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Design of Uniformly Most Powerful Alphabets for HDF 2Way Relaying Employing NonLinear Frequency Modulations
EURASIP Journal on Wireless Communications and Networking volume 2011, Article number: 128 (2011)
Abstract
HierarchicalDecodeandForward is a promising wirelessnetworkcodingbased 2way relaying strategy due to its potential to operate outside the classical multipleaccess capacity region. Assuming a practical scenario with channel state information at the receiver and no channel adaptation, there exist modulations and exclusive codes for which even nonzero channel parameters (denoted as catastrophic) cause zero hierarchical minimal distance  significantly degrading its performance. In this work, we state that nonbinary linear alphabets cannot avoid these parameters and some exclusive codes even imply them; contrary XOR does not. We define alphabets avoiding all catastrophic parameters and reaching its upper bound on minimal distance for all parameter values (denoted uniformly most powerful (UMP)). We find that binary, nonbinary orthogonal and biorthogonal modulations are UMP. We optimize scalar parameters of FSK (modulation index) and fullresponse CPM (frequency pulse shape) to yield UMP alphabets.
I. Introduction
Cooperative communication in wireless relay networks can potentially be of great benefit, offering several gains that may decrease required transmission power, increase the system capacity, improve the cell coverage or interference mitigation while balancing the quality of service and keeping relatively low deployment costs [1]. Consequently, future wireless networks are envisaged to include the cooperative relaying techniques. However, it also brings several new challenging problems such as the extra resources (e.g. frequency or time slots) taken for interferencefree relay traffic when considering practical halfduplex constraints (each node cannot send and receive at the same time). This is well demonstrated in the simplest cooperative network  2way relay channel (2WRC) comprising two terminals A and B bidirectionally communicating between themselves via a supporting relay R. Traditional protocols avoiding interference require 4 stages for every packet exchange (Figure 1a). Employing the advent of network coding [2] and wireless broadcast medium [3], the communication is more effective and reduces to 3 stages, Figure 1b). The first two stages of 3stage protocols can be considered as a multipleaccess (MAC) channel with orthogonally separated sources. Generally, setting the terminal rates to be within the MAC capacity region, we can reliably perform MAC in a single stage, resulting in a 2stage protocol. Direct estimation of network coded data from signal interference is very promising due to its ability to operate outside this MAC capacity region [4]. This strategy appears under the name denoiseandforward (DNF) or physical layer network coding [5]. Usually, a general term wireless network coding (WNC) is used to stress the fact that network codinglike operations are done in the wireless domain at the physical layer. Instead of DNF, we rather use generic term hierarchicaldecodeandforward (HDF) strategy, which is better suited specially for more complicated multisource networks [6].
HDF consists of MAC stage when both terminals transmit simultaneously to the relay with exclusively coded data decoding and broadcast (BC) stage when the relay broadcasts the exclusively coded data, Figure 1c). The exclusive code (XC) permits message decoding at the terminals using their own messages serving as a complementaryside information [7].
HDF performance in MAC stage, assuming fading channel with channel state information at the receiver side (CSIR), is unavoidably parametric. There are some modulation alphabets (e.g., QPSK) for which even nonzero channel parameters (denoted as catastrophic) cause zero hierarchical minimal distance, which significantly degrades its performance. Adaptive extendedcardinality network coding [8] and adaptive precoding technique [9] were proposed to suppress this problem with channel parametrization. However, both techniques require some form of adaptation that might not be always available.
The aim of our paper is to introduce modulation alphabets and exclusive codes resistant to the problem of parametrization on condition of CSIR and no adaptation, similar to in [10]. We define a class of alphabets avoiding all catastrophic parameters and reaching its upper bound on minimal distance for all parameter values (denoted uniformly most powerful (UMP)). The papers [11], [12] are also related, restricting, however, on noncoherent (no CSIR) complexorthogonal frequency shift keying (FSK) modulations.
The following contributions are provided:

1)
Exclusive code (XC) must fulfill certain conditions not to imply catastrophic parameters. Particularly, the XC matrix must be symmetric, and the same symbols must lie on its main diagonal. Bitwise XOR operation obeys these conditions, and it is the only solution for binary and even quaternary alphabet. Hence, it is convenient to assume fixed XOR XC.

2)
All nonbinary linear modulations with one complex dimension (e.g., PSK, QAM) have inevitably catastrophic parameters and binary modulations not even fulfilling the UMP condition. It is shown that nonbinary UMP alphabets require more than a single complex dimension.

3)
Nonbinary complexorthogonal and nonbinary complex biorthogonal modulations with XOR are UMP.

4)
Nonlinear frequency modulations FSK and fullresponse continuous phase modulation (CPM) naturally comprise multiple complex dimensions needed to obey UMP condition. We optimize a scalar parameters of FSK (modulation index) and fullresponse CPM (frequency pulse shape) to yield UMP alphabets. We find that a lower modulation index (proportional to bandwidth) than that leading to complexorthogonal alphabet fulfills UMP condition. Numerical simulations conclude that the considered frequency modulations do not have catastrophic parameters and perform close to the utmost UMP alphabets which however require more bandwidth.
II. System model
A. Constellation space model and used notation
Let both terminals A and B in 2WRC use the same modulation alphabet \mathcal{A} with cardinality \left\mathcal{A}\right={M}_{c} to be strictly a power of two. We suppose that the alphabet is formed by complex arbitrarydimensional baseband signals in the constellation space \mathcal{A}={\left\{{\mathbf{\text{s}}}_{{c}_{T}}\right\}}_{{c}_{T}=0}^{{M}_{c}1}\subset {\u2102}^{{N}_{S}}, where symbol {c}_{T}\in {\mathbb{Z}}_{{M}_{c}}=\left\{0,1,\dots ,{M}_{c}1\right\} denotes a data symbol transmitted by terminal T ∈ {A, B} and N_{ s } denotes the signal dimensionality. Linear modulations (e.g., PSK, QAM) have single complex dimension, i.e., N_{ s } = 1 and their constellation vectors are complex scalars {s}_{{c}_{t}}\in \u2102. Later in this paper, we will use nonlinear frequency modulations FSK and fullresponse CPM, which are multidimensional and its dimensionality is N_{ s } = M_{ c } ; the constellation space vectors are consequently {\mathbf{\text{s}}}_{{c}_{T}}\in {\u2102}^{{N}_{S}}. Without loss of generality, we assume memoryless constellation mapper \mathcal{M} such that it directly corresponds to the signal indexation, {\mathbf{\text{s}}}_{{c}_{T}}=\mathcal{M}\left({c}_{T}\right).
B. Model assumptions
We assume a timesynchronized scenario with full CSIR, which is obtained, for example, by preceding tracking of pilot signals. The synchronization issues are beyond the scope of this paper, and interested reader may see e.g. [13], [14] for further details. We restrict ourselves that adaptive techniques are not available either due to the missing feedback channel, increased system complexity, or unfeasible channel dynamics.
We consider persymbol relaying (avoiding delay induced at the relay) and no channel coding, which however can be additionally concatenated with our scheme [15].
C. Hierarchicaldecodeandforward strategy
HDF strategy in 2WRC consists of two stages, see Figure 2.
In the first MAC stage, both terminals A and B transmit simultaneously to the relay in the interfering manner, see Figure 3.
The received composite (interfering) signal is
where w is complex AWGN with variance 2N_{0} per complex dimension, and the channel parameters h_{ A } and h_{ B } are frequencyflat complex Gaussian random variables with unit variance and Rayleigh/Rician distributed envelope. The Rician factor K is defined as a power ratio between stationary and scattered components. We assume that the channel parameters h_{ A } , h_{ B } are known to R.
Subsequently, the relay decodes exclusively coded data symbol c_{ AB } = c_{ A } ⊕ c_{ B } from interfering signal (1). Operation ⊕ is a network codinglike exclusive (invertible) operation, which incorporates data from multiple sources via a principle of exclusivity, see Section IID for more details. We assume a minimal cardinality exclusive operation, i.e. cardinality of c_{ AB } alphabet is M_{ c }[7]. We suppose an approximated nearest neighbor twostep decoding [16]: estimate of exclusively coded data symbol c_{ AB } is obtained by joint maximum likelihood decoding
followed by exclusive encoding {\u0109}_{AB}={\u0109}_{A}\oplus {\u0109}_{B}; by, ⋆^{2}, we denote the squared vector norm.
In the BC stage, R broadcasts exclusive symbol c_{ AB } , which is sufficient for successful decoding. Particularly, the terminal A obtains desired data symbol c_{ B } with knowledge of c_{ AB } and its own data c_{ A } as c_{ B } = c_{ AB } ⊖ c_{ A } , where ⊖ denotes an inverse operation to exclusive coding and vice versa for B.
In this paper, we entirely focus on the MAC stage, which dominates the error performance, rather than BC stage due to the additional multipleaccess interference [16].
D. Exclusive coding
We are aware that the term network coding is often used [5], but we rather propose the term exclusive coding to point out some important differences. Particularly, wellknown network coding is related to the linklayer techniques, and it is often assumed as a linear commutative operation with minimal cardinality of the output alphabet. Here, we require only existence of inversion, which we refer as an exclusivity. XC is regarded as an operation not necessarily commutative but with existence of inversion, i.e., a group. The XC operation can be well described by a matrix formed by exclusively coded symbols placed on a position corresponding to relevant c_{ A } and c_{ B } . We denote this matrix as an exclusive code matrix XC given by {c}_{A}\oplus {c}_{B}={\left[\mathbf{\text{XC}}\right]}_{{c}_{A},{c}_{B}}. For example, the matrices
define binary XOR, quaternary modulo sum and quaternary bitwise XOR operation, respectively. We will use often short 'XOR' to denote bitwise XOR exclusive code. The notation resembles Sudoku game where in each row and each column, every element can appear only once. Exclusively coded symbols in XC matrix may take different values, the only demand is the existence of inversion ⊖. For instance, the inversion exists also if we replace one symbol with a new not yet introduced, which extends the cardinality of exclusively coded symbols c_{ AB } . We assume the XC matrix in a standard form where the first row is in increasing order starting from zero.
We restrict ourselves on the minimal cardinality \mathrm{\text{XC}}\left({c}_{AB}\in {\mathbb{Z}}_{{M}_{c}}\right) in order to avoid redundancy in the BC stage. It is interesting that the number of all distinct XC matrices with the minimal cardinality (a.k.a. Latin squares) grows very fast with alphabet size [17] as depicted in Table 1.
E. Parametric Hierarchical constellation
Since the relay has CSIR, we will conveniently introduce a model of hierarchical constellation, which uses instead of h_{ A } , h_{ B } only one complex parameter α always α ≤ 1 [18].
The useful received signal (1) can be normalized by h_{ A }
where \alpha ={h}_{B}\u2215{h}_{A}\in \u2102 and \text{E[\u2225}{\text{w}}^{\prime}{\text{\u2225}}^{2}]={\scriptscriptstyle \raisebox{1ex}{$2{N}_{0}{N}_{S}$}\!\left/ \!\raisebox{1ex}{${h}_{A}{}^{2}$}\right.} or by h_{ B }
with \text{E[}\parallel {{w}^{\prime}}^{\prime}{\parallel}^{2}]={\scriptscriptstyle \raisebox{1ex}{$2{N}_{0}{N}_{S}$}\!\left/ \!\raisebox{1ex}{${h}_{B}{}^{2}$}\right.} where operator E[⋆] denotes the statistical expectation. By adaptive switching between (6) and (7), we ensure that α or 1\u2215\alpha  is always lower or equal to one. The decoding processing (2) remains the same for both cases; therefore, the switching has only theoretical value that we need to focus on system performance only for α ≤ 1. The useful hierarchical (composite) signal is {u}_{{c}_{A}{c}_{B}}\left(\alpha \right)={\mathbf{\text{s}}}_{{c}_{A}}+\alpha {\mathbf{\text{s}}}_{{c}_{B}} or {u}_{{c}_{A}{c}_{B}}\left(\alpha \right)=\alpha {\mathbf{\text{s}}}_{{c}_{A}}+{\mathbf{\text{s}}}_{{c}_{B}} according to α; however, for both cases, useful parametric hierarchical signal at the relay is
because both terminals have the same \mathcal{A}.

III.
Hierarchical minimal distance as a performance metric and catastrophic parameters.
A. Hierarchical minimal distance
Let us focus on the error performance in the MAC stage. Defining a symbol error ∂ _{ AB } ≠ c_{ AB } , the symbol error probability is well approximated by sum of weighted pairwise error probabilities. The pairwise error probability is a function of distance between signals corresponding to c_{ AB } ≠ c' _{ AB } and the pairwise error probability dominating the performance for high signaltonoise ratio (SNR) is a function of the minimal distance. In our case, the minimal distance is a minimal distance between hierarchical signals {u}_{{c}_{A}{c}_{B}} and {u}_{{c}_{{A}^{\prime}}{c}_{{B}^{\prime}}} with different XC symbols,
and we will call it a hierarchical minimal distance; when it is clear, we omit the attribute hierarchical.
Note that the minimal distance is given not only by modulation alphabet but also by XC operation. In general, the minimal distance is parametrized by α and so is the error performance.
Remark 1. Facing the fact that the hierarchical constellation is randomly parametrized, we start investigation with the simplification that the error performance is given solely by minimal distance. We are aware that this is a rough approximation, since the minimal distance is relevant performance metric only asymptotically (as SNR → ∞) and the error curves are linearly proportional also to the number of signal pairs having the minimal distance.
B. Catastrophic parameters and paper motivation
In the preceding section, we have seen that the HDFMAC stage with CSIR has parametric minimal distance (asymptotic performance). For some modulation alphabets and exclusive codes, there exist such nonzero parameters (called catastrophic), which yields even zero minimal distance. This problem is well demonstrated, for example, by QPSK and complexorthonormal QFSK (modulation index κ = 1) modulation with XOR (5). Minimal distance of QPSK is depicted in Figure 4a indicating several catastrophic parameters e.g. for α_{cat} = j. On the contrary, numerical evaluation of QFSK with κ = 1, see Figure 4b, seems to have minimal distance parabolically dependent on α as
It has no catastrophic parameters, and therefore, it is robust to the parametrization. Zero distance for α = 0 is expected because it means that one of the channel is relatively zero. This paper focuses on the design of alphabets and XCs like this. We demonstrate by Figure 4c with QFSK κ = 1 and modulo sum XC (4) that not only modulation alphabet, but also exclusive code influence the parameter robustness.
Before we state the core idea of UMP alphabet, let us precisely define catastrophic parameters and state a couple of important lemmas based on them.
Definition 2 (Catastrophic parameters). A catastrophic parameter is such a nonzero parameter α_{cat} that forces two hierarchical signals corresponding to different XC symbols to the same point (equivalently, it indicates zero minimal distance). In our notation, for some α_{cat} ≠ 0, there exists {c}_{A}\oplus {c}_{B}\ne {c}_{{A}^{\prime}}\oplus {c}_{{B}^{\prime}} that
C. Exclusive code not implying catastrophic parameters
This section shows that XC must fulfill certain conditions not to imply catastrophic parameters regardless of the modulation alphabet. This reduces the number of XCs, see Table 1 involved in search for alphabets and XCs robust to the parametrization. The conditions are derived again in order to avoid catastrophic parameters.
Theorem 3. A matrix of XC with different symbols on the main diagonal implies α_{cat} = 1 and XC matrix which is not symmetric over the main diagonal has α_{cat} = 1 regardless of modulation.
Proof: Let two hierarchical signals correspond to the XC matrix main diagonal, c_{ A } = c_{ B } , c_{ A } ' = c_{ B } ' i.e. {\mathbf{\text{u}}}_{{c}_{A}{c}_{A}}, {\mathbf{\text{u}}}_{{{c}_{A}}^{\prime}{{c}_{A}}^{\prime}}, c_{ A } ≠ c_{ A } ' and their XC symbols are different c_{ A } ⊕ c_{ A } ≠ c_{ A } ' ⊕ c_{ A } '. Equation (11) is then
and α_{cat} = 1., which is similar to for nonsymmetric XC matrix. Assume {u}_{{c}_{A}{c}_{B}}, {u}_{{c}_{B}{c}_{A}} with c_{ A } ⊕ c_{ B } ≠ c_{ B } ⊕ c_{ A } , Equation (11) is
and α_{cat} = 1. We conclude that XC matrix should be symmetric with the same code symbols on its main diagonal.
Remark 4 (Suitability of bitwise XOR XC). XOR fulfills these conditions, and it is the only solution for binary and even quaternary alphabet (unfortunately it is not the only choice for e.g. octal alphabet) [17]. Once we fix XC (at least for binary and quaternary case), the only thing that influences the parameter robustness is the modulation alphabet. Therefore, from now on, we assume ⊕ is XOR for all cases and we relate the parametrization robustness only with particular modulation alphabets.
D. Nonbinary linear modulations are catastrophic
In this section, we demonstrate that any nonbinary linear modulation can never avoid catastrophic parameters, as we have seen particularly for QPSK, Figure 4a.
Lemma 5. Nonbinary linear modulations unavoidably have catastrophic parameters.
Proof: Linear modulations like QAM and PSK have dimensionality N_{ s } = 1 and signals in the constellation space are s_{ A } , s_{ B } ∈ ℂ. Considering hierarchical signals {u}_{{c}_{A}{c}_{B}}, {u}_{{{c}_{A}}^{\prime}{{c}_{B}}^{\prime}} with c_{ A } ⊕ c_{ B } ≠ c_{ A } ' ⊕ c_{ B } ' and symbols not being in the same row and column of XC matrix (c_{ A } ≠ c_{ A } ', c_{ A } ' ≠ c_{ B } '), there exists such a parameter that
this parameter equals to {\alpha}^{\prime}=\left({s}_{{{c}_{A}}^{\prime}}{s}_{{c}_{A}}\right)\u2215\left({s}_{{c}_{B}}{s}_{{{c}_{B}}^{\prime}}\right). Binary alphabets are excluded from consideration while its different hierarchical signals always lie in the same row or column of the XC matrix.
Since we assume channel model switching, it would be a catastrophic parameter if it was α' ≤ 1, but as we discussed in the previous section, the XC matrix should be symmetric and so, also for symmetric signals {u}_{{c}_{B}{c}_{A}}, {u}_{{{c}_{B}}^{\prime}{{c}_{A}}^{\prime}}, there exists a parameter α″ such
{\alpha}^{\u2033}=1\u2215{\alpha}^{\prime}. Hence, the catastrophic parameter equals to α ' or α″, whether its absolute value is lower or equal to one.
IV. Uniformly most powerful alphabet
Inspired by previous sections, we define a class of alphabets with hierarchical minimal distance of the form like in (10) avoiding all catastrophic parameters and being robust to the channel parametrization. We will show that the form (10) corresponds to alphabets reaching the minimal distance upperbound for all parameter values.
A. Minimal distance upperbound
Lemma 6. Minimal distance of any alphabet is upperbounded by
where{\delta}_{min}^{2}is a minimal distance of a single (nonhierarchical) modulation alphabet; {\delta}_{min}^{2}=\underset{{c}_{A}\ne {{c}_{A}}^{\prime}}{min}\parallel {\mathbf{\text{s}}}_{{c}_{A}}{\mathbf{\text{s}}}_{{{c}_{A}}^{\prime}}{\parallel}^{2}and{c}_{A},{{c}_{A}}^{\prime}\in {\mathbb{Z}}_{{M}_{c}}.
Proof: We obtain the upper bound by evaluating minimum operator only along the hierarchical signals corresponding to a single row of XC matrix (it means for c_{ A } = c_{ A } '),
Since we do not evaluate the minimum operator along the all possible hierarchical signal differences, we need to use inequality in (17). The minimum evaluation along a single column of XC matrix (c_{ B } = c_{ B } ' in (9)) yields
As we are considering α ≤ 1, we conclude that the bound (16) is more tight than (18).
B. UMP alphabet definition
Definition 7. Uniformly most powerful (UMP) alphabets have hierarchical minimal distance reaching the upperbound (16) for all parameter values and it equals to
where {\delta}_{min}^{2}=\underset{{c}_{A}\ne {{c}_{A}}^{\prime}}{min}\parallel {\mathbf{\text{s}}}_{{c}_{A}}{\mathbf{\text{s}}}_{{{c}_{A}}^{\prime}}{\parallel}^{2} and {c}_{A},{{c}_{A}}^{\prime}\in {\mathbb{Z}}_{{M}_{c}}. We restrict on α ≤ 1 due to the adaptive switching, Section IIE.
C. UMP alphabet properties
Lemma 8. It is important to stress that UMP alphabets do not have any catastrophic α_{cat}, and according to Lemma 5 and Remark 4, nonbinary linear modulations are never UMP and all UMP alphabets are using XOR exclusive code.
Remark 9. Extendedcardinality XC as well as minimal cardinality XC have different code symbols in each XC matrix row (Sudoku principle) and thus the bound holds for extendedcardinality XC as well; particularly, for systems with adaptive XC [8]. In the other words, the performance of adaptive XC system cannot be better than of UMP alphabet if both are using alphabets with the same {\delta}_{min}^{2}.
Remark 10. Two properties influence good HDF performance, a) being UMP and b) having large minimal distance of individual constellations {\delta}_{min}^{2}. These properties can be interpreted as follows. The property b) is proportional to robustness to AWGN. The UMP condition a) (considering the upperbound (16)) presents the best possible type of inevitable parametrization by α.
Remark 11 (Parallel with UMP statistical tests). Simplifiedly matching error performance with minimal distance (Remark 1), we state that among all alphabets with identical {\delta}_{min}^{2}, the UMP alphabets have the best performance ∀α ∈ ℂ. Based on this observation, we use the term UMP originally used in statistical detection theory due to the common principle. Composite hypothesis tests have parametrized PDFs and UMP detector, if exists, assuming knowledge of the instant value of the random parameter yields the best performance for all parameter values [19]. It resembles exactly our case, the likelihood function of joint [c_{ A } , c_{ B } ] detection is also parametrized (by h_{ A } , h_{ B } ) [10] and assuming CSIR the optimal detector of UMP alphabets has the best performance for all parameter values.
D. Binary modulation is UMP
Evaluating formula (9) with respect to the binary XC matrix (3), we straightforwardly obtain
where {\delta}_{min}^{2}=\parallel {\mathbf{\text{s}}}_{0}{\mathbf{\text{s}}}_{1}{\parallel}^{2}. It means that binary alphabets are always UMP regardless of the particular alphabet. Considering Remark 10, the optimal binary UMP alphabet is BPSK which maximizes {\delta}_{min}^{2}.
E. Nonbinary orthonormal modulation is UMP
We have seen in Figure 4b that complexorthonormal QFSK is UMP. This holds in general which describes the following lemma.
Lemma 12. Complexorthonormal modulation is UMP.
Remark 13. Before we prove the Lemma 12, it is convenient to introduce simplified UMP condition easier to verify.
The UMP condition (19) also implies that the minimum distance is formed by the hierarchical signal differences corresponding to the rows of XC matrix, see also the proof of Lemma 6. Therefore, if the squared norm of hierarchical signal differences with indices not being in the same row of XC matrix are always larger or equal than the bound,
for ∀c_{ A } ≠ c_{ A } ', c_{ B } ≠ c_{ B } ', c_{ A } ⊕ c_{ B } ≠ c_{ A } ' ⊕ c_{ B } ' and ∀α ∈ ℂ, α ≤ 1, then (19) is fulfilled. Expanded left side of (21) is
where 〈⋆, ⋆〉 denotes an inner product. Since inequality (22) must hold for all φ = arg α, it must hold for the worst case φ_{ c } , where the part with inner product is minimal and (22) becomes
This form of invariancy condition is easier to verify due to the presence of only one real variable α.
Proof: Orthonormal modulation has all distances (as well as the minimal one) for c_{ A } ≠ c_{ A } ', c_{ B } ≠ c_{ B } ' equal to 2, then (23) simplifies to
which further adjusts to
While (25) must hold for any α ≤ 1, it requires to hold for critical α = 1. Here, a critical parameter is such a parameter that if the condition is fulfilled for that one, then it is fulfilled for all other parameter values. Condition (25) with critical α = 1 is then
We prove (26) considering that any inner product of orthonormal modulation is either 0 or 1. Equation (26) is fulfilled except for the case where the r.h.s. equals to 2. It happens when \u3008{\mathbf{\text{s}}}_{{c}_{A}},{\mathbf{\text{s}}}_{{c}_{B}}\u3009=1 & \u3008{\mathbf{\text{s}}}_{{{c}_{A}}^{\prime}},{\mathbf{\text{s}}}_{{{c}_{B}}^{\prime}}\u3009=1 & \u3008{\mathbf{\text{s}}}_{{c}_{A}},{\mathbf{\text{s}}}_{{{c}_{B}}^{\prime}}\u3009=0 & \u3008{\mathbf{\text{s}}}_{{{c}_{A}}^{\prime}},{\mathbf{\text{s}}}_{{c}_{B}}\u3009=0 and when \u3008{\mathbf{\text{s}}}_{{c}_{A}},{\mathbf{\text{s}}}_{{c}_{B}}\u3009=0 & \&\u3008{\mathbf{\text{s}}}_{{{c}_{A}}^{\prime}},{\mathbf{\text{s}}}_{{{c}_{B}}^{\prime}}\u3009=0 & \&\u3008{\mathbf{\text{s}}}_{{c}_{A}},{\mathbf{\text{s}}}_{{{c}_{B}}^{\prime}}\u3009=1 & \u3008{\mathbf{\text{s}}}_{{{c}_{A}}^{\prime}},{\mathbf{\text{s}}}_{{c}_{B}}\u3009=1. Let us consider the first case, \u3008{\mathbf{\text{s}}}_{{c}_{A}},{\mathbf{\text{s}}}_{{c}_{B}}\u3009=1 & \u3008{\mathbf{\text{s}}}_{{{c}_{A}}^{\prime}},{\mathbf{\text{s}}}_{{{c}_{B}}^{\prime}}\u3009=1 entails that {\mathbf{\text{s}}}_{{c}_{A}}={\mathbf{\text{s}}}_{{c}_{B}} & {\mathbf{\text{s}}}_{{{c}_{A}}^{\prime}}={\mathbf{\text{s}}}_{{{c}_{B}}^{\prime}} thus c_{ A } = c_{ B } & c_{ A } ' = c_{ B } ', which corresponds to hierarchical signals from the main diagonal of the XC matrix. Thus, using the XC code suitable for UMP, see Remark 4, this case is excluded. Similarly, the second condition \&\u3008{\mathbf{\text{s}}}_{{c}_{A}},{\mathbf{\text{s}}}_{{{c}_{B}}^{\prime}}\u3009=1 & \u3008{\mathbf{\text{s}}}_{{{c}_{A}}^{\prime}},{\mathbf{\text{s}}}_{{c}_{B}}\u3009=1 implies c_{ A } = c_{ B } ' & c_{ B } = c_{ A } ' and is excluded by XC with symmetrical XC matrix, again excluded by XOR.
V. Design of ump frequency modulations
In this section, we consider nonlinear frequency modulations that naturally possess multidimensional alphabets, according to Lemma 5 needed to avoid catastrophic parameters. We will conclude that the considered frequency modulations avoid catastrophic parameters and are close to meet the UMP condition. We propose and use simple scalar alphabet parametrization easy to meet the UMP condition. Based on the error simulations, we will find that existence of catastrophic parameters is much more detrimental than not being UMP. In case of frequency modulations (without catastrophic parameters), UMP alphabets are important since according to Remark 11, they form a performance benchmark.
A. UMPFSK design
1) FSK definition and basic properties: We assume the following unit energy FSK signals of one symbol duration
where t ∈ [0, T_{ s } ) is a temporal variable, T_{ s } is a symbol duration and c\in {\mathbb{Z}}_{{M}_{c}} denotes a data symbol. Its constellation space alphabet is N_{ s } dimensional \mathcal{A}={\left\{{\mathbf{\text{s}}}_{c}\right\}}_{c=0}^{{M}_{c}1}\subset {\u2102}^{{N}_{s}}, where N_{ s } = M_{ c } . Its signal correlation as well as minimal distance is determined by modulation index κ which also roughly corresponds to the occupied bandwidth [20]. It is wellknown fact that FSK is complexorthonormal for integer modulation index κ ∈ ℕ and with minimal κ = 1 is often used in noncoherent detection. In coherent detection (with CSIR), it has maximal minimal distance {\delta}_{min}^{2}=2 for \kappa =1\u22152 also often denoted as minimum shift.
2) Design of UMPQFSK modulation by index optimization: According to Lemma 12, FSK κ = 1 is UMP. Yet, we try to answer a question whether full complexorthogonality is required to meet UMP. To investigate the nonorthogonal case, we assume κ < 1, which also means a modulation roughly with narrower bandwidth, see more detailed discussion of bandwidth requirements in Section VIC.
Let us assume quaternary M_{ c } = 4 (binary is UMP regardless of alphabet, Section IVD) FSK (QFSK) to consider this question where we optimize modulation index κ to meet the UMP condition. The following lemma is true.
Lemma 14. QFSK\kappa =5\u22156 is UMP, see Figure 5b. The same approach may be used for any alphabet cardinality  the results for octal UMPFSK require\kappa =13\u221514which leads to a conjecture that\kappa \left({M}_{c}\right)=2{M}_{c}3\u22152{M}_{c}2is sufficient.
Proof: We have seen in Section IVE that the condition implying UMP property (23) is
for c_{ A } ≠ c_{ A } ', c_{ B } ≠ c_{ B } ', c_{ A } ⊕ c_{ B } ≠ c_{ A } ' ⊕ c_{ B } ' and ∀α ∈ ℂ, α ≤ 1. The following steps further adjust (28) to a suitable 2nd order polynomial form
where auxiliary constants
and c={b}^{2}\text{+\u2225}{s}_{{c}_{A}}{s}_{{c}_{A}\text{'}}\text{\u2225}2/\text{\u2225}{s}_{{c}_{B}}{s}_{{c}_{B}\text{'}}{\text{\u2225}}^{2}{\delta}_{\mathrm{min}}^{2} are not functions of α. Thus, the condition (28) has a critical α, which either equals to b if b ≤ 1 or limits value 1 if b ≥ 1. In Figure 6, we plot the constant b for all indices c_{ A } ≠ c_{ A } ', c_{ B } ≠ c_{ B } ', c_{ A } ⊕ c_{ B } ≠ c_{ A } ' ⊕ c_{ B } ' for QFSK and XOR XC. We conclude that constant b is always greater than 1 for roughly κ ≳ 0.3. For practical purposes, we restrict on \kappa \ge 1\u22152 because then the minimal distance {\delta}_{min}^{2} is reasonably high. The restriction implies that constant b ≥ 1 and so the critical α = 1. L.h.s of (28) for critical α = 1 are depicted in Figure 7 by thin light blue color. In the same figure, we chart their minimum (thick blue) and the minimal distance of QFSK {\delta}_{min}^{2} (thick green). The lowest modulation index leading to UMPQFSK is \kappa =5\u22156.
B. Biorthonormal modulation is UMP
According to the results from the preceding section, we see that UMP property does not require an accurate complexorthonormal alphabet. Inspired also by [10], we have a conjecture that biorthonormal modulation is UMP. The Appendix proves the following lemma.
Lemma 15. Biorthonormal modulation is UMP.
Remark 16. Interestingly, the symmetrical XC matrix with the same main diagonal is not sufficient in this case, and an extra kind of symmetry, which obeys XOR as well, is required.
C. UMPCPM design
1) CPM basic properties: CPM is a constant envelope modulation (suitable for satellite communication) with more compact spectrum in compare to the linear modulations with constant envelope (with rectangular (REC) modulation pulse). It has a multidimensional alphabet and better spectral properties than FSK (no Dirac pulses in the spectrum and faster asymptotic spectrum attenuation due to the continuous phase). Bandwidth requirements of the considered schemes are investigated in Section VIC. CPM includes memory [21] and its modulator consists of the discrete part including memory and the nonlinear memoryless part [22]. Denominator of CPM modulation index κ is proportional to the number of modulator states described by its trellis and the optimal decoder need to perform Viterbi decoding.
CPM possess several degrees of freedom; for simplicity, we restrict on the fullresponse (i.e. the frequency pulse is of the symbol length) and minimum shift \kappa =1\u22152 case for which constellation space alphabet is N_{ s } = M_{ c } dimensional.
2) Design of fullresponse\kappa =1\u22152UMPCPM by pulse shape optimization: In the same way, we have excluded channel coding from design of UMP alphabet, we do not need to consider modulation memory of CPM. In our case, the nonlinear memoryless part is determining.
Assumed fullresponse \kappa =1\u22152 CPM has the modulation trellis with only two states and the nonlinear memoryless alphabet consists of 2M_{ c } signals of which the first half starting from the first state have opposite sign than the other half starting from the latter state, see, for example, the trellis of binary scheme in Figure 8.
Our design is based on Lemma 15, utilizing the above mentioned symmetries, we design a biorthonormal UMP modulation simply keeping orthonormal signals starting from the first state.
3) CPM signals notation: Let us denote positivesign alphabet (signals starting from the zero state) {\mathcal{A}}^{+} and negativesign alphabet {\mathcal{A}}^{}={\mathcal{A}}^{+}. The overall alphabet (nonlinear memoryless part) is \mathcal{A}=\left\{{\mathcal{A}}^{+},{\mathcal{A}}^{}\right\}. Assuming unit energy signals, fullresponse h=1\u22152 CPM has
where data symbol c ∈ { (M_{ c }  1),  (M_{ c }  3),..., (M_{ c }  1)}, t is normalized to one symbol duration t ∈ [0, 1) and β(t) is a phase pulse.
4) Proposed pulse parametrization: The remaining degree of freedom which we exploit to set the signal correlation is a phase pulse shape. We introduce a simple shaping form obtained as a linear parametrization of Raised Cosine (RC) pulse which we denote as a Scaled RC (SRC) pulse. The proposed parametric SRC phase pulse is
where p is a real parameter. The phase pulse correspond to REC pulse for p = 0 and to RC pulse for p = 1, see Figure 9.
This parametrization has number of advantages, it does not influence the number of modulator states/signal alphabet cardinality, and it has known analytical formula for bandwidth [23] (roughly the higher p the wider bandwidth).
5) Design of binary UMPCPM
Lemma 17. Binary fullresponse CPM with\kappa =1\u22152and parametric SRC pulse (34) with p ≃ 2.35 is UMP.
Proof: Let us consider a binary case, the positivesign alphabet is
The correlation coefficient \rho =\u3008{s}_{0}\left(t\right),{s}_{1}\left(t\right)\u3009={\int}_{0}^{1}{s}_{0}\left(t\right){s}_{1}^{*}\left(t\right)dt has an analytic expression in the case of SRC pulse. The expression consists of generalized hypergeometric functions with a zero real part, see ρ in Figure 10.
We conclude that p ≃ 2.35 leads to the orthonormal signals, and the lemma is true.
Remark 18. The proposed pulse parametrization has an extra advantage that the squared norm of the signal difference of binary alphabet is always 2 for any p. The reason is simply given by zero real part of ρ for any p, as has been mentioned in the proof above, then s_{0}(t)  s_{1}(t)^{2} = 2(1  ℜ{ρ}) = 2. Hence, we can adjust the correlation required for UMP condition without affecting the minimal distance {\delta}_{min}^{2}.
We evaluate parametric minimal distance in Figure 11 to confirm the UMP property of the proposed scheme. We conclude that minimal distance of nonUMP schemes with REC and RC are close to be UMP. In the last section with numerical results, we will see that the error performance of these schemes are practically identical. However, in the case of quaternary/higherorder alphabet, the differences are more significant.
6) Design of quaternary UMPCPM
Lemma 19. Quaternary fullresponse CPM with\kappa =1\u22152and parametric SRC pulse (34) with p ≃ 7 or p ≃ 10.2 is UMP.
Proof: The above derivation for binary alphabet can be generalized for any alphabet cardinality; for simplicity, we focus on the quaternary case. Let us consider 4ary fullresponse CPM \kappa =1\u22152 and SRC pulse; the modulation trellis has the same number of states, see Figure 12.
The positivesign alphabet is
where c ∈ {3, 1, 1, 3} a data symbol. Our target is by variation of parameter p make set {\mathcal{A}}^{+} orthonormal. There are 4 signals in the set; thus, there are six different signal pairs that must be mutually orthonormal. Let us assume a sum of squared absolute values of individual correlation coefficients (of every signal pairs from {\mathcal{A}}^{+}) ∑_{ i }ρ_{ i } ^{2} as an indication function. This indication function is zero only for orthogonal alphabet. In Figure 13, we depict the indicating function against parameter p, concluding that orthonormal set is obtained, for example, for p ∈ {7, 10.2}, which proves the lemma.
To demonstrate the UMP property of the proposed scheme, we evaluate parametric minimal distance with REC, RC and proposed SRC pulse, see Figure 14. Contrary to binary case, the minimal distance of nonUMP schemes with REC and RC are far to be UMP, and scheme with RC pulse has even lower {\delta}_{min}^{2}=1, which is in correspondence with the later presented error simulations.
VI. Numerical results
A. Error performance of memoryless modulations
In this section, we numerically evaluate symbol error rate (SER) in HDFMAC stage for several alphabets. We assume simple AWGN channel, frequencyflat uncorrelated Rayleigh/Rice (with Rician factor K = 10 dB) fading. As we discussed in Section IIB, we assume uncoded communication and complete channel state information available at the receiving side (CSIR). In Figure 15, we depict following memoryless modulations using XOR XC: nonUMP QPSK, QFSK \kappa =1\u22152, UMPQFSK κ = 1 and proposed UMPQFSK \kappa =5\u22156. All modulation alphabets have the same minimal distance of scalar modulation {\delta}_{min}^{2}; therefore, the performance in AWGN channel is almost similar. The advantage of UMP alphabets is more evident in fading channels, particularly nonUMP QPSK has roughly ~ 2 dB penalty in Rayleigh fading and even about ~ 10 dB penalty in Rice K = 10 dB over UMP alphabets for sufficiently large SNR. It is interesting that nonUMPQFSK \kappa =1\u22152 is also well robust to the channel parametrization. Therefore, we may expect that nonlinear modulations avoiding all catastrophic parameters are generally more robust to the channel parametrization than the linear one. We supplement the error simulations by related endtoend throughput simulations including BC stage using for simplicity the same alphabet. We evaluate the throughputs as a relative number of bits of correctly detected 256bit long packets, see Figure 16.
B. Error performance of fullresponse CPM
Here, SER in the MAC stage of nonlinear fullresponse CPM \kappa =1\u22152 with optimized modulation pulses are shown. We have seen that the presence of discrete memory does not influence the UMP property, although it cannot be ignored at the receiver side. We use a joint [∂_{ A }, ∂_{ B }] decoding algorithm based on the vector Viterbi algorithm [24] describing the structure of receiving signals by a supertrellis with superstates. Each superstate is a vector of states that join together the actual state at the node A with the state at the node B. Then, the joint estimate of [∂_{ A }, ∂_{ B }] is obtained by the sequence Viterbi algorithm over the supertrellis. Thereafter, the exclusively coded data symbols are obtained as ∂ _{ AB } = ∂ _{ A } ⊕ ∂_{ B }.
In Figure 17, it is depicted SER of binary fullresponse CPM \kappa =1\u22152 with REC, RC and proposed SRC pulse and also UMPBPSK modulation as a reference. We describe binary fullresponse CPM \kappa =1\u22152 as a minimum shift keying (MSK) modulation to shorten the notation, though MSK strictly use the REC pulse. We conclude that nonUMP schemes with different pulses have almost the same performance, as we expected from Figure 11. Therefore, choosing a pulse with the narrowest spectra (REC) is appropriate. In this case, the proposed SRC pulse has only theoretical value as a performance benchmark.
The proposed SRC pulse is more advantageous for quaternary alphabet, see Figure 18. We shortly denote quaternary fullresponse CPM \kappa =1\u22152 as a QMSK modulation. We observe that performance with REC pulse is close to the UMP alphabet performance, but contraintuitively the performance with RC pulse is by several dBs worse even in the AWGN channel. The REC pulse, in this case, is a practical choice because the proposed SRC pulse requires more bandwidth, see the bandwidth comparison in the following section.
C. Bandwidth comparison
Bandwidth requirement of the considered modulations is presented in Table 2. The first part of the table describes bandwidth of linear memoryless alphabets (denoted shortly as 'Linear \mathcal{A}') which is solely given by used pulse function [20]. We present wellknown rootraised cosine (RRC) pulse parametrized by rolloff factor λ whose compact and finite bandwidth is W={\scriptscriptstyle \raisebox{1ex}{$\text{(1+}\lambda \text{)}$}\!\left/ \!\raisebox{1ex}{$2{T}_{s}$}\right.}. Due to the finiteness of the bandwidth, ideal RRC has infinite time duration. Vice versa, rectangular (REC) pulse finite in temporal domain has infinite bandwidth and in this case, we use the fractional powercontainment bandwidth definition where W_{99%} is a bandwidth containing 99% of the total signal power. In this paper, all linear modulations (with one complex dimension including BPSK) use the same bandwidth. Note the significant difference of error performance between BPSK and QPSK in Figure 15 and 17. BPSK is UMP only if it uses a single entire complex dimension (entire bandwidth); thus, it needs one real dimension more than in pointtopoint communication to be UMP.
The other part of the table is dedicated to nonlinear modulations. We denote fullresponse CPM \kappa =1\u22152 with SRC pulse by acronym 'MSK'. Its fractional bandwidth, analytically described in [23], shows that biorthogonal case require more bandwidth than e.g. the case with p = 0. The similar trend has also bandwidth of QMSK which we obtained according to [21], see average power spectral densities in Figure 19.
VII. Conclusion and discussions
In this paper, we have investigated modulations and exclusive codes (XC; network codeslike) robust to the parametrization for hierarchicaldecodeandforward (HDF) with channel state information at the receiver (CSIR) not requiring any channel adaptation techniques. We have found that such modulation is BPSK and definitely not the other linear modulations (PSK, QAM,...) with higher cardinality than 2, because they possess socalled catastrophic parameters. The catastrophic parameter is such a nonzero channel parameter that causes zero hierarchical minimal distance strongly degrading overall average performance. It is interesting that minimal cardinality XC (Latin square) should have symmetrical XC matrix with the same diagonal not to imply catastrophic parameters. These conditions fulfills bitwise XOR, which is the unique solution for binary and quaternary alphabet. It has been shown that to avoid the catastrophic parameters, modulations with more than a single complex dimension have to be considered. This inspired us to assume nonlinear frequency modulations naturally having multidimensional waveforming alphabet. We have precisely defined uniformly most powerful (UMP) alphabets, which not only obviate all catastrophic parameters but also reach the minimal distance upper bound for all parameter values. UMP can be interpreted as the most suitable type of unavoidable channel parametrization, and among all alphabets with identical minimal distance of the single alphabet, the UMP alphabets have the best performance serving as a performance benchmark. It is proved that any binary, nonbinary complexorthogonal and nonbinary complex biorthogonal alphabets are UMP.
We have found that the considered frequency shift keying (FSK) and fullresponse \kappa =1\u22152 continuous phase modulation (CPM) avoid catastrophic parameters, and we have optimized their parameters (modulation index for FSK and phase pulse shape for CPM) to meet UMP. Based on the numerical simulations, we conclude that existence of catastrophic parameters is much more detrimental to the average performance than violation of UMP condition. The proposed alphabets are summarized in Table 3; the note 'close to UMP' means that despite the alphabet is not UMP, it performs very close to the UMP benchmark.
We have analyzed error and bandwidth performance of the used modulations in Section VI. However, the optimal modulation choice will depend, besides errorbandwidth performance, on the other properties such as complexity or hardware requirements. For instance, BPSK with RRC pulse and MSK p = 0 have comparable errorbandwidth performance, but MSK possess constant envelope (which allows more efficient power amplifier) at the price of more complex decoding processing (includes e.g. Viterbi algorithm). If we insist on constant envelope feature than MSK p = 0 needs to be compared with BPSK with REC which requires much more bandwidth. BPSK with REC pulse and two times shorter pulse duration (to deliver 2 bits per channel use) have similar error performance but two times wider bandwidth than QFSK \kappa =1\u22152. On the other hand, QFSK receiver consists of parallel bank of matched filters (number of filters equals to the dimensionality) while BPSK receiver has only one filter. QMSK p = 0 is more preferable than QFSK \kappa =1\u22152 since it owns narrower bandwidth, if we can afford slightly more complex decoding processing (e.g., Viterbi algorithm).
Appendix
Proof (Biorthonormal modulation is UMP): Let us consider the modified condition (28) which we have used in UMPFSK derivation. It states that UMP alphabet modulation fulfills
for c_{ A } ≠ c_{ A } ', c_{ B } ≠ c_{ B } ', c_{ A } ⊕ c_{ B } ≠ c_{ A } ' ⊕ c_{ B } ', where {c}_{A},{c}_{B},{{c}_{A}}^{\prime},{{c}_{B}}^{\prime},\in {\mathbb{Z}}_{{M}_{c}} and ∀α ∈ ℂ, α ≤ 1. Assume a biorthonormal modulation for which two biorthonormal signals have one of the three possible mutual relations. They are orthonormal (described by symbol ⊥), or the same ⇇ or have the opposite sign ⇆. According to the mutual relation, its inner product and squared norm of its difference is
In the first step, we analyze all possible values of the inner product term
Without taking into account the restrictions c_{ A } ≠ c_{ A } ', c_{ B } ≠ c_{ B } ' and c_{ A } ⊕ c_{ B } ≠ c_{ A } ' ⊕ c_{ B } ', the term z ∈ {0, 1, 2, 3, 4}. Note, if the term z = 0, then the inequality (37) is fulfilled because by definition \parallel {\mathbf{\text{s}}}_{{C}_{B}}{\mathbf{\text{s}}}_{{{c}_{B}}^{\prime}}{\parallel}^{2}\ge {\delta}_{min}^{2}. There are four inner products in (39) that can take three different geometrical arrangements (⊥, ⇇, ⇆) altogether there is 3^{4} = 81 combinations. We will show by exhaustive listing of all cases where z ≠ 0 that such a case
a) cannot happen because the inner products give such results that geometrically this situation cannot exist (e.g. two vectors cannot be orthonormal and perpendicular at the same time) or
b) the inequality (37) is still fulfilled or c) this situation is excluded by XOR symmetries concluding that always z = 0 which proves the lemma.
In the second step, we show that z ≠ 1 because always the case a) happen. Due to the space limitation, we will not list all that cases; in fact, considering geometrical properties, it is sufficient to list only those schemes which have different geometrical arrangement. Let us denote the geometrical arrangement by a vector of inner products. The ordered configurations leading to z = 1 are
For example, let us assume the situation where \left(\u3008{\mathbf{\text{s}}}_{{c}_{A}},{\mathbf{\text{s}}}_{{c}_{B}}\u3009,\u3008{\mathbf{\text{s}}}_{{{c}_{A}}^{\prime}},{\mathbf{\text{s}}}_{{{c}_{B}}^{\prime}}\u3009,\u3008{\mathbf{\text{s}}}_{{c}_{A}},{\mathbf{\text{s}}}_{{{c}_{B}}^{\prime}}\u3009,\u3008{\mathbf{\text{s}}}_{{{c}_{A}}^{\prime}},{\mathbf{\text{s}}}_{{c}_{B}}\u3009\right)=\left(0,0,0,1\right). In Figure 20, we draw all possible configurations corresponding to the first three inner products \left(\u3008{\mathbf{\text{s}}}_{{c}_{A}},{\mathbf{\text{s}}}_{{c}_{B}}\u3009\u3008{\mathbf{\text{s}}}_{{{c}_{A}}^{\prime}},{\mathbf{\text{s}}}_{{{c}_{B}}^{\prime}}\u3009\u3008{\mathbf{\text{s}}}_{{c}_{A}},{\mathbf{\text{s}}}_{{{c}_{B}}^{\prime}}\u3009\right)=\left(0,0,0\right), by that we demonstrate that it implies the last term to be \u3008{\mathbf{\text{s}}}_{{{c}_{A}}^{\prime}},{\mathbf{\text{s}}}_{{c}_{B}}\u3009=0. In this case, it is not true; hence, this example situation would never occur. Hereby, the situations with any permutation of the inner products (0, 0, 1, 0), (0, 1, 0, 0), etc. are also excluded. In Figure 21 and 22,
we similarly show that configuration (1, 1, 1) ⇒ 1 and (1, 1, 1) ⇒ 1. We denote these three situations above as a (x, x, x) ⇒ x law. The last remaining configurations of z = 1, (40), which are not excluded by the (x, x, x) ⇒ x law, are (0, 1, 1, 1) and (0, 1, 1, 1). They are again excluded by a), see Figures 23 and 24. Since we excluded all situations where z = 1 then z ≠ 1.
In the last third step, we will conclude that α = 1 is critical and check all the remaining situations/configurations where z ∈ {2, 3, 4}. We will see that either a), b) or c) happen and thus always z = 0. We have seen in the Section VA that α = 1. is critical if the term b=\left\u3008{\mathbf{\text{s}}}_{{c}_{A}}{\mathbf{\text{s}}}_{{{c}_{A}}^{\prime}},{\mathbf{\text{s}}}_{{c}_{B}}{\mathbf{\text{s}}}_{{{c}_{B}}^{\prime}}\u3009\right/{\u2225{\mathbf{\text{s}}}_{{c}_{B}}{\mathbf{\text{s}}}_{{{c}_{B}}^{\prime}}\u2225}^{2}{\delta}_{min}^{2} is b ≥ 1. In our case, b=z\u22152 and assuming z ∈ {2, 3, 4} always b ≥ 1 and so a critical parameter is α = 1.
The configurations corresponding to the case z = 2 are
The configuration (0, 0, 1, 1) entails that {\mathbf{\text{s}}}_{{c}_{A}}={\mathbf{\text{s}}}_{{c}_{B}} & {\mathbf{\text{s}}}_{{{c}_{A}}^{\prime}}={\mathbf{\text{s}}}_{{{c}_{B}}^{\prime}} thus implies k = n & l = m and is excluded by XC with symmetrical XC matrix which we already assume. The configuration (0, 0, 1, 1) is rather problematic, see its configuration in Figure 25.
Without deep analysis, we conclude that whatever mapping between waveforms {\mathbf{\text{s}}}_{{c}_{A}} and data symbols c we use, XOR will always exclude this situation. In other words, the hierarchical signals of this situation have always the same XC symbol. The configuration (0, 1, 1, 0) falls into the case b), where the situation is geometrically possible, but it implies that \parallel {\mathbf{\text{s}}}_{{c}_{A}}{\mathbf{\text{s}}}_{{{c}_{A}}^{\prime}}{\parallel}^{2}=4 and the condition (37) is fulfilled. The similar situation occurs for configurations
The case of z = 3 are excluded in a very similar way by a) as for z = 1. The last case z = 4 have possible configurations
The former case means {\mathbf{\text{s}}}_{{c}_{A}}={\mathbf{\text{s}}}_{{c}_{B}} & {\mathbf{\text{s}}}_{{{c}_{A}}^{\prime}}={\mathbf{\text{s}}}_{{{c}_{B}}^{\prime}} thus k = l & m = n, which corresponds to hierarchical signals from the main diagonal of XC matrix. This is excluded, because such a condition we already assume. The latter case implies k = n & m = l and is again excluded by symmetrical XC matrix.
By the above three steps, we have proved z = 0, which proves the lemma.
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Acknowledgements
This work was supported by the FP7ICT SAPHYRE project, by Grant Agency of the Czech Republic grant 102/09/1624; and by Grant Agency of the Czech Technical University in Prague, grant SGS 10/287/OHK3/3T/13.
Endnotes
^{a} For the sake of notation simplicity, we assume orthonormal modulations rather than orthogonal with constant symbol energy. The results here shown for orthonormal modulation are true also in orthogonal case with constant symbol energy.
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Hekrdla, M., Sykora, J. Design of Uniformly Most Powerful Alphabets for HDF 2Way Relaying Employing NonLinear Frequency Modulations. J Wireless Com Network 2011, 128 (2011). https://doi.org/10.1186/168714992011128
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DOI: https://doi.org/10.1186/168714992011128