IEEE J. Select. Areas Commun., vol. 19, no. 7, pp. 1211–1221, July 2001. [16] ——, “Software-defined-radio-assisted adaptive broadband frequency hopping multicarrier DS-CDMA,” IEEE Commun. Mag., vol. 40, no. 3, pp. 174–183, Mar. 2002. [17] C.-F. Hong and G.-C. Yang, “Multicarrier FH codes for multicarrier FH-CDMA wireless systems,” IEEE Trans. Commun., vol. 48, no. 10, pp. 1626–1630, Oct. 2000. [18] S. H. Kim and S. W. Kim, “Frequency-hopped multiple-access communications with multicarrier on-off keying in Rayleigh fading channels,” IEEE Trans. Commun., vol. 48, no. 10, pp. 1692–1701, Oct. 2000. [19] S. Sharma, G. Yadav, and A. K. Chaturvedi, “Multicarrier on-off keying for fast frequency hopping multiple access systems in Rayleigh fading channels,” IEEE Trans. Wireless Commun., vol. 6, no. 3, pp. 769–774, Mar. 2007. [20] J. Wang and H. Huang, “MC DS/SFH-CDMA systems for overlay systems,” IEEE Trans. Wireless Commun., vol. 1, no. 3, pp. 448–455, July 2002. [21] M. Elkashlan and C. Leung, “Performance of frequency-hopping multicarrier CDMA on an uplink with correlated Rayleigh fading,” in Proc. IEEE Global Telecommun. Conf. (GLOBECOM), Dec. 2003, vol. 6, pp. 3407–3411. [22] L. Hanzo, M. Münster, B. J. Choi, and T. Keller, OFDM and MC-CDMA for Broadband Multi-User Communications, WLANs and Broadcasting. New York: John Wiley & Sons, 2003. [23] O.-S. Shin and K. B. Lee, “Performance comparison of FFH and MCFH spread-spectrum systems with optimum diversity combining in frequency-selective Rayleigh fading channels,” IEEE Trans. Commun., vol. 49, no. 3, pp. 409–4126, Mar. 2001. [24] M. Ebrahimi and M. Nasiri-Kenari, “Performance analysis of multicarrier frequency-hopping (MC-FH) code-division multiple-access systems: Uncoded and coded schemes,” IEEE Trans. Veh. Technol., vol. 53, no. 4, pp. 968–981, July 2004. [25] Z. Taghavi and M. Nasiri-Kenari, “Multiuser performance analysis of MC-FH and FFH systems in the presence of partial-band interference,” in Proc. IEEE Int. Symp. Personal, Indoor and Mobile Radio Commun. (PIMRC), Sep. 2003, vol. 1, pp. 578–582. [26] ——, “Iterative multiuser receiver for coded MC-FH multiple access systems in the presence of partial-band interference,” in Proc. IEEE Veh. Technol. Conf. (VTC)-Fall, Sep. 2004, vol. 3, pp. 1899–1903. [27] K. Yang and G. L. Stüber, “Throughput analysis of a slotted frequencyhop multiple-access network,” IEEE J. Select. Areas Commun., vol. 8, no. 4, pp. 588–602, May 1990. [28] K. Cheun and T. Jung, “Performance of asynchronous FHSS-MA networks under Rayleigh fading and tone jamming,” IEEE Trans. Commun., vol. 49, no. 3, pp. 405–408, Mar. 2001. [29] Y. Yoon, K. Lee, D. Kim, and K. Kim, “Performance improvement of a fast FH-FDMA system by the clipped-linear combining receiver,” in Proc. IEEE Military Commun. Conf (MILCOM)., Oct. 2001, vol. 2, pp. 1345–1349. [30] L. Xiao, J. Lu, and Y. Yao, “Diversity combining for FFH system

[31] [32] [33] [34]

897

with worst-case partial-band noise jamming,” in Proc. IEEE Int. Conf. Commun. Technol. (ICCT), Aug. 2000, vol. 1, pp. 720–724. S. Lin and D. J. Costello, Error Control Coding, 2nd ed. Upper Saddle River, NJ: Prentice Hall, 2004. J. K. Omura and B. K. Levitt, “Coded error probability evaluation for antijam communication systems,” IEEE Trans. Commun., vol. 30, no. 5, pp. 896–903, May 1982. A. J. Viterbi, CDMA: Principles of Spread-Spectrum Communications, Indianapolis, IN: Addison Wesley, 1995. J. G. Proakis, Digital Communications, 4th ed. New York: McGrawHill, 2001.

Reza Nikjah (S’05) received the B.Sc. degree in electrical engineering from Isfahan University of Technology, Isfahan, Iran, in September 2000, and the M.Sc. degree in communication engineering from Sharif University of Technology, Tehran, Iran, in January 2003. He was a research associate in the wireless communication research laboratory in Sharif University of Technology, Tehran, Iran, from January 2003 to December 2004. He has been with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB, Canada, since January 2005, working toward the Ph.D. degree in wireless communications in the Alberta Informatics Circle of Research Excellence (iCORE) Wireless Communications Laboratory. His current research interests include cooperative communications, spread spectrum communications, channel coding, and information theory. Mr. Nikjah has been a recipient of the Alberta Ingenuity Student Scholarship and the iCORE Post Graduate Scholarship Award since May 2005. Norman C. Beaulieu (S’82-M’86-SM’89-F’99) received the B.A.Sc. (honors), M.A.Sc., and Ph.D. degrees in electrical engineering from the University of British Columbia, Vancouver, BC, Canada, in 1980, 1983, and 1986, respectively. He was a Queen’s National Scholar Assistant Professor with the Department of Electrical Engineering, Queen’s University, Kingston, ON, Canada, from September 1986 to June 1988, an Associate Professor from July 1988 to June 1993, and a Professor from July 1993 to August 2000. In September 2000, he became the iCORE Research Chair in Broadband Wireless Communications at the University of Alberta, Edmonton, AB, Canada, and in January 2001, the Canada Research Chair in Broadband Wireless Communications. His current research interests include broadband digital communications systems, ultrawide bandwidth systems, fading channel modeling and simulation, diversity systems, interference prediction and cancellation, importance sampling and semi-analytical methods, decision-feedback equalization, and space-time coding. Dr. Beaulieu is a Member of the IEEE Communication Theory Committee and served as its Representative to the Technical Program Committee of the 1991 International Conference on Communications and as Co-Representative to the Technical Program Committee of the 1993 International Conference on Communications and the 1996 International Conference on Communications. He was General Chair of the Sixth Communication Theory Mini-Conference in association with GLOBECOM’97 and Co-Chair of the Canadian Workshop on Information Theory 1999. He has been an Editor for Wireless Communication Theory of the IEEE T RANSACTIONS ON C OMMUNICATIONS since January 1992, and was Editor-in-Chief from January 2000 to December 2003. He served as an Associate Editor for Wireless Communication Theory of IEEE C OMMUNICATIONS L ETTERS from November 1996 to August 2003. He served on the Editorial Board of the P ROCEEDINGS OF T HE IEEE from November 2000 to December 2006. He was awarded the University of British Columbia Special University Prize in Applied Science in 1980 as the highest standing graduate in the faculty of Applied Science. He received the Natural Science and Engineering Research Council of Canada (NSERC) E. W. R. Steacie Memorial Fellowship in 1999. He was elected a Fellow of the Engineering Institute of Canada in 2001 and was awarded the Médaille K. Y. Lo Medal of the Institute in 2004. He was elected Fellow of the Royal Society of Canada in 2002, and was awarded the Thomas W. Eadie Medal of the Society in 2005. He was also awarded the Alberta Science and Technology Leadership Foundation ASTech Outstanding Leadership in Alberta Technology Award in 2005. In 2006, he was elected Fellow of the Canadian Academy of Engineering. He is the 2006 recipient of the J. Gordin Kaplan Award for Excellence in Research, the University of Alberta’s most prestigious research prize. Professor Beaulieu is the recipient of the IEEE Communications Society Edwin Howard Armstrong Achievement Award for 2007.