ap-4-10.dvi Acta Polytechnica Vol. 50 No. 4/2010 Parameter-Invariant Hierarchical Exclusive Alphabet Design for 2-WRC with HDF Strategy T. Uřičář Abstract Hierarchical eXclusive Code (HXC) for the Hierarchical Decode and Forward (HDF) strategy in the Wireless 2-Way Relay Channel (2-WRC)has the achievable rate region extended beyond the classical MAC region. Although directHXCdesign is in general highly complex, a layered approach toHXCdesign is a feasible solution. While the outer layer code of the layered HXCcanbeany state-of-the-art capacity approaching code, the inner layermust bedesigned in suchaway that the exclusive property of hierarchical symbols (received at the relay) will be provided. The simplest case of the inner HXC layer is a simple signal space channel symbol memoryless mapper called Hierarchical eXclusive Alphabet (HXA). The proper design of HXA is important, especially in the case of parametric channels, where channel parametrization (e.g. phase rotation) can violate the exclusive property of hierarchical symbols (as seen by the relay), resulting in significant capacity degradation. In this paper we introduce an example of a geometrical approach to Parameter-Invariant HXA design, and we show that the corresponding hierarchical MAC capacity region extends beyond the classical MAC region, irrespective of the channel parametrization. Keywords: physical layer network coding, wireless 2-way relay channel, HDF strategy, HXA design. 1 Introduction 1.1 Background and related work Communication scenarios based on principles similar toNetworkCoding (NC) [1] are expected tohavegreat potential for wireless communication networks. Al- though pure NC operates with a discrete (typically binary) alphabet over lossless discrete channels, its principles can be extended into the wireless domain. Such an extension is however non-trivial, because sig- nal space link models (e.g. the MAC phase in relay communications) lack the simple finite field properties found and used in pure discrete NC. NC-based ap- proaches in the signal space domain are called Physi- cal Network Coding (PNC) or Network Coded Modu- lation (NCM). The major benefit of NCM is the possibility to increase the throughput in the MAC phase of bi- directional communication,which is believed to be the bottleneck in the overall system. The strategy where the relay decodes only hierarchical symbols (code- words), which jointly represent information received from both sources, is called the Hierarchical Decode and Forward (HDF) strategy [2, 3]. The increased MAC phase throughput of the HDF strategy provides a performance improvement over standard techniques based on the Amplify & Forward or Joint Decode & Forward paradigms. The authors of [4] present the simplest realizationofHDFstrategywithminimal car- dinalitymapping, which they call “modulo decoding”. More general relay output mapping, which also takes into account the possibility of extended cardinality, is introduced in [2]. Only limited code design and capacity region re- sults are available even for the simplest possible sce- nario of the parametric 2-Way Relay Channel (2- WRC), see [5, 6, 7, 8]. Lattice-based code construc- tion [4, 9] using the principles from [10] is limited to non-parametric Gaussian channels. References [2, 3] present a layered approach to Hierarchical eXclusive Code (HXC) design for parametric 2-WRC, based on theHierarchical eXclusiveAlphabet (HXA).Hierarchi- cal MAC capacity regions for various alphabets, con- stellation point indexing and channel parametrization are evaluated in [3]. Significant capacity degradation is causedbychannel parametrization,whichhighlights the importance of HXA design resistant to channel parametrization. 1.2 Goals and contribution of this paper Theproper design ofHXA is critical for overall system performance, especially in the case of parametric 2- WRC, where the channel parametrization (e.g. phase rotation) can cause significant capacity degradation. The parametrization should be taken into account in the design process, and HXA resistant to the effects of channel parametrization should be found. One way to achieve this is by designing the HXA with param- eter invariant Hierarchical Decision Maps (HDM) at the relay. The first design approaches of this kind (E-PHXC design criteria [11, 12]) have led so far only 79 Acta Polytechnica Vol. 50 No. 4/2010 to orthogonal or non-zeromeanHXA with unsatisfac- tory performance (capacity limited by classicalMAC). In this paper, we introduce an example of a geo- metrical approach to the design of amulti-dimensional HXAwhich extends thehierarchicalMAC capacity re- gion beyond the classical MAC region irrespective of the channel parametrization. 2 System model and definitions We adopt the system model presented in [3]. We con- sider a parametric wireless 2-WRC system (Fig. 1), which has 3 physically separated nodes (sources A, B and relay R) supporting two-way communication througha common shared relayR.The source for data A is co-located with the destination for data B, and vice-versa. The transmitted data of each source serves at the same time as Side Information (SI) for the re- verse link. The system is wireless, and all transmitted and received symbols are signal space symbols. The channel is assumed to be a linear frequency flat with Additive White Gaussian Noise (AWGN). The whole system operates in a half-duplex mode (one node can- not simultaneously receive and transmit). The overall bi-directional communication is split into a Multiple Access (MAC) phase and a Broadcast (BC) phase. MAC phase BC phase SI SI A R B Fig. 1: Model of 2-WRC with side information 2.1 MAC phase Now, we define all formal details. Subscripts A and B denotes variables associated with node A and B, respectively. The source data messages are dA, dB, and they are composed of data symbols dA, dB ∈ Ad = {0,1, . . . , Md − 1}, with alphabet cardinality |Ad| = Md. For notational simplicity, we omit the sequence number indices of the individual symbols. The source node codewords are cA,cB with code sym- bols cA, cB ∈ Ac, |Ac| = Mc. The encoding opera- tion is performed by the encoders CA, CB with code- books cA ∈ CA and cB ∈ CB. A signal space rep- resentation (with an orthonormal basis) of the trans- mitted channel symbols is sA = s(cA), sB = s(cB), sA, sB ∈ As ⊂ CN. We assume a common channel symbol mapper As(•). A signal space representation of the overall coded frame is sA(cA) and sB(cB). The received useful signal is u = sA + αsB. (1) It equivalently represents the parametric channel with both links parametrized according to the flat fading, which is assumed to be constant over the frame. It is obtainedbyproper commonrescalingof the true chan- nel response u′ = hAsA +hBsB by 1/hA and denoting α = hB/hA, hA, hB, α ∈ C1. The received signal at the relay is x = hAu + w (2) where the circularly symmetric complex Gaussian noise w has variance σ2w per complex dimension. 2.2 BC phase The relay receives signal x and processes it using aHi- erarchicalDecode and Forward (HDF) strategy. More details will be given in section 3. The output code- word and its code symbols are cR and cR. These are mapped into signal space channel symbols v ∈ AR and signal space codewords v with the codebook v ∈ CR andare broadcast to destinations A and B. At node B (the destination for data A), the received signal space symbols are yA = v + wA (3) where the complex circularly symmetric AWGN wA has variance σ2A per complex dimension. We denote the signal space symbols at node A (a destination for data B) similarly yB = v + wB. 3 Hierarchical exclusive code 3.1 Hierarchical decode and forward strategy The HDF strategy is based on relay processing, which fully decodes the Hierarchical Data (HD) message dAB(dA,dB) and sends out the corresponding code- word v = v(dAB) which represents the original data messagesdA anddB only throughthe exclusive law [6] v(dAB(dA,dB)) �=v(dAB(d′A,dB)) , ∀dA �=d ′ A, (4) v(dAB(dA,dB)) �=v(dAB(dA,d′B)) , ∀dB �=d ′ B .(5) The hierarchical data is a joint representation of the data fromboth sources such that they uniquely repre- sent one data source given full knowledge of the other source. Assuming that destination node B has per- fect Side Information (SI) on the node’s own data dB it can then decode the message dA (and similarly for node A). Data dB will be called complementary data fromtheperspectiveof thedatadA operations. TheSI on the complementarydatawill bedenotedasComple- mentary SI (C-SI) [3]. A code (codebook) satisfying the exclusive law at the signal space codeword level (for the HD messages) is called a Hierarchical eXclu- sive Code (HXC) [3]. We will denote the mapping satisfying the exclusive law by the operator X(•, •). 80 Acta Polytechnica Vol. 50 No. 4/2010 Unlike the standard relaying techniques based on the JointDecode & Forward paradigm,where the NC approach is used mainly for a BC phase of communi- cation, in the hierarchical approach (HDF) the hier- archical data is directly obtained from the received signal observations without the need to decode the individual data streams. The fact that only the hi- erarchical data (not the individual data streams) is decoded at the relay in theMACphase allows the sys- tem throughput to be increased above the constraints given by the classical MAC region. To facilitate this potential throughputbenefit, theMACstage encoding must be such that the observation at the relay allows directHXCmapping to the hierarchicaldata. In other words, alongside the complete signal path (MAC and BC) the coding must always be HXC w.r.t. the hier- archical data [3]. Throughout this paper we assume only the HXC withminimal cardinalityof the relayhierarchical code- book (|CR| = max(|CA|, |CB|)), which requires that both nodes (A, B) have perfect C-SI on the comple- mentary data. A general discussion on the relay hier- archical codebook cardinality can be found in [3]. 3.2 Layered HXC design for perfect C-SI Direct design of the HXC codebook CR providing the mappingv(dAB(dA,dB)) is evidentlyhighly complex. An alternative approachbased on layered HXC design is presented in [2, 3]. Layered HXC (Fig. 2) consists of the outer layer (error correcting) code and the inner layer (closer to the channel symbols), which provides the exclusive property of thehierarchical symbols. The outer layer code can be an arbitrary state-of-the-art capacity achieving code (e.g. turbo code or LDPC), and the inner layer can be designed (in the simplest case) as a simple signal space channel symbol memo- ryless mapper. An alphabet memoryless mapper cAB = Xc(cA, cB) (6) fulfilling the exclusive lawwill be called aHierarchical eXclusive Alphabet (HXA). The entire model of the layered system can be found e.g. in [3]. Fig. 2: Layered Hierarchical eXclusive Code Theorems in [2] show the viability of the layered approach to HXC design. The capacity is alphabet constrained by Hierarchical-MAC (H-MAC) channel hierarchical symbols and is achievable by the outer standard channel code CA = CB = C combined with inner symbol-wise HXA As. The H-MAC rate region has a rectangular shape RA = RB = RAB ≤ I(cAB;x). (7) The exclusive property at the symbol level allows simple determination of the soft per-symbol measure decoding metric for hierarchical symbols at the relay. It can either be directly used by the full hierarchical relay decoder or properly source encoded and sent on a per-symbol basis without decoding. 4 Parameter-invariant HXA The complexity of the proper HXA design increases in the case of layered HXC design for a parametric channel. Some specific channel parameter values can cause the signal space points corresponding to differ- ent hierarchical symbols to fall into the same useful signal (1) and thereby break the exclusive property, resulting in significant capacity degradation [3]. One possible solution to this inconvenience is to take the channel parametrization into account inherently from the beginning of the HXA design, e.g. by forcing the Hierarchical Decision Maps (HDM) at the relay to be invariant to channel parameter α: Xs(α)(sA, sB)= Xs(sA, sB), ∀α (8) HXAs that have the HDM invariant to channel parametrization will be called Parameter-Invariant HXAs (PI-HXA). 4.1 E-PHXC design criteria A first attempt to design the PI-HXA for the layered HXC in 2-WRC was given by the Extended Paramet- ric HXC (E-PHXC) design criteria [11]. These crite- ria utilize the criterion for the α-invariant hierarchical decision region pairwise boundary Rkl (i.e. the deci- sion region boundary between the useful signal pair uk(iA ,iB) = siA + αsiB and u l(i′A,i ′ B) = si′ A + αsi′ B ): 〈 siA − si′A;siB − si′B 〉 = 0, (9)〈 siB − si′B;siB + si′B 〉 = 0, (10) to force some subset of the decision region boundaries to be invariant to channel parametrization (see [13] for details). As shown in [12], theE-PHXCdesign criteria result in HXAs which have all permissible decision re- gion boundaries invariant to channel parametrization, hence the condition for a parameter invariantHDM is naturally satisfied. Although the E-PHXC design criteria provide a feasible way to design the HXA with parameter in- variant HDM, the resulting PI-HXA has a number of drawbacks and only limited performance (see [12] for details). The strictness of the E-PHXCdesign criteria forces sources (A, B) to use different channel symbol mappers (AA(•) and AB(•)) and the solution leads to mutually orthogonal alphabets or alphabetswith non- zeromeanandnon-equaldistance (which is apparently 81 Acta Polytechnica Vol. 50 No. 4/2010 not optimal). The solution with mutually orthogonal alphabets has the MAC capacity region equivalent to the classical-MAC decoding [12]. 4.2 Generalized approach to the design The strictness of the E-PHXC design criteria (which in turn causes only the orthogonal solution toPI-HXA design to be feasible) is due to the fact that all permis- sible pairwise boundaries are forced to be parameter invariant. This is obviously not necessary since some decision region boundaries can be “overlaid” by other boundaries or remain somehow“hidden” inside the hi- erarchical decision region. In such cases, the resulting final shapeof theHDMwill remainunaffectedby these boundaries. Hence these boundaries do not have to be considered by the PI-HXA design criteria. This ap- proach toPI-HXAdesign should relax the strictness of the design criteria (compared to E-PHXC) and hence non-orthogonal PI-HXAs with the rate region extend- ing the classicalMAC region can possibly be found. A comparison of this “generalized”design approachwith E-PHXC based design is shown in Fig. 3. 5 PI-HXA design 5.1 Principles of geometrical design The derivation of the systematic design criteria (de- sign algorithm) for PI-HXA is still relatively complex. The particular constellation space boundaries of the HDM at the relay result from the selected PI-HXA constellation (i.e. alphabet mapper As), and the de- sign criteria for invariant decision region boundaries directly affect the requirements given on the PI-HXA constellation. This mutual relationship increases the complexity of the systematic solution to PI-HXA de- sign. We show the viability of the layered HXC solu- tion in parametric channels (i.e. the possibility to find PI-HXA) by introducing some major simplifications which allow a geometric interpretation of the PI-HXA design problem. The idea of the geometric approach to PI-HXAdesign is based on the “constellation space patterns” of the useful signal u = sA + αsB. Definition 1: A constellation space pattern UiA is the subspace spannedbytheuseful signal u = sA+αsB for sA = siA, ∀sB ∈ As and ∀α ∈ C 1. The absolute value of the channel parameter |α| ∈ (0;∞) causes the constellation space patterns to be potentially unbounded. This is the only remaining in- convenience for a simple geometric interpretation of PI-HXA design. As we will show in the following sub- section, the constellation space patterns can be effec- tively bounded by simple processing at the relay. 5.2 Two-mode relay processing The received (useful) signal u is obtained by rescal- ing the true channel response (u′ = hAsA + hBsB) by 1/hA. The only purpose of this rescaling is to obtain a simplified expression of the useful signal (1), which is (after rescaling) parametrized only by a single com- plex channel parameter α = hB/hA. It is obvious that the true channel response canalternativelybe rescaled by 1/hB, hence we can obtain two alternative models of the useful signal u: M1 : uM1 = sA + αsB, (11) M2 : uM2 = 1 α sA + sB. (12) Fig. 3: The final shape of the HDM at the relay is affected only by the pairwise boundaries between different hierarchical codewords (given by a different colour of the region in the figure). The generalized approach to the PI-HXA design requires only these particular boundaries to be invariant to the channel parametrization (unlike E-PHXC) 82 Acta Polytechnica Vol. 50 No. 4/2010 This corresponds to two alternative models of the re- ceived signal at the relay: xM1 = hAuM1 + w, (13) xM2 = hBuM2 + w. (14) The relay can potentially swap these two channel models (respectivemodels of the useful signal) in such a way that the absolute value of the channel parame- ter (α formodel M1 and 1 α for M2) is always less than (or equal to) one. This processing at the relay will be called 2-mode relay processing. If the hierarchical mapping at the relay is “symmetric”: c ij AB = Xc(c i A, c j B)= Xc(c j A, c i B), ∀i, j ∈ {1, . . . |Ac|} , (15) then 2-mode relay processing can be used transpar- ently to both sources A,B (i.e. the sources are not aware which channel model is in use at the relay for the current transmission), and hence it is feasible for the HDF strategy with HXC. The symmetry of the relay hierarchicalmapper al- lows the relay to swap these two equivalent models of the useful signal (11), (12) transparently to both sources. In this way the relay can ensure that the value of the channel parameter in the useful signal model remains bounded, which in turn affects the sub- spaces spannedby the useful signals uM1, uM2, i.e. the constellation space patterns. Definition 2: A constellation space pattern UiB is the subspace spanned by the useful signal uM2 = 1 α sA + sB for sB = siB, ∀sA ∈ As and ∀α ∈ C 1. Definition 3: A bounded constellation space pat- tern U′i is the subspace given by: U′i = { UiA for |α| ≤ 1 UiB for |α| > 1 . (16) Fig. 4: Example of the constellation space patterns for 2-mode relay processing (|As| = |Ac| =4). The particular line style corresponds to the particular αsB, ∀sB ∈ As Table 1: Principles of 2-mode relay processing channel |α| ∣∣∣∣1α ∣∣∣∣ u U′i |hA| ≥ |hB| ≤ 1 ≥ 1 uM1 U iA |hA| < |hB| > 1 < 1 uM2 U iB Constellation space patterns U′i are effectively bounded (Fig. 4) by simple swapping of the useful signal models at the relay. The only requirement for this 2-mode relay processing is symmetry of the relay hierarchical output mapper (15). We summarize the principles of 2-mode relay processing in Table 1. 5.3 An example of 2-dimensional PI-HXA We assume a real-valued channel symbol memoryless mapper As ⊂ R2 (common to both sources). The channel parameter is complex α ∈ C1, and the use- ful signals are uM1, uM2 ∈ C 2. The assumption of a real-valued alphabet (As ⊂ R2) allows the following simple interpretation of the real and imaginary part of the received useful signal uM1 ∈ C 2 (similarly for uM2): � {uM1} = sA + � {α} sB, (17) � {uM1} = � {α} sB, (18) which corresponds to the following vector notation: � {[ uM1,1 uM1,2 ]} = [ sA,1 sA,2 ] +� {α} [ sB,1 sB,2 ] , (19) � {[ uM1,1 uM1,2 ]} = � {α} [ sB,1 sB,2 ] . (20) It is obvious that the imaginary part of the useful sig- nal � {uMi } depends solely on the channel symbols fromone source(sourceB formode M1 (11)andsource A for mode M2 (12)). This can be viewed as an ad- ditional side information transmission from the corre- sponding source. The relay employs 2-mode processing, hence the corresponding constellation space patterns U′i are bounded. To simplify the design example even more, only the real part of the constellation space patterns (U′iRe = � { U′i } ) will be considered here. These as- sumptions allow a simple geometric interpretation of thePI-HXAdesignproblem in R2. The commonchan- nel symbol mapper As cause that for iA = iB the corresponding patterns UiA and UiB define identical subspaces. Hence it is sufficient to consider only UiA for the case |α| ≤ 1 (it is equivalent to the analysis of UiB for the case ∣∣∣∣1α ∣∣∣∣ < 1). By defining the particular channel symbol mapper (As)we also directly define the corresponding constel- 83 Acta Polytechnica Vol. 50 No. 4/2010 lation space patterns UiA. In this case, the geomet- rical PI-HXA design example turns into a puzzle-like problem of the constellations space pattern (UiARe for iA ∈ {1,2, . . . |As|}) arrangement in R2. The main goal of this “puzzle” is to find a suitable channel sym- bol mapper As and a proper hierarchical eXclusive mapper cijAB = Xc(c i A, c j B)= X s(s i A, s j B), which will jointly prevent the possibility of violation of the ex- clusive law for arbitrary channel parameter α. s1 s2 s3 s4 As (−1,1) (1,1) (−1, −1) (1, −1) Fig. 5: Channel symbol mapper (As) and the resulting constellation space patterns (U′iRe) for the example of PI- HXA Here we show an example of a two-dimensional 4- ary (|As| = |Ac| = 4) PI-HXA, designed according to the assumptions given in this section. The selected constellation space symbols (i.e. the chosen channel symbol mapper As) and the resulting constellation space patterns (U′iRe) are depicted in Fig. 5. The fi- nal task of the example of geometrical PI-HXAdesign is aproper choiceof thehierarchical eXclusivemapper. The selection of a suitable hierarchical eXclusivemap- per can be visualized as a “colouring” (partitioning) process of the constellation space patterns. A visu- alization of this colouring process for the example of PI-HXA from Fig. 5 is presented in Fig. 6. The re- sulting hierarchical eXclusive mapper Xs(siA, s j B) cor- responds to a bit-wise XOR of the symbol indices (i ∈ {1,2, . . . |As|}). 6 Numerical results To show the viability of the layered HXC design in a parametric 2-WRC with HDF strategy we present some numerical evaluations of the mutual information (capacity) and the minimum squared distance of the example of PI-HXA design from Fig. 5. 6.1 Mutual information (capacity) We evaluate the hierarchical and single-user (alpha- bet limited cut-set bound) rates (Fig. 7) for an ex- ample alphabet As (see Fig. 5) and various channel sA sB sA sB sA sB sA sB Fig. 6: Design of a suitable hierarchical eXclusive mapper Xs(siA, s j B) for the example of PI-HXA from Fig. 5. The resulting hierarchical eXclusive mapper corresponds to a bit-wise XOR of the symbol indices (i ∈ {1,2, . . . |As|}) 84 Acta Polytechnica Vol. 50 No. 4/2010 −10 −5 0 5 10 15 20 0 0.5 1 1.5 2 2.5 3 SNR γ x [dB] [b it/ sy m b o l] H−MAC capacity for alphabets M2N4−HXA−c1 E ψ [C AB (ψ)] min ψ [C AB (ψ)] max ψ [C AB (ψ)] C AB (ψ=0) C 0 cut−set bound finite alph C s sum−rate/user bound finite alph C u cut−set bound Gauss alph Fig. 7: Capacity (mutual information) for the example of PI-HXA from Fig. 5 parametrization. The Signal-to-Noise Ratio (SNR) is defined as the ratio of the real base-band symbol en- ergy of one source (e.g. A, to have a fair compari- son for reference cases) to the noise power spectrum density ratio γ = ( ĒsA /2 ) /N0. Assuming orthonor- mal basis signal space complex envelope representa- tion of the AWGN, we have σ2w = 2N0 and thus γ = E [ ‖sA‖2 ] /σ2w. The alphabet As is indexed by symbols cA, cB ∈ {0, . . . , Mc − 1}. The exclusive hi- erarchical mapping corresponds to a bit-wise XOR of the symbol indices (Fig. 6). The graph (Fig. 7) shows the classical MAC cut- set bounds (1st and 2nd order) related to one user in comparison to the capacity of the HDF strategy with the example of PI-HXA. The HDF capacity is parametrized by the actual relative phase shift of the source-relay channels, while the amplitude is kept con- stant |α| in our setup (to respect the symmetry of the rates from A and B). We show the minimal, maximal and mean values of the HDF capacity. The results were obtained by the technique shown in [3], where details can be found. It is obvious from Fig. 7 that the HDF capac- ity approaches the alphabet constrained cut-set bound limit for medium to high SNR. For SNR values above approximately 2 dB the capacity outperforms the classical MAC capacity, irrespective of the channel parametrization (relative phase shift of the source- relay channel), which has only a minor impact on the resulting performance. 6.2 Minimum distance Theminimumdistanceperformance is quite closely connected with the error rate of the whole system [8]. We define the minimum squared distance as: d2min = minXs(sA ,sB) �=Xs(ŝA ,ŝB) d2(sA ,sB)−(ŝA,ŝB), (21) where d2 is the squared Euclidean distance between the useful signal u = sA + αsB and its candidate û = ŝA + αŝB: d2(sA,sB)−(ŝA ,ŝB) = ‖(sA − ŝA)+ α(sB − ŝB)‖ 2 . (22) Fig. 8depicts the squaredminimumdistance asa func- tionof channelparameter α. It is obvious fromthis fig- ure, that theminimumsquareddistance ishighly resis- tant to the relativephase shift (� α) of the source-relay channels. Note that distance shortening at |α| → 0 is inevitable. ℜ {α} ℑ { α } d min 2 (α) (for example of PI−HXA) −1.5 −1 −0.5 0 0.5 1 1.5 −1.5 −1 −0.5 0 0.5 1 1.5 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Fig. 8: Minimum squared distance as a function of the channelparameter (for theexampleofPI-HXAfromFig. 5) 85 Acta Polytechnica Vol. 50 No. 4/2010 7 Conclusion The importance of finding HXA resistant to channel parametrization(PI-HXA)wasstated in [3]. Itwasob- servedthere that someparticularchannelparametriza- tion values have disastrous effects on the system per- formance (significant capacitydegradationcausedbya violationof the exclusiveproperty). Thefirstapproach to PI-HXA design (E-PHXC design criteria [11]) has until now led only to orthogonal or non-zero mean HXA with unsatisfactory performance (capacity lim- ited by classical MAC). In this paper we have presented an example of a geometrical approach to PI-HXA design. Though the design was based on many simplifying assumptions, the numerical results show relatively high resistance of the capacity (Fig. 7) and the minimum distance (Fig. 8) to the channel parametrization. In addition, our setup requiresnoadaptationof the hierarchical ex- clusive mapping (unlike [6, 8]). Hence the processing at the relay can always be kept transparent to both sources. The numerical results presented in this paper show the viability of the layered approach toHXC design in 2-WRC with the HDF strategy, even for the case of parametric channels. Deriving systematic criteria for PI-HXA design is a topic for future work. Acknowledgement This workwas supported by the FP7-ICTSAPHYRE project, by Grant Agency of the Czech Republic, project 102/09/1624, and by Ministry of Education, Youth and Sport of the Czech Republic, programme MSM6840770014, grant OC188 and by the Grant Agency of the Czech Technical University in Prague, grant No. SGS10/287/OHK3/3T/13. 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[11] Uricar, T., Sykora, J.: Extended design crite- ria for hierarchical exclusive code with pairwise parameter-invariantboundaries forwireless2-way relay channel, in COST 2100 MCM, (Vienna, Austria), p. 1–8, Sept. 2009. TD-09-952. [12] Uricar, T., Sykora, J.: Design criteria for hier- archical exclusive code with parameter invariant decision regions for wireless 2-way relay channel, Submitted for publication, 2010. [13] Sykora, J.: Design criteria for parametric hierar- chical exclusive constellation space code for wire- less 2-way relay channel, in COST 2100 MCM, (Valencia, Spain), p. 1–6, May 2009. TD-09-855. Tomáš Uřičář E-mail: uricatom@fel.cvut.cz Dept. of Radioelectronics Faculty of Electrical Engineering Czech Technical University in Prague Technicka 2, 166 27 Prague, Czech Republic 86