Efficient OOK/DS-CDMA Detection Threshold Selection

Transcription

1 2013 American Control Conference ACC) Washington, DC, USA, June 17-19, 2013 Efficient OOK/DS-CDMA Detection Threshold Selection Dimitrios Katselis, Carlo Fischione and Håan Hjalmarsson Abstract A major constraint in deployments of wireless sensor networs WSNs) is the energy consumption related to the battery lifetime of the networ nodes. To this end, power efficient digital modulation techniques such as On- Off eying OOK) are highly attractive. However, the OOK detection thresholds, namely the thresholds against which the received signals are compared to detect which bit is transmitted, must be carefully selected to minimize the bit error rate. This is challenging to accomplish in resource-limited nodes with constrained computational capabilities. In this paper, the system scenario considers simultaneously transmitting nodes in Rayleigh fading conditions. Various iterative algorithms to numerically select the detection thresholds are established. Convergence analysis accompanies these algorithms. Numerical simulations are provided to support the derived results and to compare the proposed algorithms. It is concluded that OOK modulation is beneficial in resource constrained WSNs provided that efficient optimization algorithms are employed for the threshold selection. I. INTRODUCTION Minimizing the energy consumption is a crucial tas in the deployment of WSNs. To this end, on off eying OOK) has been proposed as an effective technique for these networs 1] 3]. To minimize the bit error rate BER) at the receiver, OOK requires an optimal real-time computation of the detection thresholds. However, such an operation is difficult or impossible to perform in resourcelimited nodes, because accurate channel state information CSI) for both the useful and the interfering signals must be acquired. The difficulty is exacerbated when nodes have high mobility, where due to moving metal objects, or movements of the nodes themselves, the nodes experience severe fading conditions. Vehicular applications are also a typical example, where the node or obstacle mobility give quic and deep fading fluctuations in addition to induced Doppler spread and time-selectivity of the channels. In all these situations, the minimization of the average BER is of relevant interest 4]. This yields the need for proper threshold selection, if OOK is employed. We assume OOK modulation over direct sequence code division multiple access DS-CDMA) with randomly selected signatures for each node. This modulation, together with the DS-CDMA, is commonly employed in popular WSN standards such as IEEE ]. To minimize the computations performed in each node, the derivation This wor was supported by the European Research Council under the advanced grant LEARN, the Swedish Research Council under the contract and by EU projects Hydrobionets and Hycon2. D. Katselis, C. Fischione and H. Hjalmarsson are with the Automatic Control Lab, ACCESS Linnaeus Center, School of Electrical Engineering, KTH Royal Institute of Technology , Stocholm, Sweden. dimitri@th.se, carlofi@ee.th.se, hjalmars@th.se. of closed-form expressions for the average bit error probability BEP) would be of interest. The threshold selection problem becomes challenging due to the absence of such expressions. Instead, accurate approximations of acceptable computational complexity for the average BEP may be used to serve this goal. In this paper, we first propose an approximation to the average BEP. Then, we derive three iterative numerical algorithms offering different performance. The algorithms minimize the average BEP in every transmitter-receiver pair, while they run locally in the receiving nodes, providing them with the detection thresholds. For the first algorithm, we show how its time-varying step sizes have to be selected to minimize the number of iterations, which reduces the computational complexity. Then, a second algorithm, which is a variation of the first algorithm, is developed based on a constant stepsize. Finally, to shed more light into the detection threshold selection problem, we propose a third algorithm that uses a heavy ball version of the second algorithm 11]. The first and second algorithms are of firstorder, in the sense that each new threshold, produced at the end of an iteration, only depends on the previous threshold. The heavy ball formulation of the third algorithm introduces memory to the threshold selection, maing the order of the algorithm equal to 2. This approach is similar to the well-nown Successive Over-Relaxation SOR) method in numerical analysis 12]. To give the approximation of the average BEP, assumptions have to be imposed on the nature of fast fading and shadowing impairing the communication lins between the desired transmitter and receiver and between the interferers and the desired receiver. However, the focus of this paper is mainly on the problem of detection threshold selection. Therefore, we consider only fast fading as in 7], 8] and we examine the Rayleigh case 13], 14]. The rest of the paper is organized as follows: Section II presents the system model occurring by using the OOK modulation over DS-CDMA with random signatures. The BEP is analytically approximated in Section III. Algorithms for selecting the detection thresholds are derived in Section IV. Supporting simulations verifying the validity of our analysis are provided in Section V, whereas Section VI concludes the paper. II. SYSTEM MODEL Consider the scenario where K transmitter-receiver pairs of nodes are communicating. The bits of the ith transmitter, i = 1,...,K, are transmitted using an OOK modulation: only bits having value one are transmitted over the wireless /$ AACC 3523

2 channel, while no signal is transmitted when the bit is zero. Assuming DS-CDMA, each bit is transmitted via a spreading sequence. Considering the realistic scenario of imperfect synchronization among distributed nodes, the asynchronous version of DS-CDMA is considered. The same fixed bandwidth W is allocated to each transmitter-receiver pair. Let N = T b /T c be the processing gain, where T b and T c are the bit and chip durations, respectively. All the communication lins in the networ are considered narrowband, therefore the fading is assumed to be flat. These assumptions are representative of the IEEE standard. We will only consider the case of slow flat fading, therefore all channels are considered quasi-static with respect to the duration of two spreading sequences, i.e, two bit symbol) durations. To elaborate more the signal model, we assume that the transmitter of the ith lin transmits the signal 8] s i t) = 2p i b i t)a i t)cosω c t+θ i ), 1) where p i represents the transmitted signal power, b i t) is the data signal data waveform), a i t) the spreading waveform, ω c the carrier frequency and θ i the phase angle of the carrier for the ith transmitter 1. The data signal b i t) is a random process expressed as b i t) = + = b i) dt T b), 2) where dt) = 1,0 t < T b and dt) = 0 otherwise. Additionally, b i) denotes the th bit of the ith transmitter, taing values in {0,1} with probabilities Prb i) = 1] = π i),prb i) = 0] = 1 πi) for all. The spreading signal can be expressed as a i t) = + = a i) ct T c), 3) where ct) is the chip waveform time-limited to 0,T c ). Usually, the chip waveform is assumed to be normalized so that T c c 2 t)dt = T 0 c, while the th chip of the ith user,, taes on values in { 1,+1}. All signatures {ai) } are randomly generated in the sense that every chip polarity is determined by flipping an unbiased coin. Each signal s i t) is transmitted through a flat fading channel. Let h ij t) be the wireless channel coefficient associated to the path from the transmitter of the ith lin to the receiver of the jth lin. Then, h ij t) can be expressed as 8], 14] a i) h ij t) = A ij e jβij δt τ ij ), 4) where δt) represents the Dirac delta function. The fading random variables A ij s are independent and they follow a Rayleigh distribution. Furthermore, A ij s correspond to the envelope of complex Gaussian random processes and {β ij } K i,j=1 correspond to random phases introduced by the channel, considered to be uniformly distributed in 0,2π) 14]. Finally, {τ ij } K i,j=1 correspond to propagation delays 1 transmitter of the ith lin through the wireless medium and lac of synchronism amongst the transmitters. They are assumed to be random variables uniformly distributed in 0,T b ). The received signal at the input of the jth correlation matched filter) receiver is r j t) = = K s i t) h ij t)+n j t) i=1 K 2pi A ij b i t τ ij )a i t τ ij )cosω c t+φ ij ) i=1 +n j t), 5) where denotes convolution. Easily, φ ij = θ i + β ij ω c τ ij and is uniformly distributed in 0,2π). Furthermore, n j t) is assumed to be zero mean, white Gaussian noise with two-sided power spectral density N 0 /2. The random variables, defined above, are generated by different physical sources, thus {A ij } K i,j=1,{φ ij} K i,j=1,{bi) }K i=1,{ai) }K i=1 are mutually independent 7], 8]. In the next section, we characterize the bit error probability experienced at a receiver node. III. BIT ERROR PROBABILITY In this section, we give the expression of the BER that will be used in the development of the iterative detection threshold algorithms. Due to the basic assumptions and properties of the spreadspectrum multiple access systems, our setup possesses symmetry with respect to the desired receiver. Therefore, we can focus on the received signal of the jth receiver. Additionally, only relative time delays and phase angles are important. Thus, without loss of generality we can set φ jj = τ jj = 0. The parameters {τ ij },{φ ij },i j are now the time delays and phase differences of the ith transmitter relative to the jth transmitter at the jth receiver 6], 9]. The decision statistic at the output of the jth coherent correlation receiver for the th transmitted bit of the jth transmitter, is Z j) = D j) +I j) +N j), 6) where D j) is the information bearing signal for the jth pair, I j) is the multiple access interference MAI) due to the presence of multiple transmitting nodes and N j) is the AWGN noise having zero mean and variance N 0 T b /4. Then and I j) K i=1,i j D j) B j) bj) = pj 2 A jjt b }{{} B j) b j), 7) pi 2 A ij b i) 1 Rij) τ ij )+b i) ˆR ] ij) τ ij ) cosφ ij ), 8) 3524

3 where R ij) τ ij ), ˆR ij) τ ij ) are continuous-time partial correlation functions given by R ij) τ ij ) = ˆR ij) τ ij ) = τij 0 Tb a i t τ ij )a j t)dt, τ ij a i t τ ij )a j t)dt 9) and 0 τ ij < T b. We denote b i) 1 Rij) τ ij )+b i) ˆR ij) τ ij ) by W ij), where we neglect the dependence on the subscript due to symmetry of our problem with respect to this subscript. A complete probabilistic description of W ij) is very complicated and long, thus it is omitted here due to lac of space 2. To setup the algorithms for the threshold selection, we need to have an accurate expression or approximation of the BEP. For the case of BPSK, two approaches to obtain such an approximation appear in 8], 16], namely the standard Gaussian approximation SGA) and the improved Gaussian approximation IGA). It has been observed that the IGA is much more accurate than the SGA, especially for small K. Furthermore, in 16], Holtzman has proposed a much simpler but still accurate BEP approximation based on the Stirling formula. We will pursue an analogous approximation here for the OOK modulation. Let A j) be the set of indices corresponding to chip boundaries without transitions in the signature waveform of the jth transmitter and B j) the set of indices corresponding to chip boundaries with transitions in the same signature waveform 6]. Clearly, A j) B j) =, A j) B j) = {0,1,...,N 1}, A j) + B j) = N 1, while according to our assumptions both A j) and B j) are uniformly distributed on {0,1,...,N 1}. Here, C denotes the cardinality of the set C, the set intersection, the set union and the empty set. Furthermore, we assume that G ij) = p i /2A ij cosφ ij ) and we define the set G j) = {G 1j),G 2j),...,G j 1j),G j+1j),...,g Kj) }. Similarly, S j) = {S 1j),S 2j),...,S j 1j),S j+1j),...,s Kj) }, where S ij) = τ ij τ ij /T c T c U0,T c ) models the fractional chip displacement of the ith transmitter relative to the jth transmitter at the jth receiver. Denoting by E ] the expectation operator and using a similar analysis as in 8], 16] for the IGA, it can be proved that 3 E I j) +N j) Gj),S j),b j)] = 0. Furthermore, we may define ϑ j as follows: ) ] 2 G ϑ j = E I j) +N j) j),s j),b j). 10) Based on the fact that G ij) are zero mean Gaussian random variables with variance EA 2 ij ]p i/4, it can be shown that all 2 This analysis is beyond the scope of this paper. It will be presented together with the complete proofs of lemmas and propositions in this paper in the corresponding journal version. 3 This result is a direct consequence of the aforementioned probabilistic description of W ij) omitted here due to lac of space. the moments, hence the probability density functions, of ϑ j s are nown. Remar 1: The above is exploited in IGA. Nevertheless, Stirling formula can provide a tight approximation of the probability of error 16]. The derivation of the BER distinguishes two cases of error: the decision variable 6) is decoded as a zero bit, when a one bit was transmitted; or 6) is decoded as one, when no bit was transmitted. Conditioning on the transmitted bits of the jth transmitter, its channel amplitude A jj and G j),s j),b j), we denote these probabilities by P j 0,Ajj,G j),s j),bj) and P j 1,Ajj,G j),s j),bj), respectively, where P j 0,Ajj,G j),s j),b =Pr Z j) j) δ j b j) = 0,A jj,g j),s j), ) B j)] δ j = Q, 11) ϑj P j 1,Ajj,G j),s j),b =Pr Z j) j) < δ j b j) = 1,A jj,g j),s j), ) B j)] B j) δ j = Q, 12) ϑj where δ j is the decision threshold for Z j), Qx) = 1/ 2π x exp t2 /2)dt is the complementary standard Gaussian distribution function. In 12), we have used the even symmetry of the Gaussian bell. By averaging with respect to A jj,g j),s j),b j), we obtain )] P j δ j ) = 1 π j)) δ j E Ajj,G j),s j),b Q j) ϑj +π j) E Ajj,G j),s j),b j) Q )] B j) δ j. ϑj 13) The minimization of 13) with respect to δ j is difficult, because the resulting probability function is non linear and no closed-form is available for the expectations. Hence, we resort to the Stirling approximation proposed by Holtzman 16], as we mentioned earlier. With this goal in mind, let us define the random variables ζ j0 δ j ) = δ j / ϑ j, ζ j1 δ j ) = B j) δ j )/ ϑ j. Let µ ζjb δ j ) and σ ζjb δ j ) be the mean value and standard deviation of ζ jb for b = 0,1, respectively, with respect to A jj,g j),s j),b j). Then, by 14], 16] we have EQζ jb )] f b δ j ) 2 3 Q µ ζjb δ j ) ) + 1 µ 6 Q ζjb δ j )+ ) 3σ ζjb δ j ) + 1 µ 6 Q ζjb δ j ) ) 3σ ζjb δ j ). 14) Eq. 14) can be used to compute an approximate expression of the BEP 13): P j δ j ) 1 π j) )f 0 δ j )+π j) f 1 δ j ). 15) 3525

4 The nowledge of the probability density functions of ϑ j s can be used to compute the mean values of 1/ ϑ j and 1/ϑ j present in µ ζjb δ j ) and σ ζjb δ j ), b = 0,1, e.g., by employing Riemann integrations over the essential support of the involved random variables. In the next section, we present algorithms to efficiently select the detection thresholds. IV. OPTIMAL DETECTION THRESHOLDS The optimal value of δ j, which we denote by δj, is given by the minimization of 15). Instantaneously, the threshold should be placed between the minimum and maximum value of esssupp{d j) }, where esssupp{x} stands for the essential support of the variable x. Since we are ] averaging over A jj,g j),s j),b j), δj j), where j) min, j) max min = 2 q T b σ Ajj pj /2 and j) max = µ j) B +2 q T b σ Ajj pj /2. Here, µ j) B = p i /2EA jj ]T b and σa 2 jj the variance of A jj. Moreover, the integers q, have to be empirically selected. An empirical rule of thumb corresponds to q 2, = 1. We now state the following important result: Proposition 1: There is a unique minimum of the BEP 15), which is attained when 1 π j)) df 0 δ j ) +π j)df 1δ j ) = 0. 16) Setch of Proof : The BEP 15) is given by the sum of a strictly decreasing function and ] a strictly increasing function in the region j) min, j) max under very mild conditions. Therefore, there is a unique minimum in this interval. To prove that this minimum is attained at points where the BEP derivative ] is zero, note that there are values of δ j in j) min, j) max such that df b δ j )/,b = 0,1 are both positive or negative. Since these derivatives are continuous functions of δ j, it follows that there are δ j s for which the BEP derivative is zero. Remar 2: It can be shown that the BEP probability is a quasiconvex function. Furthermore, dp j 0)/ 0, P j δ j ) 1 π j) ) as δ j and P j δ j ) π j) as δ j +. Therefore,δ j 0) = 0 is a good initialization point for the following detection threshold selection algorithms based on the general notion of descent directions. From Proposition 1, it follows that the minimum of the BEP is given by solving 16). Unfortunately, the derivatives of the functions f 0 δ j ) and f 1 δ j ) are highly non-linear and a closed-form solution cannot be achieved. Numerical techniques have to be used. Simple algorithms such as the steepest-descent, the bisection or a straightforward gridding 17] can be applied to compute the solution of 16). These algorithms, however, are not optimized for our function at hand. In the following, we propose an algorithm for a quic numerical computation of the optimal detection thresholds. A. Algorithm 1: Optimal Contractive Threshold Selector OCTS) Proposition 1 suggests the following iterative algorithm: δ j +1) = δ j ) β) 1 π j)) df 0 δ j )) +π j)df ] 1δ j )), 17) where is the iteration step, and β) is any scalar. When β) is such that the mapping is contractive, then it is easy to show that the fixed point of the mapping belongs in the set interval) of possible optimal detection thresholds 18]. The important question is how to choose β) so that the convergence of the algorithm is quic. We have the following result: Proposition 2: Let the convergence rate of the algorithm in 17) be given by the Lipschitz constant of the iteration and βj) 1 = ). 18) 1 π j) d 2 f 0 δ j )) +π dδj 2 j)d2 f 1 δ j )) dδj 2 Then the convergence rate is maximized. By the previous results, we can set up an optimal algorithm for the determination of the detection thresholds, given by the following iteration: δ j +1) =δ j ) β j) 1 π j)) df 0 δ j )) +π j)df ] 1δ j )), 19) In the following, we will call this algorithm Optimal Contractive Threshold Selector OCTS). Remar 3: Clearly, the OCTS is a first-order algorithm in the sense that every new threshold computation depends only on the previous threshold. Note that in every step of the algorithm, the optimal stepsize βj ), as well as the factor in the bracets, have to be re-evaluated. Additionally, OCTS is a time-varying step size algorithm. In the sequel, we propose an alternative version of this algorithm that uses a constant stepsize till the algorithm converges. This is motivated by the fact that a constant stepsize is generally computationally easier to achieve. B. Algorithm 2: Constant Contractive Threshold Selector CCTS) The OCTS has the form of the standard gradient method. The study of this standard version will lead to many useful results, conclusions and variants for the OCTS. In essence, we want to minimize 15) and we set up an iteration of the form: δ j +1) = δ j ) β) dp jδ j )) 20) for some initial threshold δ j 0) and some step size β). An alternative way to examine if 20) converges and at what speed, is to chec if P j δ j ) is twice differentiable, strongly convex and if dp j δ j )/ is Lipschitz continuous. 3526

5 Clearly, P j δ j ) is twice differentiable. Furthermore, P j δ j ) is strongly convex and has a Lipschitz derivative ] if 0 < l d 2 P j δ j )/dδj 2 L for all δ j j) min, j) max. ] The last condition may not hold for all δ j j) min, j) max. Nevertheless, in our context ] it is enough to show ] that there is a subinterval Γ j) min,γj) max j) min, j) max such that this condition holds. Since there is a threshold point nulling the first derivative of BEP, there is a neighborhood around this point that the second derivative is positive. This implies the existence of the aforementioned desired subinterval. If l,l are nown beforehand, then it is not hard to see that the best convergence rate is attained for β j ) = 2/L + l), which results to ) L l δ j +1) δj δ j ) δ L+l j 21) for the gradient method. We may also proceed even further by isolating the function gl,l) = L l)/l + l). It can be easily checed that g/ L > 0 and g/ l < 0. The convergence { speed is therefore maximized ]} when l = inf d 2 P j δ j )/dδj 2 δ j Γ j) min,γj) max and L = { ]} sup d 2 P j δ j )/dδj 2 δ j Γ j) min,γj) max. The constants l and L can be determined via line search as follows: { l = inf d 2 P j δ j )/dδj 2 > 0 δ j { L = sup d 2 P j δ j )/dδj 2 > 0 δ j j) min, j) max j) min, j) max ]}, ]} 22) Equation 21) shows that for given l,l we can setup the algorithm: Clearly, the gradient algorithm for the tas of the threshold computation is first-order, while the heavy ball method yields a second-order algorithm. This is done by introducing memory to the computation process. If P j δ j ) were strongly convex and had Lipschitz continuous derivatives everywhere, then the heavy ball method would converge 11]. The inequality describing its convergence properties would be: δ j +1) =δ j ) 2 1 π j)) df 0 δ j )) +π j)df ] 1δ j )) δ j +1) δj, L+l 23) which is a contraction with a fixed stepsize. We call this algorithm Constant Contractive Threshold Selector CCTS). Remar 4: Clearly, CCTS requires a line search to determine l,l. The complexity of such a search is comparable with that of the determination of the optimal detection thresholds via line search or gridding. Nevertheless, such a choice is justified in two alternative setups: either when l,l are a priori nown to the receiver or if l,l are approximately determined via a much coarser gridding than the one required to determine the optimal thresholds. C. Algorithm 3: Constant Second-Order Threshold Selector CSOTS) The Heavy Ball Method for our problem can be expressed in two steps of state transitions. Setting an auxiliary state vector θ j ), we have θ j ) = 1 π j)) df 0 δ j )) +π j)df ] 1δ j )) + ζ j )θ j 1) 24) δ j +1) = δ j )+ξ j )θ j ) 25) 3527 Thresholds OCTS CCTS CSOTS TABLE I K = 5,T b = 1,N = 8,N 0 = 1,π 1) = 0.4,π 2) = 0.2,π 3) = 0.3,π 4) = 0.5,π 5) = 0.6,{p i = 1} K i=1 : OPTIMAL THRESHOLDS VALUES OBTAINED BY OCTS, CCTS AND CSOTS WITH A COARSE GRIDDING IN THE DETERMINATION OF l,l. for some initial points θ j 0),δ j 0) and some positive sequences ζ j ),ξ j ). Typically, we set θ j 0) = 0. This algorithm can be rewritten as the iteration: δ j +1) =δ j ) ξ j ) 1 π j)) df 0 δ j )) +π j)df 1δ j )) dδ ] j +ζ j )δ j ) δ j 1)). 26) L l L+ L δ j ) δ j 27) which is achieved for ξ j ) = 4/ L+ l) 2 and ζ j ) = max{ 1 ξ j )l, 1 ξ j )L } 2 11]. Since it holds L l)/ L+ L) L l)/l+l) the heavy ball method would converge faster than the CCTS. Nevertheless, in our case P j δ j ) is strongly convex and has Lipschitz Γ j) min,γj) max ]. Convergence continuous derivatives only in can be guaranteed as shown in Remar 5 below, but not always faster than CCTS. In the following, we will call the algorithm 26) with the aforementioned constant optimal ξ j ) and ζ j ) Constant Second-Order Threshold Selector CSOTS). Remar 5: Whenever P j δ j + 1)) P j δ j )) fails, a safe way to ensure convergence is to set ζ j ) = 0 and ξ j ) = 2/L+l) and to iterate a reduction of ξ j ) by, e.g., a factor of 1/2 till the desired BEP reduction is achieved. In this way, CSOTS behaves as a heavy-ball/ccts hybrid algorithm with guaranteed convergence for a quasiconvex average BEP. V. SIMULATIONS In this section, numerical examples are provided to validate the derived theoretical results and compare the proposed

6 Iterations OCTS CCTS CSOTS TABLE II K = 5,T b = 1,N = 8,N 0 = 1,π 1) = 0.4,π 2) = 0.2,π 3) = 0.3,π 4) = 0.5,π 5) = 0.6,{p i = 1} K i=1 : NUMBER OF ITERATIONS FOR OCTS, CCTS AND CSOTS WITH A COARSE GRIDDING IN THE Threshold Value DETERMINATION OF l,l. Tb=1, Tc=0.125, N=8, K=5, Eq. Pows= Iterations OCTS CCTS CSOTS Fig. 1. K = 5,T b = 1,N = 8,N 0 = 1,π 1) = 0.4,π 2) = 0.2,π 3) = 0.3,π 4) = 0.5,π 5) = 0.6,{p i = 1} K i=1 : Numerical illustration of the obtained first transmit/receive pair thresholds for the OCTS, CCTS and CSOTS with a coarse gridding in the determination of l,l vs. the number of iterations. algorithms. First, we give an illustrative example for the convergence of the algorithms presented in this paper. We assume K = 5,T b = 1,N = 8,N 0 = 1,π 1) = 0.4,π 2) = 0.2,π 3) = 0.3,π 4) = 0.5,π 5) = 0.6,{p i = 1} K i=1. In Table I, the obtained optimal thresholds for the OCTS, CCTS and CSOTS are given, whereas the corresponding number of iterations are presented in Table II. Fig. 1 gives a graphical representation of the algorithm trajectories for the first transmit/receive pair till convergence. In all cases, the halting step corresponds to the case that δ j +1) δ j ) falls below ǫ = We observe that all algorithms give approximately the same optimal threshold values and in a few iterations. The parameters l,l have been determined with a coarse gridding relatively to the one that would be needed to determine the optimal thresholds. We observe that this does not affect the numerical performance of CCTS and CSOTS. Furthermore, CSOTS as a hybrid scheme is not always faster 4 than CCTS or even OCTS as these specific numerical results illustrate. VI. CONCLUSIONS In this paper, we presented various algorithms of different complexity and performance to numerically select the detection thresholds in OOK/CS-CDMA resource-limited networs, and we analytically characterized their convergence properties. Numerical simulations illustrated our results and showed that all the proposed algorithms for the optimal detection threshold determination correspond to good choices for practical deployments of such networs. REFERENCES 1] C. Fischione, K. H. Johansson, A. Sangiovanni-Vincentelli, and B. Zurita Ares, Minmum Energy Coding in CDMA Wireless Sensor Networs, IEEE Transactions on Wireless Communications, vol. 8, no. 2, pp , February ] C. Erin, H. H. Asada, and K.Y. Siu, Minimum Energy Coding for RF Transmission, Proc. WCNC-99, New Orleans, LA, Sept ] M. A. M. Viera, C. N. J. Choelo, D. C. J. da Silva, and J. M. da Mata, Survey on Wireless Sensor Networs Devices, Proc. ETFA- 03, Lisbon, Portugal, Sept ] A. F. Molish, Wireless Communications, Wiley, ] P. Par, P. Di Marco, P. Soldati, C. Fischione, and K. H. Johansson, A generalized Marov chain model for effective analysis of slotted IEEE , Proc. MASS-09, Macau S.A.R.), China, Oct ] J. S. Lehnert, M. B. Pursley, Error Probabilities for Binary Direct- Sequence Spread-Spectrum Communications with Random Signature Sequences, IEEE Transactions on Communications, vol. 35, no, 1, pp , January ] B. Smida, L. Hanzo, S. Affes, Accurate BER of MC-DS-CDMA over Rayleigh Fading Channels, Proc. MC-SS-07, Herrsching, Germany, 7-9 May ] J. Cheng, N. C. Beaulieu, Accurate DS-CDMA Bit-Error Probability Calculation in Rayleigh Fading Channels, IEEE Transactions on Wireless Communications, vol. 1, no. 1, pp. 3-15, January ] M. B. Pursley, D. V. Sarwate, W. E. Star, Error Probability for Direct-Sequence Spread Spectrum Multiple-Access CommunicationsPart I: Upper and Lower Bounds, IEEE Transactions on Communications, vol. 30, no. 5, pp , May ] S-M. Tseng, M. R. Bell, Asynchronous Multicarrier DS-CDMA Using Mutually Orthogonal Complementary Sets of Sequences, IEEE Transactions on Communications, vol. 48, no. 1, pp , January ] J. Nocedal and S. J. Wright, Numerical Optimization, Springer Series in Operations Research, Springer-Verlag, New Yor, 2nd edition), ] R. Janti, S-L. Kim, Second-Order Power Control with Asymptotically Fast Convergence, IEEE Journal on Selected Areas in Communications, vol. 18, no. 3, pp , March ] T. S. Rappaport, J. C. Liberti, Smart Antennas for Wireless Communications : IS-95 and Third Generation CDMA Applications, Prentice Hall Communications Engineering and Emerging Technologies Series, ] G. L. Stuber, Principles of Mobile Communication, Kluwer Academic Publishers, ] S. Verdu, Multiuser Detection, Cambridge University Press, ] J. M. Holtzman, A Simple, Accurate Method to Calculate Spread- Spectrum Multiple-Access Error Probabilities, IEEE Transaction on Communications, vol. 40, no. 3, pp , March ] S. Boyd and L. Vandenberghe Convex Optimization, Cambridge, University Press ] D. P. Bertseas and J. N. Tsitsilis, Parallel and Distributed Computation: Numerical Methods, Athena Scientific, It is faster in most cases. 3528