Maximum-likelihood receivers for synchronous fast

www.ietdl.org Published in IET Communications Received on 9th April 2012 Revised on 1st August 2012 doi: 10.1049/iet-com.2012.0189

ISSN 1751-8628

Maximum-likelihood receivers for synchronous fast frequency-hopped multiple-access M-ary frequency-shift-keying systems over frequency-selective Rician-fading channels L.-M.-D. Le K.C. Teh K.H. Li School of Electrical and Electronic Engineering, Nanyang Technological University, Nanyang Avenue 639798, Singapore E-mail: [email protected]

Abstract: In this study, the performance of a synchronous fast frequency-hopped multiple-access (MA) M-ary frequency-shiftkeying system employing maximum-likelihood (ML) method over either frequency-selective Rician- or Rayleigh-fading channels is investigated. The ML receiver structures are derived and analysed. It is assumed that all the signals arrive at a receiver with the same power strength and all the users operating in the system undergo independent frequency-selective fading. The performance of the ML receivers is compared with that of the linear-combining and the product-combining receivers. The bit-error rate results of the two conventional receivers are obtained via simulation. The proposed analysis, validated by simulation results, shows that the proposed ML receivers can suppress the MA interference more effectively than other existing receivers under different fading conditions. Moreover, the proposed ML receivers are shown to be robust against the inaccuracy in estimation of the required side information.

1

Introduction

Over the past few decades, the spread-spectrum (SS) technique has attracted considerable interest in various military and commercial applications. The most practical SS methods are direct-sequence (DS) modulation and frequency hopping (FH). DS code-division multiple-access (DS-CDMA) is widely used in personal communications network and microcellular mobile communications [1]. As a multipleaccess (MA) method, FH multiple-access (FH-MA) is less popular than DS-CDMA [2]. However, FH systems have certain advantages over DS systems such as low probability of detection and interception, reduced near–far problem, less stringent power control, better resistance to multiple-access interference (MAI), not requiring a contiguous and much lower portable power consumption [3–7]. In particular, FH technique has been used for various MA applications. For commercial applications, FH-MA technology is used in mobile cellular communications, personal communications, wireless local area network (IEEE 802.11) and bluetooth (wireless personal area network) [6]. Owing to the difficulty of maintaining phase coherence between consecutive hops, a non-coherent modulation format such as M-ary frequencyshift-keying (MFSK) is often employed for FH communication systems [8–10]. However, the use of MFSK receivers degrades the system performance because of the non-coherent diversity-combining loss [8, 11]. In FH-MA systems, the interference caused by simultaneous transmission is called MAI. This interference IET Commun., pp. 1–9 doi: 10.1049/iet-com.2012.0189

occurs when the reference user and a non-reference user transmit their signals at the same frequency band. The effects of MAI on the performance of FH-MA systems employing non-coherent frequency-shift keying (FSK) modulation have been studied in [12 – 17]. In [12], the authors reported the bit-error rate (BER) performance comparison among the linear-combining, productcombining and clipper receivers for synchronous fast FH (FFH)-MA/MFSK systems in a frequency-non-selective Rician-fading environment. The Fourier – Bessel series was employed in the analysis of [12]. The maximum-likelihood (ML) structure for asynchronous systems over frequencynon-selective Rayleigh-fading channels was presented in [13]. In [14], the authors estimated the symbol error probabilities of synchronous and asynchronous FH-MA/ MFSK systems via semi-analytic Monte Carlo simulations over non-fading and frequency-non-selective Rayleighfading channels. In [12 – 14], the channel was modelled as a frequency-nonselective fading process. However, in practice, most wireless channels are frequency selective [18]. When the bandwidth of the transmitted signal is greater than the coherence bandwidth of the channel, the frequency components of the transmitted signal with frequency separation exceeding the coherence bandwidth are subject to different gains and phase shifts. In such a case, the channel is said to be frequency selective [19]. Frequency-selective multipath fading is common in urban and indoor environments and is a significant source of potential degradation in a wideband mobile 1

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www.ietdl.org communication system [20]. The performance of FFH-MA binary FSK (BFSK) systems employing equal-gain diversity technique has been analysed in [15] over frequencyselective Rayleigh-fading channels. In [16], the author studied the performance of both MFSK and differentially phase-shift keying systems using the ML method in a frequency-selective Rayleigh-fading environment. The channel was modelled as N independent Rayleigh-fading channels. It should be noted that the characteristic function (CF) derived in [16] is not a closed-form expression and thus the optimum non-linearity was approximated by a soft limiter. The performance of an FFH-MA/MFSK system employing Reed-Solomon coding with linear-combining receiver was investigated in [17] over both Rayleigh- and Rician-fading channels. From the literature survey, it appears that no prior work has been done for performance analysis of an FFH-MA/MFSK system with ML receiver over frequency-selective Ricianfading channels. Thus, in this paper, the analytical BER expressions for a synchronous FFH-MA system employing MFSK modulation scheme with the ML receivers will be derived over both frequency-selective Rician and Rayleighfading channels. It is assumed that all the active users are chip-synchronous [21]. In mobile radio, this is the case for base-to-user transmission. We will derive the probability density function (pdf) of the ML decision variable. Following that, the discrete convolution approach will be applied to obtain the pdf of the final decision statistics. For the special case of frequency-selective Rician fading, that is, frequency-selective Rayleigh fading, the exact closedform expression of the CF of the ML decision variable will be derived. The ML receiver structures will be proposed and analysed. The remainder of this paper is organised as follows. The system model and the channel model will be presented in the next section. In Section 3, the BER expressions for the ML receivers are derived. Numerical results and discussions are presented in Section 4. The conclusion is given in the last section.

2

System model

The block diagram of the FFH-MA/MFSK receiver employing ML method is depicted in Fig. 1, where the desired signal is contaminated by MAI and additive white Gaussian noise (AWGN). We assume that the received signal is perfectly dehopped by a frequency synthesiser, which is controlled by a pseudo-noise code generator. The dehopped signal is then detected by a bank of M square-law detectors. The squarelaw detector outputs rml are then processed by the ML receivers, denoted by the function F(.) in Fig. 1, to produce the resultant outputs zml. Note that m ¼ 1, 2, . . . , M and l ¼ 1, 2, . . . , L, where M represents the modulation order of MFSK and L is the diversity order of FFH. Following that, these resultant outputs are combined to form M decision statistics ym. The symbol corresponding to the largest combiner output is chosen as the decision. Let K be the number of active users operating in this FFH-MA/MFSK system. It is assumed that all the signals arrive at a receiver with the same power level [21]. In the FFH-MA/MFSK system, there are N non-overlapping FH bands and thus the probability of a particular user being jammed by another user is 1/N. Without loss of generality, we assume that the first user is the desired user and there are Kl interfering users in the lth hop, where Kl , K. The dehopped signal in the lth hop can be expressed as rl (t) = sl (t) +

Kl 

Jk (t) + v(t)

(1)

k=1

Kl Jk (t) is the MAI where sl(t) denotes the desired signal, k=1 and v(t) represents the noise term because of AWGN with variance of s2v = N0 B, where N0 is the one-sided power spectral density level. Note that B ¼ 1/Th is the bandwidth of each frequency slot and Th is the hop interval. The Rician-fading channel can be modelled as the sum of a dominant line-of-sight (LOS) component and a

Fig. 1 Block diagram of an FFH-MA/MFSK receiver 2

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IET Commun., pp. 1–9 doi: 10.1049/iet-com.2012.0189

www.ietdl.org Rayleigh-fading component [22]. We model the frequencyselective Rayleigh-fading component as a correlated twopath model within one hopping duration Th by following [23, 24]. Thus, we can rewrite (1) as rl (t) =

√ √ 2As cos (2pfn t + fs,l ) + 2|as1 ,l | cos (2pfn t + us1 ,l ) √ + 2|as2 ,l | cos (2pfn (t − ts ) + us2 ,l ) +

respectively. Note that xs1 ,l , xs2 ,l , ys1 ,l , ys2 ,l , xk1 ,l , xk2 ,l , yk1 ,l and yk2 ,l are independent Gaussian random variables with zero mean and variance of one; ls and lk are functions of the cross-correlation coefficients (rs and rk) of the two main diffused paths of the desired signal and the kth interfering user signal, given by   l2s  8 (4/p − 1)rs + 1 − 1 (7) and

Kl   √ 2Ak cos (2pfk,l t + fk,l )

l2k  8

k=1

√ + 2|ak1 ,l | cos (2pfk,l t + uk1 ,l )  √ + 2|ak2 ,l | cos (2pfk,l (t − tk ) + uk2 ,l ) + v(t)

(2)

√ √ for n ¼ 1, 2, . . . , M, where 2As and 2Ak represent the amplitudes of the LOS of the desired signal and the kth √ user signal, respectively. Note that √ interfering 2|as1 ,l | and 2|as2 ,l | denote the two main √ correlated Rayleigh amplitudes of the desired signal; 2|ak1 ,l | and √ 2|ak2 ,l | are the two main correlated Rayleigh amplitudes of the kth interfering user signal. In (2), fn and fk, l are the baseband frequencies of the desired signal and the kth interfering user signal, respectively. The terms fs, l , us1 ,l , us2 ,l and ts denote the random phases of the LOS and the two diffused paths of the desired signal in the lth hop and the time delay between the two correlated paths of the desired signal, respectively. Similar notations ( fk, l , uk1 ,l , uk2 ,l and tk) are used for the kth interfering user signal. The random variables fs, l , us1 ,l , us2 ,l , fk, l , uk1 ,l , uk2 ,l are assumed to be uniformly distributed over [0, 2p], whereas ts and tk are uniformly distributed over (0, Th]. We define the power ratios of the two main correlated diffused path of the desired signal and the kth interfering user signal as 1s = s2s2 /s2s1 and 1k = s2k2 /s2k1 , respectively. Note that 2s2s1 and 2s2s2 are the diffused power levels of the first and second main diffused paths of the desired signal, respectively; 2s2k1 and 2s2k2 denote the diffused power levels of the first and second main paths of the kth interfering user signal, respectively. Next, the two main correlated Rayleigh amplitudes of the desired signal can be generated from the two uncorrelated complex Gaussianfading amplitudes with unit variance, bs1 ,l = xs1 ,l + jys1 ,l and bs2 ,l = xs2 ,l + jys2 ,l [25]. Similarly, the two main correlated Rayleigh amplitudes of the kth interfering user signal are generated from two random variables, that is, bk1 ,l = xk1 ,l + jyk1 ,l and bk2 ,l = xk2 ,l + jyk2 ,l . Following that |as1 ,l |, |as2 ,l |, |ak1 ,l | and |ak2 ,l | can be rewritten as [23] |as1 ,l | = ss1 |xs1 ,l + jys1 ,l |

(3)

  (4/p − 1)rk + 1 − 1 ,

(8)

respectively. It is assumed that the total average power of the desired user signal PS = A2s + 2s2s1 + 2s2s2 and the average power of all the interfering users’ signals Pk = A2k + 2s2k1 + 2s2k2 , for k ¼ 1, 2, . . . , Kl, remain constant from hop to hop. The parameters A2s and A2k represent the power levels of the LOS component of the desired user signal and the kth interfering user’s signal, respectively. The specular-to-total diffused power ratios Fs = A2s /(2s2s1 + 2s2s2 ) and Fk = A2k /(2s2k1 + 2s2k2 ) determine different levels of fading conditions, for example, essentially no fading (Fs ¼ 1000 or Fk ¼ 1000), Rician fading (Fs ¼ 10 or Fk ¼ 10) and Rayleigh fading (Fs ¼ 0 or Fk ¼ 0).

3

Probability of bit-error analysis

The square-law detector output rml including the effects of MAI and receiver noise is given by 2 2 rml = rI,ml + rQ,ml

(9)

where rI, ml and rQ, ml are its in-phase and quadrature-phase components, respectively. 3.1

Frequency-selective Rician-fading channels

For the case of frequency-selective Rician-fading channel, according to [23] and after some mathematical manipulations, the expressions of the in-phase and quadrature-phase components can be obtained as   √ rI,ml = xs1 ,l ss1 1 + l2s 1s + 2ls 1s cos us,l 

+xs2 ,l ss2 1 − l2s d(fm − fn ) +

Kl 

 √ xk1 ,l sk1 1 + l2k 1k + 2lk 1k cos uk,l d(fk,l − fm )

k=1

(see (4))

 A + xk2 ,l sk2 1 − l2k d(fk,l − fm ) + √ml + nI,l 2 k=1 Kl 

|ak1 ,l | = sk1 |xk1 ,l + jyk1 ,l |

(5)

(10)

and (see (6))   

|as2 ,l | = ss2 l2s |xs1 ,l + jys1 ,l |2 + (1 − l2s )|xs2 ,l + jys2 ,l |2 + 2ls 1 − l2s (xs1 ,l xs2 ,l + ys1 ,l ys2 ,l )

(4)

  

|ak2 ,l | = sk2 l2k |xk1 ,l + jyk1 ,l |2 + (1 − l2k )|xk2 ,l + jyk2 ,l |2 + 2lk 1 − l2k (xk1 ,l xk2 ,l + yk1 ,l yk2 ,l ) IET Commun., pp. 1–9 doi: 10.1049/iet-com.2012.0189

(6) 3

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s2 √ s2ml = s2s1 + 2s2s1 ls 1s cos us,l + s2s2 d(fm − fn ) + v 2 (17) K

l  √ s2k1 + 2s2k1 lk 1k cos uk,l + s2k2 d(fk,l − fm ) +

and  √ rQ,ml = −[ys1 ,l ss1 1 + l2s 1s + 2ls 1s cos us,l  + ys2 ,l ss2 1 − l2s ]d(fm − fn ) −

Kl 

k=1

 √ yk1 ,l sk1 1 + l2k 1k + 2lk 1k cos uk,l d(fk,l − fm )

k=1

−

 A yk2 ,l sk2 1 − l2k d(fk,l − fm ) − √ml − nQ,l 2 k=1

Kl 

(11) where

us,l = us1 ,l − us2 ,l + 2pfm ts

(12)

uk,l = uk1 ,l − uk2 ,l + 2pfm tk

(13)

and

Note that d(.) denotes the Dirac delta function, the terms nI,l and nQ,l represent independent zero-mean Gaussian random variable with a variance of s2v /2. In (10) and (11), the parameter Aml is given by A2ml = A2s d(fm − fn ) +

Kl 

According to [11], the square-law detector output rml is noncentral chi-squared distributed with two degrees of freedom and A2ml is its non-centrality parameter. The corresponding conditional pdf expression of rml can be expressed as   1 rml + A2ml prml (rml /ul , fl , f l ) = 2 exp − 2sml 2s2ml (18)  √ Aml rml U (rml ) × I0 s2ml

ul = [us,l , u1,l , ... , uKl ,l ] and where the vectors fl = [fs1,l , fs2,l , ... , fsKl ,l , f11,l , f12,l , ... , f1Kl ,l , f21,l , f23,l , ..., f2Kl ,l , ... , fKl 1,l , ... , fKl Kl −1,l ] represent the phases of all active users’ signals in the lth hop, as defined in (12), (13) and (15), (16), respectively. The vector f l = [f1,l , f2,l , ... , fKl ,l ] represents the frequencies of all the Kl interfering users’ signals in the lth hop. On the basis of the pdf expression derived in (18) and the ML criteria, the ML receiver structure can be derived as √ 2 zml = Dml rml + Bml +Eml (19) where

A2k d(fk,l − fm )

k=1

+

Kl 

2As Ak cos fsk,l d(fm − fn )d(fk,l − fm )

(14) Bml =

k=1

+

Kl Kl  

1 1 − 2s2ml,NS 2s2ml,S

(20)

Aml,NS s2ml,S − Aml,S s2ml,NS s2ml,NS − s2ml,S

(21)

Dml =

Ai Aj cos fij,l d(fi,l − fm )d(fj,l − fm )

and

i=1 j=1(i=j)

Eml = Cml − Dml B2ml

where

fsk,l = fs,l − fk,l

(15)

fij,l = fi,l − fj,l

(16)

and

(22)

The detailed derivation is provided in the appendix. In the subsequent analysis, this receiver structure is called suboptimum ML-A, which is used to study the system performance over frequency-selective Rician-fading channels. With the aid of [11], the conditional pdf of zml, when signal is present or not present, can be derived as (see (23)) and (see (24))

The derived in-phase component rI,ml and quadrature-phase component rQ,ml are Gaussian distributed with mean of √ Aml / 2 and a common variance given by

 respectively, where Oml = ((zml − Eml )/(Dml )) − Bml and  Qml = ((zml − Eml )/(Dml )) + Bml .

    O2ml + A2ml,S Aml,S Oml Oml pzml (zml /ul , fl , f l , fm = fn ) = 2  exp − I0 U (Oml ) 2s2ml,S s2ml,S 2sml,S Dml (zml − Eml )     Q2ml + A2ml,S Aml,S Qml Qml − 2  exp − I0 U (−Qml ) 2s2ml,S s2ml,S 2sml,S Dml (zml − Eml )

(23)

    O2ml + A2ml,NS Aml,NS Oml Oml pzml (zml /ul , fl , f l , fm = fn ) = 2  exp − I0 U (Oml ) 2s2ml,NS s2ml,NS 2sml,NS Dml (zml − Eml )     Q2ml + A2ml,NS Aml,NS Qml Qml − 2  exp − I0 U (−Qml ) 2s2ml,NS s2ml,NS 2sml,NS Dml (zml − Eml )

(24)

4

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www.ietdl.org The proposed suboptimum ML-A receiver structure derived in (48) requires the side information on the noise variance s2v , the frequencies fk, l and the specular-to-total diffused power ratios Fs and Fk. The Gaussian noise variance can be estimated from a ‘noise-only’ channel for noise power measurement [26], and the frequencies fk, l can be extracted from energy-detector [27]. The specular-tototal diffused power ratio can be obtained by using moment-based estimation method [28] or ML estimation method [29]. The moment-based estimator in [28] uses the first- and the second-order moments of the fading envelope. In [29], the authors proposed the ML estimator using samples of both the fading envelope and the fading phase. Following that, by taking the expectation of pzml (zml /ul , fl , f l , fm = fn ) and pzml (zml /ul , fl , f l , fm = fn ) over ul, fl and fl, we obtain the corresponding unconditional pdfs of zml as pzml (zml /fm = fn ) = Eul ,fl ,f l [pzml (zml /ul , fl , f l , fm = fn )] (25) and pzml (zml /fm = fn ) = Eul ,fl ,f l [pzml (zml /ul , fl , f l , fm = fn )] (26) respectively. Adopting the quantisation approach, we can obtain the pdf of the discrete random variable zˆml . Since all the L hops are independent of each other, the subscript l can be ignored and the pdfs of the final decision statistics  yˆ m = Ll=1 zˆml can be obtained as [13] pyˆ m ( yˆ m /fm = fn ) = [pzˆm (ˆzm /fm = fn )]⊗L

(27)

over ul, fl and fl to obtain the corresponding unconditional pdfs of zml. Following that, the discrete convolution approach is adopted to obtain the pdfs of the final statistics. Finally, the pdfs of the final statistics are used for (30). 3.2

Frequency-selective Rayleigh-fading channels

It is well known that the frequency-selective Rayleigh-fading channel is the special case of the frequency-selective Rician fading when the amplitude of the LOS component approaches zero. Thus, by setting the parameters Aml,S ¼ Aml,NS ¼ 0, the ML decision variable zml can be obtained from (47) as zml = aml rml + bml

(31)

where aml = (1/(2s2ml,NS )) − (1/(2s2ml,S )) and bml = ln((s2ml,NS )/(s2ml,S )), and the pdf expression of rml in (18) becomes   1 rml prml (rml /ul , f l ) = 2 exp − 2 U (rml ) 2sml 2sml

(32)

In the subsequent analysis, the receiver structure in (31) is denoted as optimum ML-B, which is used to investigating the system performance over frequency-selective Rayleighfading channels as a special case. From (31) and (32), the corresponding conditional pdf of zml can be obtained as pzml (zml /ul , f l ) =

  zml − bml exp − U (zml − bml ) 2aml s2ml 2aml s2ml (33) 1

The corresponding CF of zml can be derived as

and pyˆ m ( yˆ m /fm = fn ) = [pzˆm (ˆzm /fm = fn )]⊗L

(28)

where the superscript ⊗ L denotes the (L – 1)-fold discrete convolution operation. Without loss of generality, we assume that the desired signal is present in the first branch of the receiver with frequency f1. Following that, the average BER is Pb =

M P 2(M − 1) s

(29)

(34)

By taking the expectation of czml (jv/ul , f l ) over ul and fl, the unconditional CF of zml can be obtained as

czml (jv) = Eul ,f l [czml (jv/ul , f l )]

(35)

Following  that, the CF of the final decision statistics ym = Ll=1 zml can be obtained as (36)

Note that the CF of ym in (36) can be obtained for higher diversity level L without extra computational complexity. The corresponding pdf of ym can be obtained by performing the inverse Fourier transform given by

GL/2 

(30)

k

for k = 2, 3, . . . , M Note that C denotes the upper limit of quantisation range and G is the number of steps used in the quantisation process. Specifically, to obtain the BER results based on (30), we first obtain the conditional pdfs of the ML decision variable zml. Then we take the expectation of these conditional pdfs IET Commun., pp. 1–9 doi: 10.1049/iet-com.2012.0189

exp (jvbml ) 1 − j2vaml s2ml

cym (jv) = [czm (jv)]L

where the probability of symbol error Ps is given by   2g1 C pyˆ 1 Ps = 1 − G g1 =−GL/2  g  M −1 1  2gk C × pyˆ k , G g =−GL/2

czml (jv/ul , f l ) =

pym (ym ) =

1 2p

1 −1

cym (jv)e−jvym dv

(37)

Finally, the average BER is Pb =

M P 2(M − 1) s

(38)

where Ps denotes the probability of symbol error, which is 5

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www.ietdl.org given by Ps = 1 −

 1   y1 −1

−1

py2 (y2 )dy2 × · · · ×



y1 −1

pyM (yM )dyM

× py1 (y1 )dy1 (39)

4

Numerical results and discussions

In this section, we present the BER performance of the ML receivers for the FFH-MA/MFSK system over frequencyselective fading channels. For a fair comparison, the total spread-spectrum bandwidth is fixed and the total number of FH bands is set to be N ¼ 1000 for the BFSK modulation case without diversity. For subsequent numerical results, the BER values are plotted against the number of users K. In Fig. 2, both the analytical and simulated BER results of the proposed suboptimum ML-A receiver and the optimum ML-B receiver over both frequency-selective Rician and Rayleigh-fading channels are presented. The signal-to-noise ratio (SNR) is set to be 13.35 dB corresponding to a BER of 1025 for a BFSK system under no-fading and jammerfree conditions. All the users’ signals are assumed to experience independent fading. The BER results are obtained with different specular-to-total diffused power ratios Fs and Fk. The close match in both theoretical and simulation results validates the analytical expressions derived in Section 3. From Fig. 2, it can also be observed that larger specular-to-total diffused power ratios lead to better BER performance. It should be noted that the analytical BER results of the suboptimum ML-A and optimum ML-B receivers are obtained based on the BER expressions derived in Sections 3.1 and 3.2, respectively. The BER results of the optimum ML-B receiver match well with the ones of the ML-A receiver with Fs ¼ Fk ¼ 0. This shows that the CF in (34) and the ML receiver structure in (31) can be applied to study the performance of FFH-MA/ MFSK system with ML receiver under frequency-selective Rayleigh-fading channels as a special case. As mentioned earlier, to implement the proposed ML receivers, side

Fig. 2 Analytical and simulated BER results of the suboptimum ML-A and optimum ML-B receivers with M ¼ 4, L ¼ 3, Eb/N0 ¼ 13.35 dB, 1s ¼ 1k ¼ 1 and rs ¼ rk ¼ 0.5 over frequency-selective Rician- and Rayleigh-fading channels 6

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information is required. Therefore in this figure, the BER results with the imperfect side information are included to examine their effects on the system performance. The results with imperfect side information are obtained via simulation. It can be observed that these BER curves are close to the BER curves with perfect side information. This demonstrates that the proposed ML receivers are robust against the inaccuracy in estimation of the required side information. Fig. 3 shows BER results of the suboptimum ML-A receiver with different diversity levels L over frequencyselective Rician-fading channels. It can be observed that there exists an optimum diversity gain for the suboptimum ML-A receiver, that is, L ¼ 9 with K . 40. This phenomenon is due to the fact that when the diversity order is increased, the system performance degrades because of the non-coherent diversity-combining loss [11]. On the other hand, the FFH system with higher diversity orders can effectively counteract MAI effects and fading. Thus, there is an optimum diversity order. Both analytical and simulated BER results of the optimum ML-B receiver with different diversity levels L over frequency-selective Rayleigh-fading channels are provided in Fig. 4. The optimum diversity gain for the optimum ML-B receiver is observed to be L ¼ 15 with K . 60 over frequencyselective Rayleigh-fading channels. From Figs. 3 and 4, we also observe that the improvement in system performance by using diversity decreases as the number of users increase. This is because the non-coherent diversitycombining loss is accumulated when K increases. In Fig. 5, the BER results of the suboptimum ML-A receiver with different power ratios 1s , 1k and crosscorrelation coefficients rs , rk are presented under various types of fading. When all users’ signals experience frequency-selective Rician fading, the system performance is not very sensitive to the channel parameters 1s , 1k , rs and rk . On the other hand, for the case of Rayleigh fading (Fs ¼ Fk ¼ 0), the system performance under the fading channel with larger values of the power ratios 1s and 1k is slightly worse than that of the smaller values of 1s and 1k . Comparing to the power ratios 1s and 1k , increasing the values of the cross-correlation coefficients rs and rk degrades the system performance more significantly. This

Fig. 3 BER performance of the suboptimum ML-A receiver with M ¼ 8, SNR ¼ 13.35 dB, 1s ¼ 1k ¼ 1, rs ¼ rk ¼ 0.7, Fs ¼ Fk ¼ 2 and various diversity orders L over frequency-selective Rician-fading channels IET Commun., pp. 1–9 doi: 10.1049/iet-com.2012.0189

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Fig. 4 Analytical and simulated BER results of the optimum ML-B receiver with M ¼ 4, SNR ¼ 16 dB, 1s ¼ 1k ¼ 1, rs ¼ rk ¼ 0.9 and various diversity orders L over frequency-selective Rayleigh-fading channels

Fig. 6 BER performance comparison of various diversitycombining receivers with M ¼ 4, L ¼ 3, SNR ¼ 13.35 dB, 1s ¼ 1k ¼ 1, and rs ¼ rk ¼ 0.7 over frequency-selective Ricianand Rayleigh-fading channels

observation is consistent with the results shown in Fig. 3 of [23] for the BFSK system with multitone jamming over frequency-selective Rayleigh-fading channels. In Fig. 6, we compare the BER performance of the proposed suboptimum ML-A receiver with two conventional diversitycombining receivers, namely, the linear-combining and the product-combining receivers for different specular-to-total diffused power ratios (Fs, Fk). The BER results of the two conventional receivers are obtained via simulation. This figure shows that the proposed suboptimum ML-A receiver outperforms the two conventional receivers over both frequency-selective Rician- and Rayleigh-fading channels. The improvement in performance between the suboptimum ML-A receiver and the two conventional receivers can be as large as three orders of magnitude. It is also observed that this improvement becomes more significant when the number of users K increases. This is because although a higher K leads to higher MAI power, the suboptimum ML-A receiver has the perfect MAI information to counteract the MAI more efficiently.

5

In this paper, we have derived the analytical BER expressions of a synchronous FFH-MA/MFSK system with the ML receivers over both frequency-selective Rician- and Rayleighfading channels. For the case of frequency-selective Rician fading, the pdf of the ML decision variable has been derived and the discrete convolution approach has been used to obtain the pdf of the final decision statistics. For the case of frequency-selective Rayleigh fading, the closed-form expression of the CF of the ML decision variable has been derived. The ML receiver structures have been proposed and analysed. To implement the proposed ML receivers, the side information is required. However, it has been shown that the proposed ML receivers are not sensitive to the imperfect side information. Our analysis, validated by simulation results, shows that the proposed suboptimum and optimum ML receivers can suppress the MAI more effectively than the conventional receivers, that is, up to three orders of magnitude in the performance improvement.

6

Fig. 5 BER performance of the suboptimum ML-A receiver with M ¼ 4, L ¼ 5, SNR ¼ 13.35 dB, and various values of 1s , 1k , and rs , rk over frequency-selective Rician- and Rayleigh-fading channels IET Commun., pp. 1–9 doi: 10.1049/iet-com.2012.0189

Conclusion

References

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www.ietdl.org 9 Teh, K.C., Li, K.H., Kot, A.C.: ‘Performance analysis of an FFH/BFSK self-normalising receiver under multitone jamming’, IEE Proc. Commun., 1998, 145, (6), pp. 431–436 10 Al-Dweik, A.J., Sharif, B.S.: ‘Exact performance analysis of synchronous FH-MFSK wireless networks’, IEEE Trans. Veh. Technol., 2009, 58, (7), pp. 3771– 3776 11 Proakis, J.G.: ‘Digital communications’ (McGraw-Hill, New York, 2001, 4th edn.) 12 Han, Y., Teh, K.C.: ‘Performance study of linear and nonlinear diversity-combining techniques in synchronous FFH/MA communication systems over fading channels’, IET Commun., 2007, 1, (1), pp. 1 –6 13 Han, Y., Teh, K.C.: ‘Maximum-likelihood receiver with side information for asynchronous FFH-MA/MFSK systems over Rayleigh fading channels’, IEEE Commun. Lett., 2006, 10, (6), pp. 435–437 14 Cheun, K., Choi, K.: ‘Performance of FHSS multiple-access networks using MFSK modulation’, IEEE Trans. Commun., 1996, 44, (11), pp. 1514– 1526 15 Solaiman, B., Glavieux, A., Hillion, A.: ‘Equal gain diversity improvement in fast frequency hopping spread spectrum multipleaccess (FFH-SSMA) communications over Rayleigh fading channels’, IEEE J. Sel. Areas Commun., 1989, 7, (1), pp. 140– 147 16 Yue, O.C.: ‘Maximum likelihood combining for noncoherent and differentially coherent frequency-hopping multiple-access systems’, IEEE Trans. Inf. Theory, 1982, 28, (4), pp. 631–639 17 Wang, J., Moeneclaey, M.: ‘Multiple hops/symbol FFH-SSMA with MFSK modulation and Reed –Solomon coding for indoor radio’, IEEE Trans. Commun., 1993, 41, (5), pp. 793– 801 18 Schober, R., Lampe, L.H.-J., Gerstacker, W.H., Pasupathy, S.: ‘Modulation diversity for frequency-selective fading channels’, IEEE Trans. Inf. Theory, 2003, 49, (9), pp. 2268–2276 19 Biglieri, E., Proakis, J., Shamai, S.: ‘Fading channels: informationtheoretic and communications aspects’, IEEE Trans. Inf. Theory, 1998, 44, (6), pp. 2219– 2292 20 Gui, X., Ng, T.S.: ‘Performance of asynchronous orthogonal multicarrier CDMA system in frequency selective fading channel’, IEEE Trans. Commun., 1999, 47, (7), pp. 1084–1091 21 Hung, C.P., Su, Y.T.: ‘Diversity combining considerations for incoherent frequency hopping multiple access systems’, IEEE J. Sel. Areas Commun., 1995, 13, (2), pp. 333– 344 22 Ziemer, R.E., Peterson, R.L.: ‘Introduction to digital communication’ (Prentice-Hall, New Jersey, 2001, 2nd edn.) 23 Wu, T.-M.: ‘A suboptimal maximum-likelihood receiver for FFH/BFSK systems with multitone jamming over frequency-selective Rayleighfading channels’, IEEE Trans. Veh. Technol., 2008, 57, (2), pp. 1316– 1322 24 Wu, T.-M., Hung, P.C.: ‘Maximum-likelihood receivers for FFH/BFSK systems with multitone jamming over frequency-selective Rayleigh fading channels’. Proc. IEEE Int. Conf. Communication, Glasgow, Scotland, UK, 2007, pp. 815– 820 25 Beaulieu, N.C.: ‘Generation of correlated Rayleigh fading envelopes’, IEEE Commun. Lett., 1999, 3, (6), pp. 172 –174 26 Lee, J., Miller, L.E., Kim, Y.K.: ‘Probability of error analyses of a BFSK frequency-hopping system with diversity under partial-band jamming interference. II. Performance of square-law nonlinear combining soft decision receivers’, IEEE Trans. Commun., 1984, 32, (12), pp. 1243– 1250 27 Torrieri, D.J.: ‘Principles of secure communication systems’ (Artech House, Boston, 1992, 2nd edn.) 28 Tepedelenlioglu, C., Abdi, A., Giannakis, G.B.: ‘The Rician K factor: estimation and performance analysis’, IEEE Trans. Wirel. Commun., 2003, 2, (4), pp. 799 –810 29 Chen, Y., Beaulieu, N.C.: ‘Maximum likelihood estimation of the K factor in Rician fading channels’, IEEE Commun. Lett., 2005, 9, (12), pp. 1040– 1042 30 Teh, K.C., Kot, A.C., Li, K.H.: ‘Performance study of a maximumlikelihood receiver for FFH/BFSK systems with multitone jamming’, IEEE Trans. Commun., 1999, 47, (5), pp. 766– 772

7

where prml (rml /ul , fl , f l , fm = fn ) represents the pdf of the square-law detector output when the desired signal is present and prml (rml /ul , fl , f l , fm = fn ) denotes the corresponding pdf when the signal is not present. From (18), we obtain prml (rml /ul , fl , f l , fm = fn ) = 

  √ rml + A2ml,S Aml,S rml I0 U (rml ) × exp − 2s2ml,S s2ml,S

(41)

and prml (rml /ul , fl , f l , fm = fn )    √ rml + A2ml,NS Aml,NS rml 1 exp − = 2 I0 U (rml ) 2sml,NS 2s2ml,NS s2ml,NS (42) where Kl 

A2ml,S = A2s +

A2k d(fk,l − fm ) +

k=1

+

Kl Kl  

Kl 

2As Ak cos fsk,l d(fk,l − fm )

k=1

Ai Aj cos fij,l d(fi,l − fm )d(fj,l − fm )

i=1 j=1(i=j)

(43)

A2ml,NS =

Kl 

A2k d(fk,l − fm )

k=1

+

Kl Kl  

Ai Aj cos fij,l d(fi,l − fm )d(fj,l − fm )

i=1 j=1(i=j)

(44) √ s2ml,S = s2s1 + 2s2s1 ls 1s cos us,l + s2s2 +

Kl 

√ s2k1 + 2s2k1 lk 1k cos uk,l + s2k2 d(fk,l − fm )

k=1

s2 + v 2 (45) and

s2ml,NS =

Appendix

1 2s2ml,S

Kl

 √ s2k1 + 2s2k1 lk 1k cos uk,l + s2k2 d(fk,l − fm ) k=1

7.1

Derivation of ML receiver structure

The function F(.) of the ML receiver can be expressed as [24]  zml = ln

prml (rml /ul , fl , f l , fm = fn )



prml (rml /ul , fl , f l , fm = fn )

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(40)

+

s2v 2 (46)

Following that, the ML decision variable zml can be obtained by substituting (41) and (42) into (40) as IET Commun., pp. 1–9 doi: 10.1049/iet-com.2012.0189

www.ietdl.org 

zml =

1



1 r 2s2ml,NS 2s2ml,S ml     √ √ Aml,S rml Aml,NS rml + ln I0 − ln I0 (47) s2ml,S s2ml,NS   A2ml,NS A2ml,S s2ml,NS − + ln + 2 2sml,NS 2s2ml,S s2ml,S −

Since the computation of ln(I0(.)) is non-linear in nature, by following [23], we adopt the approximation lnI0(x) ¼ x, for x . 4, to simplify (47) to     Aml,S Aml,NS √ 1 1 zml = − − r + rml + Cml 2s2ml,NS 2s2ml,S ml s2ml,S s2ml,NS

where

Cml

  A2ml,NS A2ml,S s2ml,NS = 2 − + ln 2sml,NS 2s2ml,S s2ml,S

(49)

The above approximation is applicable when the SNR is high (SNR . 6 dB), which is of practical use [30]. In order to derive the pdf expressions of the suboptimum ML-A receiver output zml from (41) and (42), we perform some mathematical manipulations to simplify (48) to zml = Dml

√ 2 rml + Bml +Eml

(50)

(48)

IET Commun., pp. 1–9 doi: 10.1049/iet-com.2012.0189

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