Patent Publication Number: US-9426757-B2

Title: Method and apparatus for transmitting and receiving a signal in MIMO broadcast channel with imperfect CSIT

Description:
CROSS-REFERENCE TO RELATED APPLICATION(S) 
     This application claims the benefit under 35 U.S.C. §119(e) of a U.S. Provisional application filed on Jan. 14, 2014 in the U.S. Patent and Trademark Office and assigned Ser. No. 61/927,193, the entire disclosure of which is hereby incorporated by reference. 
    
    
     JOINT RESEARCH AGREEMENT 
     The present disclosure was made by or on behalf of the below listed parties to a joint research agreement. The joint research agreement was in effect on or before the date the present disclosure was made and the present disclosure was made as a result of activities undertaken within the scope of the joint research agreement. The parties to the joint research agreement are 1) SAMSUNG ELECTRONICS CO., LTD. and 2) IMPERIAL INNOVATIONS LTD. 
     TECHNICAL FIELD 
     The present disclosure relates to a multiple antenna multiuser radio communication system. More particularly, the present disclosure relates to transmission strategies for Multiple Input and Multiple Output (MIMO) broadcast channel with imperfect Channel State Information at the Transmitter (CSIT). 
     BACKGROUND 
     In Multi User-Multiple Input and Multiple Output (MU-MIMO) transmissions (for both Broadcast Channel and Interference Channel), the major performance drop is caused by the imperfect Channel State Information at the Transmitter (CSIT). This is because in current standards the MU-MIMO transmission strategy has been designed under the assumption of perfect CSI knowledge at the transmitter but is actually used in scenarios where CSI is imperfectly known at the transmitter. Moreover, given the CSIT feedback mechanism in current standardizations, the accuracies of the CSIT vary across subbands and users depending on the availability of wideband Precoding Matrix Indicator (PMI) and user-specific PMI. An interesting work is to design new transmission blocks that cope with the imperfect (instantaneous) CSIT and making use of the varying CSIT qualities to benefit the performance. 
     The metric considered in this disclosure is the Degrees of Freedom (DoF). It can be interpreted as the number of interference-free streams transmitted to each receiver. Mathematically, it is given by 
               d   =       lim     P   -&gt;   ∞       ⁢     R       log   2     ⁢   P           ,         
where R is the rate and P stands for the Signal-to-Noise Ratio (SNR).
 
     To investigate the impact of the imperfect CSIT on the sum DoF performance, the terminology, CSIT quality, is introduced. The CSIT quality is considered within the range of 0 to 1, representing unknown CSIT and perfect CSIT respectively. Moreover, the CSIT qualities are likely to vary across subbands and users. This setup can be interpreted as a practical deployment in line with Long Term Evolution (LTE) by connecting the CSIT quality with the availability of wideband PMI and subband PMI. 
     With the classical MU-MIMO transmission, the sum DoF performance is N 2 −N 1 +N 1 ā+N 1   b , where ā and  b  respectively stand for the average CSIT quality of receiver 1 and receiver 2 across all the subbands. If Single User-Multiple Input and Multiple Output (SU-MIMO) is performed, the maximal sum DoF performance will be N 2  if the transmitter only sends messages intended for receiver 2. The key ingredient of the achievability relies on the interference cancellation using the past CSIT and channel output. 
     Moreover, the optimal strategy for the Multiple Input and Single Output (MISO) case despite the existence of imperfect CSIT, but DoF loss is incurred if it is reused in the MIMO case. The transmitted signal is made up of private messages and common messages. Intuitively, since two receivers have different antennas, the number of common messages should be limited by N 1 , otherwise, Rx 1  is unable to decode them. This causes space resource wasted at Rx 2  as it should have decoded N 2  streams of common messages at a time. 
     The above information is presented as background information only to assist with an understanding of the present disclosure. No determination has been made, and no assertion is made, as to whether any of the above might be applicable as prior art with regard to the present disclosure. 
     SUMMARY 
     Aspects of the present disclosure are to address at least the above-mentioned problems and/or disadvantages and to provide at least the advantages described below. Accordingly, an aspect of the present disclosure is to provide a signal transmission method and apparatus embodied in such a way that a User Equipment (UE) generates feedback information in consideration of stochastic information on channel estimation error for use by a Base Station (BS) in a multi-antenna multiuser radio communication system. 
     Another aspect of the present disclosure is to provide a method and apparatus for feedback of channel information based on a channel estimation error prediction of a UE in a multi-antenna multiuser radio communication system. 
     Yet another aspect of the present disclosure is to provide a stochastic precoding design and stochastic user selection method and apparatus of a BS based on a feedback channel information received from a user in a multi-antenna multiuser radio communication system. 
     In accordance with an aspect of the present disclosure, a signal transmission/reception method of a terminal for use in a mobile communication system is provided. The signal transmission/reception method includes receiving a reference signal transmitted by a base station, estimating channel information based on the reference signal, predicting channel estimation error based on the channel information, and transmitting feedback information generated based on the channel estimation error to the base station. 
     In accordance with another aspect of the present disclosure, a signal transmission/reception method of a base station for use in a mobile communication system is provided. The signal transmission/reception method includes transmitting a reference signal to a terminal and receiving feedback information generated based on the reference signal, wherein the feedback information is generated based on channel estimation error by the terminal on the basis of the reference signal. 
     In accordance with another aspect of the present disclosure, a terminal of transmitting receiving signals for use in a mobile communication system is provided. The terminal includes a transceiver which transmits and receives signals to and from a base station and a control unit which controls the transceiver to receive a reference signal from the base station, estimates channel information based on the reference signal, predicts channel estimation error based on the channel information, and controls the transceiver to transmit feedback information generated based on the channel estimation error. 
     In accordance with another aspect of the present disclosure, a BS of transmitting receiving signals for use in a mobile communication system is provided. The BS includes a transceiver which transmits and receives signals to and from a terminal and a control unit which controls the transceiver to transmit a reference signal to the terminal and receive feedback information generated based on the reference signal, wherein the feedback information is generated based on channel estimation error by the terminal on the basis of the reference signal. 
     Other aspects, advantages, and salient features of the disclosure will become apparent to those skilled in the art from the following detailed description, which, taken in conjunction with the annexed drawings, discloses various embodiments of the present disclosure. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The above and other aspects, features, and advantages of certain embodiments of the present disclosure will be more apparent from the following description taken in conjunction with the accompanying drawings, in which; 
         FIG. 1  is a diagram illustrating a system Model of (M, N 1 , N 2 ) Multiple Input and Multiple Output (MIMO) Broadcast Channel with imperfect Channel State Information at the Transmitter (CSIT) according to an embodiment of the present disclosure; 
         FIG. 2  is a diagram illustrating a CSIT pattern according to an embodiment of the present disclosure; 
         FIG. 3  is a diagram illustrating received signals at each receiver according to an embodiment of the present disclosure; and 
         FIG. 4  is a diagram illustrating received signals at each receiver according to another embodiment of the present disclosure. 
     
    
    
     Throughout the drawings, it should be noted that like reference numbers are used to depict the same or similar elements, features, and structures. 
     DETAILED DESCRIPTION 
     The following description with reference to the accompanying drawings is provided to assist in a comprehensive understanding of various embodiments of the present disclosure as defined by the claims and their equivalents. It includes various specific details to assist in that understanding but these are to be regarded as merely exemplary. Accordingly, those of ordinary skill in the art will recognize that various changes and modifications of the various embodiments described herein can be made without departing from the scope and spirit of the present disclosure. In addition, descriptions of well-known functions and constructions may be omitted for clarity and conciseness. 
     The terms and words used in the following description and claims are not limited to the bibliographical meanings, but, are merely used by the inventor to enable a clear and consistent understanding of the present disclosure. Accordingly, it should be apparent to those skilled in the art that the following description of various embodiments of the present disclosure is provided for illustration purpose only and not for the purpose of limiting the present disclosure as defined by the appended claims and their equivalents. 
     It is to be understood that the singular forms “a,” “an,” and “the” include plural referents unless the context clearly dictates otherwise. Thus, for example, reference to “a component surface” includes reference to one or more of such surfaces. 
     Some of elements are exaggerated, omitted or simplified in the drawings and the elements may have sizes and/or shapes different from those shown in drawings, in practice. The same reference numbers are used throughout the drawings to refer to the same or like parts. 
     It will be understood that each block of the flowchart illustrations and/or block diagrams, and combinations of blocks in the flowchart illustrations and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions/acts specified in the flowchart and/or block diagram block or blocks. These computer program instructions may also be stored in a non-transitory computer-readable recording medium that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the non-transitory computer-readable recording medium produce an article of manufacture including instruction means which implement the function/act specified in the flowchart and/or block diagram block or blocks. The computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions/acts specified in the flowchart and/or block diagram block or blocks. 
     Furthermore, the respective block diagrams may illustrate parts of modules, segments or codes including at least one or more executable instructions for performing specific logic function(s). Moreover, it should be noted that the functions of the blocks may be performed in different order in several modifications. For example, two successive blocks may be performed substantially at the same time, or may be performed in reverse order according to their functions. 
     The term “module” according to the various embodiments of the disclosure, means, but is not limited to, a software or hardware component, such as a Field Programmable Gate Array (FPGA) or Application Specific Integrated Circuit (ASIC), which performs certain tasks. A module may advantageously be configured to reside on the addressable storage medium and configured to be executed on one or more processors. Thus, a module may include, by way of example, components, such as software components, object-oriented software components, class components and task components, processes, functions, attributes, procedures, subroutines, segments of program code, drivers, firmware, microcode, circuitry, data, databases, data structures, tables, arrays, and variables. The functionality provided for in the components and modules may be combined into fewer components and modules or further separated into additional components and modules. In addition, the components and modules may be implemented such that they execute one or more Central Processing Units (CPUs) in a device or a secure multimedia card. 
     The various embodiments of the present disclosure are to be regarded in an illustrative rather than a restrictive sense in order to help understand the present disclosure. It is obvious to those skilled in the art that the present disclosure is applicable to other radio communication systems with appropriate modifications and changes without departing from the broader spirit and scope of the disclosure. 
     Although the description has been made with reference to particular embodiments, the present disclosure can be implemented with various modifications without departing from the scope of the present disclosure. Thus, the present disclosure is not limited to the particular embodiments disclosed but will include the following claims and their equivalents. 
     Although various embodiments of the disclosure have been described using specific terms, the specification and drawings are to be regarded in an illustrative rather than a restrictive sense in order to help understand the present disclosure. It is obvious to those skilled in the art that various modifications and changes can be made thereto without departing from the broader spirit and scope of the disclosure. 
     According to various embodiments of the present disclosure, the method of at enhancing the sum Degrees of Freedom (DoF) performance based on the integration of private messages and common messages as that in the Multiple Input and Single Output (MISO) case is suggested. The key ingredients lie in 1) always sending common messages and private symbols intended for Rx 2  with the DoF N 2  and try to send private symbols to Rx 1  with the DoF as large as possible; 2) design the transmission (e.g., precoder, power allocation, etc.) of common messages in order to achieve the target DoF for common messages at both receivers. 
     The following notations are used in the rest of this text. Bold lower letters stand for vectors whereas a symbol not in bold font represents a scalar. (•) T  and (•) H  represent the transpose and conjugate transpose of a matrix or vector respectively. N(H) and R(H) respectively stand for the null space and the range of H. E[•] refers to the expectation of a random variable, vector or matrix. ∥•∥ is the norm of a vector. f(P)˜P B  corresponds to 
                 lim     P   -&gt;   ∞       ⁢       log   ⁢           ⁢     f   ⁡     (   P   )           log   ⁢           ⁢   P         =   B         
where P is SNR throughout the paper and logarithms are in base 2. (x) +  means max(x, 0).gcd(N 1 , N 2 ) is the greatest common devisor of integer N 1  and N 2 .
 
       FIG. 1  is a diagram illustrating a system Model of (M, N 1 , N 2 ) Multiple Input and Multiple Output (MIMO) Broadcast Channel with imperfect Channel State Information at the Transmitter (CSIT) according to an embodiment of the present disclosure. 
     2.1 System Model 
     Referring to  FIG. 1 , consider that a Tx has M antennas and two receivers, Rx 1  and Rx 2 , which respectively have N 1  and N 2  antennas. Without loss of generality, assume N 1 ≦N 2 . In any given subband j, the received signals at Rx 1  and Rx 2  are denoted by y j  and z j , where y j  is a N 1 ×1 vector and z j  is a N 2 ×1 vector. ε j1  and ε j2  are respectively the noise vector observed by Rx 1  and Rx 2 . ε j1  and ε j2  are said to have a unit covariance matrix. 
     The transmitted signal in subband j is represented by a M×1 vector, s j , subject to the power constraint ∥s j ∥≦P, where P stands for the SNR throughout this document. Similar to the MISO case, the transmitted signal consists of three kinds of messages: 
     Common messages I, denoted as c I  hereafter, are broadcast to both users and unique for each subband. They should be recovered by both users, but can be intended exclusively for user  1  or user  2 ; 
     Common messages II, denoted as c II  hereafter, should be recovered by both users, but can be intended exclusively for user  1  or user  2 . Unlike c I , c II  are broadcast twice, i.e., once in a subband where Rx 1  can decode it and once in a subband where Rx 2  is more capable of decoding it; and 
     Private message, denoted as u j  and v j  subband j, are respectively intended for user  1  or user  2  only. u j  and v j  are respectively N 1 ×1 and N 2 ×1 vectors. Private symbols intended for Rx 1  and Rx 2  are hereafter expressed as PS 1  and PS 2 . 
     H j  and G j  respectively represent the channel of Rx 1  and Rx 2  in subband j. H j  is a M×N 1  complex Gaussian matrix with identity covariance. H j  likely has full-rank. Similarly, G j  is a full-rank M×N 2  complex Gaussian matrix with identity covariance. As a start point, consider that M≧N 1 +N 2 . 
     Assume a general setup (valid for both Frequency Division Duplex (FDD) and Frequency Division Duplex (TDD)) where the transmitter obtains the CSI instantaneously, but with imperfectness, due to the estimation error and/or finite rate in the feedback link. 
     Denoting Ĥ j  and Ĝ j  as the imperfect CSI of Rx 1  and Rx 2  in subband j respectively, the CSI of user  1  and user  2  can be respectively modeled as:
 
 H   j   =Ĥ   j   +{tilde over (H)}   j   ,G   j   =Ĝ   j   +{tilde over (G)}   j   Equation 1
 
where {tilde over (H)} j  and {tilde over (G)} j  are the corresponding error vectors. Each column of {tilde over (H)} j  has the same normE[{tilde over (h)} ji   H {tilde over (h)} ji ]=σ j1   2 . Similarly, each column in {tilde over (G)} j  has the same normE[{tilde over (g)} ji   H {tilde over (g)} ji ]=σ j2   2 . Ĥ j  and Ĝ j  are respectively independent of {tilde over (H)} j  and {tilde over (G)} j . The norm of the columns in Ĥ j  and Ĝ j  scale as P 0  when P→∞. Ĥ j  and Ĝ j  are obtained by both users and the transmitter, but H j  and G j  are only known by Rx 1  and Rx 2  respectively.
 
       FIG. 2  is a diagram illustrating a CSIT pattern according to an embodiment of the present disclosure. 
     Referring to  FIG. 2 , to investigate the impact of the imperfect CSIT on the DoF performance, assume that the variance of the norm of each column in the error matrix exponentially scales with SNR, namely σ j1   2 ˜P −a     j    and σ j2   2 ˜P −b     j   . a j  and b j  are respectively interpreted as the qualities of the CSIT of Rx 1  and Rx 2  in subband j and given as follows: 
     
       
         
           
             
               
                 
                   
                     
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     a j  and b j  vary within the range of [0,1]. a j =1 (resp. b j =1) is equivalent to perfect CSIT because the full DoF performance, i.e., (d 1 , d 2 )=(N 1 , N 2 ), can be achieved by simply performing Zero-Forcing Beamforming (ZFBF). a j =0 (resp. b j =0) is equivalent to no CSIT because it means that the variance of the CSI error scales as P 0 , such that the imperfect CSIT cannot benefit the DoF when doing ZFBF. Besides, a j  and b j  vary across all the L subbands as shown in  FIG. 2 . 
     It is important to note the following quantities,
 
 E   H,N(H     j     )   [H   j   H   N ( H   j )]= E   H,N(H     j     ) [( Ĥ   j   H   +{tilde over (H)}   j   H ) N ( H   j )]= E   H,N(H     j     )   [{tilde over (H)}   j   H   N ( H   j )]˜ P   −a     j     I   N     1     Equation 3
 
 E   G,N(G     j     )   [G   j   H   N ( G   j )]= E   G,N(G     j     ) [( Ĝ   j   H   +{tilde over (G)}   j   H ) N ( G   j )]= E   G,N(G     j     )   [{tilde over (G)}   j   H   N ( G   j )]˜ P   −b     j     I   N     2     Equation 4
 
as they are frequently used in the rest of this disclosure.
 
2.2 Reusing MISO Case Scheme in the (M, N 1 , N 2 ) MIMO Broadcast Channel
 
     The main ingredient in the MISO case scheme is sending the private symbols via partial-ZFBF while transmitting common messages using the remaining power. The power and rate allocated to each private symbol and common symbol are determined based on the CSIT quality pattern. Besides, the common symbols are projected on the space spanned by the channels of both receivers. Generally, reusing the scheme in the (M, N 1 , N 2 ) case, the transmitted signal in subband j is given by: 
                     s   j     =           CS   ⁢           ⁢   1     +     CS   ⁢           ⁢   2         ︸       (     P   -     P     max   ⁡     (       a   j     ,     b   j       )           )     ⁢     I     N   1             +         N   ⁡     (     G   ^     )       ⁢     u   j         ︸       P   b     ⁢     j     I     N   1               +         N   ⁡     (     H   ^     )       ⁢     v   j         ︸       P   a     ⁢     j     I     N   2                         Equation   ⁢           ⁢   5               
where N(Ĝ) is a M×N 1  precoding matrix spanning the null space of Ĝ. Since the nullity of Ĝ is M−N 2  and N 1 &lt;M−N 2  by assumption, it is possible to send N 1  symbols to the orthogonal space of Ĝ. Similarly, N(Ĥ) is a M×N 2  precoding matrix spanning the null space of Ĥ. The received signals are expressed as:
 
     
       
         
           
             
               
                 
                   
                       
                   
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     Counting all the private symbols sent in the L subbands, u 1:L  achieve the 
                 DoF   ⁢         ∑     j   =   1     L     ⁢           ⁢       N   1     ⁢     b   j         L       =       N   1     ⁢     b   _         ,         
while v 1:L  achieve the
 
                 DoF   ⁢         ∑     j   =   1     L     ⁢           ⁢       N   2     ⁢     a   j         L       =       N   2     ⁢     a   _         ,         
because the private symbols are allocated with the power subject to the quality of the CSIT of their unintended receivers and are drowned by the noise.
 
     By treating the private symbols as noise, Rx 1  can decode common messages (including both CS 1  and CS 2 ) with the 
               DoF   ⁢           N   1     ⁢       ∑     j   =   1     L     ⁢           ⁢   1       -     b   j       L       ,         
while Rx 2  can decode them with the
 
               DoF   ⁢           N   2     ⁢       ∑     j   =   1     L     ⁢           ⁢   1       -     a   j       L       ,         
As the common messages should be decodable by both receivers, the achievable DoF is N 1 (1−max(ā,  b )).
 
     However, there are three problems limiting the sum DoF performance. 
     Space Resource Wasted at Rx 2   
     With the MISO scheme, N 1  streams of common messages are sent at a time so as to make them decodable at both receivers. This causes N 2 −N 1  dimensions of the received signals at Rx 2  unused when decoding the common messages. Intuitively, the sum DoF performance can be improved if the resource (space, frequency and power) at Rx 2  are fully employed to since Rx 2  has a larger inherent DoF (see Section 2.3 and MC 1 for details). 
     Achievable DoF of Common Messages 
     Definition of Inherent DoF for common messages: By treating the private symbols as noise, the inherent DoF for common messages at Rx 1  and Rx 2  are respectively N 1 (1− b ) and N 2 (1−ā). 
     The ideal DoF of the common messages is determined by the minimum value of them, namely min(N 1 (1− b ), N 2 (1−ā)). However, this is not achievable by reusing the MISO case scheme. Specifically, simply considering a one-subband case with N 1 (1− b )=N 2 (1−ā), Rx 1  can decode each stream with a high DoF but the number of streams is small, while Rx 2  can decode more symbols but each with a small DoF. In the MISO scheme, the common messages are projected to the space spanning the channels of both receivers, therefore the common messages can only be decoded with the DoF min(N 1 , N 2 )×min(1−a, 1−b), namely min(N 1 , N 2 ) streams are sent, each of them has the DoF min(1−a, 1−b). Otherwise, at least one receiver will fail to decode them. Hence, a challenge is to design a transmission strategy to enhance the DoF of common messages such that both receivers can decode them at the same time. 
     Transmission Strategy when Rx 2  has a Larger Inherent DoF for the Common Messages than Rx 1   
     As a reminder, the MISO scheme said that when ā&lt; b  (or 1−ā&gt;1− b , namely Rx 2  has larger inherent DoF for the common messages than Rx 1  if (N 1 , N 2 )=(1,1)) is replaced, the transmission strategy design starts with finding b′ 1:L  such that  b ′=ā and b′ j ≦b j , ∀j. This mechanism is interpreted as decreasing the power and DoF allocated to PS 1  and in turn the DoF of the common messages is increased. This mechanism does not harm the sum DoF performance since both receivers are having the same dimensions in their received signal and the common messages contribute to the sum DoF equivalently as the private symbols. 
     However, this equivalence does not hold generally in the MIMO case due to the discrepancy between the dimensions of the received signals at each receiver. Considering one-subband scenario and Rx 2  has a larger inherent DoF for the common messages, namely N 2 (1−a)&gt;N 1 (1−b), the power and DoF of PS 2  is increased rather than doing the above mechanism. Although a DoF loss might be incurred for the private symbols intended for Rx 1 , the sum DoF improves because the private symbols intended for Rx 2  contribute more than Rx 1 . 
     Moreover, in multiple-subband scenario with varying CSIT qualities, increasing the DoF of PS 2  in different subbands might cause a different DoF loss of PS 1 . Consequently, the CSIT quality pattern will have an impact on the sum DoF performance, when Rx 2  has a larger inherent DoF for the common messages, namely N 2 (1−ā)&gt;N 1 (1− b ). 
     In the next section, the main claims are provided which address the proposed problems and lead to an enhanced sum DoF performance. 
     2.3 Main Claims (MC) 
     In this section, MC 1 addresses the first and third problems in Section 2.2 while MC 2 and MC 3 address the solution to the second problem. 
     MC 1 
     A transmission strategy for two-receiver MIMO BC with imperfect CSIT comprises: 1) sending common messages (CS 1  and CS 2 ) and private messages (PS 1  and PS 2 ); 2) sending CS 1 , CS 2  and PS 2  with the DoF (d c , d 2 ), such that d c +d 2 =N 2 ; 3) sending PS 1  with the DoF as high as possible while guaranteeing d c +d 2 =N 2 ; and 4) based on the CSIT quality pattern, determining the number, power and precoder for each kind of message. 
     i. When N 1 (1− b )≧N 2 (1−ā), similar to the MISO case, the Tx 1 ) sends common messages with the DoF N 1 (1− b ); 2) sends PS 1  with the DoF N 1   b  and PS 2  with the DoF N 2 −N 1 (1− b )=N 2 (1−ā′); 3) the power allocated to PS 2  in each subband is P a′     j   , where a′ j ≦a j . 
     ii. When N 1 (1− b )&lt;N 2 (1−ā), unlike the MISO case, the DoF of common messages, PS 1  and PS 2  depend on the specific CSIT pattern. 
     iii. In MC 1.iii, the Tx sends PS 2  in each subband with power P a′     j   , a′ j ≧a j , and they are determined by finding the equality regarding the inherent DoF at each Rx, namely 
     
       
         
           
             
               
                 1 
                 L 
               
               ⁢ 
               
                 N 
                 1 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     j 
                     = 
                     1 
                   
                   L 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   ( 
                   
                     1 
                     - 
                     
                       max 
                       ⁡ 
                       
                         ( 
                         
                           
                             b 
                             j 
                           
                           , 
                           
                             
                               a 
                               j 
                             
                             - 
                             
                               a 
                               j 
                             
                           
                         
                         ) 
                       
                     
                   
                   ) 
                 
               
             
             = 
             
               
                 1 
                 L 
               
               ⁢ 
               
                 N 
                 2 
               
               ⁢ 
               
                 
                   ∑ 
                   
                     j 
                     = 
                     1 
                   
                   L 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     ( 
                     
                       1 
                       - 
                       
                         a 
                         j 
                         ′ 
                       
                     
                     ) 
                   
                   . 
                 
               
             
           
         
       
     
     iv. In MC 1.iv, the value of either side of equality is determined as the DoF for common messages. 
     v. In MC 1.iv, the calculation of a′ j  starts from the subband with lowest b j . 
     vi. Unlike the MISO case, the common messages are transmitted using the strategy given in MC 2 and MC 3 because the two receivers employ a different number of antennas to decode them. 
     Intuition 1 of MC 1: Sending PS 1  with power higher than P b     j    is equivalent with sending them with power P b     j   . Generally, the following equalities regarding the inherent DoF at each receiver and achieved DoF tuple (d c , d 1 , d 2 ) can be written as:
 
 N   1   =d   c   +d   1   +I   1   Equation 8
 
 N   2   =d   c   +d   2   +I   2   Equation 9
 
where I 2  is the interference caused by PS 1  because they are sent with power higher than P b     j   . Now the power of PS 1  is reduced to P b     j   . Consequently, there is no interference caused at Rx 2  and in turn the DoF of PS 2  is increased to d′ 2 =d 2 +I 2 . At the same time, the DoF of PS 1  is decreased to d′ 1 =(d 1 −I 2 ) + . Besides, the DoF of common messages remains. The resulted sum DoF is d c +d′ 1 +d′ 2 =d c +d 2 +I 2 +(d 1 −I 2 ) + =d 1 +d 2 +d c +(I 2 −d 1 )+≧d 1 +d 2 +d c .
 
     Intuition 2: When Rx 2  has a larger inherent DoF, increasing the power allocated to PS 2  with the DoF d′ c +d′ 2 =N 2  results in a better sum DoF performance. Here, considering Rx 2  has a larger inherent DoF, I 2  in the equation N 2 =d c +d 2 +I 2  stands for the unused resource. 
     Now, the DoF of PS 2  d′ 2  is increased, where d′ 2 =d 2 +I 2 . Since PS 2  consists of N 2  streams, the increment in each stream is 
                 I   2       N   2       .         
Moreover, as the received signal at Rx 1  has N 1 (≦N 2 ) dimensions, this increment results it
 
                 I   2       N   2       ×     N   1           
interference at Rx 1 . Hence, if
 
                 d   1     ≥         I   2       N   2       ⁢     N   1         ,         
the DoF of PS 1  becomes
 
               d   1   ′     =       d   1     -         I   2       N   2       ⁢     N   1               
and the DoF of common messages remains. The equalities are
 
               N   1     =         d   c     +     d   1   ′     +         I   2       N   2       ⁢     N   1       +       I   1     ⁢           ⁢   and   ⁢           ⁢     N   2         =       d   c     +       d   2   ′     .               
Apparently, the resulted sum DoF is improved because the increase in the DoF of PS 1  is greater than the loss of the DoF of PS 2 .
 
     If 
                 d   1     &lt;         I   2       N   2       ⁢     N   1         ,         
the Tx sends PS 2  and common messages with d′ c +d′ 2 =N 2  and d′ 2 &gt;d 2 +I 2 . In this case, the DoF of PS 1  becomes zero. The resulted sum DoF is d′ c +d′ 2 =(d c +d 2 +I 2 )+d 1 −d 1 =(d c +d 1 +d 2 )+(I 2 −d 1 )&gt;d c +d 1 +d 2 .
 
     Combining these two intuitions, the key ingredients lie in always sending common messages and PS 2  with the DoF d c +d 2 =N 2 . Refer to Section 2.4.3 and 2.5.4 for more details. 
     MC 2 
     The transmission of CS 1  comprises: properly determining the number, power and precoder of CS 1 , such that Rx 1  decodes them using N 1  antennas and at the same time Rx 2  decodes them with N 2  antennas. More specifically, 
     i. t{circumflex over (N)} 1 {circumflex over (N)} 2 CS 1  are sent, where 
     
       
         
           
             
               t 
               = 
               
                 gcd 
                 ⁡ 
                 
                   ( 
                   
                     
                       N 
                       1 
                     
                     , 
                     
                       N 
                       2 
                     
                   
                   ) 
                 
               
             
             , 
             
               
                 
                   N 
                   ^ 
                 
                 1 
               
               = 
               
                 
                   N 
                   1 
                 
                 t 
               
             
             , 
             
               
                 
                   
                     N 
                     ^ 
                   
                   2 
                 
                 = 
                 
                   
                     N 
                     2 
                   
                   t 
                 
               
               ; 
             
           
         
       
     
     ii. Each CS 1  achieves the 
                 DoF   ⁢           ⁢   Δ     =       d   c       t   ⁢       N   ^     1     ⁢       N   ^     2           ,         
where d c  is the target DoF of CS 1 ; and
 
     iii. With n=1, 2, . . . , t{circumflex over (N)} 1 {circumflex over (N)} 2  denoting the index of the CS 1 , 
     For those 
                 ⌈     n     N   1       ⌉     =     ⌈     n     N   2       ⌉       ,         
the precoder is a vector in the space spanned by the channels of both receivers. The power allocated to them is
 
               P     1   -       (       ⌈     n     N   1       ⌉     -   1     )     ⁢   Δ         ;         
and
 
     For those 
                 ⌈     n     N   1       ⌉     &gt;     ⌈     n     N   2       ⌉       ,         
the precoder is a summation of two parts: 1) a vector in the space spanned by the channels of both receivers and with power
 
               P     1   -       (       ⌈     n     N   1       ⌉     -   1     )     ⁢   Δ         ;         
2) a vector in the null space of the channel of Rx 1  and with power
 
     
       
         
           
             
               P 
               
                 1 
                 - 
                 
                   
                     ( 
                     
                       
                         ⌈ 
                         
                           n 
                           
                             N 
                             2 
                           
                         
                         ⌉ 
                       
                       - 
                       1 
                     
                     ) 
                   
                   ⁢ 
                   Δ 
                 
               
             
             . 
           
         
       
     
     As a consequence of the precoder design and power allocation, Rx 1  will observes the CS 1  in {circumflex over (N)} 2  power levels and each level contains N 1  unique CS 1 , while Rx 2  will observes the CS 1  in {circumflex over (N)} 1  power levels and each level contains N 2  unique CS 1 . Refer to Section 2.4.1 for details of the transmission design. 
     MC 3 
     Transmission of CS 2  comprises: properly determining the number and power of the CS 2 , such that Rx 1  decodes them using N 1  antennas in the subband where Rx 1  has a large inherent DoF for the common messages, and Rx 2  decodes them using N 2  antennas in the subband where Rx 2  has a large inherent DoF for the common messages. More specifically, 
     i. t{circumflex over (N)} 1 {circumflex over (N)} 2  common symbols are sent, where t=gcd(N 1 , N 2 ), 
     
       
         
           
             
               
                 
                   N 
                   ^ 
                 
                 1 
               
               = 
               
                 
                   N 
                   1 
                 
                 t 
               
             
             , 
             
               
                 
                   
                     N 
                     ^ 
                   
                   2 
                 
                 = 
                 
                   
                     N 
                     2 
                   
                   t 
                 
               
               ; 
             
           
         
       
     
     ii. Each common symbol achieves the 
                 DoF   ⁢           ⁢   δ     =       d   c       t   ⁢           ⁢       N   ^     1     ⁢       N   ^     2           ,         
where d c  is the target DoF of CS 2 ; and
 
     iii. Without loss of generality, assume that Rx 1  decodes them in subband  1  while Rx 2  recovers them in subband  2 . These t{circumflex over (N)} 1 {circumflex over (N)} 2  common symbols are sent twice: 
     In subband  1 , the CS 2  with index n (where n=1, 2, . . . , t{circumflex over (N)} 1 {circumflex over (N)} 2 ) is sent with the precoder in the space spanned by the channel of Rx 1  and allocated with power 
               P     1   -       (       ⌈     n     N   1       ⌉     -   1     )     ⁢   Δ         ;         
and
 
     In subband  2 , the CS 2  with index n (where n=1, 2, . . . , t{circumflex over (N)} 1 {circumflex over (N)} 2 ) is sent with the precoder in the space spanned by the channel of Rx 2  and allocated with power 
     
       
         
           
             
               P 
               
                 1 
                 - 
                 
                   
                     ( 
                     
                       
                         ⌈ 
                         
                           n 
                           
                             N 
                             2 
                           
                         
                         ⌉ 
                       
                       - 
                       1 
                     
                     ) 
                   
                   ⁢ 
                   Δ 
                 
               
             
             . 
           
         
       
     
     As a consequence, Rx 1  will observe the CS 2  in {circumflex over (N)} 2  power levels and each level contains N 1  unique CS 2  while Rx 2  will observe the CS 2  in {circumflex over (N)} 1  power levels and each level contains N 2  unique CS 2 . In this way, Rx 1  (resp. Rx 2 ) can decode them from subband  1  (resp. subband  2 ) and remove the CS 2  in subband  2  (resp. subband  1 ) so as to decode other symbols in subband  2  (resp. subband  1 ). Refer to Section 2.5.1 for more details. 
     Note that when N 1 =N 2 , MC 2 and MC 3 will become the same and similar as that in MISO scheme. 
     2. 4 Transmission Strategy for One Subband Scenario 
     2.4.1. N 1 (1−b)=N 2 (1−a), Two Receivers have the Same Inherent DoF for Common Messages 
     N 1  PS 1  are sent with the power P b  and the precoder N(Ĝ), while N 2 PS 2  are sent with power P a  and the precoder N(Ĥ). 
     According to MC 2, the t{circumflex over (N)} 1 {circumflex over (N)} 2  CS 1  are received by Rx 1  in N 2  power levels and each level contains N 1  unique CS 1 , while they are received by Rx 2  in {circumflex over (N)} 1  power levels and each level contains N 2  unique CS 1 . Note that higher index of the level refers to a lower received power. The gap between two adjacent power levels is Δ. 
     It is obvious that some CS 1  are received in the same power level at each receiver (namely, 
                 ⌈     n     N   1       ⌉     =     ⌈     n     N   2       ⌉       ,         
n is the index of the CS 1 ) while others are received in the difference power levels (namely,
 
                   ⌈     n     N   1       ⌉     &gt;     ⌈     n     N   2       ⌉       )     .         
To this end, the precoder of these CS 1  are designed to make use of the null space of the CSIT of Rx 1 . More specifically,
 
     {circumflex over (N)} 1 {circumflex over (N)} 2 t different CS 1  are sent, each achieves the 
     
       
         
           
             
               
                 DoF 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 Δ 
               
               = 
               
                 
                   
                     1 
                     - 
                     a 
                   
                   
                     
                       N 
                       ^ 
                     
                     1 
                   
                 
                 = 
                 
                   
                     1 
                     - 
                     b 
                   
                   
                     
                       N 
                       ^ 
                     
                     2 
                   
                 
               
             
             ; 
           
         
       
     
     For those 
                 ⌈     n     N   1       ⌉     =     ⌈     n     N   2       ⌉       ,         
the precoder is a vector in the space spanned by the channels of both receivers. The power allocated to them is
 
               P     1   -       (       ⌈     n     N   2       ⌉     -   1     )     ⁢   Δ         .         
These CS 1  can be expressed
 
     
       
         
           
             
               
                 
                   R 
                   ⁡ 
                   
                     ( 
                     
                       
                         H 
                         ^ 
                       
                       , 
                       
                         G 
                         ^ 
                       
                     
                     ) 
                   
                 
                 ⁢ 
                 
                   c 
                   n 
                 
               
               
                 ︸ 
                 
                   P 
                   
                     1 
                     - 
                     
                       
                         ( 
                         
                           
                             ⌈ 
                             
                               n 
                               
                                 N 
                                 2 
                               
                             
                             ⌉ 
                           
                           - 
                           1 
                         
                         ) 
                       
                       ⁢ 
                       Δ 
                     
                   
                 
               
             
             ; 
           
         
       
     
     For those 
                 ⌈     n     N   1       ⌉     &gt;     ⌈     n     N   2       ⌉       ,         
the precoder is a summation of two parts: 1) a vector in the space spanned by the channels of both receivers with power
 
               P     1   -       (       ⌈     n     N   1       ⌉     -   1     )     ⁢   Δ         ;         
2) a vector in the null space of Ĥ with power
 
               P     1   -       (       ⌈     n     N   2       ⌉     -   1     )     ⁢   Δ         .         
These CS 1  can be written as
 
                 (         R   ⁡     (       H   ^     ,     G   ^       )         ︸     P     1   -       (       ⌈     n     N   1       ⌉     -   1     )     ⁢   Δ             +       N   ⁡     (     H   ^     )         ︸     P     1   -       (       ⌈     n     N   2       ⌉     -   1     )     ⁢   Δ               )     ⁢     c   n       ,         
where the first term is dominant at Rx 1  while the second term is dominant at Rx 2 .
 
     The transmitted signal is expressed as: 
     
       
         
           
             
               
                 
                   s 
                   = 
                   
                     
                       ( 
                       
                         
                           ∑ 
                           
                             
                               n 
                               = 
                               1 
                             
                             , 
                             
                               
                                 ⌈ 
                                 
                                   n 
                                   
                                     N 
                                     1 
                                   
                                 
                                 ⌉ 
                               
                               = 
                               
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                                   n 
                                   
                                     N 
                                     2 
                                   
                                 
                                 ⌉ 
                               
                             
                           
                           
                             
                               
                                 N 
                                 ^ 
                               
                               1 
                             
                             ⁢ 
                             
                               
                                 N 
                                 ^ 
                               
                               2 
                             
                             ⁢ 
                             t 
                           
                         
                         ⁢ 
                         
                           
                             
                               R 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     H 
                                     ^ 
                                   
                                   , 
                                   
                                     G 
                                     ^ 
                                   
                                 
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                             ⁢ 
                             
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                                     ( 
                                     
                                       
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                                           n 
                                           
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                                       - 
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                                     ) 
                                   
                                   ⁢ 
                                   Δ 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     + 
                     
                       ( 
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 = 
                                 1 
                               
                               , 
                               
                                 
                                   ⌈ 
                                   
                                     n 
                                     
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                                   ⌉ 
                                 
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                                   ⌈ 
                                   
                                     n 
                                     
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                                       2 
                                     
                                   
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                           ⁢ 
                           
                             
                               
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                                 ⁡ 
                                 
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                       ) 
                     
                     + 
                     
                       
                         
                           N 
                           ⁡ 
                           
                             ( 
                             
                               G 
                               ^ 
                             
                             ) 
                           
                         
                         ⁢ 
                         u 
                       
                       
                         ︸ 
                         
                           
                             P 
                             b 
                           
                           ⁢ 
                           
                             I 
                             
                               N 
                               1 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         
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                           ⁡ 
                           
                             ( 
                             
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                             ) 
                           
                         
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                         v 
                       
                       
                         ︸ 
                         
                           
                             P 
                             a 
                           
                           ⁢ 
                           
                             I 
                             
                               N 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   10 
                 
               
             
           
         
       
     
     1) The received signal at Rx 1  is: 
     
       
         
           
             
               
                 
                   
                     
                       
                         y 
                         = 
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 ∑ 
                                 
                                   
                                     n 
                                     = 
                                     1 
                                   
                                   , 
                                   
                                     
                                       ⌈ 
                                       
                                         n 
                                         
                                           N 
                                           1 
                                         
                                       
                                       ⌉ 
                                     
                                     = 
                                     
                                       ⌈ 
                                       
                                         n 
                                         
                                           N 
                                           2 
                                         
                                       
                                       ⌉ 
                                     
                                   
                                 
                                 
                                   
                                     
                                       N 
                                       ^ 
                                     
                                     1 
                                   
                                   ⁢ 
                                   
                                     
                                       N 
                                       ^ 
                                     
                                     2 
                                   
                                   ⁢ 
                                   t 
                                 
                               
                               ⁢ 
                               
                                 
                                   
                                     H 
                                     H 
                                   
                                   ⁢ 
                                   
                                     R 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           H 
                                           ^ 
                                         
                                         , 
                                         
                                           G 
                                           ^ 
                                         
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     c 
                                     n 
                                   
                                 
                                 
                                   ︸ 
                                   
                                     P 
                                     
                                       1 
                                       - 
                                       
                                         
                                           ( 
                                           
                                             
                                               ⌈ 
                                               
                                                 n 
                                                 
                                                   N 
                                                   2 
                                                 
                                               
                                               ⌉ 
                                             
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                         ⁢ 
                                         Δ 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     
                                       n 
                                       = 
                                       1 
                                     
                                     , 
                                     
                                       
                                         ⌈ 
                                         
                                           n 
                                           
                                             N 
                                             1 
                                           
                                         
                                         ⌉ 
                                       
                                       &gt; 
                                       
                                         ⌈ 
                                         
                                           n 
                                           
                                             N 
                                             2 
                                           
                                         
                                         ⌉ 
                                       
                                     
                                   
                                   
                                     
                                       
                                         N 
                                         ^ 
                                       
                                       1 
                                     
                                     ⁢ 
                                     
                                       
                                         N 
                                         ^ 
                                       
                                       2 
                                     
                                     ⁢ 
                                     t 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     
                                       H 
                                       H 
                                     
                                     ⁢ 
                                     
                                       R 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             H 
                                             ^ 
                                           
                                           , 
                                           
                                             G 
                                             ^ 
                                           
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       c 
                                       n 
                                     
                                   
                                   
                                     ︸ 
                                     
                                       P 
                                       
                                         1 
                                         - 
                                         
                                           
                                             ( 
                                             
                                               
                                                 ⌈ 
                                                 
                                                   n 
                                                   
                                                     N 
                                                     1 
                                                   
                                                 
                                                 ⌉ 
                                               
                                               - 
                                               1 
                                             
                                             ) 
                                           
                                           ⁢ 
                                           Δ 
                                         
                                       
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     H 
                                     H 
                                   
                                   ⁢ 
                                   
                                     N 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         H 
                                         ^ 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     c 
                                     n 
                                   
                                 
                                 
                                   ︸ 
                                   
                                     P 
                                     
                                       1 
                                       - 
                                       
                                         
                                           ( 
                                           
                                             
                                               ⌈ 
                                               
                                                 n 
                                                 
                                                   N 
                                                   2 
                                                 
                                               
                                               ⌉ 
                                             
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                         ⁢ 
                                         Δ 
                                       
                                       - 
                                       a 
                                     
                                   
                                 
                               
                             
                             ) 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               
                                 H 
                                 H 
                               
                               ⁢ 
                               
                                 N 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     G 
                                     ^ 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               u 
                             
                             
                               ︸ 
                               
                                 
                                   P 
                                   b 
                                 
                                 ⁢ 
                                 
                                   I 
                                   
                                     N 
                                     1 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               
                                 
                                   H 
                                   H 
                                 
                                 ⁢ 
                                 
                                   N 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       H 
                                       ^ 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 v 
                               
                               + 
                               
                                 ε 
                                 1 
                               
                             
                             
                               ︸ 
                               
                                 
                                   P 
                                   0 
                                 
                                 ⁢ 
                                 
                                   I 
                                   
                                     N 
                                     1 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         ≈ 
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 ∑ 
                                 
                                   
                                     n 
                                     = 
                                     1 
                                   
                                   , 
                                   
                                     
                                       ⌈ 
                                       
                                         n 
                                         
                                           N 
                                           1 
                                         
                                       
                                       ⌉ 
                                     
                                     = 
                                     
                                       ⌈ 
                                       
                                         n 
                                         
                                           N 
                                           2 
                                         
                                       
                                       ⌉ 
                                     
                                   
                                 
                                 
                                   
                                     
                                       N 
                                       ^ 
                                     
                                     1 
                                   
                                   ⁢ 
                                   
                                     
                                       N 
                                       ^ 
                                     
                                     2 
                                   
                                   ⁢ 
                                   t 
                                 
                               
                               ⁢ 
                               
                                 
                                   
                                     H 
                                     H 
                                   
                                   ⁢ 
                                   
                                     R 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           H 
                                           ^ 
                                         
                                         , 
                                         
                                           G 
                                           ^ 
                                         
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     c 
                                     n 
                                   
                                 
                                 
                                   ︸ 
                                   
                                     P 
                                     
                                       1 
                                       - 
                                       
                                         
                                           ( 
                                           
                                             
                                               ⌈ 
                                               
                                                 n 
                                                 
                                                   N 
                                                   2 
                                                 
                                               
                                               ⌉ 
                                             
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                         ⁢ 
                                         Δ 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     ∑ 
                                     
                                       
                                         n 
                                         = 
                                         1 
                                       
                                       , 
                                       
                                         
                                           ⌈ 
                                           
                                             n 
                                             
                                               N 
                                               1 
                                             
                                           
                                           ⌉ 
                                         
                                         &gt; 
                                         
                                           ⌈ 
                                           
                                             n 
                                             
                                               N 
                                               2 
                                             
                                           
                                           ⌉ 
                                         
                                       
                                     
                                     
                                       
                                         
                                           N 
                                           ^ 
                                         
                                         1 
                                       
                                       ⁢ 
                                       
                                         
                                           N 
                                           ^ 
                                         
                                         2 
                                       
                                       ⁢ 
                                       t 
                                     
                                   
                                   ⁢ 
                                   
                                     
                                       
                                         H 
                                         H 
                                       
                                       ⁢ 
                                       
                                         R 
                                         ⁡ 
                                         
                                           ( 
                                           
                                             
                                               H 
                                               ^ 
                                             
                                             , 
                                             
                                               G 
                                               ^ 
                                             
                                           
                                           ) 
                                         
                                       
                                       ⁢ 
                                       
                                         c 
                                         n 
                                       
                                     
                                     
                                       ︸ 
                                       
                                         P 
                                         
                                           1 
                                           - 
                                           
                                             
                                               ( 
                                               
                                                 
                                                   ⌈ 
                                                   
                                                     n 
                                                     
                                                       N 
                                                       1 
                                                     
                                                   
                                                   ⌉ 
                                                 
                                                 - 
                                                 1 
                                               
                                               ) 
                                             
                                             ⁢ 
                                             Δ 
                                           
                                         
                                       
                                     
                                   
                                 
                                 + 
                               
                               ) 
                             
                             ⁢ 
                             
                               
                                 
                                   H 
                                   H 
                                 
                                 ⁢ 
                                 
                                   N 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       G 
                                       ^ 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 u 
                               
                               
                                 ︸ 
                                 
                                   
                                     P 
                                     b 
                                   
                                   ⁢ 
                                   
                                     I 
                                     
                                       N 
                                       1 
                                     
                                   
                                 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               
                                 H 
                                 H 
                               
                               ⁢ 
                               
                                 N 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     H 
                                     ^ 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               v 
                             
                             + 
                             
                               ε 
                               1 
                             
                           
                           
                             ︸ 
                             
                               
                                 P 
                                 0 
                               
                               ⁢ 
                               
                                 I 
                                 
                                   N 
                                   1 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   1 
                                 
                                 
                                   
                                     
                                       N 
                                       ^ 
                                     
                                     1 
                                   
                                   ⁢ 
                                   
                                     
                                       N 
                                       ^ 
                                     
                                     2 
                                   
                                   ⁢ 
                                   t 
                                 
                               
                               ⁢ 
                               
                                 
                                   
                                     H 
                                     H 
                                   
                                   ⁢ 
                                   
                                     R 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           H 
                                           ^ 
                                         
                                         , 
                                         
                                           G 
                                           ^ 
                                         
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     c 
                                     n 
                                   
                                 
                                 
                                   ︸ 
                                   
                                     P 
                                     
                                       1 
                                       - 
                                       
                                         
                                           ( 
                                           
                                             
                                               ⌈ 
                                               
                                                 n 
                                                 
                                                   N 
                                                   1 
                                                 
                                               
                                               ⌉ 
                                             
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                         ⁢ 
                                         Δ 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                           + 
                           
                             
                               
                                 H 
                                 H 
                               
                               ⁢ 
                               
                                 N 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     G 
                                     ^ 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               u 
                             
                             
                               ︸ 
                               
                                 
                                   P 
                                   b 
                                 
                                 ⁢ 
                                 
                                   I 
                                   
                                     N 
                                     1 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               
                                 
                                   H 
                                   H 
                                 
                                 ⁢ 
                                 
                                   N 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       H 
                                       ^ 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 v 
                               
                               + 
                               
                                 ε 
                                 1 
                               
                             
                             
                               ︸ 
                               
                                 
                                   P 
                                   0 
                                 
                                 ⁢ 
                                 
                                   I 
                                   
                                     N 
                                     1 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 
                                   
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           1 
                                         
                                         
                                           
                                             N 
                                             ^ 
                                           
                                           1 
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           
                                             H 
                                             H 
                                           
                                           ⁢ 
                                           
                                             R 
                                             ⁡ 
                                             
                                               ( 
                                               
                                                 
                                                   H 
                                                   ^ 
                                                 
                                                 , 
                                                 
                                                   G 
                                                   ^ 
                                                 
                                               
                                               ) 
                                             
                                           
                                           ⁢ 
                                           
                                             c 
                                             n 
                                           
                                         
                                         
                                           ︸ 
                                           P 
                                         
                                       
                                     
                                     + 
                                     
                                       
                                         ∑ 
                                         
                                           n 
                                           = 
                                           
                                             
                                               N 
                                               1 
                                             
                                             + 
                                             1 
                                           
                                         
                                         
                                           
                                             N 
                                             1 
                                           
                                           × 
                                           2 
                                         
                                       
                                       ⁢ 
                                       
                                         
                                           
                                             H 
                                             H 
                                           
                                           ⁢ 
                                           
                                             R 
                                             ⁡ 
                                             
                                               ( 
                                               
                                                 
                                                   H 
                                                   ^ 
                                                 
                                                 , 
                                                 
                                                   G 
                                                   ^ 
                                                 
                                               
                                               ) 
                                             
                                           
                                           ⁢ 
                                           
                                             c 
                                             n 
                                           
                                         
                                         
                                           ︸ 
                                           
                                             P 
                                             
                                               1 
                                               - 
                                               Δ 
                                             
                                           
                                         
                                       
                                     
                                     + 
                                     … 
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       ∑ 
                                       
                                         n 
                                         = 
                                         
                                           
                                             N 
                                             1 
                                           
                                           + 
                                           
                                             ( 
                                             
                                               
                                                 
                                                   N 
                                                   ^ 
                                                 
                                                 2 
                                               
                                               - 
                                               1 
                                             
                                             ) 
                                           
                                           + 
                                           1 
                                         
                                       
                                       
                                         
                                           N 
                                           1 
                                         
                                         × 
                                         
                                           
                                             N 
                                             ^ 
                                           
                                           2 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       
                                         
                                           H 
                                           H 
                                         
                                         ⁢ 
                                         
                                           R 
                                           ⁡ 
                                           
                                             ( 
                                             
                                               
                                                 H 
                                                 ^ 
                                               
                                               , 
                                               
                                                 G 
                                                 ^ 
                                               
                                             
                                             ) 
                                           
                                         
                                         ⁢ 
                                         
                                           c 
                                           n 
                                         
                                       
                                       
                                         ︸ 
                                         
                                           P 
                                           
                                             1 
                                             - 
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     
                                                       N 
                                                       ^ 
                                                     
                                                     2 
                                                   
                                                   - 
                                                   1 
                                                 
                                                 ) 
                                               
                                               ⁢ 
                                               Δ 
                                             
                                           
                                         
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               
                                 H 
                                 H 
                               
                               ⁢ 
                               
                                 N 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     G 
                                     ^ 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               u 
                             
                             
                               ︸ 
                               
                                 
                                   P 
                                   b 
                                 
                                 ⁢ 
                                 
                                   I 
                                   
                                     N 
                                     1 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               
                                 
                                   H 
                                   H 
                                 
                                 ⁢ 
                                 
                                   N 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       H 
                                       ^ 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 v 
                               
                               + 
                               
                                 ε 
                                 1 
                               
                             
                             
                               ︸ 
                               
                                 
                                   P 
                                   0 
                                 
                                 ⁢ 
                                 
                                   I 
                                   
                                     N 
                                     1 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   11 
                 
               
             
           
         
       
     
     The approximation is due to the fact that H H  R(Ĥ, Ĝ)c n  is dominant compared to H H N(Ĥ)c n  at high SNR, namely 
               1   -       (       ⌈     n     N   1       ⌉     -   1     )     ⁢   Δ       ≥     1   -       (       ⌈     n     N   2       ⌉     -   1     )     ⁢   Δ     -     a   ·     ∀     n   ∈       [     1   ,         N   ^     1     ⁢       N   ^     2     ⁢   t       ]     .                   
This can be verified by replacing
 
             Δ   =       1   -   b       N   2             
and using the fact that
 
     
       
         
           
             
               
                 
                   ⌈ 
                   
                     n 
                     
                       N 
                       1 
                     
                   
                   ⌉ 
                 
                 - 
                 
                   ⌈ 
                   
                     n 
                     
                       N 
                       2 
                     
                   
                   ⌉ 
                 
               
               ≤ 
               
                 
                   N 
                   2 
                 
                 - 
                 
                   
                     N 
                     1 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       N 
                       1 
                     
                     ⁡ 
                     
                       ( 
                       
                         1 
                         - 
                         b 
                       
                       ) 
                     
                   
                 
               
             
             = 
             
               
                 
                   N 
                   2 
                 
                 ⁡ 
                 
                   ( 
                   
                     1 
                     - 
                     a 
                   
                   ) 
                 
               
               . 
             
           
         
       
     
     From the last equation, the N 1 ×N 2  CS 1  received at Rx 1  are written in {circumflex over (N)} 2  terms. Each term consists of N 1  different CS 1  with the same power. Specifically, the m th term (mε1, 2, . . . {circumflex over (N)} 2 ) contains the common symbols with index from (m−1)×N 1 +1 to m×N 1 , and their received power scale as P 1−(M−1)Δ . 
     Decoding at Rx 1  (MMSE-SIC): Since the dimension of received signal is N 1 , Rx 1  decode N 1  CS 1  at a time by treating the r.h.s of them as noise. In this way, each CS 1  recovered in the 1 st  to the (N 2 −1)th term achieves the DoF Δ. Each CS 1  in the last term is recovered by treating H H N(Ĝ)u as noise and the DoF is 1−(N 2 −1)Δ−b=Δ. After that, all the CS 1  have been removed and the private symbols u are decoded only subject to noise. Consequently, the DoF of the common message achieved at Rx 1  is {circumflex over (N)} 1 {circumflex over (N)} 2 tΔ=N 1 (1−b)=N 2 (1−a) and the DoF of private symbols intended for Rx 1  is N 1 b. 
     2) The received signal at Rx 2  is: 
     
       
         
           
             
               
                 
                   
                     
                       
                         z 
                         = 
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 ∑ 
                                 
                                   
                                     n 
                                     = 
                                     1 
                                   
                                   , 
                                   
                                     
                                       ⌈ 
                                       
                                         n 
                                         
                                           N 
                                           1 
                                         
                                       
                                       ⌉ 
                                     
                                     = 
                                     
                                       ⌈ 
                                       
                                         n 
                                         
                                           N 
                                           2 
                                         
                                       
                                       ⌉ 
                                     
                                   
                                 
                                 
                                   
                                     
                                       N 
                                       ^ 
                                     
                                     1 
                                   
                                   ⁢ 
                                   
                                     
                                       N 
                                       ^ 
                                     
                                     2 
                                   
                                   ⁢ 
                                   t 
                                 
                               
                               ⁢ 
                               
                                 
                                   
                                     G 
                                     H 
                                   
                                   ⁢ 
                                   
                                     R 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           H 
                                           ^ 
                                         
                                         , 
                                         
                                           G 
                                           ^ 
                                         
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     c 
                                     n 
                                   
                                 
                                 
                                   ︸ 
                                   
                                     P 
                                     
                                       1 
                                       - 
                                       
                                         
                                           ( 
                                           
                                             
                                               ⌈ 
                                               
                                                 n 
                                                 
                                                   N 
                                                   2 
                                                 
                                               
                                               ⌉ 
                                             
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                         ⁢ 
                                         Δ 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     
                                       n 
                                       = 
                                       1 
                                     
                                     , 
                                     
                                       
                                         ⌈ 
                                         
                                           n 
                                           
                                             N 
                                             1 
                                           
                                         
                                         ⌉ 
                                       
                                       &gt; 
                                       
                                         ⌈ 
                                         
                                           n 
                                           
                                             N 
                                             2 
                                           
                                         
                                         ⌉ 
                                       
                                     
                                   
                                   
                                     
                                       
                                         N 
                                         ^ 
                                       
                                       1 
                                     
                                     ⁢ 
                                     
                                       
                                         N 
                                         ^ 
                                       
                                       2 
                                     
                                     ⁢ 
                                     t 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     
                                       G 
                                       H 
                                     
                                     ⁢ 
                                     
                                       R 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             H 
                                             ^ 
                                           
                                           , 
                                           
                                             G 
                                             ^ 
                                           
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       c 
                                       n 
                                     
                                   
                                   
                                     ︸ 
                                     
                                       P 
                                       
                                         1 
                                         - 
                                         
                                           
                                             ( 
                                             
                                               
                                                 ⌈ 
                                                 
                                                   n 
                                                   
                                                     N 
                                                     1 
                                                   
                                                 
                                                 ⌉ 
                                               
                                               - 
                                               1 
                                             
                                             ) 
                                           
                                           ⁢ 
                                           Δ 
                                         
                                       
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   
                                     G 
                                     H 
                                   
                                   ⁢ 
                                   
                                     N 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         H 
                                         ^ 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     c 
                                     n 
                                   
                                 
                                 
                                   ︸ 
                                   
                                     P 
                                     
                                       1 
                                       - 
                                       
                                         
                                           ( 
                                           
                                             
                                               ⌈ 
                                               
                                                 n 
                                                 
                                                   N 
                                                   2 
                                                 
                                               
                                               ⌉ 
                                             
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                         ⁢ 
                                         Δ 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               
                                 G 
                                 H 
                               
                               ⁢ 
                               
                                 N 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     G 
                                     ^ 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               u 
                             
                             
                               ︸ 
                               
                                 
                                   P 
                                   0 
                                 
                                 ⁢ 
                                 
                                   I 
                                   
                                     N 
                                     1 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               
                                 G 
                                 H 
                               
                               ⁢ 
                               
                                 N 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     H 
                                     ^ 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               v 
                             
                             
                               ︸ 
                               
                                 
                                   P 
                                   a 
                                 
                                 ⁢ 
                                 
                                   I 
                                   
                                     N 
                                     2 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               ε 
                               2 
                             
                             
                               ︸ 
                               
                                 
                                   P 
                                   0 
                                 
                                 ⁢ 
                                 
                                   I 
                                   
                                     N 
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 ∑ 
                                 
                                   
                                     n 
                                     = 
                                     1 
                                   
                                   , 
                                   
                                     
                                       ⌈ 
                                       
                                         n 
                                         
                                           N 
                                           1 
                                         
                                       
                                       ⌉ 
                                     
                                     = 
                                     
                                       ⌈ 
                                       
                                         n 
                                         
                                           N 
                                           2 
                                         
                                       
                                       ⌉ 
                                     
                                   
                                 
                                 
                                   
                                     
                                       N 
                                       ^ 
                                     
                                     1 
                                   
                                   ⁢ 
                                   
                                     
                                       N 
                                       ^ 
                                     
                                     2 
                                   
                                   ⁢ 
                                   t 
                                 
                               
                               ⁢ 
                               
                                 
                                   
                                     G 
                                     H 
                                   
                                   ⁢ 
                                   
                                     R 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           H 
                                           ^ 
                                         
                                         , 
                                         
                                           G 
                                           ^ 
                                         
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     c 
                                     n 
                                   
                                 
                                 
                                   ︸ 
                                   
                                     P 
                                     
                                       1 
                                       - 
                                       
                                         
                                           ( 
                                           
                                             
                                               ⌈ 
                                               
                                                 n 
                                                 
                                                   N 
                                                   2 
                                                 
                                               
                                               ⌉ 
                                             
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                         ⁢ 
                                         Δ 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 ∑ 
                                 
                                   
                                     n 
                                     = 
                                     1 
                                   
                                   , 
                                   
                                     
                                       ⌈ 
                                       
                                         n 
                                         
                                           N 
                                           1 
                                         
                                       
                                       ⌉ 
                                     
                                     &gt; 
                                     
                                       ⌈ 
                                       
                                         n 
                                         
                                           N 
                                           2 
                                         
                                       
                                       ⌉ 
                                     
                                   
                                 
                                 
                                   
                                     
                                       N 
                                       ^ 
                                     
                                     1 
                                   
                                   ⁢ 
                                   
                                     
                                       N 
                                       ^ 
                                     
                                     2 
                                   
                                   ⁢ 
                                   t 
                                 
                               
                               ⁢ 
                               
                                 
                                   
                                     G 
                                     H 
                                   
                                   ⁢ 
                                   
                                     N 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         H 
                                         ^ 
                                       
                                       ) 
                                     
                                   
                                   ⁢ 
                                   
                                     c 
                                     n 
                                   
                                 
                                 
                                   ︸ 
                                   
                                     P 
                                     
                                       1 
                                       - 
                                       
                                         
                                           ( 
                                           
                                             
                                               ⌈ 
                                               
                                                 n 
                                                 
                                                   N 
                                                   2 
                                                 
                                               
                                               ⌉ 
                                             
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                         ⁢ 
                                         Δ 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                           + 
                           
                             
                               
                                 G 
                                 H 
                               
                               ⁢ 
                               
                                 N 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     G 
                                     ^ 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               u 
                             
                             
                               ︸ 
                               
                                 
                                   P 
                                   0 
                                 
                                 ⁢ 
                                 
                                   I 
                                   
                                     N 
                                     1 
                                   
                                 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               
                                 G 
                                 H 
                               
                               ⁢ 
                               
                                 N 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     H 
                                     ^ 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               v 
                             
                             
                               ︸ 
                               
                                 
                                   P 
                                   a 
                                 
                                 ⁢ 
                                 
                                   I 
                                   
                                     N 
                                     2 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               ε 
                               2 
                             
                             
                               ︸ 
                               
                                 
                                   P 
                                   0 
                                 
                                 ⁢ 
                                 
                                   I 
                                   
                                     N 
                                     2 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         ≈ 
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     
                                       
                                         n 
                                         = 
                                         1 
                                       
                                       , 
                                     
                                     
                                       
                                         ⌈ 
                                         
                                           n 
                                           
                                             N 
                                             1 
                                           
                                         
                                         ⌉ 
                                       
                                       = 
                                       
                                         ⌈ 
                                         
                                           n 
                                           
                                             N 
                                             2 
                                           
                                         
                                         ⌉ 
                                       
                                     
                                   
                                   
                                     N 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     
                                       G 
                                       H 
                                     
                                     ⁢ 
                                     
                                       R 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             H 
                                             ^ 
                                           
                                           , 
                                           
                                             G 
                                             ^ 
                                           
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       c 
                                       n 
                                     
                                   
                                   
                                     ︸ 
                                     P 
                                   
                                 
                               
                               + 
                               
                                 
                                   ∑ 
                                   
                                     
                                       
                                         n 
                                         = 
                                         1 
                                       
                                       , 
                                     
                                     
                                       
                                         ⌈ 
                                         
                                           n 
                                           
                                             N 
                                             1 
                                           
                                         
                                         ⌉ 
                                       
                                       &gt; 
                                       
                                         ⌈ 
                                         
                                           n 
                                           
                                             N 
                                             2 
                                           
                                         
                                         ⌉ 
                                       
                                     
                                   
                                   
                                     N 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     
                                       G 
                                       H 
                                     
                                     ⁢ 
                                     
                                       N 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           H 
                                           ^ 
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       c 
                                       n 
                                     
                                   
                                   
                                     ︸ 
                                     P 
                                   
                                 
                               
                             
                             ) 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     
                                       
                                         n 
                                         = 
                                         
                                           
                                             N 
                                             2 
                                           
                                           + 
                                           1 
                                         
                                       
                                       , 
                                     
                                     
                                       
                                         ⌈ 
                                         
                                           n 
                                           
                                             N 
                                             1 
                                           
                                         
                                         ⌉ 
                                       
                                       = 
                                       
                                         ⌈ 
                                         
                                           n 
                                           
                                             N 
                                             2 
                                           
                                         
                                         ⌉ 
                                       
                                     
                                   
                                   
                                     
                                       N 
                                       2 
                                     
                                     × 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     
                                       G 
                                       H 
                                     
                                     ⁢ 
                                     
                                       R 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             H 
                                             ^ 
                                           
                                           , 
                                           
                                             G 
                                             ^ 
                                           
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       c 
                                       n 
                                     
                                   
                                   
                                     ︸ 
                                     
                                       P 
                                       
                                         1 
                                         - 
                                         Δ 
                                       
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   ∑ 
                                   
                                     
                                       
                                         n 
                                         = 
                                         
                                           
                                             N 
                                             2 
                                           
                                           + 
                                           1 
                                         
                                       
                                       , 
                                     
                                     
                                       
                                         ⌈ 
                                         
                                           n 
                                           
                                             N 
                                             1 
                                           
                                         
                                         ⌉ 
                                       
                                       &gt; 
                                       
                                         ⌈ 
                                         
                                           n 
                                           
                                             N 
                                             2 
                                           
                                         
                                         ⌉ 
                                       
                                     
                                   
                                   
                                     
                                       N 
                                       2 
                                     
                                     × 
                                     2 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     
                                       G 
                                       H 
                                     
                                     ⁢ 
                                     
                                       N 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           H 
                                           ^ 
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       c 
                                       n 
                                     
                                   
                                   
                                     ︸ 
                                     
                                       P 
                                       
                                         1 
                                         - 
                                         Δ 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                           + 
                           … 
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 
                                   ∑ 
                                   
                                     
                                       
                                         n 
                                         = 
                                         
                                           
                                             
                                               N 
                                               2 
                                             
                                             ⁡ 
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     N 
                                                     ^ 
                                                   
                                                   1 
                                                 
                                                 - 
                                                 1 
                                               
                                               ) 
                                             
                                           
                                           + 
                                           1 
                                         
                                       
                                       , 
                                     
                                     
                                       
                                         ⌈ 
                                         
                                           n 
                                           
                                             N 
                                             1 
                                           
                                         
                                         ⌉ 
                                       
                                       = 
                                       
                                         ⌈ 
                                         
                                           n 
                                           
                                             N 
                                             2 
                                           
                                         
                                         ⌉ 
                                       
                                     
                                   
                                   
                                     
                                       N 
                                       2 
                                     
                                     × 
                                     
                                       
                                         N 
                                         ^ 
                                       
                                       1 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     
                                       G 
                                       H 
                                     
                                     ⁢ 
                                     
                                       R 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           
                                             H 
                                             ^ 
                                           
                                           , 
                                           
                                             G 
                                             ^ 
                                           
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       c 
                                       n 
                                     
                                   
                                   
                                     ︸ 
                                     
                                       P 
                                       
                                         1 
                                         - 
                                         
                                           
                                             ( 
                                             
                                               
                                                 
                                                   N 
                                                   ^ 
                                                 
                                                 1 
                                               
                                               - 
                                               1 
                                             
                                             ) 
                                           
                                           ⁢ 
                                           Δ 
                                         
                                       
                                     
                                   
                                 
                               
                               + 
                               
                                 
                                   ∑ 
                                   
                                     
                                       
                                         n 
                                         = 
                                         
                                           
                                             
                                               N 
                                               2 
                                             
                                             ⁡ 
                                             
                                               ( 
                                               
                                                 
                                                   
                                                     N 
                                                     ^ 
                                                   
                                                   1 
                                                 
                                                 - 
                                                 1 
                                               
                                               ) 
                                             
                                           
                                           + 
                                           1 
                                         
                                       
                                       , 
                                     
                                     
                                       
                                         ⌈ 
                                         
                                           n 
                                           
                                             N 
                                             1 
                                           
                                         
                                         ⌉ 
                                       
                                       &gt; 
                                       
                                         ⌈ 
                                         
                                           n 
                                           
                                             N 
                                             2 
                                           
                                         
                                         ⌉ 
                                       
                                     
                                   
                                   
                                     
                                       N 
                                       2 
                                     
                                     × 
                                     
                                       
                                         N 
                                         ^ 
                                       
                                       1 
                                     
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     
                                       G 
                                       H 
                                     
                                     ⁢ 
                                     
                                       N 
                                       ⁡ 
                                       
                                         ( 
                                         
                                           H 
                                           ^ 
                                         
                                         ) 
                                       
                                     
                                     ⁢ 
                                     
                                       c 
                                       n 
                                     
                                   
                                   
                                     ︸ 
                                     
                                       P 
                                       
                                         1 
                                         - 
                                         
                                           
                                             ( 
                                             
                                               
                                                 
                                                   N 
                                                   ^ 
                                                 
                                                 1 
                                               
                                               - 
                                               1 
                                             
                                             ) 
                                           
                                           ⁢ 
                                           Δ 
                                         
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               
                                 G 
                                 H 
                               
                               ⁢ 
                               
                                 N 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     G 
                                     ^ 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               u 
                             
                             
                               ︸ 
                               
                                 
                                   P 
                                   0 
                                 
                                 ⁢ 
                                 
                                   I 
                                   
                                     N 
                                     1 
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               
                                 G 
                                 H 
                               
                               ⁢ 
                               
                                 N 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     H 
                                     ^ 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               v 
                             
                             
                               ︸ 
                               
                                 
                                   P 
                                   a 
                                 
                                 ⁢ 
                                 
                                   I 
                                   
                                     N 
                                     1 
                                   
                                 
                               
                             
                           
                           + 
                           
                             ε 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   12 
                 
               
             
           
         
       
     
     The approximation is due to the fact that G H N(Ĥ)c n  is dominant compared to G H R (Ĥ, {tilde over (G)})c n  since 
               1   -       (       ⌈     n     N   1       ⌉     -   1     )     ⁢   Δ       ≤     1   -       (       ⌈     n     N   2       ⌉     -   1     )     ⁢     Δ   .               
The CS 1  received by Rx 2  are written as the sum of N 1  brackets. N 2  unique CS 1  are contained in each bracket and received with the same power. Specifically, the mth bracket (mε1, 2, . . . , {circumflex over (N)} 1 ) contains the common symbols with index from (m−1)×N 2 +1 to m×N 2 , and their received power scale as P 1−(m−1)Δ .
 
     Decoding at Rx 2  is using MMSE-SIC and similar as that in Rx 1 . Consequently, the DoF of the common message achieved at Rx 2  is {circumflex over (N)} 1 {circumflex over (N)} 2 tΔ=N 2 (1−a)=N 1 (1−b), identical to that decoded by Rx 1 . Besides, the DoF of PS 2  is N 2 a. Moreover, the sum DoF performance is N 2 +N 1 b. 
     Besides, it is important to note that a≧({circumflex over (N)} 2 −{circumflex over (N)} 1 )Δ (derived from 
                 1   -       (       ⌈     n     N   1       ⌉     -   1     )     ⁢   Δ       ≥     1   -       (       ⌈     n     N   2       ⌉     -   1     )     ⁢   Δ     -   a       )         
is a necessary condition performing this transmission design. Otherwise, H H R(Ĥ, Ĝ)c n  might not be dominant compared to H H N(Ĥ)c n  at Rx 1 .
 
2.4.2. Rx 1  has Larger Inherent DoF for the Common Messages Compared to Rx 2 , Namely N 1 (1−b)&gt;N 2 (1−a)
 
     In this case, based on MC 1, the maximum achievable DoF of PS 1  is N 1 b while keeping d c +d 2 =N 2  because Rx 1  has a larger inherent DoF. The sum DoF performance N 2 +N 1   b  has been shown as the optimal result since it is consistent with the outer-bound. 
     The transmission strategy is constructed by calculating a′≦a such that N 1 (1−b)=N 2 (1−a′). Then, using the scheme introduced in Section 2.4.1, the power allocated to the private symbols intended for Rx 2  is P a′  and the DoF of each CS 1  is determined as 
             Δ   =         1   -     a   ′           N   ^     1       =         1   -   b         N   ^     2       .             
The transmitted signal is therefore written as:
 
     
       
         
           
             
               
                 
                   s 
                   = 
                   
                     
                       ( 
                       
                         
                           ∑ 
                           
                             
                               n 
                               = 
                               1 
                             
                             , 
                             
                               
                                 ⌈ 
                                 
                                   n 
                                   
                                     N 
                                     1 
                                   
                                 
                                 ⌉ 
                               
                               = 
                               
                                 ⌈ 
                                 
                                   n 
                                   
                                     N 
                                     2 
                                   
                                 
                                 ⌉ 
                               
                             
                           
                           
                             
                               
                                 N 
                                 ^ 
                               
                               1 
                             
                             ⁢ 
                             
                               
                                 N 
                                 ^ 
                               
                               2 
                             
                             ⁢ 
                             t 
                           
                         
                         ⁢ 
                         
                           
                             
                               R 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     H 
                                     ^ 
                                   
                                   , 
                                   
                                     G 
                                     ^ 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               c 
                               n 
                             
                           
                           
                             ︸ 
                             
                               P 
                               
                                 1 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       
                                         ⌈ 
                                         
                                           n 
                                           
                                             N 
                                             2 
                                           
                                         
                                         ⌉ 
                                       
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   Δ 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     + 
                     
                       ( 
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 = 
                                 1 
                               
                               , 
                               
                                 
                                   ⌈ 
                                   
                                     n 
                                     
                                       N 
                                       1 
                                     
                                   
                                   ⌉ 
                                 
                                 &gt; 
                                 
                                   ⌈ 
                                   
                                     n 
                                     
                                       N 
                                       2 
                                     
                                   
                                   ⌉ 
                                 
                               
                             
                             
                               
                                 
                                   N 
                                   ^ 
                                 
                                 1 
                               
                               ⁢ 
                               
                                 
                                   N 
                                   ^ 
                                 
                                 2 
                               
                               ⁢ 
                               t 
                             
                           
                           ⁢ 
                           
                             
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       H 
                                       ^ 
                                     
                                     , 
                                     
                                       G 
                                       ^ 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 c 
                                 n 
                               
                             
                             
                               ︸ 
                               
                                 P 
                                 
                                   1 
                                   - 
                                   
                                     
                                       ( 
                                       
                                         
                                           ⌈ 
                                           
                                             n 
                                             
                                               N 
                                               1 
                                             
                                           
                                           ⌉ 
                                         
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                     ⁢ 
                                     Δ 
                                   
                                 
                               
                             
                           
                         
                         + 
                         
                           
                             
                               N 
                               ⁡ 
                               
                                 ( 
                                 
                                   H 
                                   ^ 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               c 
                               n 
                             
                           
                           
                             ︸ 
                             
                               P 
                               
                                 1 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       
                                         ⌈ 
                                         
                                           n 
                                           
                                             N 
                                             2 
                                           
                                         
                                         ⌉ 
                                       
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   Δ 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     + 
                     
                       
                         
                           N 
                           ⁡ 
                           
                             ( 
                             
                               G 
                               ^ 
                             
                             ) 
                           
                         
                         ⁢ 
                         u 
                       
                       
                         ︸ 
                         
                           
                             P 
                             b 
                           
                           ⁢ 
                           
                             I 
                             
                               N 
                               1 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         
                           N 
                           ⁡ 
                           
                             ( 
                             
                               H 
                               ^ 
                             
                             ) 
                           
                         
                         ⁢ 
                         v 
                       
                       
                         ︸ 
                         
                           
                             P 
                             
                               a 
                               ′ 
                             
                           
                           ⁢ 
                           
                             I 
                             
                               N 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   13 
                 
               
             
           
         
       
     
     Since N 1 (1−b)≧N 2 (1−a), it can be verified that a≧({circumflex over (N)} 2 −{circumflex over (N)} 1 )Δ and both receivers decode the CS 1  with the DoF N 1 (1−b)=N 2 (1−a′). Besides, N(Ĥ)v will not cause interference at Rx 1  as a′≦a. Consequently, the sum DoF is N 2 (1−a′)+N 1 b+N 2 a′=N 2 +N 1 b. 
     2.4.3. Rx 2  has a Larger Inherent DoF for the Common Messages, Namely N 1 (1−b)&lt;N 2 (1−a) 
     The scheme introduced in Section 2.4.1 does not work here since such b′ satisfying 0≦b′≦b and N 1 (1−b′)=N 2 (1−a) does not necessarily exist. Therefore, based on MC 1, the power and DoF for PS 2  is increased so as to decrease the inherent DoF for common messages at Rx 2  till both receivers have same amount of inherent DoF for common messages. Although this will cause interference at Rx 1 , it still benefits the sum DoF performance as Rx 2  has larger number of antennas because N 2 ≧N 1  and PS 2  contribute to the sum DoF performance more than PS 1 . 
     Once the power of PS 2  is increased to P a′ , the interference caused at Rx 1  is P a′−a . The inherent DoF for common messages is found by N 1 (1−max(b, a′−a))=N 2 (1−a′). Hence, the transmission strategy depends on whether PS 1  is drowned by the interference or not, namely the relationship between b and a′−a. 
     1) d c  is found when a′−a≦b, namely d c =N 1 (1−b)=N 2 (1−a′) and this leads to 
     
       
         
           
             a 
             ≥ 
             
               
                 ( 
                 
                   
                     N 
                     2 
                   
                   - 
                   
                     N 
                     1 
                   
                 
                 ) 
               
               × 
               
                 
                   
                     1 
                     - 
                     b 
                   
                   
                     N 
                     2 
                   
                 
                 . 
               
             
           
         
       
     
     Compute a′&gt;a, such that N 1 (1−b)=N 2 (1−a′); 
     Using the scheme introduced in Section 2.4.1, the power allocated to the private symbols intended for Rx 2  is P a′  and the DoF of each CS 1  is determined as 
             Δ   =         1   -     a   ′           N   ^     1       =         1   -   b         N   ^     2       .             
The transmitted signal is therefore written as:
 
     
       
         
           
             
               
                 
                   s 
                   = 
                   
                     
                       ( 
                       
                         
                           ∑ 
                           
                             
                               n 
                               = 
                               1 
                             
                             , 
                             
                               
                                 ⌈ 
                                 
                                   n 
                                   
                                     N 
                                     1 
                                   
                                 
                                 ⌉ 
                               
                               = 
                               
                                 ⌈ 
                                 
                                   n 
                                   
                                     N 
                                     2 
                                   
                                 
                                 ⌉ 
                               
                             
                           
                           
                             
                               
                                 N 
                                 ^ 
                               
                               1 
                             
                             ⁢ 
                             
                               
                                 N 
                                 ^ 
                               
                               2 
                             
                             ⁢ 
                             t 
                           
                         
                         ⁢ 
                         
                           
                             
                               R 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     H 
                                     ^ 
                                   
                                   , 
                                   
                                     G 
                                     ^ 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               c 
                               n 
                             
                           
                           
                             ︸ 
                             
                               P 
                               
                                 1 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       
                                         ⌈ 
                                         
                                           n 
                                           
                                             N 
                                             2 
                                           
                                         
                                         ⌉ 
                                       
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   Δ 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     + 
                     
                       ( 
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 = 
                                 1 
                               
                               , 
                               
                                 
                                   ⌈ 
                                   
                                     n 
                                     
                                       N 
                                       1 
                                     
                                   
                                   ⌉ 
                                 
                                 &gt; 
                                 
                                   ⌈ 
                                   
                                     n 
                                     
                                       N 
                                       2 
                                     
                                   
                                   ⌉ 
                                 
                               
                             
                             
                               
                                 
                                   N 
                                   ^ 
                                 
                                 1 
                               
                               ⁢ 
                               
                                 
                                   N 
                                   ^ 
                                 
                                 2 
                               
                               ⁢ 
                               t 
                             
                           
                           ⁢ 
                           
                             
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       H 
                                       ^ 
                                     
                                     , 
                                     
                                       G 
                                       ^ 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 c 
                                 n 
                               
                             
                             
                               ︸ 
                               
                                 P 
                                 
                                   1 
                                   - 
                                   
                                     
                                       ( 
                                       
                                         
                                           ⌈ 
                                           
                                             n 
                                             
                                               N 
                                               1 
                                             
                                           
                                           ⌉ 
                                         
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                     ⁢ 
                                     Δ 
                                   
                                 
                               
                             
                           
                         
                         + 
                         
                           
                             
                               N 
                               ⁡ 
                               
                                 ( 
                                 
                                   H 
                                   ^ 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               c 
                               n 
                             
                           
                           
                             ︸ 
                             
                               P 
                               
                                 1 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       
                                         ⌈ 
                                         
                                           n 
                                           
                                             N 
                                             2 
                                           
                                         
                                         ⌉ 
                                       
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   Δ 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     + 
                     
                       
                         
                           N 
                           ⁡ 
                           
                             ( 
                             
                               G 
                               ^ 
                             
                             ) 
                           
                         
                         ⁢ 
                         u 
                       
                       
                         ︸ 
                         
                           
                             P 
                             b 
                           
                           ⁢ 
                           
                             I 
                             
                               N 
                               1 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         
                           N 
                           ⁡ 
                           
                             ( 
                             
                               H 
                               ^ 
                             
                             ) 
                           
                         
                         ⁢ 
                         v 
                       
                       
                         ︸ 
                         
                           
                             P 
                             
                               a 
                               ′ 
                             
                           
                           ⁢ 
                           
                             I 
                             
                               N 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   14 
                 
               
             
           
         
       
     
     Since 
               a   ≥       (       N   2     -     N   1       )     ×       1   -   b       N   2           ,         
the necessary condition a≧({circumflex over (N)} 2 −{circumflex over (N)} 1 )Δ holds. The received signals are written as:
 
     
       
         
           
             
               
                 
                   y 
                   = 
                   
                     
                       ( 
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             1 
                           
                           
                             
                               N 
                               ^ 
                             
                             2 
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               
                                 
                                   
                                     N 
                                     1 
                                   
                                   × 
                                   
                                     ( 
                                     
                                       m 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                                 + 
                                 1 
                               
                             
                             
                               
                                 N 
                                 1 
                               
                               × 
                               m 
                             
                           
                           ⁢ 
                           
                             
                               
                                 H 
                                 H 
                               
                               ⁢ 
                               
                                 R 
                                 ( 
                                 
                                   
                                     H 
                                     ^ 
                                   
                                   , 
                                   
                                     G 
                                     ^ 
                                   
                                 
                                 ) 
                               
                               ⁢ 
                               
                                 c 
                                 n 
                               
                             
                             
                               ︸ 
                               
                                 P 
                                 
                                   1 
                                   - 
                                   
                                     
                                       ( 
                                       
                                         m 
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                     ⁢ 
                                     Δ 
                                   
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     + 
                     
                       
                         
                           H 
                           H 
                         
                         ⁢ 
                         
                           N 
                           ⁡ 
                           
                             ( 
                             
                               G 
                               ^ 
                             
                             ) 
                           
                         
                         ⁢ 
                         u 
                       
                       
                         ︸ 
                         
                           
                             P 
                             b 
                           
                           ⁢ 
                           
                             I 
                             
                               N 
                               1 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         
                           H 
                           H 
                         
                         ⁢ 
                         
                           N 
                           ⁡ 
                           
                             ( 
                             
                               H 
                               ^ 
                             
                             ) 
                           
                         
                         ⁢ 
                         v 
                       
                       
                         ︸ 
                         
                           
                             P 
                             a 
                           
                           ⁢ 
                           
                               
                           
                           - 
                           
                             a 
                             
                               I 
                               
                                 N 
                                 1 
                               
                             
                           
                         
                       
                     
                     + 
                     
                       ε 
                       1 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   15 
                 
               
             
             
               
                 
                   z 
                   = 
                   
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           1 
                         
                         
                           
                             N 
                             ^ 
                           
                           1 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               ∑ 
                               
                                 
                                   
                                     n 
                                     = 
                                     
                                       
                                         
                                           N 
                                           2 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             m 
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                       
                                       + 
                                       1 
                                     
                                   
                                   , 
                                 
                                 
                                   
                                     ⌈ 
                                     
                                       n 
                                       
                                         N 
                                         1 
                                       
                                     
                                     ⌉ 
                                   
                                   = 
                                   
                                     ⌈ 
                                     
                                       n 
                                       
                                         N 
                                         2 
                                       
                                     
                                     ⌉ 
                                   
                                 
                               
                               
                                 
                                   N 
                                   2 
                                 
                                 × 
                                 m 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   G 
                                   H 
                                 
                                 ⁢ 
                                 
                                   R 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         H 
                                         ^ 
                                       
                                       , 
                                       
                                         G 
                                         ^ 
                                       
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   c 
                                   n 
                                 
                               
                               
                                 ︸ 
                                 
                                   P 
                                   
                                     1 
                                     - 
                                     
                                       
                                         ( 
                                         
                                           m 
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                       ⁢ 
                                       Δ 
                                     
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               ∑ 
                               
                                 
                                   
                                     n 
                                     = 
                                     
                                       
                                         
                                           N 
                                           2 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             m 
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                       
                                       + 
                                       1 
                                     
                                   
                                   , 
                                 
                                 
                                   
                                     ⌈ 
                                     
                                       n 
                                       
                                         N 
                                         1 
                                       
                                     
                                     ⌉ 
                                   
                                   &gt; 
                                   
                                     ⌈ 
                                     
                                       n 
                                       
                                         N 
                                         2 
                                       
                                     
                                     ⌉ 
                                   
                                 
                               
                               
                                 
                                   N 
                                   2 
                                 
                                 × 
                                 m 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   G 
                                   H 
                                 
                                 ⁢ 
                                 
                                   N 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       H 
                                       ^ 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   c 
                                   n 
                                 
                               
                               
                                 ︸ 
                                 
                                   P 
                                   
                                     1 
                                     - 
                                     
                                       
                                         ( 
                                         
                                           m 
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                       ⁢ 
                                       Δ 
                                     
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         
                           G 
                           H 
                         
                         ⁢ 
                         
                           N 
                           ⁡ 
                           
                             ( 
                             
                               G 
                               ^ 
                             
                             ) 
                           
                         
                         ⁢ 
                         u 
                       
                       
                         ︸ 
                         
                           
                             P 
                             0 
                           
                           ⁢ 
                           
                             I 
                             
                               N 
                               1 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         
                           G 
                           H 
                         
                         ⁢ 
                         
                           N 
                           ⁡ 
                           
                             ( 
                             
                               H 
                               ^ 
                             
                             ) 
                           
                         
                         ⁢ 
                         v 
                       
                       
                         ︸ 
                         
                           
                             P 
                             
                               a 
                               ′ 
                             
                           
                           ⁢ 
                           
                             I 
                             
                               N 
                               2 
                             
                           
                         
                       
                     
                     + 
                     
                       ε 
                       2 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   16 
                 
               
             
           
         
       
     
     As b≧a′−a, H H N(Ĝ)u is the dominant part compared to H H N(Ĥ)v. Hence, Both receivers can decode CS 1  and the DoF achieved by CS 1  is N 1 (1−b)=N 2 (1−a′). After removing c n , Rx 1  decodes H H N(Ĝ)u by treating H H N(Ĥ)v as noise and H H N(Ĝ)u achieves the DoF N 1 (b−a′+a). Rx 2  recovers G H N(Ĥ)v only subject to noise and the DoF of G H N(Ĥ)v is N 2 a′. Consequently, the sum DoF performance is 
     
       
         
           
             
               
                 N 
                 2 
               
               + 
               
                 
                   N 
                   1 
                 
                 ⁡ 
                 
                   ( 
                   
                     b 
                     - 
                     
                       a 
                       ′ 
                     
                     + 
                     a 
                   
                   ) 
                 
               
             
             = 
             
               
                 N 
                 2 
               
               + 
               
                 
                   
                     N 
                     1 
                   
                   ⁡ 
                   
                     ( 
                     
                       a 
                       + 
                       
                         
                           
                             
                               N 
                               2 
                             
                             - 
                             
                               N 
                               1 
                             
                           
                           
                             N 
                             2 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             b 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                     ) 
                   
                 
                 . 
               
             
           
         
       
     
     2) d c  is found when a′−a&gt;b, namely d c =N 1 (1−a′+a)=N 2 (1−a′) and this leads to 
     
       
         
           
             a 
             &lt; 
             
               
                 ( 
                 
                   
                     N 
                     2 
                   
                   - 
                   
                     N 
                     1 
                   
                 
                 ) 
               
               × 
               
                 
                   1 
                   - 
                   b 
                 
                 
                   N 
                   2 
                 
               
               ⁢ 
               
                 : 
               
             
           
         
       
     
     Compute a′&gt;a, such that N 1 (1−a′+a)=N 2 (1−a′); 
     Using the scheme introduced in Section 2.4.1, the power allocated to PS 2  is P a′  and the DoF of each CS 1  is determined as 
               Δ   =         1   -     a   ′           N   ^     1       =       1   -     a   ′     +   a         N   ^     2           ,         
while PS 1  is sent because b&lt;a′−a. The transmitted signal is therefore written as:
 
     
       
         
           
             
               
                 
                   s 
                   = 
                   
                     
                       ( 
                       
                         
                           ∑ 
                           
                             
                               n 
                               = 
                               1 
                             
                             , 
                             
                               
                                 ⌈ 
                                 
                                   n 
                                   
                                     N 
                                     1 
                                   
                                 
                                 ⌉ 
                               
                               = 
                               
                                 ⌈ 
                                 
                                   n 
                                   
                                     N 
                                     2 
                                   
                                 
                                 ⌉ 
                               
                             
                           
                           
                             
                               
                                 N 
                                 ^ 
                               
                               1 
                             
                             ⁢ 
                             
                               
                                 N 
                                 ^ 
                               
                               2 
                             
                             ⁢ 
                             t 
                           
                         
                         ⁢ 
                         
                           
                             
                               R 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     H 
                                     ^ 
                                   
                                   , 
                                   
                                     G 
                                     ^ 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               c 
                               n 
                             
                           
                           
                             ︸ 
                             
                               P 
                               
                                 1 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       
                                         ⌈ 
                                         
                                           n 
                                           
                                             N 
                                             2 
                                           
                                         
                                         ⌉ 
                                       
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   Δ 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     + 
                     
                       ( 
                       
                         
                           
                             ∑ 
                             
                               
                                 n 
                                 = 
                                 1 
                               
                               , 
                               
                                 
                                   ⌈ 
                                   
                                     n 
                                     
                                       N 
                                       1 
                                     
                                   
                                   ⌉ 
                                 
                                 &gt; 
                                 
                                   ⌈ 
                                   
                                     n 
                                     
                                       N 
                                       2 
                                     
                                   
                                   ⌉ 
                                 
                               
                             
                             
                               
                                 
                                   N 
                                   ^ 
                                 
                                 1 
                               
                               ⁢ 
                               
                                 
                                   N 
                                   ^ 
                                 
                                 2 
                               
                               ⁢ 
                               t 
                             
                           
                           ⁢ 
                           
                             
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       H 
                                       ^ 
                                     
                                     , 
                                     
                                       G 
                                       ^ 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 c 
                                 n 
                               
                             
                             
                               ︸ 
                               
                                 P 
                                 
                                   1 
                                   - 
                                   
                                     
                                       ( 
                                       
                                         
                                           ⌈ 
                                           
                                             n 
                                             
                                               N 
                                               1 
                                             
                                           
                                           ⌉ 
                                         
                                         - 
                                         1 
                                       
                                       ) 
                                     
                                     ⁢ 
                                     Δ 
                                   
                                 
                               
                             
                           
                         
                         + 
                         
                           
                             
                               N 
                               ⁡ 
                               
                                 ( 
                                 
                                   H 
                                   ^ 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               c 
                               n 
                             
                           
                           
                             ︸ 
                             
                               P 
                               
                                 1 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       
                                         ⌈ 
                                         
                                           n 
                                           
                                             N 
                                             2 
                                           
                                         
                                         ⌉ 
                                       
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   Δ 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     + 
                     
                       
                         
                           N 
                           ⁡ 
                           
                             ( 
                             
                               H 
                               ^ 
                             
                             ) 
                           
                         
                         ⁢ 
                         v 
                       
                       
                         ︸ 
                         
                           
                             P 
                             
                               a 
                               ′ 
                             
                           
                           ⁢ 
                           
                             I 
                             
                               N 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   17 
                 
               
             
           
         
       
     
     Simply replacing 
               Δ   =       1   -     a   ′     +   a         N   ^     2         ,         
the condition a≧({circumflex over (N)} 2 −{circumflex over (N)} 1 )Δ can be verified. The received signals are written as:
 
     
       
         
           
             
               
                 
                   y 
                   = 
                   
                     
                       ( 
                       
                         
                           ∑ 
                           
                             m 
                             = 
                             1 
                           
                           
                             
                               N 
                               ^ 
                             
                             2 
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               
                                 
                                   
                                     N 
                                     1 
                                   
                                   × 
                                   
                                     ( 
                                     
                                       m 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                                 + 
                                 1 
                               
                             
                             
                               
                                 N 
                                 1 
                               
                               × 
                               m 
                             
                           
                           ⁢ 
                           
                             
                               
                                 H 
                                 H 
                               
                               ⁢ 
                               
                                 R 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       H 
                                       ^ 
                                     
                                     , 
                                     
                                       G 
                                       ^ 
                                     
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 c 
                                 n 
                               
                             
                             
                               P 
                               
                                 1 
                                 - 
                                 
                                   
                                     ( 
                                     
                                       m 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                   ⁢ 
                                   Δ 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                     + 
                     
                       
                         
                           H 
                           H 
                         
                         ⁢ 
                         
                           N 
                           ⁡ 
                           
                             ( 
                             
                               H 
                               ^ 
                             
                             ) 
                           
                         
                         ⁢ 
                         v 
                       
                       
                         ︸ 
                         
                           P 
                           
                             
                               a 
                               ′ 
                             
                             - 
                             
                               a 
                               
                                 I 
                                 
                                   N 
                                   1 
                                 
                               
                             
                           
                         
                       
                     
                     + 
                     
                       ε 
                       1 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   18 
                 
               
             
             
               
                 
                   z 
                   = 
                   
                     
                       
                         ∑ 
                         
                           m 
                           = 
                           1 
                         
                         
                           
                             N 
                             ^ 
                           
                           1 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               ∑ 
                               
                                 
                                   
                                     n 
                                     = 
                                     
                                       
                                         
                                           N 
                                           2 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             m 
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                       
                                       + 
                                       1 
                                     
                                   
                                   , 
                                 
                                 
                                   
                                     ⌈ 
                                     
                                       n 
                                       
                                         N 
                                         1 
                                       
                                     
                                     ⌉ 
                                   
                                   = 
                                   
                                     ⌈ 
                                     
                                       n 
                                       
                                         N 
                                         2 
                                       
                                     
                                     ⌉ 
                                   
                                 
                               
                               
                                 
                                   N 
                                   2 
                                 
                                 × 
                                 m 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   G 
                                   H 
                                 
                                 ⁢ 
                                 
                                   R 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         H 
                                         ^ 
                                       
                                       , 
                                       
                                         G 
                                         ^ 
                                       
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   c 
                                   n 
                                 
                               
                               
                                 ︸ 
                                 
                                   P 
                                   
                                     1 
                                     - 
                                     
                                       
                                         ( 
                                         
                                           m 
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                       ⁢ 
                                       Δ 
                                     
                                   
                                 
                               
                             
                           
                           + 
                           
                             
                               ∑ 
                               
                                 
                                   
                                     n 
                                     = 
                                     
                                       
                                         
                                           N 
                                           2 
                                         
                                         ⁡ 
                                         
                                           ( 
                                           
                                             m 
                                             - 
                                             1 
                                           
                                           ) 
                                         
                                       
                                       + 
                                       1 
                                     
                                   
                                   , 
                                 
                                 
                                   
                                     ⌈ 
                                     
                                       n 
                                       
                                         N 
                                         1 
                                       
                                     
                                     ⌉ 
                                   
                                   &gt; 
                                   
                                     ⌈ 
                                     
                                       n 
                                       
                                         N 
                                         2 
                                       
                                     
                                     ⌉ 
                                   
                                 
                               
                               
                                 
                                   N 
                                   2 
                                 
                                 × 
                                 m 
                               
                             
                             ⁢ 
                             
                               
                                 
                                   G 
                                   H 
                                 
                                 ⁢ 
                                 
                                   N 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       H 
                                       ^ 
                                     
                                     ) 
                                   
                                 
                                 ⁢ 
                                 
                                   c 
                                   n 
                                 
                               
                               
                                 ︸ 
                                 
                                   P 
                                   
                                     1 
                                     - 
                                     
                                       
                                         ( 
                                         
                                           m 
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                       ⁢ 
                                       Δ 
                                     
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         
                           
                             G 
                             H 
                           
                           ⁢ 
                           
                             N 
                             ⁡ 
                             
                               ( 
                               
                                 H 
                                 ^ 
                               
                               ) 
                             
                           
                           ⁢ 
                           v 
                         
                         ︸ 
                       
                       
                         
                           P 
                           a 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           I 
                           
                             N 
                             2 
                           
                         
                       
                     
                     + 
                     
                       ε 
                       2 
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   19 
                 
               
             
           
         
       
     
     The decoding is similar as above and using MMSE-SIC. As no PS 1  is sent, the sum DoF performance is N 2 . Note that the CSIT does not make a contribution to the sum DoF performance compared to the scenario without CSIT of either receiver (the sum DoF is N 2  as well, achieved by sending private symbols to Rx 2  only), but the Tx can make use of the CSIT to send common messages with the DoF N 2 (1−a′). Since the common messages can be regarded as intended for Rx 1 , this case is more meaningful in the scenario where the two receivers have their particular QoS target or the Tx has to consider fairness between the receivers. 
     2.5 Transmission Strategy for Multiple Subband Scenario 
     2.5.1. Two-Subband Scenario N 2 (1−ā)=N 1 (1− b ) and the Transmission of CS 2   
     Here, consider a two-subband scenario with 
                 N   2     ⁡     (     1   -         a   1     +     a   2       2       )       =       N   1     ⁡     (     1   -         b   1     +     b   2       2       )             
and the transmission strategy can be easily extended to multiple-subband scenario. Besides, assume N 1 (1− b   1 )&gt;N 2 (1−a 1 ) and N 1 (1−b 2 )&lt;N 2 (1−a 2 ), namely Rx 1  has larger inherent DoF for common messages in subband  1 , while Rx 2  has larger inherent DoF for common messages in subband  2 . The case N 1 (1−b 1 )=N 2 (1−a 1 ) and N 1 (1−b 2 )=N 2 (1−a 2 ) can be solved independently and separately using the method introduced in Section 2.3. The scenario
 
                 N   2     ⁡     (     1   -         a   1     +     a   2       2       )       ≠       N   1     ⁡     (     1   -         b   1     +     b   2       2       )             
will be discussed later.
 
     Based on MC 3, the key ingredient in this scheme is to send CS 2  with the DoF N 2 (1−a 2 )−N 1 (1−b 2 ) (or equivalently N 1 (1−b 1 )−N 2 (1−a 1 )), such that Rx 2  can decode them from subband  2  and Rx 1  recovers them from subband  1 . 
     Transmission Strategy: 
     1) Private symbols: In each subband, N 1  private symbols are sent to Rx 1  with the ZF-precoder N(Ĝ j ) and the power P b     j   , while N 2  private symbols are sent to Rx 2  with the ZF-precoder N(Ĥ j ) and the power P a     j   . 
     2) CS 1  in subband  1 : The DoF is determined to be N 2 (1−a 1 ). According to MC 2 and as discussed in Section 2.4.1, {circumflex over (N)} 1 {circumflex over (N)} 2 t CS 1  are sent to both receivers. These CS 1  in subband  1  are denoted as c 1:N     1     N     2     I (1). Each CS 1  has the 
               DoF   ⁢           ⁢     Δ   1       =         1   -     a   1           N   ^     1       .           
The necessary condition, a 1 ≧({circumflex over (N)} 2 −{circumflex over (N)} 1 )Δ 1 , can be verified using the fact that N 1 (1−b 1 )&gt;N 2 (1−a 1 ).
 
     3) CS 1  in subband  2 : The DoF is determined to be N 1 (1−b 2 ). The method introduced in Section 2.4.1 is not applicable here because the necessary condition a 2 ≧({circumflex over (N)} 2 −{circumflex over (N)} 1 )Δ 2   
             (       where   ⁢           ⁢     Δ   2       =       1   -     b   2           N   ^     2         )         
does not hold in general. Hence, the joint design of CS 1  and CS 2  is considered here.
 
     A. a 2 ≧b 2 : The transmission of CS 1  in subband  2  is divided into two parts as shown in Table 1. 
     
       
         
           
               
               
               
               
               
               
               
             
               
                 TABLE 1 
               
               
                   
               
               
                   
                 Notation 
                 DoF 
                 Number 
                 DoF/symbol 
                 Power 
                 Precoder 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 Part 1 
                 c n   I,1  (2) 
                 N 1  (r − b 2 ) 
                 {circumflex over (N)} 1 {circumflex over (N)} 2 t 
                 
                   
                     
                       
                         
                           
                             
                               
                                 
                                   Δ 
                                   
                                     2 
                                     , 
                                     1 
                                   
                                 
                                 = 
                                 
                                   
                                     r 
                                     - 
                                     
                                       b 
                                       2 
                                     
                                   
                                   
                                     
                                       N 
                                       ^ 
                                     
                                     2 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 = 
                                 
                                   
                                     r 
                                     - 
                                     
                                       a 
                                       2 
                                     
                                   
                                   
                                     
                                       N 
                                       ^ 
                                     
                                     1 
                                   
                                 
                               
                             
                           
                         
                           
                       
                     
                   
                 
                 Derived based on MC 2 and Section 2.4.1 
               
               
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 Part 2 
                 c n   I,2  (2) 
                 N 1  (1 − r) 
                 {circumflex over (N)} 1 {circumflex over (N)} 1 t 
                 
                   
                     
                       
                         δ 
                         = 
                         
                           
                             1 
                             - 
                             r 
                           
                           
                             
                               N 
                               ^ 
                             
                             1 
                           
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         P 
                         
                           1 
                           - 
                           
                             
                               ( 
                               
                                 
                                   ⌈ 
                                   
                                     n 
                                     
                                       N 
                                       1 
                                     
                                   
                                   ⌉ 
                                 
                                 - 
                                 1 
                               
                               ) 
                             
                             ⁢ 
                             δ 
                           
                         
                       
                     
                   
                 
                 R(Ĥ 2 , Ĝ 2 ) 
               
               
                   
               
            
           
         
       
     
     Table 2 denotes the generation of two parts of CS 1  in subband  2  when a 2 ≧b 2  where 
             r   =           N   2     ⁢     a   2       -       N   1     ⁢     b   2             N   2     -     N   1               
and r is derived from N 1 (r−b 2 )=N 2 (r−a 2 ). c n   I,1 (2) are generated using the method given in MC 2 and Section 2.4.1. c n   I,2 (2) are sent with power higher than c n   I,1 (2). c n   I,2 (2) will be received by both receivers in {circumflex over (N)} 1  power levels and each level contains N 1  unique c n   I,2 (2).
 
     B. a 2 &lt;b 2 : The CS 1  in subband  2  are generated similar as the second part of CS 1  in the case a 2 ≧b 2 . {circumflex over (N)} 1 {circumflex over (N)} 1 t CS 1  are transmitted and the total DoF of them is N 1 (1−b 2 ). They are denoted as c n   I (2). Both receivers employ N 1  dimensions of the received signal to decode them and observe them in {circumflex over (N)} 1  power levels. The precoder is R(Ĥ 2 , Ĝ 2 ). Each of them has the 
               DoF   ⁢           ⁢     δ   1       =       1   -     b   2           N   ^     1             
and is allocates with the power
 
     
       
         
           
             
               P 
               
                 1 
                 - 
                 
                   
                     ( 
                     
                       
                         ⌈ 
                         
                           n 
                           
                             N 
                             1 
                           
                         
                         ⌉ 
                       
                       - 
                       1 
                     
                     ) 
                   
                   ⁢ 
                   
                     δ 
                     1 
                   
                 
               
             
             . 
           
         
       
     
     4) CS 2  Transmission: The DoF is Determined to be N 1 (1−b 1 )−N 2 (1−a 1 )=N 2 (1−a 2 )−N 2 (1−b 2 ). 
     A. a 2 ≧b 2 : Rx 1  decodes CS 2  using N 1  dimensions of the received signal in subband  1  while Rx 2  employs N 2  N 1  dimensions to decode them (because N 1  dimensions have been used to decode c n   I,2 (2)). Based on MC 3, {circumflex over (N)} 1 ({circumflex over (N)} 2 −{circumflex over (N)} 1 )t CS 2  are sent and they are denoted as c n   II , n=1, 2, . . . , {circumflex over (N)} 1 ({circumflex over (N)} 2 −{circumflex over (N)} 1 )t. Each of them has the target 
                 DoF   ⁢           N   2     ⁢     (     1   -     a   2       )       -       N   1     ⁡     (     1   -     b   2       )                 N   ^     1     ⁡     (         N   ^     2     -       N   ^     1       )       ⁢   t         =         1   -   r         N   ^     1       =   δ       ,         
same as c n   I,2 (2):
 
     In subband  1 , c n   II  is allocated with power 
             P     1   -         N   ^     2     ⁢     Δ   1       -       (       ⌈     n     N   1       ⌉     -   1     )     ⁢   δ             
and the precoder is R(Ĥ 1 ).
 
     In subband  2 , c n   II  is allocated with power 
             P     1   -       (       ⌈     n       N   2     -     N   1         ⌉     -   1     )     ⁢   δ             
and the precoder is R(Ĝ 2 ).
 
     B. a 2 &lt;b 2 : the transmission of CS 2  is divided into two parts as shown in Table 2. 
     
       
         
           
               
               
               
               
               
               
               
             
               
                 TABLE 2 
               
               
                   
               
               
                   
                 Notation 
                 DoF 
                 Number 
                 DoF/symbol 
                 Power 
                 Precoder 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
               
               
            
               
                 Part 1 
                 c n   II,1   
                 (N 2  − N 1 )(1 − b 2 ) 
                 {circumflex over (N)} 1  ({circumflex over (N)} 2  − {circumflex over (N)} 1 )t 
                 
                   
                     
                       
                         
                           δ 
                           1 
                         
                         = 
                         
                           
                             1 
                             - 
                             
                               b 
                               2 
                             
                           
                           
                             
                               N 
                               ^ 
                             
                             1 
                           
                         
                       
                     
                   
                 
                 Derived based on MC 3 
               
               
                   
               
               
                 Part 2 
                 c n   II,2   
                 N 2  (b 2  − a 2 ) 
                 {circumflex over (N)} 1 {circumflex over (N)} 2 t 
                 
                   
                     
                       
                         
                           δ 
                           2 
                         
                         = 
                         
                           
                             
                               b 
                               2 
                             
                             - 
                             
                               a 
                               2 
                             
                           
                           
                             
                               N 
                               ^ 
                             
                             1 
                           
                         
                       
                     
                   
                 
               
               
                   
               
            
           
         
       
     
     Table 3 denotes CS 2  generation when a 2 &lt;b 2 . Note that c n   II,1  is decoded by Rx 1  in subband  1  using N 1  dimensions and decoded by Rx 2  in subband  2  using N 2 −N 1  dimensions of the received signal (because N 1  dimensions have been used to decode c n   I (2)). 
     In subband  1 , is allocated with power 
               P     1   -         N   ^     2     ⁢     Δ   1       -       (     ⌈       n     N   1       -   1     ⌉     )     ⁢     δ   1           .         
They will be received by Rx 1  in {circumflex over (N)} 2 −{circumflex over (N)} 1  power levels and each level contains N 1  unique c n   II,1 ; c n   II,2  is allocated with power
 
             P     1   -         N   ^     2     ⁢     Δ   1       -       (         N   ^     2     -       N   ^     1       )     ⁢     δ   1       -       (       ⌈     n     N   1       ⌉     -   1     )     ⁢     δ   2               
and observed by Rx 1  in {circumflex over (N)} 2  power levels. Both c n   II,1  and c n   II,2  are transmitted with the precoder R(Ĥ 1 ).
 
     In subband  2 , c n   II,1  is allocated with power 
               P     1   -       (       ⌈     n       N   2     -     N   1         ⌉     -   1     )     ⁢     δ   1           .         
They will be received by Rx 2  in {circumflex over (N)} 1  power levels and each level contains N 2 −N 1  unique C n   II,1 ; c n   II,2  is allocated with power
 
             P     1   -         N   ^     1     ⁢     δ   1       -       (       ⌈     n     N   2       ⌉     -   1     )     ⁢     δ   2               
and observed by Rx 2  in {circumflex over (N)} 1  power levels. Both c n   II,1  and c n   II,2  are transmitted with the precoder R(Ĝ 2 ).
 
     With the proposed strategy, the received signals at each receiver are illustrated in  FIGS. 3 and 4  for a 2 ≧b 2  and a 2 &lt;b 2  respectively. 
       FIG. 3  is a diagram illustrating received signals at each receiver when a 2 ≧b 2  according to an embodiment of the present disclosure. The upper-left, upper-right, lower-left and lower-right blocks respectively refer to y 1 , z 1 , y 2  and z 2 . 
     Referring to  FIG. 3 , 
     a) The private symbols are received with the power subject to the CSIT quality of their unintended receiver; 
     b) According to step  2 , Rx 1  receives c 1:{circumflex over (N)}     1     {circumflex over (N)}     2     t   I (1) with N 1  dimensions and {circumflex over (N)} 2  power levels, while Rx 2  receives with N 2  dimensions and {circumflex over (N)} 1  power levels; 
     c) Following step  3 .A, c 1:{circumflex over (N)}     1     {circumflex over (N)}     2     t   I,1 (2) are received by Rx 1  in N 1  dimensions and r−b 2  channel use (namely {circumflex over (N)} 2  power levels), while they are received by Rx 2  in N 2  dimensions and r−a 2  channel use (namely {circumflex over (N)} 1  power levels); 
     d) Based on step  3 .A, c 1:{circumflex over (N)}     1     {circumflex over (N)}     1     t   I,2 (2) are received by both receivers in 1−r channel use ({circumflex over (N)} 1  power levels) but only spanning N 1  dimensions; and 
     e) As in step  4 .A, c 1:{circumflex over (N)}     1     ({circumflex over (N)}     2     −{circumflex over (N)}     1     )t   II  are received by Rx 2  in 1−r channel use ({circumflex over (N)} 1  power levels) and spanning the remaining N 2 −N 1  dimensions in subband  2 . At Rx 1 , c 1:{circumflex over (N)}     1     ({circumflex over (N)}     2     −{circumflex over (N)}     1     )t   II  are received in N 1  dimensions and {circumflex over (N)} 2 −{circumflex over (N)} 1  power levels in subband  1 . 
     Decoding (MMSE-SIC): 
     Rx 1  decodes c 1:{circumflex over (N)}     1     {circumflex over (N)}     2     t   I (1), c 1:{circumflex over (N)}     1     ({circumflex over (N)}     2     −{circumflex over (N)}     1     )t   II  and u 1  from y 1  using SIC; With the knowledge of c 1:{circumflex over (N)}     1     ({circumflex over (N)}     2     −{circumflex over (N)}     1     )t   II , Rx 1  recovers c 1:{circumflex over (N)}     1     {circumflex over (N)}     1     t   I,2 (2), c 1:{circumflex over (N)}     1     {circumflex over (N)}     2     t   I,1 (2) and u 2  from y 2 ; and 
     Rx 2  decodes c 1:{circumflex over (N)}     1     ({circumflex over (N)}     2     −{circumflex over (N)}     1     )t   II  and c 1:{circumflex over (N)}     1     {circumflex over (N)}     1     t   I,2 (2) from z 2  by treating c 1:{circumflex over (N)}     1     {circumflex over (N)}     2     t   I,1 (2) and  2   i  as noise. After that, c 1:{circumflex over (N)}     1     {circumflex over (N)}     2     t   I,1 (2) and v 2  are decoded using SIC; With the knowledge of c 1:{circumflex over (N)}     1     ({circumflex over (N)}     2     −{circumflex over (N)}     1     )t   II , c 1:{circumflex over (N)}     1     {circumflex over (N)}     2     t   I (1) and v 1  are decoded. 
       FIG. 4  is a diagram illustrating received signals at each receiver when a 2 &lt;b 2  according to another embodiment of the present disclosure. The upper-left, upper-right, lower-left and lower-right blocks respectively refer to y 1 , z 1 , y 2  and z 2 . 
     Referring to  FIG. 4 , 
     a) The private symbols and c 1:{circumflex over (N)}     1     {circumflex over (N)}     2     t   I (1) are received similar as in  FIG. 3 ; 
     b) As in step  3 .B, c 1:{circumflex over (N)}     1     {circumflex over (N)}     1     t   I (2) are received by both receivers in 1−b 2  channel use ({circumflex over (N)} 1  power levels) but only spanning N 1  dimensions; 
     c) According to step  4 .B, c 1:{circumflex over (N)}     1     ({circumflex over (N)}     2     −{circumflex over (N)}     1     )t   II,1  are received by Rx 2  in 1−b 2  channel use ({circumflex over (N)} 1  power levels) and spanning the remaining N 2 −N 1  dimensions in subband  2 . At Rx 1 , c 1:{circumflex over (N)}     1     ({circumflex over (N)}     2     −{circumflex over (N)}     1     )t   II,1  is received in N 1  dimensions and N 2 −N 1  power level in subband  1 ; and 
     d) Also based on step  4 .B, c 1:{circumflex over (N)}     1     {circumflex over (N)}     2     t   II,2  are received by Rx 1  in subband  1  with N 1  dimensions and {circumflex over (N)} 2  power levels, while they are received by Rx 2  in N 2  dimensions and {circumflex over (N)} 1  power levels in subband  2 . 
     Decoding: 
     Rx 1  decodes c 1:{circumflex over (N)}     1     {circumflex over (N)}     2     t   I (1), c 1:{circumflex over (N)}     1     ({circumflex over (N)}     2     −{circumflex over (N)}     1     )t   II,1 , c 1:{circumflex over (N)}     1     {circumflex over (N)}     2     t   II,2  and u 1  from y 1  using SIC; With the knowledge of c 1:{circumflex over (N)}     1     ({circumflex over (N)}     2     −{circumflex over (N)}     1     )t   II,1  and c 1:{circumflex over (N)}     1     {circumflex over (N)}     2     t   II,2 , Rx 1  recovers c 1:{circumflex over (N)}     1     {circumflex over (N)}     1     t   I (2) and private symbols from y 2 ; and 
     Rx 2  decodes and c 1:{circumflex over (N)}     1     ({circumflex over (N)}     2     −{circumflex over (N)}     1     )t   II,1  and c 1:{circumflex over (N)}     1     {circumflex over (N)}     1     t   I (2) from z 2  by treating c 1:{circumflex over (N)}     1     {circumflex over (N)}     2     t   II,2  and v 2  as noise. After that, c 1:{circumflex over (N)}     1     {circumflex over (N)}     2     t   II,2  and v 2  are decoded using SIC; With the knowledge of c 1:{circumflex over (N)}     1     ({circumflex over (N)}     2     −{circumflex over (N)}     1     )t   II,1  and c 1:{circumflex over (N)}     1{circumflex over (N)}       2     t   II, , c 1:{circumflex over (N)}     1     {circumflex over (N)}     2     t   I (1) and v 1  are decoded from z 1 . 
     2.5.2. Extension to Multiple-Subband Scenario with N 2 (1−ā)=N 1 (1− b ) 
     The private symbols and CS 1  are transmitted in each subband following the footsteps 1) to 3) in Section 2.5.1. The DoF achieved by PS 1  and PS 2  are respectively N 1   b  and N 2 ā, while CS 1  achieves the 
             DoF   ⁢           ∑     j   =   1     L     ⁢     min   ⁡     (         N   1     ⁡     (     1   -     b   j       )       ,       N   2     ⁡     (     1   -     a   j       )         )         L     .           
The CS 2  are generated to achieve the
 
             DoF   ⁢             1   2     ⁢       ∑     j   =   1     L     ⁢     max   ⁡     (         N   1     ⁡     (     1   -     b   j       )       ,       N   2     ⁡     (     1   -     a   j       )         )           -     min   ⁡     (         N   1     ⁡     (     1   -     b   j       )       ,       N   2     ⁡     (     1   -     a   j       )         )         L     .           
Rx 1  decodes them from the subbands with N 1 (1−b j )&gt;N 2 (1−a j ) while Rx 2  decodes them from the subbands N 1 (1−b j )&lt;N 2 (1−a j ). To this end, the procedure introduced in MISO scheme can be reused.
 
     Briefly, the transmitter randomly pairs the subbands with N 1 (1−b j1 )&gt;N 2 (1−a j1 ) and those with N 1 (1−b j2 )&lt;N 2 (1−a j2 ), then CS 2  are generated using step  4 ) in Section 2.5.1 and achieving the DoF min(q j1   + , q j2   − ), where q j1   + =N 1 (1−b j1 )−N 2 (1−a j1 ) and q j2   + =N 2 (1−a j2 )−N 1 (1−b j2 ). After that, update q j1   + =q j1   + −min(q j1   + , q j2   − ) and q j2   − =q j2   − −min(q j1   + , q j2   − ). Then repeat the procedure till all of q j1   +  and q j2   −  are zero. 
     2.5.3. Scenario N 2 (1−ā)&lt;N 1 (1− b ) (Rx 1  has a Larger Inherent DoF for Common Messages) 
     Similar as the discussion in Section 2.4.2 and based on MC 1, the transmission strategy is designed by 1) Finding a′ j ≦a j , ∀j, such that N 2 (1−ā′)=N 1 (1− b ); 2) Reusing the design introduced in Section 2.5.2 and replacing a j  with a′ j . The sum DoF performance is N 2 +N 1   b , which has been shown consistent with the outer-bound. 
     2.5.4. Scenario N 2 (1−ā)&gt;N 1 (1− b ) (Rx 2  has a Larger Inherent DoF for Common Messages) 
     The transmission strategy in this scenario follows MC 1. Similar as the discussion in Section 2.4.3, d c +d 2 =N 2  is guaranteed by increasing the power and DoF of PS 2  and the power allocated to PS 1  is fixed to be P b     j   . At the same time, in order to send PS 1  with a DoF as high as possible, the increment of PS 2  is started from the subband with the lowest b 1 . In this way, the DoF loss of PS 2  can be minimized while the gap between the inherent DoF for common messages at each receiver is reducing. Specifically, the footsteps are given as follows: 
     Ordering the subband by b j  in ascending order. π(j) is used to denote the subband with the jth lowest b j ; 
     i=1; 
     Increasing the power and DoF of PS 1  in subband π(i) to a′ π(i) , where a π(i) &lt;a′ π(i) ≦1, 
     If Σ j=1   i−1  N 1  min(a π(j) , 1−b π(j) )+Σ j=i+1   L  N 1 (1−b π(j) )+N 1 (1−max(b π(i) , a′ π(i) −a π(i) ))=Σ j=i+1   L  N 2 (1−a π(j) )+N 2 (1−a′ π(i) ), go to step d and set a′ π(j&gt;i) =a π(j&gt;i) ; 
     Else, set a′ π(i) =1 and i=i+1. 
     Construct the transmission strategy with a′ 1:L  and b j  as discussed in Section 2.5.2. 
     The l.h.s and r.h.s of the equation in step c respectively stand for the inherent DoF for common messages at Rx 1  and Rx 2  after increasing a π(1:i)  to a′ π(1:i) . Since increasing a i  to a′ i  strictly reduces the gap between inherent DoF at Rx 2  and Rx 1 , a new equality for the common messages will be found. Finally, the sum DoF performance is given by: 
     
       
         
           
             
               N 
               2 
             
             + 
             
               
                 N 
                 1 
               
               ⁢ 
               
                 
                   
                     
                       ∑ 
                       
                         j 
                         = 
                         1 
                       
                       L 
                     
                     ⁢ 
                     
                       
                         ( 
                         
                           
                             b 
                             j 
                           
                           - 
                           
                             a 
                             j 
                             ′ 
                           
                           + 
                           
                             a 
                             j 
                           
                         
                         ) 
                       
                       + 
                     
                   
                   L 
                 
                 . 
               
             
           
         
       
     
     Note that in the multiple subbands scenario with the CSIT quality a i:L =a and b i:L =b across the subbands, the transmission strategy proposed in this section results in a better DoF performance compared to using the single-subband transmission (see Section 2.4.3) individually in each subband. Especially for the case 
                 a   +             N   2     -     N   1       ⁢               N   2       ⁢     (     b   -   1     )         &lt;   0     ,         
the sum DoF is N 2  in Section 2.4.3, while the multiple-subband transmission strictly outperforms this as the Tx can always transmit PS 1  in some subbands. Hence, unlike the MISO case, the specific CSIT pattern not only impact the transmission strategy, but also the sum DoF performance.
 
2.6 Signaling Mechanisms Needed to Operate the Transmission Strategies
 
     CSIT Quality Feedback 
     As in Section 2.1, the CSIT quality in MIMO case is defined based on the norm of each column in the channel matrix, namely the channel vector from the transmission antenna array to each receive antenna. This definition is similar as that in MISO case and moreover assume the channel estimation and quantization at the receive antenna are statistically equivalent. Hence, the similar signaling mechanisms regarding the CSIT quality feedback in MISO case can be reused. 
     Specifically, depending on the PMI report mode, if wideband PMI (PUCCH 1-1-1, PUCCH 1-1-2 and PUSCH 3-1) and/or all subband PMI (PUSCH 3-2 and PUSCH 1-2) are available, each receiver reports a single value of the CSI accuracy as it is assumed the same across all subbands; if one or several subband PMI is reported (but not all), one or several estimates of the CSIT accuracies are reported. 
     Transmission Mode Indication 
     Once the Tx collects the CSI and their qualities from all the receivers, it has to perform user-selection and subband grouping, then makes a decision to the transmission strategy. The similar DCI format as the MISO case can be reused, regarding the size of the transmission block (as it depends on CSIT pattern of the co-scheduled users), the kinds of messages (CS 1 , CS 2  and PS 1 , PS 2 ), and the number of messages, the modulation and coding scheme of all CS and PS intended for the user, information about whether common message is intended for the user or not, the transmit power of each message. In addition, in the decision of the transmission mode, the following issue should be taken into account: 
     The format of CS: Based on MC 2 and MC 3, the transmission of CS consists of several power levels depending on the value of N 1  and N 2 . Hence, to perform the decoding, each receiver should know: 1) the number of power levels of the common messages they have observed (for instance, Rx 1  should know N 2  and Rx 2  should know {circumflex over (N)} 1  according to MC 2, MC 3 and Section 2.4.1) as this relates to the number of stages in the SIC; 2) the received power at each level (for instance, the power received in the mth level P 1−(m−1)Δ  according to Section 2.4.1) as this impacts the MMSE filter used in each stage of SIC. 
     While the present disclosure has been shown and described with reference to various embodiments thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without departing from the spirit and scope of the present disclosure as defined by the appended claims and their equivalents.