Multiuser Power and Bandwidth Allocation in Ad Hoc Networks with Type-I HARQ under Rician Channel with Statistical CSI

Transcription

1 Mutiuser Power and Bandwidth Aocation in Ad Hoc Networks with Type-I HARQ under Rician Channe with Statistica CSI Xavier Leturc Thaes Communications and Security France Christophe J. Le Martret Thaes Communications and Security France christophe.e Phiippe Cibat Téécom ParisTech & Université Paris-Sacay France Abstract In this paper, we address the probem of joint power and bandwidth aocation for Type-I hybrid automatic repeat request in miitary ad hoc networks based on orthogona frequency division mutipe access when ony statistica channe state information is avaiabe to perform the Resource Aocation (RA) and when considering practica moduation and coding schemes. The novety of our work ies in the consideration of the Rician fast fading channe during the RA process, which is a genera fading mode which can represent the Rayeigh fading channe and the additive white Gaussian noise channe as specia cases. The objective of this paper is to propose an agorithm to minimize the sum of the transmit power of each user under different quaity of service constraints. An accurate approximation of the coded packet error rate over the Rician fading channe is first proposed, and an efficient agorithm to sove the probem is deveoped. The effectiveness of the agorithm is demonstrated though simuations. Our resuts point out a significant energy saving when the Rician distribution is expicity taken into account when performing the RA instead of considering the Rayeigh channe. Index Terms HARQ, Resource aocation, Rician Fading, Optimization I. INTRODUCTION Hybrid Automatic Repeat request (HARQ) consists in the combination of Forward Error Correction (FEC) and Automatic Repeat request (ARQ) protoco which aows to increase the reiabiity of wireess communications. This paper focuses on Resource Aocation (RA) for Type-I HARQ in miitary ad hoc networks assuming the use of Orthogona Frequency Division Mutipe Access (OFDMA) as the mutipe access technoogy. We consider that ony statistica Channe State Information (CSI) is avaiabe to perform the RA since miitary ad hoc Medium Access Contro (MAC) schemes based on Time Division Dupexing (TDD) cannot hande instantaneous feedback, conversey to modern civiian ceuar network systems. In addition, we take into account the effect of practica Moduation and Coding Scheme (MCS) in our derivations, which gives a practica interest to our work. Our objective is to minimize the tota transmit power (the sum of the transmit power of each user) under severa Quaity of Service (QoS) constraints, which is motivated by the inherenty imited battery ifetime in miitary ad hoc networks. In this context, the main novety of our work is that we consider the Rician channe mode in our RA agorithm, which is a generic channe mode which can represent the Rayeigh fading channe and the Additive White Gaussian Noise (AWGN) channe as specia cases. Considering this channe mode appears to be interesting since miitary ad hoc networks can be depoyed in various theatres with different propagation conditions. For instance, in rura area, it may happen that a strong Line of Sight (LOS) exists between the transmitter and the receiver and, in this case, the channe is we modeed by the so caed Rician channe mode [1]. Aso, it is known that the error probabiity is ower on the Rician channe than on the Rayeigh one and hence, we expect a power saving by expicity considering the Rician mode during the RA. The probem of the minimization of the transmit power for HARQ in the singe user context has been investigated in numerous works, such as [2] where the authors minimize the average transmit power for Chase Combining (CC)-HARQ over the Rayeigh bock fading channe considering capacity achieving codes. A simiar work was done in [3], where the authors considered practica MCS over the Rayeigh bock fading channe. An extension of this work has been presented in [4] for reaying systems. In [5], the average power consumption of Type-I HARQ over the Rayeigh bock fading channe was minimized in the finite bockength regime. On the other hand, the RA probem for HARQ-based OFDMA systems with power concern was treated in [6], where the authors minimize the transmit power for Incrementa Redundancy (IR)-HARQ in a mutiuser OFDMA system when considering queuing constraints. The authors of [7] maximize the energy efficiency for IR-HARQ when channes are perfecty known by the transmitter. The Energy Efficiency (EE) of IR-HARQ in the mutiuser context is aso investigated in [8]. A the above mentioned works consider the use of capacity achieving codes, which gives upper bound concerning the achievabe performance with practica MCS. Practica MCS were taken into account in [9] where the authors maximize the EE of Type-I HARQ when the transmitter is assumed to have fu CSI information. In [10], the authors minimize the transmit power for Type-I HARQ with QoS constraints on the

2 Rayeigh channe with statistica CSI. This work is extended in [11] for Type-II HARQ, again considering the Rayeigh channe and statistica CSI. To the best of our knowedge, these are the ony works which consider the RA probem for HARQ when the objective is to minimize the transmit power with practica MCS and statistica CSI. From the above discussion, it is cear that there exist no work in the iterature which address the probem of the tota transmit power minimization of HARQ-based OFDMA systems over the Rician fading channe with statistica CSI and practica MCS. The contribution of this paper is to study this probem for Type-I HARQ. In detai, we start studying the PER of convoutiona codes over the Rician fast fading channe. We then formuate the probem of the minimization of the transmit power with QoS constraints. This probem is simiar to the one of [12] but, in our work, we consider the more genera Rician fading channe, and we add a constraint on the maximum aowed transmit power. We propose an agorithm to optimay sove this probem and finay, the effectiveness of the proposed approach is investigated through numerica simuations. The remainder of the paper is organized as foows. In Section II, we present the system mode whie in Section III we study the PER of convoutiona codes over the Rician fast fading channe. Section IV is devoted to the probem formuation, and Section V to the optima probem resoution of this probem. Numerica resuts are discussed in Section VI and finay, concuding we draw some concuding remarks in Section VII. II. SYSTEM MODEL We consider a miitary custered ad hoc network with a tota bandwidth B which is divided in N c subcarriers, and we focus on intra custer RA. The number of inks in the custer is L, and we consider the use of OFDMA as the muti access technoogy. Moreover, to perfom the RA, we assume that a Custer Head (CH) is eected in each custer. The CH centraizes the statistica CSI of the different inks, and aocate the physica resources. The channe of each ink is modeed as constant for the duration of an OFDMA symbo, and it changes independenty between consecutive symbos. This assumption is justified by the use of a we designed frequency hopping pattern aong with bit intereaved coded moduation [13]. The samped channe impuse response of ink during the jth OFDMA symbo is defined by h (j) = [h (j, 0),..., h (j, M 1)] T with (.) T denoting the transposition operator. The main originaity of our work comes from the consideration of the (possibe) existence of a LOS between the transmitter and the receiver, that is, the first tap of the channe impuse response, h (j, 0), is in genera modeed as a compex Gaussian random variabes with a non zero mean denoted by µ and variance ζ,0 2 [1]. The other taps are considered as compex Gaussian random variabes too, but their mean is zero and their variance is ζ,i 2, i {1,..., M 1}. Finay, we make the common assumption that the different taps are uncorreated. A these assumptions can be summarized as h (j) CN (µ, Σ ), where CN (µ, Σ ) corresponds to the muti-variate compex norma distribution with mean µ = [µ, 0,..., 0] T and covariance matrix Σ := diag M M (ζ,0 2,..., ζ2,m 1 ). With these notations, we can express the signa received on the nth subcarrier of ink at OFDMA symbo i as Y (i, n) = P L H (i, n)x (i, n) + Z (i, n), (1) where P L is a deterministic power attenuation coefficient which depends on the distance between the emitter and the receiver, H (i) := [H (i, 0),..., H (i, N c 1)] T denotes the Fourier transform of h (i), X (i, n) is the transmitted symbo on the nth subcarrier of the ith OFDMA symbo and Z (i, n) CN (0, N 0 B/N c ), with N 0 the noise power spectra density. The eements of H (i) are identicay distributed random variabes H (i, n) CN (µ F,,n, ζ 2) where ζ 2 := Tr(Σ ) and µ F,,n := µ e j2πn/nc. Therefore, it is known that H (i, n) foows a Rician distribution with parameters Ω := µ 2 + ζ 2 and K := µ 2 /ζ 2 [14]. The Rician K parameter is known to be an important indicator of the ink quaity, indeed, K = 0 corresponds to the Rayeigh fading whie K corresponds to the AWGN channe. Then, the higher the vaue of K, the better the quaity of the channe. We define the average Gain-to-Noise ratio (GNR) as: G := P L E[ H (i, n) 2 ] N 0 = P L Ω N 0. (2) Without any oss of generaity, we assume that for a, Ω = 1. As said previousy, we assume that the CH ony has statistica knowedge on the channe, more precisey, the CH is assumed to know ony the average GNR and the Rician K factor of each ink to perform the RA. Due to this assumption, the power aocated on each subcarrier is identica since the subcarriers are identicay distributed, and we define P := E[ X (i, n) 2 ] as the power aocated to ink to transmit one moduated symbo on one subcarrier. Moreover, we define n as the number of subcarrier aocated to ink and γ := N c /n o as the associated proportion of bandwidth. It can then easiy be seen that the average power consumed by ink to transmit one OFDMA symbo is N c γ E, where E := N c P B. (3) The average Signa-to-Noise Ratio (SNR) of ink is given by snr := G E. (4) We assume that each ink in the custer uses a Type-I HARQ mechanism, that is, for the ink, the stream of information bits is arranged into packets of L bits, and L 0, overhead bits, incuding a Cycic Redundancy Check (CRC), are added in each packet. After that, a Medium Access Contro (MAC) packet is obtained by encoding these L M, := L +L 0, bits by a FEC of rate R, and the MAC packet is moduated. We assume the use of either the BSPK moduation, of a 2 m -QAM moduation on

3 ink. The moduated packet is then sent on the channe and, after reception, the receiver decodes the packet, and check the vaidity of the received bits using the CRC, which is assumed to be error-free. If an error is detected, the receiver discards the packet, and sends a Negative ACKnowedgement (NACK) to the transmitter, who retransmits the same packet. Otherwise, if the packet is successfuy received, an ACKnowedgement (ACK) is sent to the transmitter, who sends another MAC packet. The mechanism is repeated unti an ACK is sent by the receiver, or the maximum number of transmissions, denoted by P, is reached. Notice that we consider the ACK/NACK as error free, which is a common assumption in the iterature [15], [16]. III. APPROXIMATION OF THE PACKET ERROR RATE One of the major difficuty when deaing with HARQ with practica MCS is the absence of cosed-form expression for the coded PER. In the iterature, some approximations can be found [17], [18], but they are in genera vaid ony on frequency-fat bock fading channe i.e. when the channe is modeed as a singe coefficient which remains constant within the duration of a packet. In [15], [16], the notion of effective SNR is expoited to perform the RA, but these effective SNR methods require either instantaneous CSI, or at east outdated CSI, which is not avaiabe in our study. In [19], it is proposed to upper bound the PER to perform the RA based on statistica CSI in the case of the Rayeigh channe. In the present paper, we want to extend this study by considering the more genera case of the Rician fading channe. To our best knowedge, this probem has never been addressed in the iterature. To approximate the PER in the Rician channe, we resort to an approach originay proposed in [20], where two observations are drawn concerning the reation between the PER at the output of the Viterbi decoder and the uncoded Bit Error Rate (BER) at the input of the decoder: i) this reation is amost independent of the considered moduation and more importanty ii) the reation is approximatey inear in the ogarithmic domain. From these two observations, we can approximate the actua PER of ink, q (E ), by q (E ) expressed as foows q (snr ) = (BER (snr )) a e b, (5) where a and b are fitting coefficients which depend on the packet ength, on the convoutiona code parameters and on the Rician K parameter, BER is the BER of the th ink over the Rician fading channe with parameter K and snr is given by (4). Notice that this type of approximation has been used for the RA in the singe user context in [9], where fu CSI is considered to be avaiabe at the transmitter side, whereas our work considers that ony statistica CSI is avaiabe. From (5), we see that the expression of q invoves the BER of the moduation of the th ink in the Rician fading channe. It is we known that the BER over a fading channe is obtained by averaging the BER in the AWGN channe over a the possibe vaues of the SNR. To perform this operation, we first express the instantaneous SNR on one subcarrier of the th ink which is given by snr := h 2 snr, (6) where h CN (µ, ζ 2 ). Hence, from the above discussion, the average BER of ink writes BER (snr ) = E snr [BER,AWGN (snr )], (7) where E snr [.] is the mathematica expectation taken over the possibe vaues of snr, and BER,AWGN is the BER of the th ink in the Additive White Gaussian Noise (AWGN) channe. A common approximation of BER,AWGN is BER,AWGN (snr ) ψ Q( β snr ), (8) where ψ and β are moduation-dependent parameters whose vaues can be found in Tabe 6.1 in [21], and Q(.) is the Q- function. Cacuating the exact vaue of the expectation in (7) appears to be difficut since it woud invove numerica integrations due to the presence of the Q-function. For this reason, we propose to approximate the Q-function by a combination of exponentias as suggested for exampe in [22] or [23] Q(x) i max i=1 δ i e θix2, (9) where δ i and θ i are fitting coefficients and i max is the number of exponentias in the sum. The arger the vaue of i max, the better the approximation. In this paper, we used the coefficients proposed in [23], where i max = 4. The expectation in (7) can be then approximated as BER (snr ) ψ i 4 i=1 δ i E snr [e θiβ snr ]. (10) The expectation in (10) is exacty the moment generating function of the distribution of snr evauated in θ i β. We can easiy prove that Y := 2/(snr ζ 2)snr foows a noncentra chi-square distribution with 2 degrees of freedom and noncentraity parameter µ 2 /ζ2, which yieds [24]: BER (snr ) ψ i 4 δ i i=1 e A,i 1 + θ i β snr ζ 2. (11) where A,i := θ i β snr µ 2 /(1 + θ i β snr ζ 2 ). The PER can be then approximated by pugging (11) into (5). The accuracy of the proposed PER approximation is checked in Section VI where we can observe that the approximation is quiet accurate, and therefore can be used to predict the PER of the system with an anayticay tractabe expression. IV. PROBLEM FORMULATION In this section, we formuate the optimization probem that we want to sove. The objective of this paper is to sove the probem of the minimization of the tota transmit power under two QoS constraints: a constraint on the goodput (which is the minimum usefu data rate) and a constraint on the maximum

4 aowed transmit power. For Type-I HARQ, the goodput η of the th user is given by [15] η (G E, γ ) := γ m R α (1 q (G E )), (12) where α := L M, /L. The probem that we want to sove can be then written as Probem 1. min E,γ γ E (13) =1 s.t. γ m R α (1 q (G E )) η (0),, (14) γ E Q Max,,, (15) γ 1, (16) =1 γ > 0, E > 0,, (17) where E := [E 1,, E K ] and γ := [γ 1,, γ K ]. In this probem, (14) is the minimum goodput constraint, (15) is the constraint on the maximum transmit energy, (16) is a structura constraint on the bandwidth and (17) are positivity constraints. One can see that this probem is identica to the one in [12], except that we add the constraint (15) and that we consider a different expression of the PER which takes into account the Rician channe. As discussed in Section III, there is no anaytica expression for q (G E ) in the coded case, therefore, we can not directy sove Probem 1. For this reason, we wi not directy tacke this probem but an approximation, where we repace q (G E ) by q (G E ) given in (5). The resuting probem writes Probem 2. min E,γ γ E (18) =1 s.t. m R γ α (1 q (G E )) η (0),, (19) (15), (16), (17), Our objective is now to sove Probem 2. A feasibiity condition for this probem is given in [11] and, for the rest of this paper, we consider that this feasibiity condition is satisfied. Actuay, Probem 2 appears to be difficut to sove since the objective function (18) and constraint (15) are neither concave nor convex. Therefore, to render it convex, we first perform the foowing change of variabe:, Q = γ E. (20) After this change of variabes, Probem 2 can be equivaenty rewritten in the foowing form, which wi be proved to be convex Probem 3. min Q,γ s.t. Q (21) =1 Q m R γ α (1 q(g )) η (0), γ (22) Q Q Max,, (23) γ 1 (24) =1 where Q = [Q 1,..., Q L ] Q > 0, γ > 0, (25) This probem is characterized by the foowing emma. Lemma 1. Probem 3 is the minimization of a convex function over a convex set. Proof: First, we compute the second order derivative of og(f(x)) where f(x) := exp( ax/(1 + 2bx))/(1 + 2bx) where a and b are non negative constants: og(f(x)) = 4ab (1 + 2bx) 3 + 4b 2 > 0. (26) (1 + 2bx) 2 Therefore, f is stricty og convex and hence stricy convex [25]. It foows that q is convex since it can be expressed as g(x) u where g(x) is a non negative inear combination of convex function, and u 1. Then, since η (Q, γ ) := γ (1 q (G Q /γ )) is the so-caed perspective function of q(g E ) [25] and since the perspective of a concave function is concave, it resuts that η (Q, γ ) is concave. Finay, the other constraints (15), (16), (17) and the objective function (21) are inear and as a consequence convex. Thanks to the change of variabes (20), our Probem 3 fas within the convex optimization framework. In the next section, we wi expain how to optimay sove this probem. V. OPTIMAL PROBLEM RESOLUTION Since Probem 3 is a standard convex minimization probem due to Lemma 1, we know that the Karush-Kuhn-Tucker (KKT) conditions are necessary and sufficient to find its

5 optima soution [25]. These conditions write ( ) 1 + κ + µ G q G Q = 0, (27) ( ( )) ( )) Q Q λ + µ ( 1 q G q Q G = 0, γ γ γ (28) κ (Q Q Max, ) = 0, (29) ( ( ))) µ η (0) Q γ m R α (1 q G = 0, γ (30) λ( γ 1) = 0, =1 γ (31) where λ, µ = [µ 1,..., µ L ] and κ = [κ 1,..., κ L ] are the non negative Lagrangian mutipiers. To sove this probem, we wi proceed in two steps: first, we wi consider λ as known, and we wi express the optima soution as a function of this unique mutipier. Second, we wi find the optima vaue of λ. Let us begin with the first step. A. Soution for fixed λ In what foows, our objective is to express the soution of the KKT conditions as a function of the unique mutipier λ. Given (27), we can see that µ > 0 since κ 0 and, x 0, q (x) 0. Then, (30) yieds ( γ Q m R α 1 q (G )) γ = η (0), (32) which gives us a first reation between γ and Q. To continue the resoution, we have to discuss about the possibe vaues of κ. Case 1): κ = 0 In this case, from (27), we obtain µ = G q 1 ( Q G γ ). (33) By pugging (33) into (28) and using (32), we obtain γ (λ) = η (0) m R α (1 q (F 1 (λg ))), (34) Q (λ) = γ (λ) F 1 (λg ), (35) G where F (x) := (1 q (x))/ q 1 (x) x and F is the inverse of F with respect to the composition. We denote by I κ the set of the index of the inks for which κ = 0. Case 2): κ > 0: The compementary sackness condition (29) gives us Q = Q Max,, (36) and we can obtain γ thanks to (32). More precisey, et us define the foowing function ( ( )) G Q Max, F (x) := m R α x 1 q η (0). (37) x Let γ (0) be the smaest zero of F on (0, 1). Due to (32), we have γ = γ (0) when κ > 0. Moreover, combining (27) with (28), we get the foowing inequaity ( ) Q Max, λ > F G. (38) γ (0) The index of the inks for which κ > 0 (i.e. the inks for which (38) hods) are grouped into I κ. Combining the two cases, the optima soution of Probem 3 is given by (γ (λ), Q (λ)) = { ((34), (35)) if I κ, (γ (0), Q Max, ) Otherwise. We have then exhibited the optima soution of Probem 3 as a function of the optima Lagrangian mutipier λ. In what foows, our objective wi be to find the optima vaue of λ. B. Search for the optima vaue of λ To find the optima vaue of λ, we wi use the compementary sackness condition (31). In detai, we form the sum of the optima vaues of the bandwidth sharing variabes γ, {1, L}, for a given vaue of the Lagrangian mutipiers λ, which can be formuated mathematicay as Γ(Λ) := I κ γ (0) + I κ G (Λ), (39) where G (x) := η (0) /(m R α (1 q (F 1 (xg )))). We can prove that Γ is continuous and non increasing. We use this property to find the optima vaue of λ. Since λ is non negative, there are two possibiities: i) Γ(0) < 1: in this case, we know that the optima vaue of λ is zero since increasing λ wi ony decrease the vaue of Γ, which means that (31) wi never be satisfied, and ii) Γ(0) > 1: since Γ is non increasing, the optima vaue of λ must be such that (31) is verified, i.e. Γ(λ) = 1. Finay, putting a pieces together, the optima soution of Probem 3 is expressed in the foowing theorem. Theorem 1. If Γ(0) < 1, the optima soution of Probem 3 is given by γ = G (0) and Q = γ /G F 1 (0) for a. Otherwise, this optima soution is given by γ = G (λ) and Q = γ /G F 1 (λg ) for the inks in I κ i.e. such that λ F (G Q Max, /γ (0) ), and for the other inks γ = γ (0) and Q = Q Max,, where λ is the soution of Γ(λ) = 1 on R +. Theorem 1 states that the optima soution of Probem 3 reduces to a ine search of the optima vaue of λ, indeed, either this optima vaue is zero and the optima soution of Probem 3 can be computed directy, or this optima vaue is stricty positive and hence a ine search has to be performed to sove Γ(λ) = 1.

6 K 0 10 a b TABLE I: Fitting parameters for two vaues of K VI. NUMERICAL RESULTS AND DISCUSSION In this section, we first study the accuracy of the PER approximation proposed in Section III, and second, we investigate the resuts of the agorithm proposed in Section V. The accuracy of the proposed PER approximation is iustrated on Fig. 1 and 2, where we impemented the convoutiona code of generators poynomia [171, 133] 8 over a Rician fast fading channe, aong with both the BPSK and the QPSK moduation. The fitting parameters we used are given in Tabe I. We can see that the PER approximation is cose to the PER obtained by simuation, which confirms the usefuness of this approximation. drawn in [50 m, 1 km]. The bandwidth B is set to 5 MHz, the noise power spectra density N 0 to 170 dbm/hz and the packet ength L is 128. For simpicity, we consider α = 0. The maximum aowed transmit power is set to 28 dbm and the required goodput is equa for a ink, and is given by G p /(BL), where G p is the target sum of the goodput. The carrier frequency is f c = MHz and we put P L = (4πf c /c) 2 D 3 where c is the speed of ight in vacuum. We performed 100 Monte Caro simuations for each point. Finay, we use the same convoutiona code as for Fig. 1 and 2 aong with QPSK moduation. To study the resuts of the proposed agorithm, we consider three scenarios: in the first one, the RA is performed by considering that every ink experiences a Rayeigh channe, whie, in the second scenario, we suppose that two of the L inks experience a Rician channe with parameter K = 10 for the both. In the third scenario, a the inks experience a Rician channe with parameter K = Simuated PER, K=0 Predicted PER, K=0 Simuated PER, K=10 Predicted PER, K= Packet Error Rate Transmitted Power (dbm) SNR (db) 21 First scenario Second scenario third scenario G p (bps) x 10 6 Fig. 1: PER obtained by simuation and with the approximation, BPSK moduation. Fig. 3: Resuts of the proposed agorithm for various vaues of K Packet Error Rate Simuated PER, K=0 Predicted PER, K=0 Simuated PER, K=10 Predicted PER, K= SNR (db) It can be seen from Fig. 3 that considering the Rician mode in the RA process aows a significant energy saving. For instance, for G p = kbps, 0.4 dbm (i.e. 10%) can be saved when two inks are Rician (scenario 2) compared with scenario 1, whie scenario 3 aows to save 1.9 dbm (i.e. 35%) compared with scenario 1. This can expained by the fact that the RA agorithm satisfies the goodput constraint with equaity and, in the Rayeigh channe, more energy is required to achieve the minimum goodput constraint. Therefore, performing the RA by considering that the channe is Rayeigh distributed whie its actua distribution is the Rician one induces a oss of energy, that is, considering the appropriate channe mode is of great importance when performing the RA with statistica CSI. Fig. 2: PER obtained by simuation and with the approximation, QPSK moduation. Concerning the RA agorithm, we set the number of inks to 8, and each ink is separated by a distance D uniformay VII. CONCLUSION In this contribution, we have proposed an agorithm for the optima RA probem for Type-I HARQ in mutiuser miitary ad hoc networks based on OFDMA when the channe is Rician. We first proposed a simpe yet accurate approximation of the

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