- Research
- Open Access
Design of Uniformly Most Powerful Alphabets for HDF 2-Way Relaying Employing Non-Linear Frequency Modulations
- Miroslav Hekrdla^{1}Email author and
- Jan Sykora^{1}
https://doi.org/10.1186/1687-1499-2011-128
© Hekrdla and Sykora; licensee Springer. 2011
- Received: 28 January 2011
- Accepted: 11 October 2011
- Published: 11 October 2011
Abstract
Hierarchical-Decode-and-Forward is a promising wireless-network-coding-based 2-way relaying strategy due to its potential to operate outside the classical multiple-access 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 non-zero channel parameters (denoted as catastrophic) cause zero hierarchical minimal distance -- significantly degrading its performance. In this work, we state that non-binary 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, non-binary orthogonal and bi-orthogonal modulations are UMP. We optimize scalar parameters of FSK (modulation index) and full-response CPM (frequency pulse shape) to yield UMP alphabets.
Keywords
- Network Code
- Continuous Phase Modulation
- Pairwise Error Probability
- Physical Layer Network Code
- Uniformly Much Powerful
I. Introduction
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 complementary-side 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 non-zero channel parameters (denoted as catastrophic) cause zero hierarchical minimal distance, which significantly degrades its performance. Adaptive extended-cardinality 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 non-coherent (no CSIR) complex-orthogonal frequency shift keying (FSK) modulations.
- 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. Bit-wise 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 non-binary 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 non-binary UMP alphabets require more than a single complex dimension.
- 3)
Non-binary complex-orthogonal and non-binary complex bi-orthogonal modulations with XOR are UMP.
- 4)
Non-linear frequency modulations FSK and full-response continuous phase modulation (CPM) naturally comprise multiple complex dimensions needed to obey UMP condition. We optimize a scalar parameters of FSK (modulation index) and full-response CPM (frequency pulse shape) to yield UMP alphabets. We find that a lower modulation index (proportional to bandwidth) than that leading to complex-orthogonal 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 2-WRC 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 arbitrary-dimensional 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 non-linear frequency modulations FSK and full-response 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 time-synchronized 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 per-symbol relaying (avoiding delay induced at the relay) and no channel coding, which however can be additionally concatenated with our scheme [15].
C. Hierarchical-decode-and-forward strategy
where w is complex AWGN with variance 2N_{0} per complex dimension, and the channel parameters h_{ A } and h_{ B } are frequency-flat 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.
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 multiple-access interference [16].
D. Exclusive coding
define binary XOR, quaternary modulo sum and quaternary bit-wise XOR operation, respectively. We will use often short 'XOR' to denote bit-wise 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.
Number of minimal cardinality exclusive codes in the standard notation as a function of the alphabet size.
M _{ c } | 2 | 4 | 8 |
---|---|---|---|
Number of XCs | 1 | 24 | ~ 10^{16} |
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].
- III.
Hierarchical minimal distance as a performance metric and catastrophic parameters.
A. Hierarchical minimal distance
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
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.
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.
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 bit-wise 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. Non-binary linear modulations are catastrophic
In this section, we demonstrate that any non-binary linear modulation can never avoid catastrophic parameters, as we have seen particularly for QPSK, Figure 4a.
Lemma 5. Non-binary linear modulations unavoidably have catastrophic parameters.
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.
${\alpha}^{\u2033}=1\u2215{\alpha}^{\prime}$. 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 upper-bound for all parameter values.
A. Minimal distance upper-bound
where${\delta}_{min}^{2}$is a minimal distance of a single (non-hierarchical) 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${c}_{A},{{c}_{A}}^{\prime}\in {\mathbb{Z}}_{{M}_{c}}$.
As we are considering |α| ≤ 1, we conclude that the bound (16) is more tight than (18).
B. UMP alphabet definition
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 II-E.
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, non-binary linear modulations are never UMP and all UMP alphabets are using XOR exclusive code.
Remark 9. Extended-cardinality XC as well as minimal cardinality XC have different code symbols in each XC matrix row (Sudoku principle) and thus the bound holds for extended-cardinality 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 upper-bound (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
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. Non-binary orthonormal modulation is UMP
We have seen in Figure 4b that complex-orthonormal QFSK is UMP. This holds in general which describes the following lemma.
Lemma 12. Complex-orthonormal modulation is UMP.
Remark 13. Before we prove the Lemma 12, it is convenient to introduce simplified UMP condition easier to verify.
This form of invariancy condition is easier to verify due to the presence of only one real variable |α|.
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 non-linear 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. UMP-FSK design
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 well-known fact that FSK is complex-orthonormal for integer modulation index κ ∈ ℕ and with minimal κ = 1 is often used in non-coherent 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 UMP-QFSK modulation by index optimization: According to Lemma 12, FSK κ = 1 is UMP. Yet, we try to answer a question whether full complex-orthogonality is required to meet UMP. To investigate the non-orthogonal case, we assume κ < 1, which also means a modulation roughly with narrower bandwidth, see more detailed discussion of bandwidth requirements in Section VI-C.
Let us assume quaternary M_{ c } = 4 (binary is UMP regardless of alphabet, Section IV-D) FSK (QFSK) to consider this question where we optimize modulation index κ to meet the UMP condition. The following lemma is true.
B. Bi-orthonormal modulation is UMP
According to the results from the preceding section, we see that UMP property does not require an accurate complex-orthonormal alphabet. Inspired also by [10], we have a conjecture that bi-orthonormal modulation is UMP. The Appendix proves the following lemma.
Lemma 15. Bi-orthonormal 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. UMP-CPM 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 VI-C. CPM includes memory [21] and its modulator consists of the discrete part including memory and the non-linear 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 full-response (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 full-response$\kappa =1\u22152$UMP-CPM 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.
Our design is based on Lemma 15, utilizing the above mentioned symmetries, we design a bi-orthonormal UMP modulation simply keeping orthonormal signals starting from the first state.
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.
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 UMP-CPM
Lemma 17. Binary full-response CPM with$\kappa =1\u22152$and parametric SRC pulse (34) with p ≃ 2.35 is UMP.
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 ${\delta}_{min}^{2}$.
6) Design of quaternary UMP-CPM
Lemma 19. Quaternary full-response CPM with$\kappa =1\u22152$and parametric SRC pulse (34) with p ≃ -7 or p ≃ 10.2 is UMP.
VI. Numerical results
A. Error performance of memoryless modulations
B. Error performance of full-response CPM
Here, SER in the MAC stage of non-linear full-response 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 super-trellis with super-states. Each super-state 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 super-trellis. Thereafter, the exclusively coded data symbols are obtained as ∂ _{ AB } = ∂ _{ A } ⊕ ∂_{ B }.
C. Bandwidth comparison
Bandwidth comparison.
◊ Linear $\mathcal{A}$ | RRC λ= 0 | RRC λ= 1 | REC |
---|---|---|---|
WT _{ s } | 0.5 | 1 | 10.3 |
◊ QFSK | $\kappa =1\u22152$ | $\kappa =5\u22156$ | κ = 1 |
W _{99%} T _{ s } | 10.9 | 15 | 21 |
◊ MSK | p = 0 | p = 1 | p = 2.35 |
W _{99%} T _{ s } | 0.6 | 1.1 | 1.5 |
◊ QMSK | p = 0 | p = 1 | p = -7 |
W _{99%} T _{ s } | 1.4 | 1.8 | 5.9 |
VII. Conclusion and discussions
In this paper, we have investigated modulations and exclusive codes (XC; network codes-like) robust to the parametrization for hierarchical-decode-and-forward (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 so-called catastrophic parameters. The catastrophic parameter is such a non-zero 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 bit-wise 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 non-linear 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, non-binary complex-orthogonal and non-binary complex- bi-orthogonal alphabets are UMP.
Summary of the proposed alphabets.
Alphabet | UMP relation | Notes |
---|---|---|
BPSK | UMP | optimal binary alphabet |
QFSK $\kappa =1\u22152$ | close to UMP | not optimal spectra |
QFSK $\kappa =5\u22156$ | UMP | more bandwidth than QFSK $\kappa =1\u22152$ |
MSK p = 0 | close to UMP | better spectrum than FSK; include memory |
QMSK p = 0 | close to UMP | better spectrum than FSK; include memory |
MSK p ≃ 2.35 | UMP | more bandwidth than MSK p = 0 |
QMSK p ≃ -7 | UMP | more bandwidth than QMSK p = 0 |
We have analyzed error and bandwidth performance of the used modulations in Section VI. However, the optimal modulation choice will depend, besides error-bandwidth performance, on the other properties such as complexity or hardware requirements. For instance, BPSK with RRC pulse and MSK p = 0 have comparable error-bandwidth 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
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 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 V-A 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 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.
Declarations
Acknowledgements
This work was supported by the FP7-ICT 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.
Authors’ Affiliations
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