Patent Publication Number: US-2023140226-A1

Title: Communication devices and methods

Description:
BACKGROUND 
     Field of the Disclosure 
     The present disclosure relates to communication devices and methods using a novel modulation scheme. 
     Description of Related Art 
     Quadrature Amplitude Modulation (QAM) has become one of the predominant modulation schemes used to improve spectral efficiency of both wired and wireless communications. In QAM, two real-valued signal components (either analog or digital) are transmitted over the same carrier frequency. 
     QAM is limited to only two components, generally called in-phase and quadrature components. Multi-dimensional QAM can be generated by artificially grouping several inphase and quadrature components, e.g. grouping two QAM subcarriers of an OFDM, thereby generating a four-dimensional signal. Still, the underlying concept of QAM is based on only two signal components, using cosine and sine modulators. 
     The “background” description provided herein is for the purpose of generally presenting the context of the disclosure. Work of the presently named inventor(s), to the extent it is described in this background section, as well as aspects of the description which may not otherwise qualify as prior art at the time of filing, are neither expressly nor impliedly admitted as prior art against the present disclosure. 
     SUMMARY 
     It is an object to extend the concept of QAM. 
     According to an aspect there is provided a communication device comprising circuitry configured to
         modulate a four-dimensional input signal by combining the four real-valued signal components of the input signal into a transformed signal and multiplying the transformed signal by carrier signals using two different carrier frequencies to obtain a transmit signal, and   transmit the transmit signal.       

     According to a further aspect there is provided a communication device comprising circuitry configured to
         receive a transmit signal, and   demodulate the transmit signal into a four-dimensional output signal by multiplying the transmit signal by carrier signals using two different carrier frequencies to obtain a retransformed signal and decomposing the retransformed signal into the four real-valued signal components of the output signal.       

     According to still further aspects corresponding communication methods, a computer program comprising program means for causing a computer to carry out the steps of the method disclosed herein, when said computer program is carried out on a computer, as well as a non-transitory computer-readable recording medium that stores therein a computer program product, which, when executed by a processor, causes the method disclosed herein to be performed are provided. 
     Embodiments are defined in the dependent claims. It shall be understood that the disclosed communication method, the disclosed computer program and the disclosed computer-readable recording medium have similar and/or identical further embodiments as the claimed communication devices and as defined in the dependent claims and/or disclosed herein. 
     The present disclosure presents a modulation scheme to transmit signals (analog or digital) via a new signal processing chain. It extends the concept of QAM, in which two signal components are transmitted via a cosine and sine wave of the same frequency. While QAM can be described in the complex domain (using real and imaginary numbers), the new scheme will process a four-dimensional signal. In an embodiment, quaternions (using real and three imaginary numbers) may be used to process and transmit the four independent signal components of the four-dimensional signal. 
     In an embodiment, two quaternions may be used in conjunction with two carrier frequencies at the transmitter side and at the receiver side. Frequency diversity can thus be exploited by the presented modulation scheme, since four independent signal components are spread over two frequency bands. Further, depending on the choice of the parameters of the modulation, e.g. of modulating quaternions, new four-dimensional constellations may be formed, allowing for spectral shaping to optimize the robustness of the signal. 
     The foregoing paragraphs have been provided by way of general introduction and are not intended to limit the scope of the following claims. The described embodiments, together with further advantages, will be best understood by reference to the following detailed description taken in conjunction with the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWING 
       A more complete appreciation of the disclosure and many of the attendant advantages thereof will be readily obtained as the same becomes better understood by reference to the following detailed description when considered in connection with the accompanying drawings, wherein: 
         FIG.  1    shows a schematic diagram illustrating Quadrature Amplitude Modulation. 
         FIG.  2    shows a schematic diagram of first and second communication devices according to an aspect of the present disclosure. 
         FIG.  3    shows a schematic diagram of an embodiment of a Quaternion Amplitude Modulation (Quat-AM) transmitter architecture according to the present disclosure. 
         FIG.  4    shows a schematic diagram of an embodiment of a Quat-AM receiver architecture according to the present disclosure. 
         FIG.  5    shows a schematic diagram of another embodiment of a Quat-AM transmitter architecture according to the present disclosure. 
         FIG.  6    shows a schematic diagram of another embodiment of a Quat-AM receiver architecture according to the present disclosure. 
         FIG.  7    shows diagrams illustrating the in-phase and the negative quadrature phase over frequency for the disclosed Quaternion Amplitude Modulation. 
         FIG.  8    shows a schematic diagram of another embodiment of a Quat-AM transmitter. 
         FIG.  9    shows a schematic diagram of another embodiment of a Quat-AM receiver. 
         FIG.  10    shows a schematic diagram of another embodiment of a Quat-AM transmitter. 
         FIG.  11    shows a schematic diagram of another embodiment of a Quat-AM receiver. 
         FIG.  12    shows a schematic diagram of another embodiment of a Quat-AM transmitter. 
         FIG.  13    shows a schematic diagram of another embodiment of a Quat-AM receiver. 
         FIG.  14    shows a schematic diagram of another embodiment of a Quat-AM receiver. 
         FIG.  15    shows a schematic diagram of another embodiment of a Quat-AM receiver. 
         FIG.  16    shows a schematic diagram of another embodiment of a Quat-AM receiver. 
         FIG.  17    shows a schematic diagram of another embodiment of a Quat-AM receiver. 
         FIG.  18    shows a schematic diagram of a first embodiment of a generator  300  of a digital representation of the input signals. 
         FIG.  19    shows a schematic diagram of a second embodiment of a generator  300  of a digital representation of the input signals. 
         FIG.  20    shows a schematic diagram of a transmitter as described in IEEE802.11. 
         FIG.  21    shows a schematic diagram of a receiver as described in IEEE802.11. 
     
    
    
     DETAILED DESCRIPTION OF THE EMBODIMENTS 
     Referring now to the drawings, wherein like reference numerals designate identical or corresponding parts throughout the several views,  FIG.  1    shows a schematic diagram illustrating Quadrature Amplitude Modulation (QAM). In QAM, two real-valued signal (either analog or digital) components are transmitted over the same carrier frequency f 0 =ω 0 /(27c), where ω 0  is called angular frequency. The signal on the upper branch is called inphase component and is modulated by a cosine carrier of frequency f 0 , whereas the signal on the lower branch is called quadrature-phase component and is modulated by a (negative) sine carrier of the same frequency f 0 , (sine having a 90° phase shift compared to cosine in the mathematical sense, resulting in the name quadrature component). If the signal is digital, a pulse shaper may be needed to convert discrete-time, discrete-value (=digital) values to some analog signals s l (t) and s Q (t), resp. Pulse shapers may apply a certain filter g tx (t), e.g., a root-raised-cosine (RRC) filter, allowing both band-limited and inter-symbol-interference (ISI)-free signals (after matched filtering at receiver side, resulting in a raised cosine (RC) filter, which is ISI-free). The factors of √2 are used to normalize the signal power, but can be neglected in the following. The superposition (sum) of the two modulated signal components is transmitted as signal s(t). If the components s l (t) and s Q (t) are both bandlimited (with cutoff frequency f c ), then the signal components can be recovered at receiver side. Both components are transmitted within the same frequency band, centered around carrier frequency f 0 . 
     Alternatively, the modulation from  FIG.  1    can be described by complex numbers, where a complex input signal is formed: s complex (t)=s i (t)+i*s Q (t), with i=√{square root over (−1)} being the imaginary unit of complex numbers. The up-conversion to carrier frequency f 0  is then described by multiplying s complex (t) with the complex phasor e iω     0     t . Finally, the real-part of the signal needs to be considered as the transmit signal, i.e., s(t)=√{square root over (2)} * Re{e iω     0     t *s complex (t)), where the factor √{square root over (2)} has been used as normalization again. The receiver architecture will symmetrically apply this factor, such that an overall multiplication with a factor of 2 has been performed (otherwise, the receiver will lose half the signal power due to low-pass filtering). 
     QAM is limited to only two components. Multi-dimensional QAM can be generated by artificially grouping several in-phase and quadrature phase components, e.g. grouping two QAM subcarriers of an OFDM, thereby generating a four-dimensional signal. Still, the underlying concept of QAM is based on only two signal components, using cosine and sine modulators. 
     The present disclosure proposes to extend the QAM concept.  FIG.  2    shows a schematic diagram illustrating a (first) communication device  100  (e.g. an Access Point (AP)) according to an aspect of the present disclosure for communicating with, e.g. transmitting data to, another (second) communication device  200  (e.g. a station (STA)). Each of the communication devices  100 ,  200  comprises respective circuitry  101 ,  201  configured to perform particular operations. The circuitries may be implemented by a respective processor or computer, i.e. in hardware and/or software, or by dedicated units or elements. For instance, respectively programmed processors may represent the respective circuitries  101 ,  201 . 
     According to the present disclosure the first communication device  100  comprises circuitry  101  configured to modulate a four-dimensional input signal by combining the four real-valued signal components of the input signal into a transformed signal and multiplying the transformed signal by carrier signals using two different carrier frequencies to obtain a transmit signal, and to transmit the transmit signal. The second communication device  200  comprises circuitry  201  configured to receive a transmit signal and to demodulate the transmit signal into a four-dimensional output signal by multiplying the transmit signal by carrier signals using two different carrier frequencies to obtain a retransformed signal and decomposing the retransformed signal into the four real-valued signal components of the output signal. 
     In an embodiment the QAM concept is extended to so-called quaternion based algebra. The resulting scheme will herein be denoted as Quaternion Amplitude Modulation, or Quat-AM in short. While complex numbers extend the (real) number space from a one-dimensional special unitary (SU(1)) group to a two-dimensional group (SU(2)), quaternions extend the number space to a four-dimensional special unitary (SU(4)) group: x=x 0 + x =x 0 +ix 1 +jx 2 +kx 3 , where □ 0  is the real part, and  x  is a pure quaternion (often expressed as 3D vector  x =(x 1  x 2  x3) T ), consisting of three (independent) imaginary (fundamental quaternion) units i, j and k. (.) T  is the transpose operation, transforming a row vector into a column vector (or vice versa). 
     They define a (strict) skew field, in which basic rules apply, except multiplication not being commutative. The Hamilton product of two quaternions a, b is a·b=a 0 ·b 0 −   α , b   +a 0   b  +b 0   α + α × b . Hereby, a 0 ·b 0 −   α , b   +a 0    b +b 0   α  is known from complex numbers (   α , b    being the scalar product of  α  and  b ) and  α × b  is the additional non-commutative complex part, expressed by a vector product. 
     Some further Quaternion basics are as follows:
         ijk=i 2 =j 2 =k 2 =−1   cyclic properties: ij=k;jk=i; ki=j and ji=−k;kj=−i;ik=−j   norm: |x| 2 =x 0   2 +x 1   2 +x 2   2 +x 3   2      inverse:       

     
       
         
           
             
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     In the following, a general quaternion rotation will be used, defined by x rotated =e   α   xe   b   with  b  not necessarily− α  (in fact, this special case, will be excluded). 
     The idea of Quat-AM is to consider quaternion based algebra, to form a TX/RX architecture, allowing to transmit four independent, real-valued signal components. This will be illustrated in the following by way of several embodiments for TX and RX architectures. Subscripts (e.g. “01”) indicate the description with two carrier frequencies f 0  and f 1 , and various subtypes are denoted by different super-indices (e.g., “a” vs. “b” implementation). 
       FIG.  3    shows a schematic diagram of an embodiment of a Quat-AM transmitter  110  (i.e. of an embodiment of a first communication device  100  according to the present disclosure), also designated with TX a   01,q , with four independent input signals x 0, . . . 3 , all assumed to be bandlimited with cutoff frequency f c . The signal x is obtained by grouping the real-valued inputs to one quaternion signal using multipliers  111  and adders  112 . The upcoming multiplication with a quaternion phasor e   α ω     0     t  is indicated by an arrow from left to (upper) right, meaning that the signal x is to be multiplied in a multiplier  113  from left-hand side with this phasor (the phasor itself is a quaternion). It shall be noted that multiplication in SU(4) is not commutative, i.e., for two quaternions a and b, a*b is not the same as b*a, due to the definition of the so called Hamilton product. The next multiplication with a quaternion phasor e   β ω     1     t  in a multiplier  114  is performed from right-hand side. Finally, a projection, using a real-part projector  115 , towards the real part of the signal results in a real-valued signal s (having a physical representation, e.g. modulating an electromagnetic wave): s=Re {e   α ω     0     t * x*e   β ω     1     t }. Two frequencies f 0  and f 1  and two pure quaternions (with unit power)  α  and  β  are used. 
     The transmit signal s consists of two spectra/bands, centered around the “sum and the difference frequencies” ω Σ =ω 0 +ω 1  and ω Δ =ω 0 −ω 1 . In both bands, all four signal components x 0 . . . 3  are transmitted. This offers frequency diversity in frequency selective fading channels. In general, the presented communication devices and methods allow a novel way to apply channel bonding (if two channels at ω Σ  and ω Δ  are to be grouped together, even when the bands are non-adjacent). 
     For the simple example of using f 1 =0 Hz and  α =i (conventional imaginary unit), the output is a normal QAM. In this case, only x 0  and x 1  can be recovered at the receiver and the components x 2  and x 3  cannot be resolved (mathematically, the matrix Q shown below will have only rank two, i.e., only two independent rows/columns). 
     In summary, the following relationships and abbreviations hold: ω 0 ≥ω 1 ≥ω c ; define ω Σ =ω 0 +ω 1 ;ω Δ =ω 0 −ω 1 ; ω c  being the cutoff frequency of an ideal lowpass filter 
     
       
         
           
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     c v =cos(ω v t); s v =sin(ω v t);v∈{0,1Δ,Σ};  x =(x 1 x 2 X 3 ) T    
     s=Re{e   α ω     0     t ·x·e   β ω     1     t }=c 0 c 1 x 0 −s 0 c 1     x ,α −c 0 s 1     x ,β −s 0 s 1 x 0 ϵ+s 0 s 1     x ,γ   
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       FIG.  4    shows a schematic diagram of an embodiment of a Quat-AM receiver  210  (i.e. of an embodiment of a second communication device  200  according to the present disclosure), also designated with RX a   01,q . The receiver  210  reverts the order, similar as in a conventional QAM. The factor of four (applied by a multiplier  211 ) is used for normalization (similar as √{square root over (2)} * √{square root over (2)} =2 for normal QAM, whereas Quat-AM requires twice the correction, i.e., 2*2=4). Then, multiplication with quaternion phasor e − α ω     0     t  and quaternion phasor e − β ω     1     t  in multipliers  212  and  213  is applied. After projection to four dimensions by a real-part projector  214  and three imaginary-part projectors  215 , a low pass filter (LPF)  216  may be used, similar as in conventional QAM, to remove unwanted images at higher frequencies. The matrix inversion of the four-dimensional square matrix Q is applied by a matrix inverter  217 , since the four input signals x 0 . . . 3  are combined during transmission and are not simply recovered by projection towards the four dimensions. The analysis assumes an ideal channel for ease of description, i.e., received signal r=s, without any distortions or noise. 
       FIG.  5    shows a schematic diagram of another embodiment of a Quat-AM transmitter  120  (i.e. of an embodiment of a first communication device  100  according to the present disclosure), also designated with TX b   01,q , using a quaternion assembler  121 .  FIG.  6    shows a schematic diagram of another embodiment of a Quat-AM receiver  220  (i.e. of an embodiment of a second communication device  200  according to the present disclosure), also designated with RX b   01,q , using a quaternion disassembler  221 . 
     In the following some comments on the choice of modulating quaternions  α  and  β  are provided. It can be shown that  α  and  β  must not be parallel, i.e., linear dependent. Otherwise, the four signal components cannot be fully recovered. For the special case of  α  and  β  being orthogonal, it can further be shown that the Q matrix is the identity matrix, i.e., no matrix inversion is needed for such settings. 
     If  α  ⊥ β , i.e., a and § being orthogonal, the unit dyadic forming matrix Q′ will produce the identity matrix I 3  and thus Q=I 4 . If α=±β, then Q is singular, else regular (and can be inverted). For the ideal channel it holds: {tilde over (r)}=r=s→output {circumflex over (x)}=Q −1  {tilde over (r)}=x (else potential noise enhancement). Instead of matrix inversion (=zero forcing), better methods are possible (ML, MMSE, . . . ). 
     A simple example is as follows: 
     
       
         
           
             	 
             
               
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     Then, QAMω Δ  uses I: x 0 +x 3  and Q: x 1 −x 2  and QAMω Σ  uses I: x 0 −x 3  and Q: x 1 +x 2 . This is illustrated in  FIG.  7    showing diagrams illustrating the in-phase I and the negative quadrature phase Q over frequency. 
     The presented receiver architectures are linear receivers with suboptimum performance (but reduced complexity). Matrix inversion corresponds to zero forcing (ZF) detection, which would in general increase noise power. A better linear detector would be minimum mean squared error (MMSE), which in addition considers the noise statistics during linear matrix inversion. Optimum detection considers the full observation of the four-dimensional signal {tilde over (r)} after low pass filtering, by performing maximum likelihood (ML) detection (i.e., {tilde over (r)} is compared against all possible transmit permutations and the most likely one is chosen as estimation {tilde over (x)}). Other detectors may be considered as well, such as successive interference cancellation (SIC) or suboptimum ML detectors, such as sphere decoding. 
     In the following, the frequency constraints, e.g. allowing alias-free representations of the TX and RX signals, are discussed. Further, comments on the choice of the input signals x 0 . . . 3  in case of digital modulation are provided. They can be e.g. simple BPSK signals (taking on either value−1 or +1 per component) or any other discrete real-valued signals. 
     In particular, the following holds for the frequency components (see also  FIG.  7   ):
         ω 0 ≥ω c  and ω 1 ≥ω c      |ω 0 −ω 1 |≥ω c  to avoid aliasing in TX signal   ideal LPF: 1 for |ω|&lt;ω c  and 0 for |ω|&gt;ω c,LPF ≥2ω 1 −ω c      digital representation of Quat-AM
           TX side: max. freq. component at ω max,TX =Σ+ω c ; min. sampling rate: ω s,TX &gt;2ω max,TX =2Σ+2ω c      RX side: max. freq. component at ω max,RX =2Σ+ω c ; sampling rate: ω s,RX &gt;2ω max,RX =4Σ+2ω c      for simulations: constantly sample with ω s,RX =4Σ+2ω c      
               

     The input quaternion elements x i  for digital communication are either generated by four independent PAMs, or by two QAM mappings, or by a four-dimensional QAM (or 3D+PAM). Further, the ML demapper should include P/T/Q matrix. 
       FIG.  8    shows a schematic diagram of another embodiment of a Quat-AM transmitter  130  (i.e. of an embodiment of a first communication device  100  according to the present disclosure), also designated with TX a   ΔΣ .  FIG.  9    shows a schematic diagram of another embodiment of a Quat-AM receiver  230  (i.e. of an embodiment of a second communication device  200  according to the present disclosure), also designated with RX a   ΔΣ . 
       FIG.  8    presents an alternative representation as two superimposed QAMs operating at ω Σ , and ω Δ  applied in QAM modulators  132 ,  133 . The QAMs are precoded by a precoder  131  using a four-dimensional matrix P. The optimum choice of the input components should be four-dimensional non-uniform constellations (NUCs), either before precoding using matrix P or after precoding using matrix P. The outputs of the QAM modulators  132 ,  133  are summed up in an adder  134  and multiplied by 0.5 by a multiplier  135  to obtain the output signal s. 
     If linear detection is used at the Quat-AM receiver  230  shown in  FIG.  9   , e.g. inverting the P matrix (ZF or MMSE) by an inverter  234  after QAM demodulation in QAM demodulators  232 ,  233  (receiving the signal  2   s  obtained from the receive signal s by multiplication of 2 by a multiplier  231 ), then the resulting four-dimensional signal, relating to the transmit signal x 0 . . . 3 , should be an optimum four-dimensional NUC. If ML or suboptimum forms thereof are used at the Quat-AM transmitter  130 , the precoded signal y 0 . . . 3  instead should be an optimum four-dimensional NUC. 
     In the embodiments shown in  FIGS.  8  and  9    the following holds: 
     
       
         
           
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     To summarize the modulations for AM QAM and Quat-AM can mathematically be formulated as follows: 
     AM: a 0 (t)·cos ω 0 t 
     QAM: a 0 (t)·cos ω 0 t−a 1 (t)·sin ω 0 t 
     Quat-AM: a 0 (t)·cos ω 0 t·cos ω 1 t+a 1 (t)·cos ω 0 t·sin ω 1 t+a 2 (t)·sin ω 0 t·cos ω 1 t+a 3 (t)·sin ω 0 t·sin ω 1 t=b 0 (t)−cos ω Δ  t−b 1 (t)·sin ω Δ  t+b 2 (t)·cos ω Σ  t−b 3 (t)·sin ω Σ  t 
       FIG.  10    shows a schematic diagram of an embodiment of a Quat-AM transmitter  140  representing a realization of the Quat-AM transmitter shown in  FIG.  8    (i.e. of an embodiment of a first communication device  100  according to the present disclosure), also designated with TX b   ΔΣ . The two QAM modulators  132 ,  133  shown in  FIG.  8    may be realized by multipliers with cosine and (negative) sine waveforms with frequencies ω Δ  and ω Σ , respectively. In the same manner,  FIG.  11    shows a schematic diagram of an embodiment of a Quat-AM receiver  240  representing a realization of the Quat-AM receiver shown in  FIG.  9    (i.e. of an embodiment of a second communication device  200  according to the present disclosure), also designated with RX b   ΔΣ . The two QAM demodulators  232 ,  233  shown in  FIG.  9    may be realized by multipliers with cosine and sine waveforms with frequencies ω Δ  and ω Σ , respectively, followed by low-pass filters (LPF). 
     In the Quat-AM transmitter  140  the output signal y 0 . . . 3  of the precoder  131  are multiplied by cosine and (negative) sine waveforms with frequencies ω Δ  and ω Σ , by multipliers  141 . As introduced above in the description of  FIG.  3   , the following short-hand notation has been used: c v =cos ((ω v t); s v =sin ((ω v t);v ∈ {0,1,Δ,Σ}, e.g., c 0 =cos ((ω 0 t); s Σ =sin (ω Σ t). The results are then stepwise summed up by adders  142 , before the multiplication by 0.5 is carried out by the multiplier  135 . In the Quat-AM receiver  240  the inverse multiplications by cosine and sine waveforms with frequencies ω Δ  and ω Σ , are carried out by multipliers  241 , the results of which being subjected to LPF filtering by filters  216  and inversion by inverter  234 . 
     Quat-AM may alternatively be interpreted as using four orthogonal waveforms, consisting of mixed carriers (four cosine and sine product combinations at frequencies f 0  and f 1 ). A corresponding embodiment of a Quat-AM transmitter  150  (i.e. of an embodiment of a first communication device  100  according to the present disclosure), also designated with TX 01,g , is shown in  FIG.  12   . Corresponding embodiments of a Quat-AM receiver  250 ,  255  (i.e. of embodiments of a second communication device  200  according to the present disclosure), also designated with RX 01,g , and RX 01,MF  are shown in  FIGS.  13  and  14   . 
     In the Quat-AM transmitter  150  shown in  FIG.  12    the input signals x 0 . . . 3  are precoded by a matrix T by a precoder  151 . The output signals ω 0 . . . 3  are each multiplied by a respective orthogonal waveform signal g 0 . . . 3 (t) by multipliers  152 . The results are then stepwise summed up by adders  153  to obtain the output signal s. 
     In the Quat-AM receiver  250  shown in  FIG.  13   , the inverse multiplications by orthogonal waveform signals g 0 . . . 3 (−t) by multipliers  251  are carried out, the results of which being subjected to LPF filtering by filters  216  and inversion by inverter  252  using the inverse matrix T −1 . In the Quat-AM receiver  255  shown in  FIG.  14   , the received signal r multiplied by 4 is subjected to filtering by matched filters MF{g 0 . . . 3 (t)}  256 , the results of which are subject to inversion by inverter  252  using the inverse matrix T −1 . 
     In the embodiments shown in  FIGS.  12 - 14    the following holds: 
     
       
         
           
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     The waveform signals are 
     
       
         
           
             
               
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     with b v   (μ)  being v th  bit of bin2dec(μ), e.g., b 0   (0) =0, b 1   (0) =0, b 0   (1) =0, b 1   (1) =1, b 0   (2) =1 b 1   (2) =0, . . . 
     In more detail: 
         g   0 ( t )=cos(ω 0   t )·cos(ω 1   t )
 
         g   1 ( t )=·cos(ω 0   t )·sin(ω 1   t )
 
         g   2 ( t )=−sin(ω 0   t )·cos(ω 1   t )
 
         g   3 ( t )=sin(ω 0   t )·sin(ω 1   t )
 
     If α=(0 1 0) T ;β=(1 0 0) T →γ=(0 0 1) T , then T=I 4 , i.e., the identity matrix, which means that no precoding is needed. 
     Generally, the multiplication w=Tx means that the four components of w are derived from input x via matrix multiplication. 
       FIGS.  15 - 17    show schematic diagrams of further embodiments of Quat-AM receivers  260 ,  265  and  270  (i.e. of embodiment of a second communication device  200  according to the present disclosure), also designated with RX a   01,q  (receiver  260 ), RX b   01,q  (receiver  265 ), or RX d   01,q  (receiver  270 ). 
     The Quat-AM receivers  260 ,  265 ,  270  shown in  FIGS.  15 - 17    are based on the above described Quat-AM receiver  210  shown in  FIG.  4   , however having pre-calculated the contributions to the projections onto the four dimensions. The received input signal is real-valued, so that the projection onto the four dimensions does only depend on the demodulation kernel D. 
     The Quat-AM receiver  265  multiplies the received signal (multiplied by 4) with the kernel contributions D 0 . . . 3  by multipliers  266 , before the results are subjected to LPF filtering and inversion using the inverse matrix Q −1  by the matrix inverter  217 . 
     The Quat-AM receiver  270  shows a relation to the four orthogonal waveforms” receiver  250  shown in  FIG.  13   . It applies a precoder  271  using a precoding matrix Z after LPF filtering and before matrix inversion. 
     In the embodiments shown in  FIGS.  15 - 17    the following holds: Demodulation kernel D=D 0 +iD 1 +jD 2 +kD 3  with D 0 =c 0 c 1 −s 0 s 1 ϵ and D μ =−s 0 c 1  a μ  −c 0 s 1 β μ +s 0 s 1 y μ  for μ ϵ {1,2,3}. 
     
       
         
           
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     So far, the input signals x 0 . . . 3  have been assumed to be in general analog signals with bandwidth ω c . They are then subjected to precoding by a precoder  131  or  151  as shown in  FIGS.  8 ,  10  and  12   .  FIG.  18    shows a schematic diagram of a first embodiment of a generator  300  of a digital (discrete in both time and value) representation of the input signals x 0 . . . 3 . The source bits are subjected to error correction in a FEC encoder  301 , interleaving in an interleaver  302 , modulation in a modulator  303  and serial-to-parallel conversion in a serial-to-parallel converter  304 .  FIG.  19    shows a schematic diagram of a second embodiment of a generator  300  of a digital representation of the input signals x 0 . . . 3 , according to which each symbol x k  may come from a different modulation scheme since the modulation is applied by modulators  305  per symbol x k  after serial-to-parallel conversion. 
       FIGS.  20  and  21    show schematic diagrams of a transmitter  400  and a receiver  500  as described in IEEE802.11. The above-described disclosure of modulators, in particular Quat-AM modulators, may be applied in two such transmitters  400 , when two different channels (channel bonding or channel aggregation) is used (instead of the independent IQ modulators) and receiver  500  (instead of the independent IQ determination). For conventional channel bonding/aggregation, two independent transmitters and receivers are used. In case of frequency selective fading, one of the channels may suffer severe attenuation or other effects (bursty frequency selective noise). In such cases, the information from the weaker channel is usually lost and needs to be retransmitted. In contrast, the modulators as disclosed herein combine the signals to be transmitted over two different channels. In the embodiment shown in  FIG.  8    this is visualized by precoder matrix P. Thus, a weak channel (at angular carrier frequency ω Δ  or ω Σ ) will affect both complex signals (or all four real-valued signals x 0 . . . 3 ,  130 ). The detrimental effect of selective fading is mitigated by averaging with the other less weaker channel. This frequency diversity effect is one of the main benefits of the present disclosure, resulting in diversity gains of up to 8 dB (as confirmed by simulation). 
     According to the present disclosure frequency diversity can be exploited by the novel modulation, in particular the novel Quat-AM modulation, since four independent signals are spread over two bands. Further, depending on the choice of the modulating quaternions  α  and  β , new four-dimensional constellations can be formed. Analyzing the superimposed QAM structure, e.g., shows that the resulting constellations can form two-dimensional non-uniform constellations (NUCs), one NUC per QAM. For particular settings, even probabilistic amplitude shaping (PAS) can result from the Quat-AM structure.  α =i and β=j may be chosen, resulting in a mutated QAM for the upper QAM branch (operating at ω Δ ), in which points occur with different probabilities. Even though ambiguities arise, they are resolved with the second QAM. In simulations over fading channels, diversity gains up to 8 . . . 9 dB have already been demonstrated. 
     Thus, the foregoing discussion discloses and describes merely exemplary embodiments of the present disclosure. As will be understood by those skilled in the art, the present disclosure may be embodied in other specific forms without departing from the spirit or essential characteristics thereof. Accordingly, the disclosure of the present disclosure is intended to be illustrative, but not limiting of the scope of the disclosure, as well as other claims. The disclosure, including any readily discernible variants of the teachings herein, defines, in part, the scope of the foregoing claim terminology such that no inventive subject matter is dedicated to the public. 
     In the claims, the word “comprising” does not exclude other elements or steps, and the indefinite article “a” or “an” does not exclude a plurality. A single element or other unit may fulfill the functions of several items recited in the claims. The mere fact that certain measures are recited in mutually different dependent claims does not indicate that a combination of these measures cannot be used to advantage. 
     In so far as embodiments of the disclosure have been described as being implemented, at least in part, by software-controlled data processing apparatus, it will be appreciated that a non-transitory machine-readable medium carrying such software, such as an optical disk, a magnetic disk, semiconductor memory or the like, is also considered to represent an embodiment of the present disclosure. Further, such a software may also be distributed in other forms, such as via the Internet or other wired or wireless telecommunication systems. 
     The elements of the disclosed devices, apparatus and systems may be implemented by corresponding hardware and/or software elements, for instance appropriated circuits or circuitry. A circuit is a structural assemblage of electronic components including conventional circuit elements, integrated circuits including application specific integrated circuits, standard integrated circuits, application specific standard products, and field programmable gate arrays. Further, a circuit includes central processing units, graphics processing units, and microprocessors which are programmed or configured according to software code. A circuit does not include pure software, although a circuit includes the above-described hardware executing software. A circuit or circuitry may be implemented by a single device or unit or multiple devices or units, or chipset(s), or processor(s). 
     It follows a list of further embodiments of the disclosed subject matter: 
     1. Communication device comprising circuitry configured to
         modulate a four-dimensional input signal by combining the four real-valued signal components of the input signal into a transformed signal and multiplying the transformed signal by carrier signals using two different carrier frequencies to obtain a transmit signal, and   transmit the transmit signal.       

     2. Communication device as defined in embodiment 1, wherein the circuitry is configured to combine the four real-valued signal components of the input signal into an input quaternion as transformed signal. 
     3. Communication device as defined in embodiment 2, wherein the circuitry is configured to interpret one of the four real-valued signal components as the real part of a quaternion-based transformed signal and to interpret the remaining three of the four real-valued signal components as the pure quaternion part of said quaternion-based transformed signal. 
     4. Communication device as defined in embodiment 2 or 3, wherein the circuitry is configured to multiply the transformed signal by a first quaternion phasor, as first carrier signal, using a first carrier frequency and to multiply the result of the first multiplication by a second quaternion phasor, as second carrier signal, using a second carrier frequency. 
     5. Communication device as defined in embodiment 4, wherein the circuitry is configured to multiply the transformed signal from one side by the first quaternion phasor and to multiply the result of the first multiplication from the other side by the second quaternion phasor. 
     6. Communication device as defined in embodiment 4 or 5, wherein the circuitry is configured to take a projection of the result of the second multiplication to any one of the four dimensions as transmit signal. 
     7. Communication device as defined in any one of the preceding embodiments, wherein the circuitry is configured to
         multiply the four-dimensional input signal by a four-dimensional precoding matrix to obtain a four-dimensional precoded signal,   modulate the first two signal values of the four-dimensional precoded signal by a first QAM modulator using a frequency difference between the first carrier frequency and the second carrier frequency and the second two signal values of the four-dimensional precoded signal by a second QAM modulator using a frequency sum of the first carrier frequency and the second carrier frequency, and   add the outputs of the first and second QAM modulators to obtain the transmit signal.       

     8. Communication device as defined in embodiment 7, wherein the circuitry is configured to use as four-dimensional precoding matrix the matrix 
     
       
         
           
             
               
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     and wherein α and β are quaternions from first and second quaternion phasors, respectively. 
     9. Communication device as defined in embodiment 8, wherein α and β are not parallel. 
     10. Communication device as defined in any one of the preceding embodiments, wherein the circuitry is configured to
         multiply the four-dimensional input signal by a four-dimensional precoding matrix to obtain a four-dimensional precoded signal,   multiply each of the four signal values of the four-dimensional precoded signal by a different waveform signal, and   add the output of second multiplication to obtain the transmit signal.       

     11. Communication device as defined in embodiment 10, wherein the four waveform signals used for multiplying each of the four signal values of the four-dimensional precoded signal are four combinations of products of cosine and sine carriers at the two carrier frequencies. 
     12. Communication device as defined in embodiment 10, wherein the circuitry is configured to use as four-dimensional precoding matrix the matrix 
     
       
         
           
             
               
                 T 
                 = 
                 
                   
                     ( 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             β 
                             1 
                           
                         
                         
                           
                             β 
                             2 
                           
                         
                         
                           
                             β 
                             3 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             α 
                             1 
                           
                         
                         
                           
                             α 
                             2 
                           
                         
                         
                           
                             α 
                             3 
                           
                         
                       
                       
                         
                           
                             - 
                             ϵ 
                           
                         
                         
                           
                             γ 
                             1 
                           
                         
                         
                           
                             γ 
                             2 
                           
                         
                         
                           
                             γ 
                             2 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           1 
                         
                         
                           
                             0 
                             T 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             β 
                             T 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             α 
                             T 
                           
                         
                       
                       
                         
                           
                             - 
                             ϵ 
                           
                         
                         
                           
                             γ 
                             T 
                           
                         
                       
                     
                     ) 
                   
                 
               
               , 
               
 
               
                 
                   
                     wherein 
                     ⁢ 
                         
                     α 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           α 
                           1 
                         
                         ⁢ 
                         
                           α 
                           2 
                         
                         ⁢ 
                         
                           α 
                           3 
                         
                       
                       ) 
                     
                     T 
                   
                 
                 ; 
               
             
             ⁢ 
             
 
             
               
                 β 
                 = 
                 
                   
                     ( 
                     
                       
                         β 
                         1 
                       
                       ⁢ 
                       
                         β 
                         2 
                       
                       ⁢ 
                       
                         β 
                         3 
                       
                     
                     ) 
                   
                   T 
                 
               
               ; 
             
             ⁢ 
             
 
             
               
                 γ 
                 = 
                 
                   
                     
                       ( 
                       
                         
                           γ 
                           1 
                         
                         ⁢ 
                         
                           γ 
                           2 
                         
                         ⁢ 
                         
                           γ 
                           3 
                         
                       
                       ) 
                     
                     T 
                   
                   = 
                   
                     α 
                     × 
                     β 
                   
                 
               
               ; 
             
             ⁢ 
             
 
             
               
                 ϵ 
                 = 
                 
                   
                     〈 
                     
                       α 
                       , 
                       β 
                     
                     〉 
                   
                   = 
                   
                     
                       ∑ 
                       
                         v 
                         = 
                         1 
                       
                       3 
                     
                     
                       
                         α 
                         v 
                       
                       ⁢ 
                       
                         β 
                         v 
                       
                     
                   
                 
               
               ; 
             
             ⁢ 
             
 
             
               
                 
                   
                     ❘ 
                     &#34;\[LeftBracketingBar]&#34; 
                   
                   α 
                   
                     ❘ 
                     &#34;\[RightBracketingBar]&#34; 
                   
                 
                 = 
                 
                   
                     
                       ❘ 
                       &#34;\[LeftBracketingBar]&#34; 
                     
                     β 
                     
                       ❘ 
                       &#34;\[RightBracketingBar]&#34; 
                     
                   
                   = 
                   1 
                 
               
               , 
             
           
         
       
     
     and wherein α and β are quaternions from first and second quaternion phasors, respectively. 
     13. Communication device as defined in embodiment 10, wherein the circuitry is configured to use as carrier signals products of cosine and sine carriers at the two carrier frequencies. 
     14. Communication device as defined in any one of the preceding embodiments, wherein the two different carrier frequencies are larger than a cutoff frequency of each of the four real-valued signal components of the input signal. 
     15. Communication device as defined in any one of the preceding embodiments, wherein the absolute value of the difference of the two different carrier frequencies is larger than the cutoff frequency each of the four real-valued signal components of the input signal. 
     16. Communication device as defined in any one of the preceding embodiments, wherein the difference of the two different carrier frequencies is zero. 
     17. Communication device as defined in any one of the preceding embodiments, wherein the circuitry is configured to apply a quaternary modulation of four input signals to generate a real-valued output signal based on a transformation, which is fully defined by two three-dimensional vectors α=(α 1 α 2 α 3 ) T  and β=(β 1 β 2 β 3 ) T . 
     18. Communication device as defined in any one of the preceding embodiments, wherein the circuitry is configured to perform the modulation by quaternion-based multiplications by treating the input signal as input quaternion, multiplying said input quaternion from left-side with quaternion e   α ω     0     t , multiplying the result from right-side with quaternion e   β ω     1     t , and projecting the result to any of the four dimensions. 
     19. Communication device as defined in any one of the preceding embodiments, wherein the circuitry is configured to perform the modulation by linear transformation of four input signals with a P matrix, 
     wherein said P matrix is defined as 
     
       
         
           
             
               P 
               = 
               
                 
                   ( 
                   
                     
                       
                         
                           1 
                           - 
                           ϵ 
                         
                       
                       
                         
                           γ 
                           1 
                         
                       
                       
                         
                           γ 
                           2 
                         
                       
                       
                         
                           γ 
                           3 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         
                           
                             - 
                             
                               α 
                               1 
                             
                           
                           + 
                           
                             β 
                             1 
                           
                         
                       
                       
                         
                           
                             - 
                             
                               α 
                               2 
                             
                           
                           + 
                           
                             β 
                             2 
                           
                         
                       
                       
                         
                           
                             - 
                             
                               α 
                               3 
                             
                           
                           + 
                           
                             β 
                             3 
                           
                         
                       
                     
                     
                       
                         
                           1 
                           + 
                           ϵ 
                         
                       
                       
                         
                           - 
                           
                             γ 
                             1 
                           
                         
                       
                       
                         
                           - 
                           
                             γ 
                             2 
                           
                         
                       
                       
                         
                           - 
                           
                             γ 
                             3 
                           
                         
                       
                     
                     
                       
                         0 
                       
                       
                         
                           
                             - 
                             
                               α 
                               1 
                             
                           
                           - 
                           
                             β 
                             1 
                           
                         
                       
                       
                         
                           
                             - 
                             
                               α 
                               2 
                             
                           
                           - 
                           
                             β 
                             2 
                           
                         
                       
                       
                         
                           
                             - 
                             
                               α 
                               3 
                             
                           
                           - 
                           
                             β 
                             3 
                           
                         
                       
                     
                   
                   ) 
                 
                 = 
                 
                   ( 
                   
                     
                       
                         
                           1 
                           - 
                           ϵ 
                         
                       
                       
                         
                           γ 
                           T 
                         
                       
                     
                     
                       
                         0 
                       
                       
                         
                           
                             - 
                             
                               α 
                               T 
                             
                           
                           + 
                           
                             β 
                             T 
                           
                         
                       
                     
                     
                       
                         
                           1 
                           + 
                           ϵ 
                         
                       
                       
                         
                           - 
                           
                             γ 
                             T 
                           
                         
                       
                     
                     
                       
                         0 
                       
                       
                         
                           
                             - 
                             
                               α 
                               T 
                             
                           
                           - 
                           
                             β 
                             T 
                           
                         
                       
                     
                   
                   ) 
                 
               
             
             , 
           
         
       
     
     and
 
wherein the result of the transformation is processed by two QAM modulation schemes with carrier frequencies ωΣ=ω 0 +ω 1  and ω Δ =ω 0 −ω 1 .
 
     20. Communication device comprising circuitry configured to
         receive a transmit signal, and   demodulate the transmit signal into a four-dimensional output signal by multiplying the transmit signal by carrier signals using two different carrier frequencies to obtain a retransformed signal and decomposing the retransformed signal into the four real-valued signal components of the output signal.       

     21. Communication device as defined in embodiment 20, wherein the circuitry is configured to decompose an output quaternion as retransformed signal into the four real-valued signal components of the output signal. 
     22. Communication device as defined in embodiment 21, wherein the circuitry is configured to interpret one of the four real-valued signal components as the real part of a quaternion-based retransformed signal and to interpret the remaining three of the four real-valued signal components as the pure quaternion part of said quaternion-based retransformed signal. 
     23. Communication device as defined in embodiment 21 or 22, wherein the circuitry is configured to multiply the transmit signal by a first quaternion phasor, as first carrier signal, using a first carrier frequency and to multiply the result of the first multiplication by a second quaternion phasor, as second carrier signal, using a second carrier frequency. 
     24. Communication device as defined in embodiment 23, wherein the circuitry is configured to multiply the transmit signal from one side by the first quaternion phasor and to multiply the result of the first multiplication from the other side by the second quaternion phasor. 
     25. Communication device as defined in any one of embodiments 20 to 24, wherein the circuitry is configured to
         demodulate the transmit signal by a first QAM demodulator using a frequency difference between the first carrier frequency and the second carrier frequency and by a second QAM demodulator using a frequency sum of the first carrier frequency and the second carrier frequency, and   multiply the output of the first and second QAM demodulator by the inverse of a four-dimensional precoding matrix to obtain a four-dimensional output signal.       

     26. Communication device as defined in embodiment 25, wherein the circuitry is configured to use as four-dimensional precoding matrix the matrix 
     
       
         
           
             
               
                 P 
                 = 
                 
                   
                     ( 
                     
                       
                         
                           
                             1 
                             - 
                             ϵ 
                           
                         
                         
                           
                             γ 
                             1 
                           
                         
                         
                           
                             γ 
                             2 
                           
                         
                         
                           
                             γ 
                             3 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             
                               - 
                               
                                 α 
                                 1 
                               
                             
                             + 
                             
                               β 
                               1 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 α 
                                 2 
                               
                             
                             + 
                             
                               β 
                               2 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 α 
                                 3 
                               
                             
                             + 
                             
                               β 
                               3 
                             
                           
                         
                       
                       
                         
                           
                             1 
                             + 
                             ϵ 
                           
                         
                         
                           
                             - 
                             
                               γ 
                               1 
                             
                           
                         
                         
                           
                             - 
                             
                               γ 
                               2 
                             
                           
                         
                         
                           
                             - 
                             
                               γ 
                               3 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             
                               - 
                               
                                 α 
                                 1 
                               
                             
                             - 
                             
                               β 
                               1 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 α 
                                 2 
                               
                             
                             - 
                             
                               β 
                               2 
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 α 
                                 3 
                               
                             
                             - 
                             
                               β 
                               3 
                             
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             1 
                             - 
                             ϵ 
                           
                         
                         
                           
                             γ 
                             T 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             
                               - 
                               
                                 α 
                                 T 
                               
                             
                             + 
                             
                               β 
                               T 
                             
                           
                         
                       
                       
                         
                           
                             1 
                             + 
                             ϵ 
                           
                         
                         
                           
                             - 
                             
                               γ 
                               T 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             
                               - 
                               
                                 α 
                                 T 
                               
                             
                             - 
                             
                               β 
                               T 
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               , 
               
 
               
                 
                   
                     wherein 
                     ⁢ 
                         
                     α 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           α 
                           1 
                         
                         ⁢ 
                         
                           α 
                           2 
                         
                         ⁢ 
                         
                           α 
                           3 
                         
                       
                       ) 
                     
                     T 
                   
                 
                 ; 
               
             
             ⁢ 
             
 
             
               
                 β 
                 = 
                 
                   
                     ( 
                     
                       
                         β 
                         1 
                       
                       ⁢ 
                       
                         β 
                         2 
                       
                       ⁢ 
                       
                         β 
                         3 
                       
                     
                     ) 
                   
                   T 
                 
               
               ; 
             
             ⁢ 
             
 
             
               
                 γ 
                 = 
                 
                   
                     
                       ( 
                       
                         
                           γ 
                           1 
                         
                         ⁢ 
                         
                           γ 
                           2 
                         
                         ⁢ 
                         
                           γ 
                           3 
                         
                       
                       ) 
                     
                     T 
                   
                   = 
                   
                     α 
                     × 
                     β 
                   
                 
               
               ; 
             
             ⁢ 
             
 
             
               
                 ϵ 
                 = 
                 
                   
                     〈 
                     
                       α 
                       , 
                       β 
                     
                     〉 
                   
                   = 
                   
                     
                       ∑ 
                       
                         v 
                         = 
                         1 
                       
                       3 
                     
                     
                       
                         α 
                         v 
                       
                       ⁢ 
                       
                         β 
                         v 
                       
                     
                   
                 
               
               ; 
             
             ⁢ 
             
 
             
               
                 
                   
                     ❘ 
                     &#34;\[LeftBracketingBar]&#34; 
                   
                   α 
                   
                     ❘ 
                     &#34;\[RightBracketingBar]&#34; 
                   
                 
                 = 
                 
                   
                     
                       ❘ 
                       &#34;\[LeftBracketingBar]&#34; 
                     
                     β 
                     
                       ❘ 
                       &#34;\[RightBracketingBar]&#34; 
                     
                   
                   = 
                   1 
                 
               
               , 
             
           
         
       
     
     and wherein α and β are quaternions from first and second quaternion phasors, respectively. 
     26. Communication device as defined in embodiment 25, wherein α and β are not parallel. 
     27. Communication device as defined in any one of the embodiments 20 to 26, wherein the circuitry is configured to
         multiply the transmit signal, in parallel, by each four different waveform signals, and   multiply the result by the inverse of a four-dimensional precoding matrix to obtain a four-dimensional output signal.       

     28. Communication device as defined in embodiment 27, wherein the four waveform signals are four combinations of products of cosine and sine carriers at the two carrier frequencies. 
     29. Communication device as defined in embodiment 27 or 28, wherein the circuitry is configured to use as four-dimensional precoding matrix the matrix 
     
       
         
           
             
               
                 T 
                 = 
                 
                   
                     ( 
                     
                       
                         
                           1 
                         
                         
                           0 
                         
                         
                           0 
                         
                         
                           0 
                         
                       
                       
                         
                           0 
                         
                         
                           
                             β 
                             1 
                           
                         
                         
                           
                             β 
                             2 
                           
                         
                         
                           
                             β 
                             3 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             α 
                             1 
                           
                         
                         
                           
                             α 
                             2 
                           
                         
                         
                           
                             α 
                             3 
                           
                         
                       
                       
                         
                           
                             - 
                             ϵ 
                           
                         
                         
                           
                             γ 
                             1 
                           
                         
                         
                           
                             γ 
                             2 
                           
                         
                         
                           
                             γ 
                             2 
                           
                         
                       
                     
                     ) 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           1 
                         
                         
                           
                             0 
                             T 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             β 
                             T 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           
                             α 
                             T 
                           
                         
                       
                       
                         
                           
                             - 
                             ϵ 
                           
                         
                         
                           
                             γ 
                             T 
                           
                         
                       
                     
                     ) 
                   
                 
               
               , 
               
 
               
                 
                   
                     wherein 
                     ⁢ 
                         
                     α 
                   
                   = 
                   
                     
                       ( 
                       
                         
                           α 
                           1 
                         
                         ⁢ 
                         
                           α 
                           2 
                         
                         ⁢ 
                         
                           α 
                           3 
                         
                       
                       ) 
                     
                     T 
                   
                 
                 ; 
               
             
             ⁢ 
             
 
             
               
                 β 
                 = 
                 
                   
                     ( 
                     
                       
                         β 
                         1 
                       
                       ⁢ 
                       
                         β 
                         2 
                       
                       ⁢ 
                       
                         β 
                         3 
                       
                     
                     ) 
                   
                   T 
                 
               
               ; 
             
             ⁢ 
             
 
             
               
                 γ 
                 = 
                 
                   
                     
                       ( 
                       
                         
                           γ 
                           1 
                         
                         ⁢ 
                         
                           γ 
                           2 
                         
                         ⁢ 
                         
                           γ 
                           3 
                         
                       
                       ) 
                     
                     T 
                   
                   = 
                   
                     α 
                     × 
                     β 
                   
                 
               
               ; 
             
             ⁢ 
             
 
             
               
                 ϵ 
                 = 
                 
                   
                     〈 
                     
                       α 
                       , 
                       β 
                     
                     〉 
                   
                   = 
                   
                     
                       ∑ 
                       
                         v 
                         = 
                         1 
                       
                       3 
                     
                     
                       
                         α 
                         v 
                       
                       ⁢ 
                       
                         β 
                         v 
                       
                     
                   
                 
               
               ; 
             
             ⁢ 
             
 
             
               
                 
                   
                     ❘ 
                     &#34;\[LeftBracketingBar]&#34; 
                   
                   α 
                   
                     ❘ 
                     &#34;\[RightBracketingBar]&#34; 
                   
                 
                 = 
                 
                   
                     
                       ❘ 
                       &#34;\[LeftBracketingBar]&#34; 
                     
                     β 
                     
                       ❘ 
                       &#34;\[RightBracketingBar]&#34; 
                     
                   
                   = 
                   1 
                 
               
               , 
             
           
         
       
     
     and wherein α and β are quaternions from first and second quaternion phasors, respectively. 
     30. Communication device as defined in embodiment 27, 28 or 29, wherein the circuitry is configured to use as carrier signals products of cosine and sine carriers at the two carrier frequencies. 
     31. Communication device as defined in any one of the preceding embodiments 20 to 30, 
     wherein the two different carrier frequencies are larger than a cutoff frequency of each of the four real-valued signal components of the input signal. 
     32. Communication device as defined in any one of the embodiments 20 to 31, wherein the absolute value of the difference of the two different carrier frequencies is larger than the cutoff frequency each of the four real-valued signal components of the input signal. 
     33. Communication device as defined in any one of the preceding embodiments 20 to 32, 
     wherein the difference of the two different carrier frequencies is zero. 
     34. Communication device as defined in embodiment 26, wherein the result of the demodulation is processed by two QAM modulation schemes with carrier frequencies ω Σ =ω 0 +ω 1  and ω Δ =ω 0 −ω 1 . 
     35. Communication method comprising:
         modulating a four-dimensional input signal by combining the four real-valued signal components of the input signal into a transformed signal and multiplying the transformed signal by carrier signals using two different carrier frequencies to obtain a transmit signal, and   transmitting the transmit signal.       

     36. Communication method comprising:
         receiving a transmit signal, and   demodulating the transmit signal into a four-dimensional output signal by multiplying the transmit signal by carrier signals using two different carrier frequencies to obtain a retransformed signal and decomposing the retransformed signal into the four real-valued signal components of the output signal.       

     37. A non-transitory computer-readable recording medium that stores therein a computer program product, which, when executed by a processor, causes the method according to embodiment 35 or 36 to be performed. 
     38. A computer program comprising program code means for causing a computer to perform the steps of said method according to embodiment 35 or 36 when said computer program is carried out on a computer.