Patent Publication Number: US-8995295-B2

Title: Using maximal sum-rate mutual information to optimize JCMA constellations

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a U.S. National Phase Application under 35 U.S.C. 371 of PCT International Application No. PCT/IB2010/054991 which has an international filing date of Nov. 4, 2010, and which claims priority benefit of U.S. Provisional Patent Application No. 61/258,328, filed Nov. 5, 2009, which is incorporated herein by reference. 
    
    
     FIELD AND OF THE INVENTION 
     The present invention generally relates to communication systems and methods, and, more particularly, but not exclusively to methods and systems for multiple access communications. 
     BACKGROUND OF THE INVENTION 
     The proliferation of wireless communication interfaces in mobile and stationary devices rendered the frequency spectrum scarce. Efficient use of allocated bandwidth for communication is a critical issue for the viability of wireless communication systems. 
     Frequency Division Multiple Access (FDMA) is a widely used method for allowing multiple transmitters access to a common wireless channel. An example is cellular telephony where multiple mobile users access the same base station. Specifically, FDMA was recently adopted for uplink in the Long Term Evolution (LTE) extension of the Universal Mobile Telecommunication Systems (UMTS) standard. 
     The use of Joint Constellation Multiple Access (JCMA) is known. US patent application 20090135926 is believed to represent the most relevant prior art. US patent application 20090135926 is incorporated herein by reference 
     The following documents are believed to represent further relevant prior art:
         J. G. Proakis, Digital Communication, McGraw Hill, 3rd edition, 1995.   T. M. Cover and J. Thomas. Elements of Information Theory. John Wiley and Sons Inc., second edition, 2006.   A. Goldsmith, Wireless Communications, Cambridge University Press, 2006.   C. Johnson, Radio Access Networks for UMTS, Wiley, 2008.   G. R. Tsouri and D. Wulich, “Securing OFDM over Wireless Time-Varying Channels using Sub-Carrier Over-Loading with Joint Signal Constellations”, Eurasip Journal on Wireless Communications &amp; Networking, 2009.       

     In the classical FDMA approach the total frequency spectrum is divided into sub-bands. The receiver allocates a sub-band to each transmitter, and each transmitter has exclusive use of its allocated sub-band. This converts the allocated total bandwidth into a set of orthogonal channels connecting the transmitters and receiver. 
     JCMA suggests coupling N multiple transmitters under a Superposition Modulation (SM) scheme. This allows the joining of their multiple sub-bands to a single sub-band N times larger than an original sub-band. The large bandwidth would allow transmission at a rate N times larger than the original rate. However, the energy per transmitted symbol would decrease by a factor of N as well, resulting in an increase of the Bit Error Rate (BER) upon decoding at the receiver. The increase in BER would be mitigated by applying source coding prior to transmission and by synchronizing the transmitters to form a superimposed constellation which is robust to noise at the receiver. Applying coding would result in throughput decrease. If the decrease in throughput due to coding is less than the increase in transmission rate due to bandwidth expansion, the throughput with synchronized SM would be larger than the throughput of classical FDMA. 
     As described above, JCMA is based on synchronous Superposition Modulation (SM) and coding for increasing the spectral efficiency of practical FDMA systems. SM is known to achieve maximal aggregated capacity. However, SM is largely avoided under the premise that capacity increase offered by SM is not substantial compared to the complexity involved in obtaining accurate synchronization and power control required for its operation. 
     There is thus a recognized need for, and it would be highly advantageous to have, a communication system, and particularly a JCMA communication system having increased capacity, preferably by optimizing the immunity of the constellation used to SNR. 
     SUMMARY OF THE INVENTION 
     According to one aspect of the present invention there is provided a method for calculating a JCMA constellation for use in a JCMA communication system where the method includes a step of using maximum sum-rate mutual information criterion. 
     According to another aspect of the present invention there is provided a method for calculating a JCMA constellation including calculating the maximum mutual information according to: 
                 I   ⁡     (     X   ;   Y     )       =       ∫     x   ∈   X               ⁢       ∫     y   ∈   Y               ⁢       p   ⁡     (     x   ,   y     )       ⁢     log   2     ⁢       p   ⁡     (     x   ,   y     )           p   ⁡     (   x   )       ⁢     p   ⁡     (   y   )           ⁢     ⅆ   x     ⁢     ⅆ   y             ,         
where X is a transmitter transmitting at least a part of the JCMA constellation and Y is a receiver receiving the JCMA constellation, where I(X, Y) is the mutual information function, and where p(x, y) is the joint probability density function for X and Y, and p(·) is the marginal probability density functions for X and Y respectively.
 
     According to still another aspect of the present invention there is provided a method for calculating a JCMA constellation including calculating the maximum sum-rate mutual information according to: 
                 [     P   ,   θ     ]     =     arg   ⁢           ⁢       max         P   n     ∈     [     0   ,   1     ]       ,       θ   n     ∈     [     0   ,     180   ⁢   °       ]       ,     n   =   1     ,   2   ,   …   ,   N       ⁢     I   ⁡     (       X   1     ,     X   2     ,   …   ⁢           ,       X   N     ;   Y       )             ,         
where N is the number of transmitters using the constellation, where P n  is the average transmission power for transmitter n, where θ n  right rotation of the constellation for transmitter n, where I is the mutual information function, where X n  is the transmitted constellation for transmitter n, and where Y is the receiver.
 
     According to yet another aspect of the present invention there is provided a method for calculating a JCMA constellation including selecting the JCMA constellation from a plurality of JCMA constellations where for the selected JCMA constellation the value of S is maximal, where S is given by: 
                 S   ⁡     [         x   _       (   1   )       ,       x   _       (   2   )       ,   …   ⁢           ,       x   _       (   N   )         ]       =       ∑     n   =   1     N     ⁢       I   n     ⁡     (       x   _       (   n   )       )           ,         
where
 
                   I   n     ⁡     (       x   _       (   n   )       )       =       ∑     i   =   1     M     ⁢       ∫     -   ∞     ∞     ⁢       p   ⁡     (     y   /     x   i     (   n   )         )       ⁢     P   ⁡     (     x   i     (   n   )       )       ⁢     log   2     ⁢       p   ⁡     (     y   /     x   i     (   n   )         )         p   ⁡     (   y   )         ⁢           ⁢     ⅆ   y             ,         
and where
 
     
       
         
           
             
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     Further according to another aspect of the present invention there is provided a method for calculating a JCMA constellation where the constellation calculated using the maximum sum-rate mutual information criterion provides at least one of: optimal spectral efficiency, optimal bit-rate, and optimal transmission power. 
     Still further according to another aspect of the present invention there is provided a method for calculating a JCMA constellation the method additionally including the steps of: using the maximum sum-rate mutual information criterion, calculating a plurality of JCMA constellations, where each constellation is optimal for a combination including at least two of: a number of transmitters participating in the JCMA constellation, a value of Signal-to-Noise Ratio (SNR) at a receiver of the JCMA constellation, and at least one modulation scheme. 
     Yet further according to another aspect of the present invention there is provided a method for calculating a JCMA constellation, the method additionally including the step of distributing the plurality of JCMA constellations to at least one of a receiver and a transmitter. 
     Even further according to another aspect of the present invention there is provided a method for calculating a JCMA constellation, the method including the steps of: at a receiver of the JCMA constellation performing the steps of: selecting a plurality of transmitters to form a group of transmitters for jointly transmitting a JCMA constellation, measuring the combined SNR value for the group of transmitters, selecting a JCMA constellation from the plurality of JCMA constellations according to at least two of: the number N of transmitters in the group of transmitters, the SNR value, and at least one modulation scheme, and informing the selected constellation the to the group of transmitters. 
     Additionally according to another aspect of the present invention there is provided a method for calculating a JCMA constellation where the step of selecting a JCMA constellation includes calculating the JCMA constellation using the maximum sum-rate mutual information criterion at the receiver. 
     Also according to another aspect of the present invention there is provided a method for calculating a JCMA constellation where the step of selecting a JCMA constellation includes the steps of: calculating a plurality of the JCMA constellations offline, using the maximum sum-rate mutual information criterion, distributing the plurality of JCMA constellations to at least one of a receiver and a transmitter, and in at least one of the receiver and the transmitter, selecting a JCMA constellation from the plurality of JCMA constellations according to at least two of: a number N of transmitters in a group of transmitters, the SNR value, and at least one modulation scheme. 
     Further according to another aspect of the present invention there is provided a method for calculating a JCMA constellation, the method including the steps of: receiving a constellation from the receiver, receiving a pilot signal from the receiver, calculating at a transmitter of the group of transmitters at least one of transmission power P and constellation rotation angle θ according to the pilot signal, and transmitting at least one JCMA constellation component to the receiver using the P and the θ. 
     Still further according to another aspect of the present invention there is provided a method for calculating a JCMA constellation, where the receiver is at least one of a base-station, an access point, a satellite, and a satellite ground station. 
     Yet further according to another aspect of the present invention there is provided a communication system using the method for calculating a JCMA constellation according to the maximal sum-rate mutual information criterion. 
     Even further according to another aspect of the present invention there is provided a receiver in a JCMA communication system using the method for calculating a JCMA constellation according to the maximal sum-rate mutual information criterion. 
     Additionally according to another aspect of the present invention there is provided a receiver in a JCMA communication system where the receiver is at least one of a base-station, an access point, a satellite, and a satellite ground station. 
     Also according to another aspect of the present invention there is provided a transmitter in a JCMA communication system using the method for calculating a JCMA constellation according to the maximal sum-rate mutual information criterion. 
     Further according to another aspect of the present invention there is provided a method of communication including the steps of: sending a pilot signal from a receiver to a group of transmitters, receiving the pilot signal by at least one of the receivers, calculating power level and rotation angle in the transmitter according to the pilot signal, and sending a data signal from the transmitter to the receiver using the calculated power level and rotation angle, where the pilot signal is transmitted at a power level calculated at the receiver according to a number of transmitters in the group of transmitters. 
     Yet further according to another aspect of the present invention there is provided a receiver in a communication system using the method of communication including the steps of: sending a pilot signal from a receiver to a group of transmitters, receiving the pilot signal by at least one of the receivers, calculating power level and rotation angle in the transmitter according to the pilot signal, and sending a data signal from the transmitter to the receiver using the calculated power level and rotation angle, where the pilot signal is transmitted at a power level calculated at the receiver according to a number of transmitters in the group of transmitters. 
     Yet further according to another aspect of the present invention there is provided a receiver in a communication system using the method of communication including the steps of: sending a pilot signal from a receiver to a group of transmitters, receiving the pilot signal by at least one of the receivers, calculating power level and rotation angle in the transmitter according to the pilot signal, and sending a data signal from the transmitter to the receiver using the calculated power level and rotation angle, where the pilot signal is transmitted at a power level calculated at the receiver according to a number of transmitters in the group of transmitters, and where the receiver is at least one of a base-station, an access point, a satellite, and a satellite ground station. 
     Even further according to another aspect of the present invention there is provided a non-transitory computer readable media including JCMA constellations calculated using maximum sum-rate mutual information criterion. 
     Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention belongs. The materials, methods, and examples provided herein are illustrative only and not intended to be limiting. Except to the extend necessary or inherent in the processes themselves, no particular order to steps or stages of methods and processes described in this disclosure, including the figures, is intended or implied. In many cases the order of process steps may vary without changing the purpose or effect of the methods described. 
     Implementation of the method and system of the present invention involves performing or completing certain selected tasks or steps manually, automatically, or any combination thereof. Moreover, according to actual instrumentation and equipment of preferred embodiments of the method and system of the present invention, several selected steps could be implemented by hardware or by software on any operating system of any firmware or any combination thereof. For example, as hardware, selected steps of the invention could be implemented as a chip or a circuit. As software, selected steps of the invention could be implemented as a plurality of software instructions being executed by a computer using any suitable operating system. In any case, selected steps of the method and system of the invention could be described as being performed by a data processor, such as a computing platform for executing a plurality of instructions. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The invention is herein described, by way of example only, with reference to the accompanying drawings. With specific reference now to the drawings in detail, it is stressed that the particulars shown are by way of example and for purposes of illustrative discussion of the embodiments of the present invention only, and are presented in order to provide what is believed to be the most useful and readily understood description of the principles and conceptual aspects of the invention. In this regard, no attempt is made to show structural details of the invention in more detail than is necessary for a fundamental understanding of the invention, the description taken with the drawings making apparent to those skilled in the art how the several forms of the invention may be embodied in practice. 
       In the drawings: 
         FIG. 1  is a block diagram of a generalized JCMA system incorporating a Max-I constellation calculator; 
         FIG. 2  is a simplified graphical description of a synchronized superimposed constellation; 
         FIG. 3  is a simplified graphical representation of a BER to SNR dependency for the scenario of  FIG. 2 ; 
         FIG. 4  is a simplified block diagram of a JCMA receiver, a JCMA transmitter, and a Max-I JCMA constellation calculator; 
         FIG. 5  is a simplified illustration of a JCMA-based cellular communication system incorporating a Max-I constellation calculator; 
         FIG. 6  is a simplified block diagram of a process executed by the JCMA-based cellular communication system of  FIG. 5 ; 
         FIG. 7  is a simplified flow chart of a process for calculating a JCMA constellation using Max-I; 
         FIGS. 8   a ,  8   b ,  8   c , and  8   d , which are simplified illustrations of optimal superimposed (JCMA) constellations for N=3, BPSK and SNR values of 5 dB, 10 dB, 12 dB, and 15 dB, respectively; 
         FIGS. 9   a ,  9   b , and  9   c , which are simplified illustrations of optimal superimposed (JCMA) constellations for N=4, BPSK and SNR values of 10 dB, 12 dB, and 15 dB, respectively; 
         FIGS. 10   a ,  10   b ,  10   c , and  FIG. 10   d , which are simplified illustrations of optimal superimposed (JCMA) constellations for N=5, BPSK (2QAM) and SNR values of 2 dB, 5 dB, 10 dB, and 16 dB, respectively; 
         FIG. 11  is a simplified illustration of maximum sum-rate spectral efficiency vs. SNR for various Max-I based JCMA constellations; 
         FIGS. 12   a , and  12   b , which are simplified illustrations of optimal superimposed (JCMA) constellations for N=2, QPSK and SNR values of 15 dB and 20 dB, respectively; 
         FIGS. 13   a ,  13   b , and  13   c , which are simplified illustrations of optimal superimposed (JCMA) constellations for N=3, QPSK and SNR values of 5 dB, 10 dB, and 20 dB, respectively; and 
         FIG. 14  is a simplified illustration of maximum sum-rate spectral efficiency vs. SNR for various Max-I based JCMA constellations. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     The principles and operation of a positioning system and method according to the present invention may be better understood with reference to the drawings and accompanying description. 
     Before explaining at least one embodiment of the invention in detail, it is to be understood that the invention is not limited in its application to the details of construction and the arrangement of the components set forth in the following description or illustrated in the drawings. The invention is capable of other embodiments or of being practiced or carried out in various ways. Also, it is to be understood that the phraseology and terminology employed herein is for the purpose of description and should not be regarded as limiting. 
     In this document, an element of a drawing that is not described within the scope of the drawing and is labeled with a numeral that has been described in a previous drawing has the same use and description as in the previous drawings. Similarly, an element that is identified in the text by a numeral that does not appear in the drawing described by the text, has the same use and description as in the previous drawings where it was described. 
     Reference is now made to  FIG. 1 , which is a view of a simplified block diagram of a generalized communication system  10  incorporating a Max-I constellation calculator  11  according to an embodiment of the present invention. 
       FIG. 1  preferably demonstrates a method for calculating a constellation for a modulation scheme for use by the communication system  10  where the method comprises the step of using the maximum sum-rate mutual information criterion. 
     As seen in  FIG. 1 , the communication system  10  preferably includes a plurality of transmitters  12  communicating with a receiver  13  via a communication medium  14  and the Max-I constellation calculator  11 . The Max-I constellation calculator  11  provides the communication system  10  with an optimized symbol constellation  15 . The optimization is preferably based on the maximal sum-rate mutual information criterion (Max-I) according to the number of transmitters N (element  16 ), and/or the signal-to-noise ratio (SNR) (element  17 ), and/or the modulation scheme used (element  18 ). Modulation schemes may be BPSK, QPSK, QAM, or any other symbol coding method. 
     As seen in  FIG. 1 , each transmitter  12  preferably includes a coder  19 , such as a BCH coder or any other error correcting coder, and a modulator  20  preferably producing a constellation  21 , or a constellation component  21  according to the optimized constellation  15 , and according to data  22  as coded by the coder  19 . 
     Preferably, the communication system  10  is a joint constellation multiple access (JCMA) communication system and the communication method used by the communication system  10  is a joint constellation multiple access (JCMA) communication method. In the case of the JCMA communication system and method the constellation signals  21  are components of a JCMA constellation  23 . Hence, for JCMA, the constellation components  21  are joined in the communication medium to form the JCMA constellation  23 . In the case of the JCMA communication system and method the JCMA constellation  23  is preferably compatible with the Max-I optimized constellation  15 . 
     As seen in  FIG. 1 , the communication medium  14  adds noise, such as additive white Gaussian noise (AWGN)  24  to the signal transmitted by the transmitters  11  and received by the receiver  13 . 
     As seen in  FIG. 1 , the receiver  13  preferably includes a signal demodulator  25  and decoders  26 , together demodulating and decoding the received signal and producing data channels  27  corresponding to data  22 . 
     Throughout this document the term receiver, such as receiver  13 , usually refer to a wireless concentrator station such as a cellular base-station, a WiFi access point, a satellite, and/or a satellite ground station, etc., and/or a concentrator in a wired network such as a hub, a head-end, etc. It is appreciated that such receiver, such as receiver  13 , is also capable of transmitting. 
     Throughout this document the term transmitter, such as transmitter  12 , usually refer to a wireless station such as a cellular mobile station, a WiMAX user station, a WiFi unit, a satellite, and/or a satellite ground station, etc., and/or a node in a wired network such as a terminal node. It is appreciated that such transmitter, such as transmitter  12 , is also capable of receiving 
     To calculate the Max-I optimized constellation  15  the Max-I constellation calculator  11  uses the maximum sum-rate mutual information criterion. The maximum sum-rate mutual information is calculated by the Max-I constellation calculator  11  according to Eq. 1: 
                       I   ⁡     (     X   ;   Y     )       =       ∫     x   ∈   X       ⁢       ∫     y   ∈   Y       ⁢       p   ⁡     (     x   ,   y     )       ⁢     log   2     ⁢       p   ⁡     (     x   ,   y     )           p   ⁡     (   x   )       ⁢     p   ⁡     (   y   )           ⁢           ⁢     ⅆ   x     ⁢           ⁢     ⅆ   y             ,           Eq   .           ⁢   1               
where:
         X is a transmitter transmitting at least a part of said JCMA constellation and Y is a receiver receiving said JCMA constellation;   I(X, Y) is said mutual information function; and   p(x, y) is the joint probability density function for X and Y, and p(·) is the marginal probability density functions for X and Y respectively.       

     It is appreciated that the constellation calculated using the maximum sum-rate mutual information criterion as described above provides at least one of: optimal spectral efficiency, optimal bit-rate, and optimal transmission power. 
     Reference is now made to  FIG. 2 , which is a simplified graphical description of a synchronized superimposed constellation  28 , according to a preferred embodiment of the present invention. 
     As seen in  FIG. 2 , the scenario assumes that the number of transmitters  11  is N=2 and that transmitters  11  are using Binary Phase Shift Keying (BPSK) (element  29  and  30 ). Each transmitter  11  uses a power of P watts and a sub-band of W/2 Hz. The corresponding symbol duration is T. The BER for this scenario is given by Eq. 2: 
     
       
         
           
             
               
                 
                   
                     
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     When using SM the two transmitters use the whole bandwidth W, which allows doubling of the symbol rate. The symbol duration is now T/2. The two transmitters use BPSK at double the rate, but with an energy per bit of E/2. A superimposed constellation  28  is constructed at the receiver comprising four constellation points in a Quadrature Phase Shift Keying (QPSK) like setting. If the transmitters are not symbol-synchronized the superimposed constellation would be sub-optimal for decoding. However when the transmitters are synchronized so that each transmitter occupies a single carrier component (quadrature or in-phase) the resulting constellation is always robust for decoding. 
     Since the transmitters occupy orthogonal dimensions and the SNR per bit is now SNR/2 uncoded BER is given by Eq. 3:
 
 P′   b   =Q (√{square root over ( SNR )})  Eq. 3
 
     Noting that P′ b &gt;P b . P′ b  is preferably reduced by applying a practical coding scheme. 
     To have a throughput higher than that without synchronized SM we limit the coding rate to be higher or equal to 0.5. 
     For this illustrative scenario we use a simple BCH code (n, k, d)=(127, 64, 10) with hard decoding resulting in a code rate of 0.504. For perfect codes the coded BER is according to Eq. 4: 
     
       
         
           
             
               
                 
                   
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     In this case Eq. 4 is an upper bound on the coded BER. 
     Reference is now made to  FIG. 3 , which is a simplified graphical representation of a BER to SNR dependency for the scenario of  FIG. 2 , according to a preferred embodiment of the present invention. 
       FIG. 3  depicts BER of classical FDMA and the upper bound on BER with synchronized SM and the BCH code. As shown, SNR gains of at least 2 dB for SNR higher than 8 dB. The SNR gain can be translated to reduction of power emissions, increased throughput and increased cell size. 
     Reference is now made to  FIG. 4 , which is a simplified block diagram of a JCMA receiver  31 , a JCMA transmitter  32  and a Max-I JCMA constellation calculator  33 , according to a preferred embodiment of the present invention. 
     It is appreciated that:
         the JCMA receiver  31  is equivalent to the receiver  12  of  FIG. 1  adapted for JCMA communication method and system:   the JCMA transmitter  32  is equivalent to the transmitter  13  of  FIG. 1  adapted for JCMA communication method and system; and   the Max-I JCMA constellation calculator  33  is equivalent to a Max-I constellation calculator  11  of  FIG. 1  adapted for JCMA communication method and system.       

     As seen in  FIG. 4 , the Max-I JCMA constellation calculator  33  is preferably separated from both the receiver  31  and the transmitter  32 , preferably calculating optimal JCMA constellations in offline mode, and preferably providing the Max-I calculated JCMA constellations as a look-up table (LUT) to receivers  31  and transmitters  32 . 
     It is appreciated that alternatively the Max-I JCMA constellation calculator  33  can be incorporated, preferably in the receivers  31 . 
     Preferably, the LUT contains a plurality of JCMA constellation where each JCMA constellation contains a plurality of JCMA constellation component types. Each JCMA constellation component type is preferably allocated, preferably by the receiver  31 , to a particular transmitter  32 . The JCMA constellation component type is the group of constellation points that are allocated to a particular transmitter  32 . Preferably, the JCMA constellation component type reflects the modulation scheme used by the particular transmitter  32 . 
     It is appreciated that the JCMA constellations calculated by the Max-I JCMA constellation calculator  33  and provided in the LUT enables a JCMA communication system including one or more receivers  31  and transmitters  32  to reach optimal spectral efficiency, and/or optimal bit-rate, and/or optimal transmission power. 
     As seen in  FIG. 4 , the receiver  31  preferably includes the following components:
         A receiver module  34  receiving JCMA constellations (JCMA signals or JCMA symbols) via antenna  35 . The receiver  34  is connected to a JCMA demodulator  36 , which is connected to a decoder  37  (such as a BCH decoder) outputting a data channel  38 .   A storage module  39 , such as a Flash memory module, storing a look-up table (LUT) of Max-I optimized JCMA constellations preferably calculated by the Max-I JCMA constellation calculator  33 .   A Max-I JCMA constellation selector  40  using measured SNR value  41 , number of transmitters N ( 42 ) in a JCMA group, and the modulation schemes  43  (such as BPSK, QPSK, QAM, etc.) assigned to the transmitters in the JCMA group to select and retrieve an appropriate Max-I JCMA constellation  44  from the LUT in memory module  39 .   A transmitter module  45  for transmitting a message  46  containing the selected and/or a pilot signal  47  to the transmitters  32 .       

     It is appreciated that in alternative configuration where the Max-I JCMA constellation calculator  33  is incorporated in the receivers  31  the Max-I JCMA constellation calculator  33  is preferably incorporate in the Max-I JCMA constellation selector, or the Max-I JCMA constellation selector is the Max-I JCMA constellation calculator  33 . 
     As seen if  FIG. 4 , the transmitter  32  includes the following components:
         A coder  48  (such as a BCH coder or other) receiving the data  49  for transmission. The coder  48  provides the coded data to a JCMA modulator  50 . The JCMA modulator  50  uses a modulation scheme (such as BPSK, QPSK, QAM, etc.) and a JCMA constellation  51  to produce a JCMA constellation component  52 . The JCMA constellation component is transmitted via transmitter module  53  and antenna  54  to the receiver  31 .   A storage module  55 , such as a Flash memory module, storing a look-up table (LUT) of Max-I optimized JCMA constellations preferably calculated by the Max-I JCMA constellation calculator  33 .   A receiver module  56  receiving the message  46  containing the selected constellation (or an identification of the selected constellation), and/or the pilot signal  47 .   A power and phase correction calculator  57  providing power and phase correction data  58  to the modulator  50 .       

     Preferably, the transmitter  32  receives from the receiver  31  the message  46  containing identification of the JCMA constellation selected by the receiver  31 . The transmitter  32  then retrieves the selected JCMA constellation  51  from the LUT in the storage module  55 . The selected JCMA constellation  51  includes identification of the JCMA constellation and the JCMA constellation component type allocated to the particular transmitter. Using the JCMA constellation  51  the modulator  50  creates the JCMA constellation component  52 . 
     Preferably, the transmitter  32  receives from the receiver  31  a sequence of pilot signals  47 . Using the pilot signals  47  the power and phase correction calculator  57  calculates the power and phase correction data  58 . Using the power and phase correction data  58  the modulator modifies the power and the phase of the JCMA constellation component  52  to provide an optimal joint constellation at the receiver  31 . 
     In the mode of operation described in  FIG. 4  the Max-I calculated JCMA constellations are calculated offline and are provided as a look-up table (LUT) to receivers  31  and transmitters  32 . The receiver  31  preferably sends a message (not shown) to a group of transmitters  32  selected to participate in a particular JCMA group. The message indicates a particular JCMA constellation selected from the LUT. The JCMA constellation is preferably selected per the number N of transmitters  32  in the JCMA group, and/or the SNR measured by the receiver  31  for this group of transmitters  32 , and/or modulation schemes assigned to each of the transmitters  32  of this group. 
     Reference is now made to  FIG. 5 , which is a simplified illustration of a JCMA-based cellular communication system  59  incorporating a Max-I constellation calculator  11  according to a preferred embodiment of the present invention. 
     As seen in  FIG. 5 , the JCMA-based cellular communication system  59  preferably includes a cellular base-station  60 , a plurality of cellular mobile stations (e.g. mobile telephones)  61 , and an offline JCMA Constellations calculator by Max-I  11 . 
     It is appreciated that alternatively the JCMA Constellations calculator by Max-I  11  may be incorporated in the cellular base-station  60 . It is also appreciated that the cellular base-station  60  is preferably equivalent to the receiver  13  of  FIG. 1  or to receiver  31  of FIG.  4 . Cellular base-station  60  may therefore also represent a WiFi access point or any other communication concentrator. It is further appreciated that the cellular mobile stations  61  are preferably equivalent to the transmitters  12  of  FIG. 1  or to transmitter  32  of  FIG. 4 . Cellular mobile stations  61  may therefore also represent WiFi terminals or any other type of communication nodes. 
     Element  62  of  FIG. 5  is a block diagram of a part of the base-station  60 , and element  63  is a block diagram of a part of the mobile stations  61 . 
     As seen in  FIG. 5 , the JCMA Constellations calculator  11  preferably generates a look-up table (LUT)  64  of JCMA constellation using maximal sum-rate mutual information criterion. The LUT  64  is then distributed and/or loaded to base-stations  60  and mobile stations  61 . 
     Preferably, the LUT contains a plurality of JCMA constellation where each JCMA constellation contains a plurality of JCMA constellation component types. The JCMA constellation component type is the group of constellation points to be allocated to a particular transmitter of the JCMA transmitters group forming the JCMA constellation. Preferably, the JCMA constellation component type reflects the modulation scheme used by the particular transmitter. 
     A constellation selector  65  in the base-station  60  selects a JCMA constellation  66  for the LUT according to SNR  67 , number N  68  of mobile stations  61  and type of modulation scheme  69  (e.g. BPSK, QPSK, QAM, etc.). The base-station  60  sends a message  70  to each of the mobile stations  61  informing them of the selected JCMA constellation  66  and the particular JCMA constellation component type (group of constellation points) allocated to the particular mobile stations  61 . 
     The base-station  60  then preferably transmits pilot signals  71 . Preferably, the power level of the pilot signal is adjusted according to the number of transmitters. 
     Upon accepting the message  70  A constellation selector  72  at the mobile station  61  retrieves the appropriate JCMA constellation  66  from the LUT  64  stored in the storage module  73 . In this respect the JCMA constellation  66  includes the particular JCMA constellation component type (group of constellation points) allocated to the mobile stations  61 . Using the JCMA constellation  66  a modulator  74  generates a component  75  of the JCMA constellation  66 . 
     According to the pilot signal  71 , power and phase correction module  76  calculates power P and constellation rotation angle θ. Using these power P and phase θ correction values  77  the modulator  74  transmits JCMA constellation components  75  to the receiver. Thus the JCMA constellation  78  is created from the combination of the JCMA constellation components  75  of the transmitters  61 . 
     Reference is now made to  FIG. 6 , which is a simplified flow chart of a process  79  executed by the JCMA-based cellular communication system  59  of  FIG. 5 , according to a preferred embodiment of the present invention. 
     It is appreciated that the process  79  or a similar process can also be executed by the system described in accordance with  FIG. 1  and/or  FIG. 4 . 
     As seen in  FIG. 6 , the process  79  preferably includes a LUT generation process  80 , a receiver process  81 , and a transmitter process  82 . The LUT generation process  80  is preferably executed by the Max-I constellation calculator  11  of  FIG. 5 . The receiver process  81  is preferably executed by the base station  60  of  FIG. 5 . The transmitter process  82  is preferably executed by the mobile station  61  of  FIG. 5 . 
     As seen in  FIG. 6 , the LUT generation process  80  preferably starts with step  83  by selecting or preparing the following sets:
         A set of values of signal-to-noise ratio (SNR) (e.g. 5 dB, 10 dB, 15 dB, . . . ).   A set of values (e.g. 2, 3, 4, . . . ) of the number of transmitters participating in a joint constellation multiple access (JCMA) transmitters group to form a JCMA constellation. Such as transmitters  12  of  FIG. 1 , or mobile stations  61  of  FIG. 5 .   A set of modulation schemes such as BPSK, QPSK and various QAM schemes.       

     The LUT generation process  80  preferably continues with step  84  by selecting at one value from at least two sets of the sets created in step  83 , namely an SNR value, a value for N=number of transmitters participating in a JCMA transmitters group, and one or more modulation schemes. The LUT generation process  80  then generates a JCMA constellation according the maximal sum-rate mutual information criterion. That is, a JCMA constellation for which the sum of mutual information value is maximal. Preferably, each of the JCMA constellations includes a plurality of JCMA constellation component types according to the number of transmitters in the JCMA group and the modulation schemes allocated to these transmitters. 
     The LUT generation process  80  preferably continues with step  85  to create a loop-up table (LUT) of JCMA constellations by repeating step  84  for a plurality of combinations of values of the sets of step  83 . 
     The LUT generation process  80  preferably continues with step  86  to distribute the LUT created in step  85  to receiver and transmitters such as base station  60  and mobile stations  61  of  FIG. 5 . The LUT may be distributed using any type of non-transitory computer readable media. 
     The receiver process  81  preferably starts with step  87  preferably by receiving the LUT from the LUT generation process  80 . Receiver process  81  then preferably continues with step  88  preferably by selecting a group of transmitters to participate in a JCMA constellation (a.k.a JCMA group or JCMA transmitter group). Receiver process  81  then preferably continues with step  89  preferably by selecting the modulation mechanisms for the transmitters in the JCMA group. Receiver process  81  then preferably continues with step  90  preferably by measuring SNR value for signals transmitted by the JCMA transmitter group. 
     Receiver process  81  then preferably continues with step  91  preferably by selecting from the LUT received in step  87  a JCMA constellation that fits the values of N, SNR and modulation schemes as selected in steps  88 ,  90  and  89 . Receiver process  81  then preferably continues with step  92  preferably by informing the transmitters of the JCMA group of the selected JCMA constellation and their respective constellation component type. 
     It is appreciated that alternatively, step  91  may include calculating a JCMA constellation using Max-I such as in step  84  using the values of N, SNR and modulation schemes as selected in steps  88 ,  90  and  89 . 
     Optionally but preferably the receiver process  81  continues with step  93  preferably by sending a pilot signal. Preferably, the transmission power level of the pilot signal is calculated according to a number of transmitters in said group of transmitters. Step  93  is preferably repeated, preferably indefinitely. 
     Receiver process  81  then preferably continues with step  94  preferably by receiving JCMA constellations transmitted by the JCMA transmitter group and preferably demodulating and decoding the JCMA constellations to form a data channel per each of the transmitters. Step  94  is preferably repeated, preferably indefinitely. 
     Preferably, steps  90 ,  91  and  92  are preferably repeated, preferably indefinitely. 
     The transmitter process  82  preferably starts with step  95 , preferably by receiving the LUT from the LUT generation process  80 . Transmitter process  82  then preferably continues with step  96  preferably by receiving the identification of the selected JCMA constellation as sent by the receiver in step  92 . Transmitter process  82  then preferably continues with step  97  preferably by retrieving the selected JCMA constellation and the allocated constellation component type from the LUT received in step  95 . 
     Optionally but preferably the transmitter process  82  continues with step  98  preferably by receiving the pilot signal transmitted by the receiver in step  93 . Transmitter process  82  preferably continues with step  99  preferably by calculating transmission power P and/or constellation rotation angle θ (a.k.a phase rotation, phase correction) according to the received pilot signal. Steps  98 - 99  are preferably repeated, preferably indefinitely. 
     Transmitter process  82  preferably continues with step  100  preferably by transmitting a JCMA constellation component using the selected JCMA constellation and/or JCMA constellation component type according to the LUT and the JCMA selection information received from the receiver in step  97 . Preferably the power and phase values of the JCMA constellation component are corrected by the values obtained in step  99 . Step  100  preferably repeats, preferably indefinitely. 
     It is appreciated that the transmitters may use the same modulation schemes or various mixes of modulation schemes. 
     Reference is now made to  FIG. 7 , which is a simplified flow chart of a process  101  for calculating a JCMA constellation using Max-I, as preferably executed by step  84  of  FIG. 6 , according to a preferred embodiment of the present invention. 
     As seen in  FIG. 7 , the process  101  for calculating a JCMA constellation using Max-I starts with step  102  preferably by selecting a number of transmitters and an SNR value from the sets prepared in step  83  of the of  FIG. 6 . 
     Process  101  preferably continues with step  103  by selecting a constellation component type for each of the transmitters N. 
     Process  101  preferably continues with step  104  to compute the mutual information function I for the values selected in step  102  and according to the constellation components selected in step  103 . The mutual information between n-th transmitter and the receiver is given by Eq. 5: 
                         I   n     ⁡     (       x   _       (   n   )       )       =       ∑     i   =   1     M     ⁢       ∫     -   ∞     ∞     ⁢       p   ⁡     (     y   /     x   i     (   n   )         )       ⁢     P   ⁡     (     x   i     (   n   )       )       ⁢     log   2     ⁢       p   ⁡     (     y   /     x   i     (   n   )         )         p   ⁡     (   y   )         ⁢           ⁢     ⅆ   y             ,           Eq   .           ⁢   5               
where
         the vector  x   (n) =[x 1   (n) , x 2   (n) , . . . , x M   (n) ], n=1, 2, . . . , N represent the possible M outcomes (constellation points) of the n-th transmitter, where   M is the number of possible symbols for the selected modulation scheme (e.g. 2 for BPSK, 4 for QPSK, etc.);   P(x i   (n) ) is the probability distribution of the possible outcomes of the n-th transmitter;   p(y/x i   (n) ) is the conditional probability density function of the channel outcome given x i   (n) . For Gaussian channel noise p(y/x i   (n) ) is represented by Eq. 6       

                       p   ⁡     (     y   /     x   i     (   n   )         )       =       1       2   ⁢     πσ   2           ⁢   exp   ⁢     {     -         (     y   -     x   i     (   n   )         )     2       2   ⁢     σ   2           }         ,           Eq   .           ⁢   6               
where
         σ 2  is reciprocal of the SNR.       

     Process  101  preferably continues with step  105  to compute the sum-rate mutual information, defined by Eq. 7: 
     
       
         
           
             
               
                 
                   
                     S 
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             x 
                             _ 
                           
                           
                             ( 
                             1 
                             ) 
                           
                         
                         , 
                         
                           
                             x 
                             _ 
                           
                           
                             ( 
                             2 
                             ) 
                           
                         
                         , 
                         … 
                         ⁢ 
                         
                             
                         
                         , 
                         
                           
                             x 
                             _ 
                           
                           
                             ( 
                             N 
                             ) 
                           
                         
                       
                       ] 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         1 
                       
                       N 
                     
                     ⁢ 
                     
                       
                         I 
                         n 
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             _ 
                           
                           
                             ( 
                             n 
                             ) 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   7 
                 
               
             
           
         
       
     
     Process  101  preferably repeats steps  104 - 105  for different sets of vectors  x   (1) ,  x   (2) , . . . ,  x   (N) , and when the sets of vectors are exhausted (step  106 ) Process  101  preferably continues to step  107 . 
     In step  107  process  101  (the Max-I calculator) looks for set of vectors  x   (1) ,  x   (2) , . . . ,  x   (N)  such that S[ x   (1) ,  x   (2) , . . . ,  x   (N) ]→max under constraint represented by Eq. 8: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         M 
                       
                       ⁢ 
                       
                         
                            
                           
                             x 
                             i 
                             
                               ( 
                               n 
                               ) 
                             
                           
                            
                         
                         2 
                       
                     
                     = 
                     1 
                   
                   , 
                   
                     n 
                     = 
                     1 
                   
                   , 
                   2 
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   N 
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   8 
                 
               
             
           
         
       
     
     thus selecting for the LUT the constellation for which S is maximal. 
     Or, in more compact notation, as depicted by Eqs. 9: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           x 
                           _ 
                         
                         opt 
                         
                           ( 
                           1 
                           ) 
                         
                       
                       , 
                       
                         
                           x 
                           _ 
                         
                         opt 
                         
                           ( 
                           2 
                           ) 
                         
                       
                       , 
                       … 
                       ⁢ 
                       
                           
                       
                       , 
                       
                         
                           x 
                           _ 
                         
                         opt 
                         
                           ( 
                           N 
                           ) 
                         
                       
                     
                     ] 
                   
                   = 
                   
                     arg 
                     ⁢ 
                     
                       { 
                       
                         
                           max 
                           
                             
                               
                                 x 
                                 _ 
                               
                               
                                 ( 
                                 1 
                                 ) 
                               
                             
                             , 
                             
                               
                                 x 
                                 _ 
                               
                               
                                 ( 
                                 2 
                                 ) 
                               
                             
                             , 
                             … 
                             ⁢ 
                             
                                 
                             
                             , 
                             
                               
                                 x 
                                 _ 
                               
                               
                                 ( 
                                 N 
                                 ) 
                               
                             
                           
                         
                         ⁢ 
                         
                           [ 
                           
                             S 
                             ⁡ 
                             
                               [ 
                               
                                 
                                   
                                     x 
                                     _ 
                                   
                                   
                                     ( 
                                     1 
                                     ) 
                                   
                                 
                                 , 
                                 
                                   
                                     x 
                                     _ 
                                   
                                   
                                     ( 
                                     2 
                                     ) 
                                   
                                 
                                 , 
                                 … 
                                 ⁢ 
                                 
                                     
                                 
                                 , 
                                 
                                   
                                     x 
                                     _ 
                                   
                                   
                                     ( 
                                     N 
                                     ) 
                                   
                                 
                               
                               ] 
                             
                           
                           ] 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   9 
                 
               
             
           
         
       
     
     under constraint as depicted by Eq. 10: 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         M 
                       
                       ⁢ 
                       
                         
                            
                           
                             x 
                             i 
                             
                               ( 
                               n 
                               ) 
                             
                           
                            
                         
                         2 
                       
                     
                     = 
                     1 
                   
                   , 
                   
                     n 
                     = 
                     1 
                   
                   , 
                   2 
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   N 
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   10 
                 
               
             
           
         
       
     
     The vectors  x   opt   (1) ,  x   opt   (2) , . . .  x   opt   (N)  constitute the nominal constellation of the transmitter 1, 2, . . . , N respectively. The number of points of the joint constellation equals to M N . 
     Reference is now made to  FIGS. 8   a ,  8   b ,  8   c , and  8   d , which are simplified illustrations of optimal superimposed (JCMA) constellations for N=3, BPSK and SNR values of 5 dB, 10 dB, 12 dB, and 15 dB, respectively, according to a preferred embodiment of the present invention. 
     The optimization of the superimposed constellation for N users (mobile stations) is calculated using the maximum sum-rate mutual information criterion as follows: 
     Let P=[P 1 , P 2 , . . . , P N ], θ=[θ 1 , θ 2 , . . . , θ N ], where (P n ,θ n ) are the average power and right rotation (in degrees) of constellation of user n. Using x=[−1,1] for BPSK and x=[−1,1,j,−j] for QPSK, the transmitted constellations can be written by Eq. 11: 
     
       
         
           
             
               
                 
                   
                     x 
                     n 
                   
                   = 
                   
                     x 
                     ⁢ 
                     
                       
                         P 
                         n 
                       
                     
                     ⁢ 
                     
                       ⅇ 
                       
                         j 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           θ 
                           n 
                         
                         ⁢ 
                         
                           π 
                           
                             180 
                             ⁢ 
                             ° 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Eq 
                   . 
                   
                       
                   
                   ⁢ 
                   11 
                 
               
             
           
         
       
     
     The design criterion is maximizing the sum-rate mutual information and is defined by Eq. 12:
 
 [P,θ]= arg max P     n     ε[0,1],θ     n     ε[0,180°],n=1,2, . . . , N   I ( X   1   ,X   2   , . . . , X   N   ;Y )  Eq. 12
 
     Thus,  FIGS. 8   a ,  8   b ,  8   c , and  8   d  depict superimposed constellations for x=[−1,1], for three mobile stations (N=3) and for SNR values of 5 dB, 10 dB, 12 dB, and 15 dB, respectively. 
     Table 1 shows the minimum distance d min  and the next smallest distance d min,2  for three users and x=[−1,1]. Table II shows the magnitude and relative rotations of the constellations of transmitters  2  and  3  for three users and x=[−1,1]. The power of transmitter  1  is set it 1 and his rotation is reference rotation and is set to 0. As can be seen from table II, for SNRs greater than 5 dB, the relative phases are fixed and the minimum distances are determined only by varying the power of the second user whose angle of rotation is between those of the two other users. In fact, this shows that increasing the minimum distance is achieved by decreasing the power. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 d min  and d min,2  for SNR 2, 5, 10, 12, 15 dB. 
               
            
           
           
               
               
               
               
               
            
               
                 SNR 
                 d min   
                   
                 d min,2   
                   
               
            
           
           
               
               
               
               
               
            
               
                 [dB] 
                 Value 
                 # 
                 value 
                 # 
               
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 2 
                 0.0000 
                 1 
                 2.0000 
                 18 
               
               
                 5 
                 0.9701 
                 1 
                 1.4994 
                 4 
               
               
                 10 
                 1.3036 
                 1 
                 1.4183 
                 4 
               
               
                 12 
                 1.3474 
                 1 
                 1.4159 
                 4 
               
               
                 15 
                 1.3787 
                 1 
                 1.3985 
                 2 
               
               
                   
               
            
           
         
       
     
     Note that for any distance the right-hand column indicates how many pairs are characterized by that distance. 
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 Magnitude and Relative Rotation of the 
               
               
                 Constellation of Transmitters 2 and 3 
               
            
           
           
               
               
               
               
               
            
               
                   
                 d min   
                   
                 d min,2   
                   
               
            
           
           
               
               
               
               
               
            
               
                 SNR 
                 θ 2   
                   
                 θ 2   
                   
               
               
                 [dB] 
                 [deg] 
                 P 2   
                 [deg] 
                 P 2   
               
               
                   
               
            
           
           
               
               
               
               
               
            
               
                 2 
                 60 
                 1 
                 120 
                 1 
               
               
                 5 
                 44.8 
                 0.9656 
                 89.6 
                 1 
               
               
                 10 
                 45 
                 0.7627 
                 90 
                 1 
               
               
                 12 
                 45 
                 0.7404 
                 90 
                 1 
               
               
                 15 
                 45 
                 0.7267 
                 90 
                 1 
               
               
                   
               
            
           
         
       
     
     Based on the optimization results we make the following observations:
         Recall that the sum-rate is always the dominant rate constraint (it is never loose).   In order to take advantage of the maximum available rate the mobile units need to coordinate both power and rotation as seen in Table 2   It is noted that the optimal superimposed constellation for high SNR does not use the maximum power available at all transmitters as seen in Table 2. In fact by backing off the power of one user it is possible to obtain higher sum-rate. This means that one user is “sacrificing” its performance and its maximum achievable rate is less than its individual constraint. For example, if the user with reduced power is user 2 then       

     
       
         
           
             
               I 
               ⁡ 
               
                 ( 
                 
                   
                     
                       X 
                       2 
                     
                     ; 
                     
                       Y 
                       | 
                       
                         X 
                         1 
                       
                     
                   
                   , 
                   
                     X 
                     3 
                   
                 
                 ) 
               
             
             &lt; 
             
               
                 max 
                 
                   
                     P 
                     3 
                   
                   ∈ 
                   
                     [ 
                     
                       0 
                       , 
                       1 
                     
                     ] 
                   
                 
               
               ⁢ 
               
                 I 
                 ⁡ 
                 
                   ( 
                   
                     
                       
                         X 
                         2 
                       
                       ; 
                       
                         Y 
                         | 
                         
                           X 
                           1 
                         
                       
                     
                     , 
                     
                       X 
                       3 
                     
                   
                   ) 
                 
               
             
           
         
       
         
         
           
             It can be seen from  FIGS. 8   c ,  8   b ,  8   c , and  8   d , that as the SNR increases, the distance between the inner points increases as well. This brings the inner points closer to the outer points therefore reducing d min,2  and increasing d min . 
             It is noted that the superimposed constellations at high SNR coincides with the superimposed constellation, obtained by maximizing d min  rather than sum-rate mutual-information. 
             At low SNR, all users make the best effort in the sense that they transmit with the maximum available power. From  FIGS. 8   a ,  8   b ,  8   c , and  8   d , it is seen that when the SNR is 2 dB the constellations are spread at equal angular distances of 60 degrees. The superimposed constellation points form a lattice and for one pair d min  is zero, however, d min,2  is maximized (see also Table I). As the SNR increases the minimum distance increases. At the same time, the next smallest distance d min,2  decreases. The explanation lies in the dominating error event: at high SNR the error event is dominated by the closest pair of constellation points. Hence for high SNR we aim at maximizing the minimum distance. For low SNR, error events between constellation points which are more than d min  apart become a dominant factor. Therefore, the optimal constellation has a zero d min  but the largest d min,2 . This decreases the probability of error for constellation points except the nearest pair. Since only a single pair is located at a distance d min  and many other pairs are located at a distance d min,2  apart, the overall probability of error is decreased. 
             This process also optimizes phase and power. This optimization obtains d min =1.1321 at SNR=10 dB (cf. 1.3036 when optimizing over power as well) and d min =0.8723 at SNR=5 dB (cf. 0.9071 obtained when optimizing over power as well). Comparing the maximum achievable sum-rate spectral efficiencies obtained when optimizing only the phase to those obtained when optimizing both phase and power, it is possible to conclude that the increase in the achievable rate associated with the increased minimum distances is only of the order of 0.5%. It follows that the main advantage in optimizing over power is the ability to achieve the same rate while using less transmission power, hence extending the lifetime of the mobile stations without sacrificing rate. 
           
         
       
    
     Reference is now made to  FIGS. 9   a ,  9   b , and  9   c , which are simplified illustrations of optimal superimposed (JCMA) constellations for N=4, BPSK and SNR values of 10 dB, 12 dB, and 15 dB, respectively, according to a preferred embodiment of the present invention. 
       FIGS. 9   a ,  9   b , and  9   c  depict superimposed constellations for x=[−1,1], for four mobile stations (N=4) and for SNR values of 10 dB, 12 dB, and 15 dB, respectively. 
     Reference is now made to  FIGS. 10   a ,  10   b ,  10   c , and  FIG. 10   d , which are simplified illustrations of optimal superimposed (JCMA) constellations for N=5, BPSK (2QAM) and SNR values of 2 dB, 5 dB, 10 dB, and 16 dB, respectively, according to a preferred embodiment of the present invention. 
       FIGS. 10   a ,  10   b , and  10   c  depict superimposed constellations for x=[−1,1], for five mobile stations (N=5) and for SNR values of 2 dB, 5 dB, 10 dB, and 16 dB, respectively. 
     Reference is now made to  FIG. 11 , which is a simplified illustration of maximum sum-rate spectral efficiency vs. SNR for various Max-I based JCMA constellations according to a preferred embodiment of the present invention. 
       FIG. 11  shows the maximum sum-rate spectral efficiency vs. SNR for x=[−1,1].  FIG. 11  shows the overall sum-rate spectral efficiency in bits/sec/Hz vs. SNR for the proposed multiuser modulation as well as for the standard frequency-division multiplexing (FDMA). In FDMA the average power is kept constant assuming averaging is taken over all three users frames. 
     There are many examples in the literature for the multiple-access channel that demonstrate the advantage of simultaneous transmission over time-sharing. Indeed, the superiority of superposition modulation over frequency-sharing is clearly evident from  FIG. 11 : the FDMA sum-rate cannot exceed the single user rate which is 1 bit/sec/Hz for BPSK or 2 bit/sec/Hz for QPSK (4QAM). By appropriate alignment of the constellations the sum rate, at high SNR, increases by factor of N. 
     Reference is now made to  FIGS. 12   a , and  12   b , which are simplified illustrations of optimal superimposed (JCMA) constellations for N=2, QPSK and SNR values of 15 dB and 20 dB, respectively, according to a preferred embodiment of the present invention. 
       FIGS. 12   a ,  12   b  depict superimposed constellations for x=[−1,1,j,−j], for two mobile stations (N=2) and for SNR values of 15 dB, and 20 dB, respectively. 
     Reference is now made to  FIGS. 13   a ,  13   b , and  13   c , which are simplified illustrations of optimal superimposed (JCMA) constellations for N=5, BPSK and SNR values of 5 dB, 10 dB, and 20 dB, respectively, according to a preferred embodiment of the present invention. 
       FIGS. 13   a ,  13   b , and  13   c  depict superimposed constellations for x=[−1,1,j,−j], for three mobile stations (N=3) and for SNR values of 5 dB, 10 dB, and 20 dB, respectively. 
     Reference is now made to  FIG. 14 , which is a simplified illustration of maximum sum-rate spectral efficiency vs. SNR for various Max-I based JCMA constellations according to a preferred embodiment of the present invention. 
       FIG. 14  shows the maximum sum-rate spectral efficiency vs. SNR for x=[−1,1,j,−j].  FIG. 14  shows the overall sum-rate spectral efficiency in bits/sec/Hz vs. SNR for the proposed multiuser modulation as depicted in  FIGS. 12   a ,  12   b  and  FIGS. 13   a ,  13   b , and  13   c  as well as for the standard frequency-division multiplexing (FDMA) 
     It is shown that the use of synchronized superposition-modulation with coding in FDMA systems results in an increase of spectral efficiency. Three example scenarios were investigated: a two-user scenario with an arbitrary QPSK superimposed constellation coupled with BCH coding, three, four and five-user BPSK scenario with mutual-information optimization and two, three-users QPSK scenario with mutual-information optimization. 
     For the two-user scenario an SNR gain of 2 dB for SNR higher than 8 dB was observed. 
     For the sum-rate mutual-information optimization was shown to be achieved when both power and phase of the different users are synchronized. The obtained sum-rate was shown to depend on SNR with a possible N-fold gain in sum-rate compared to FDMA. For high SNR mutual-information optimization scenario was numerically shown to be equivalent to optimizing the maximum minimal Euclidean distance. In addition, achieving maximum sum-rate required some of the users to sacrifice their individual rate. However, this also allows for extending their battery life while maintaining maximal sum-rate. For low SNR the optimal superimposed constellation was shown to be a lattice where performance is governed by the second smallest Euclidean distance. 
     Spectral efficiency is expected to grow substantially if more transmitters would be synchronized to form larger superimposed signal constellations and advanced coding techniques such as turbo codes and low density parity check codes would be used. Our ongoing research includes practical means of implementing JCMA based on synchronized SM with advanced coding schemes and high order superimposed signal constellations in emerging wireless standards such as LTE of UMTS. 
     It is further noted that pilot signal energy should be sent over the uplink sub-band from receiver to transmitters and the energy of the pilot signal should be directly proportional to the number of transmitters. Recent work (G. R. Tsouri and D. Wulich, referenced above) provided simple signal piloting rules for achieving accurate synchronization and power control of transmitters in SM scenarios.