A MAXIMUM-LIKELIHOOD DECODER FOR JOINT PULSE POSITION AND AMPLITUDE MODULATIONS

Transcription

1 The 18th Annual IEEE International Syposiu on Personal, Indoor and Mobile Radio Counications (PIMRC 07) A MAXIMUM-LIKELIHOOD DECODER FOR JOINT PULSE POSITION AND AMPLITUDE MODULATIONS Chadi Abou-Rjeily, Meber IEEE Departent of Electrical and Coputer Engineering Lebanese Aerican University (LAU), Byblos, Lebanon chadi.abourjeily@lau.edu.lb ABSTRACT In this paper, we consider the proble of Maxiu- Likelihood (ML) detection with ulti-antenna Ipulse Radio Ultra-Wideband (IR-UWB) counications in the cases where the inforation is odulated onto either the positions or the positions and aplitudes of the transitted UWB pulses. The proposed solution assures good convergence ties since the structure of these sparse and ulti-diensional constellations is taken into consideration. While all the known ML decoders can not be applied with these constellations that do not have a lattice structure, the proposed solution assures an optial detection. I INTRODUCTION In the literature, space-tie (ST) block codes were principally associated with PAM and QAM constellations [1, 2]. For PAM constellations, decoding a P T ST block code transitting at a rate of n sybols PCU is equivalent to the detection in a signal-space of diension nt. For QAM, the real and iaginary parts of each inforation sybol can be decoded separately and the decoding proble corresponds to the detection in a 2nT -diensional space where all the coponents of the inforation vectors are independent fro each other. For these linear odulations, the linearity of a ST code facilitates the design of axiu-likelihood (ML) decoders. The sphere decoder algoriths [3 6] constitute possible practical ipleentations of these ML decoders. On the other hand, there is a growing interest in applying ST coding techniques on Ipulse Radio Ultra-Wideband (IR- UWB) [7]. For IR-UWB, Pulse Position Modulations (PPM) and hybrid Pulse Position and Aplitude Modulations (PPAM) are appealing since they take advantage fro the high teporal resolution to deliver higher data rates with lower coplexity [8]. For M-PPM and hybrid M-PPM-M -PAM, the coponents of the inforation vectors are not independent fro each other. Aong the the M coponents of the vector representation of each inforation sybol, only one coponent can be different fro zero (this coponent corresponds to the position of the transitted pulse). Therefore, even when linear ST codes are used (for exaple the codes proposed in [7] for IR-UWB), the sphere decoders can not be applied when these codes are associated with PPM or PPAM. The sphere decoding algoriths are based either on the Pohst enueration strategy [3, 4] or on the Schnorr-Euchner enueration strategy [5, 6]. These enueration strategies correspond to two different techniques adopted for spanning the set of adissible values at each layer of the decoded inforation vector. Initially proposed for decoding hypercubes carved fro lattices, non of the above strategies is adapted to PPM or PPAM constellations. In this paper, we consider the proble of ML detection with ulti-diensional and non-linear PPM and PPAM constellations. Inspired fro the Schnorr-Euchner enueration strategy, we propose non-trivial odifications of the sphere decoders based on this strategy. While ML detection with PPAM was previously considered in [9], the solution that we propose in this paper adits better convergence ties. Notations: stands for the Kronecker product. The function round(x) rounds x to the nearest integer while x rounds x to the nearest odd integer. X :,k corresponds to the k-th colun of the atrix X. X (p) corresponds to the eleents (p 1)M + 1,..., pm of the MP -diensional vector X for p = 1,..., P. In the sae way, X (p,p ) corresponds to the M M atrix coposed fro the eleents X i,j of the P M P M atrix X for i = (p 1)M + 1,..., pm and j = (p 1)M + 1,..., p M for p, p = 1,..., P. I :, stands for the -th colun of the M M identity atrix I M. 0 M and 1 M stand for the M-diensional vectors whose coponents are all equal to 0 and 1 respectively. For a vector x and a atrix X, diag(x) constructs a square atrix whose diagonal eleents are equal to x while Diag(X) returns a vector coposed fro the diagonal eleents of X. II PROBLEM FORMULATION Consider a hybrid M-PPM-M -PAM constellation where each odulated pulse can occupy M positions with an aplitude that can take M possible values. This is a M-diensional constellation where each inforation sybol is represented by a M-diensional vector belonging to the set: C ={(2 1 M )I :, ; = 1,..., M ; = 1,..., M} (1) In what follows, PPM will be treated as a special case of PPAM. Consider a ulti-antenna Tie-Hopping (TH) UWB syste with P transit antennas, Q receive antennas and a Rake equipped with L fingers. For M-diensional constellations, the linear dependence between the baseband inputs and outputs of the channel can be expressed as: X = HA + N (2) where the QLM-diensional vectors X and N stand for the decision and noise vectors respectively. A C P is the P M- diensional inforation vector whose ((p 1)M + )-th coponent corresponds to the aplitude of the pulse (if any) /07/$25.00 c 2007 IEEE

2 The 18th Annual IEEE International Syposiu on Personal, Indoor and Mobile Radio Counications (PIMRC 07) transitted at the -th position of the p-th antenna for = 1,..., M and p = 1,..., P. H is the QLM P M channel atrix given by H = [H1 T HQ T ]T where H q = [Hq,1 T Hq,L T ]T for q = 1,..., Q. The atrix H q,l is given by H q,l = [H q,l,1 H q,l,p ] for l = 1,..., L. H q,l,p is a M M atrix for p = 1,..., P. The (, )-th eleent of H q,l,p corresponds to the ipact of the signal transitted during the -th position of the p- th antenna on the -th correlator (corresponding to the -th position) placed after the l-th Rake finger of the q-th receive antenna. For exaple, for TH-UWB [7]: H q,l,p (, ) = r q,p (( )δ + l ) (3) where δ stands for the odulation delay. l is the l-th finger delay and r q,p corresponds to the frequency selective channel between antennas p and q. For linear ST codes applied over T sybol durations and transitting at a rate of n sybols PCU, A becoes a nt M- diensional vector. In eq. (2), the atrix H ust be replaced by (I T H)Φ where Φ is a T P M nt M atrix describing the linear dependence between the coded sybols and the inforation sybols. In what follows, we consider the decoding of vectors coposed of P sub-vectors each having M coponents. We consider the case where QL P. In this case, H has full-rank and eq. (2) can be written in an equivalent way as: X P M 1 = H P M P M A P M 1 + N P M 1 (4) where the indices indicate the corresponding atrices diensions and H = ( H T H ) 1 2. The ML detection corresponds to deciding in the favor of the vector  verifying:  = arg in A C P X HA 2 (5) For M-PPM-M -PAM, the coordinates of the P M- diensional inforation vector A in eq. (4) are not independent. A is coposed of P sub-vectors that are ultiples of the coluns of the M M identity atrix. Taking this fact into consideration, the transit antenna array can be seen as coposed of P sub-arrays each having M virtual antennas fro which only one antenna is active at a tie. But because of the co-channel interference and the frequency selectivity of the UWB channels, each one of the P M data streas will interfere with all the other streas. Moreover, the PPAM constellations are sparse. In fact, the generated lattice has a cardinality of (M + 1) P M since the aplitude in each position can be equal to zero in addition to the M nonzero values. However, aong these points only (MM ) P points are valid. Therefore, applying the sphere decoders without any odifications results in non-valid points. III DECODING ALGORITHM The Schnorr-Euchner (SE) enueration strategy was first applied in [5] for searching for the closest lattice point. This enueration enhances the spanning of the adissible interval at each level of the received sybol vector. Consider for exaple the k-th level for a certain value of k {1,..., P }. Designate by e k,k the value of the received signal after applying a zero-forcing decision feedback equalization (ZF-DFE) (in other words, e k,k is considered in the subspace of the transitted constellation). Designate by R the upper triangular atrix obtained by applying a QR decoposition on the channel atrix and denote by r i,j its (i, j)-th coponent. When deterining the squared Euclidean distance between the received vector and the current node (the point we are checking in a search algorith), the influence of the k-th level is deterined by: ( ) 2 d 2 k = (r k,k (e k,k x k )) 2 ek,k x k (6) V k,k where V = R 1 (iplying that V k,k = r 1 k,k since R is upper triangular), x k is the k-th coponent of the checked lattice point and the squared Euclidean distance is given by: d 2 = P d 2 k (7) k=1 where the interference of the levels k + 1,..., P on the k-th level is eliinated recursively by applying a ZF-DFE. Suppose that R has positive diagonal eleents, d 2 k is a second degree function of x k. The integer that iniizes d 2 k is given by x k = round(e k,k ). The second closest lattice point is given by x k +1 (resp. x k 1) when e k,k > x k (resp. e k,k < x k ). In other words, spanning the interval corresponding to the values of the k-th coordinate of the lattice point in the order of increasing distance is equivalent to the following enueration: x k,k, x k,k + ρ, x k,k ρ, x k,k + 2ρ, x k,k 2ρ,... (8) where ρ = sign(e k,k x k ) (since V k,k > 0 for k = 1,..., P ). For finite constellations, this spanning can be readily odified in order not to check lattice points that are outside the boundaries of the transitted constellation. The Schnorr- Euchner enueration with a 4-PAM constellation is represented scheatically in Fig. 1 where the black points correspond to the 4-PAM aplitude levels ({±1, ±3}) while the gray points correspond to the infinite extension of 4-PAM. The plotted parabola corresponds to the squared Euclidean distance between the received point and the checked lattice point. The enueration is the sae as in eq. (8) but now ρ is replaced by 2ρ. Moreover, iρ, for a certain integer i, is replaced by (i + 1)ρ each tie one of the upper or lower boundaries is exceeded. The first tie when both iρ and (i + 1)ρ exceed the constellation s boundaries, the search process is terinated and the resulting point (±(M 1) for M -PAM) corresponds to the point that has the largest distance fro the received point. Consider the case of ulti-diensional PPM or PPM-PAM constellations. The received vector of length P M is now seen as coposed of P layers. However, in this case, each layer is coposed of a hypercube of diension M rather than a finite interval of points (of diension 1) as with PAM constellations. The M coponents of each hypercube can not span different intervals independently because only one coponent can have

3 The 18th Annual IEEE International Syposiu on Personal, Indoor and Mobile Radio Counications (PIMRC 07) Figure 1: The Schnorr-Euchner enueration with a 1- diensional 4-PAM constellation. a nonzero value. Suppose that, for a given layer (coposed of M diensions), we can efficiently order all the valid M- diensional sub-vectors of the PPM or PPM-PAM constellation as a function of their increasing distances fro a certain received point. In this case, the sphere decoder based on the Schnorr-Euchner enueration can be readily odified to decode these constellations. Unfortunately, ordering the MM eleents of a M-PPM- M -PAM constellation is not a siple task and the coplexity of this ordering ight increase with the value of M. In what follows, we present a odified algorith that perfors a joint M-diensional span. It is not necessary to sort all the valid MM eleents, it is sufficient to develop an efficient way that perits to generate the M-diensional vectors, at each layer, in a recursive anner (just like the Schnorr-Euchner enueration). In this case, the recursion is stopped if the point falls outside the current sphere. The coplexity of the proposed solution does not increase with M. As all the other decoders, the coplexity increases with M because this results in increasing the diensionality of the signal subspace. The ain difference between the Schnorr-Euchner enueration for PAM constellations and the proposed enueration strategy for PPM-PAM constellations is as follows. For the enueration with finite PAM constellations, falling outside the boundaries of the constellation can be siply reedied by perforing a jup in the opposite sense. This is possible because we are sure that this double jup results in an increase in the Euclidean distance as shown in Fig. 1. For PPM-PAM, we ust be sure that the next candidate node is a valid node before passing to this node. In fact, the relative increase in the iniu distance resulting fro this jup (at a certain coponent of a given layer) ust be copared with the increase induced by passing to a next candidate node at the other coponents of the sae layer. To be ore explicit, we further highlight this point by an exaple. Consider a 2-PPM-4-PAM constellation given by: {( 3, 0), ( 1, 0), (1, 0), (3, 0), (0, 3), (0, 1), (0, 1), (0, 3)} (9) and represented in Fig. 2. In this exaple, the closet lattice point to the received point is given by (3, 0). Applying the SE enueration on the x-axis, the next lattice node in the sense of increasing the iniu distance is the point (1, 0). Continuing the SE enueration on this axis, the next candidate node is (5, 0). This is not a valid 4-PAM point and the SE enueration will pass to the point ( 1, 0). Denote by d A B the relative increase in the iniu distance when passing fro point A to point B. While d (1,0) (5,0) < d (1,0) (0,1), we have d (1,0) ( 1,0) > d (1,0) (0,1) and therefore, unlike the SE, the next node is (0, 1) rather than the node obtained by perforing a double SE jup fro (1, 0) (the point ( 1, 0)). Therefore, a joint 2-diensional enueration is needed and the values of d A B at the two different levels of the sae layer ust be only evaluated (and copared) for valid points A and B. An exaple of a 2-diensional span is shown in Fig. 2. In general, a M-diensional span ust be applied and a new decoding algorith for PPM-PAM is needed. We first give the pseudo code of the function decide that will be used in the decoding algorith. Denote by e the M-diensional vector obtained by applying ZF on the levels 1,..., k 1, k + 1,..., P and DFE on the levels k + 1,..., P. Consider the -th coponent of the k-th layer with {1,..., M} and k {1,..., P }. If the -th coordinate of the M-diensional sub-vector at the k-th layer is given by x, then the influence of the k-th layer on the squared Euclidean distance is given by: d 2 ( ) k = R e x 2 E I :, x 2 R :, = (E ) T E 2x (R :,) T E + (x ) 2 (R :,) T R :, (E ) T E + D (10) Therefore, when the -th level of the k-th layer is considered alone (the levels 1,..., 1, + 1,..., M have zero values and the -th level is considered as a PAM constellation), this level can be spanned in the following way: x where: +2ρ X x = 2ρ = (R :,) T E (R:,) T R :, X ; ρ +4ρ ( = sign X 4ρ,... (11) ) x (12) (13) At the k-th layer, the input of the function decide consists of R and E. The outputs consists of the M-diensional vectors D, X and x whose -th coponents are given in eq. (10), eq. (12) and eq. (13) respectively. x contains the coordinates of the first node to be visited at the M levels of the k-th layer (if these layers are consider separately). D stores the distances corresponding to these nodes while X is used to deterine the SE span at the M levels. At this stage, we know siply the starting point of each one of the M levels as well as their corresponding SE enuerations without taking into account the possibility of passing fro one level to another during the joint M-diensional span (for exaple, in Fig. 2, when passing fro (2) (3), (5) (6) and (6) (7)). In addition to D, X and x, the function decide returns the scalars pos and ap that correspond to the position

4 The 18th Annual IEEE International Syposiu on Personal, Indoor and Mobile Radio Counications (PIMRC 07) Figure 2: The enueration with a 2-diensional 2-PPM-4- PAM constellation. and the aplitude of the closest lattice point respectively. The pseudo code of the function decide is given by: decide (Input: [Y, R, M ]. Output: [pos, ap, x, X, D]) Y 1 = R T Y ; Ỹ 1 = diag (Y 1 ) Y 2 = Diag ( R T R ) ; Ỹ 2 = diag (Y 2 ) ) ) 1 X = Diag (Ỹ 1 (Ỹ2, x = X x = ax (1 M, x) ; x = in (M 1, x) ) D = 2Diag (Ỹ1 diag (x) + Diag (Ỹ2 (diag (x)) 2) pos = arg in (D) ; ap = x(pos) In what follows, the variables step, flag1 and flag are M P atrices. At the k-th layer step :,k is a M-diensional vector that perits to deterine, at each one of the M levels, the difference between the next generated node and the current node. Throughout the algorith, step is peranently odified to assure that x,k + step,k (the next node) is always a valid PAM sybol when x,k (the current node) is valid. This constraint is added since, as explained earlier, only valid points ust be considered. Note that step,k can be equal to zero. In this case, perforing x,k + step,k perits to pass to the -th level (fro anyone of the other M 1 levels). Once the -th coponent is allowed to have a nonzero value, step,k is updated according to: {2ρ, 4ρ, 6ρ... iρ, 2sign(iρ )... 2sign(iρ )} (14) where ρ is defined in eq. (13). i is the integer for which x,k + iρ falls outside the boundaries of the PAM constellation for the first tie. When this happens, flag1,k is set to zero. In this case, for testing the reaining nodes, we fix step,k = 2sign(iρ ). When x,k + step,k falls outside the boundaries for the second ties, the spanning is stopped at the -th level and we set flag,k to one. All the levels of the k-th layer are spanned when flag :,k has all of its coponents equal to 1. The pseudo code of the sphere decoder with M-PPM-M -PAM constellations is given by: Decoding algorith (Input: [z, R, P, M, M, C]. Output: â) Step 1: Set k = P + 1, dist = 0, V = R 1, ρ = 0 M, bestdist = C (sphere squared radius). Let C 0 >> 1. Step 2: newdist = dist + ρ T ρ, if (newdist < bestdist)&(k 1) go to Step 3 else go to Step 4 endif. Step 3: if k = P + 1, e :,k 1 = V z else for i = 1,..., k 1, e (i) :,k 1 = e(i) :,k V (i,k) ρ endfor endif. k = k 1, dist = newdist, E :,k = R(k,k) e :,k [pos, ap, x :,k, X :,k, D :,k ] = decide(e :,k, R(k,k), M ) X :,k = api :,pos step :,k = 0 M, step pos,k = 2sign ( ) X step,k X step,k flag :,k = 0 M, flag1 :,k = 1 M step pos,k = 2sign ( ) step pos,k, flag1 pos,k = 0 endif ρ = E :,k apr(k,k) :,pos go to Step 2. Step 4: if newdist < bestdist for i = 1,..., P, â (i) = X :,i endfor, bestdist = newdist elseif k = P, terinate else k = k + 1 endif. if M =1 flag,k = 1 go to Step 4 endif generate the next node: R = R (k,k) :,pos, D step,k = 2R T E ( ) :,k xpos,k + step pos,k + ( R T R ) ( ) 2 x pos,k +step pos,k + C0 flag pos,k pos = arg in =1,...,M (D,k ) x pos,k = x pos,k + step pos,k. if step pos,k = 0 step pos,k = 2sign ( ) X pos,k x pos,k else step pos,k = ( 1) flag1 pos,k steppos,k 2flag1 pos,k sign ( ) step pos,k endif. if flag1 pos,k 0 flag1 pos,k = 0, step pos,k = 2sign(step pos,k ) else flag pos,k = 1 endif. flag pos,k = 1 endif endif. ap = x pos,k, X :,k = api :,pos, ρ = E :,k apr(k,k) :,pos go to Step 2.

5 The 18th Annual IEEE International Syposiu on Personal, Indoor and Mobile Radio Counications (PIMRC 07) , 4 PPM 2 PAM 3 3, 4 PPM 4 PAM 6 6, 4 PPM 2 PAM 6 6, 4 PPM 4 PAM SNR per bit (db) Figure 3: Convergence w.r.t. algorith 2 in [9] as a function of the SNR for 4-diensional constellations. Lines of Step 3 as well as lines of Step 4 assure that the next node does not fall outside the boundaries of the PAM constellation. This is why D pos,k can be evaluated at the eighth and ninth lines of Step 4. At these lines, having flag pos,k equal to 1 avoids spanning the level pos of the k-th layer. Updating step pos,k as in lines of Step 4 perits to generate the enueration given in eq. (14). IV SIMULATIONS AND RESULTS Siulations show the perforance over the IEEE a channel odel recoendation CM2 that corresponds to nonline-of-sight propagation [10]. The antennas of the transit and the receive arrays are supposed to be sufficiently spaced so that each one of the P Q sub-channels can be generated independently fro the other sub-channels. The second derivative of the Gaussian pulse with a duration of 0.5 ns is used. The odulation delay is chosen to verify δ = T w. In Fig. 3, we copare the convergence ties of the proposed algorith and [9] as a function of the SNR with 4-diensional constellations. Siulations are perfored with P P uncoded systes and a 1-finger Rake. The initial sphere radius is set to infinity in both cases. The ordinate corresponds to the factor by which the relative convergence tie (with respect to algorith 2 in [9]) is reduced. A siilar coparison is perfored in Fig. 4 for different hybrid constellation sizes with P P ultiantenna UWB systes with a 1-finger Rake for P = 2, 4. The SNR is fixed at 15 db and the initial sphere radius is set to infinity. The superiority of the proposed solution is evident. Moreover, results show that the gap between the proposed solution and algorith 2 in [9] increases with the diensionality of the constellation. V CONCLUSION We proposed a ML decoding algorith for M-diensional PPM and PPAM constellations (M 2). Inspired fro the one diensional Schnorr-Euchner enueration strategy, we pro- 2 2, M =2 2 2, M =4 4 4, M =2 4 4, M = M: nuber of odulation positions Figure 4: Convergence w.r.t. algorith 2 in [9] at a SNR of 15 db with M-PPM-M -PAM for M = 2, 4. posed a solution that enhances the span of each one of the M- diensional subspaces in which the transitted sybols are distributed. Because of this enhanced span that is adapted to the PPM and PPAM constellations, the proposed algorith outperfors previously announced solutions of this proble. REFERENCES [1] S. M. Alaouti, A siple transit diversity technique for wireless counications, IEEE J. Select. Areas Coun., vol. 16, pp , October [2] F. Oggier, G. Rekaya, J.-C. Belfiore, and E. Viterbo, Perfect space tie block codes, IEEE Trans. Infor. Theory, vol. 52, no. 9, pp , Septeber [3] E. Viterbo and J. Boutros, A universal lattice code decoder for fading channels, IEEE Trans. Infor. Theory, vol. 45, pp , July [4] M. O. Daen, H. E. Gaal, and G. Caire, On axiu-likelihood detection and the search for the closest lattice point, IEEE Trans. Infor. Theory, vol. 49, pp , October [5] E. Agrell, T. Eriksson, A. Vardy, and K. Zeger, Closest point search in lattices, IEEE Trans. Infor. Theory, vol. 48, pp , August [6] Z. Guo and P. Nilsson, Reduced coplexity schnorr-euchner decoding algoriths for io systes, IEEE Coun. Lett., vol. 8, pp , May [7] C. Abou-Rjeily, N. Daniele, and J.-C. Belfiore, Space tie coding for ultiuser ultra-wideband counications, IEEE Trans. Coun., vol. 54, pp , Noveber [8] H. Zhang, W. Li, and T. A. Gulliver, Pulse position aplitude odulation for tie-hopping ultiple-access UWB counications, IEEE Trans. Coun., vol. 53, pp , August [9] C. Abou-Rjeily, N. Daniele, and J.-C. Belfiore, MIMO UWB counications using odified Herite pulses, in Proceedings IEEE Int. Conf. on Personal, Indoor and Mobile Radio Coun., Septeber [10] J. Foerster, Channel odeling sub-coittee Report Final, Technical report IEEE /490, IEEE a Wireless Personal Area Networks, 2002.