Abstract:
A receiver in a multiple input, multiple output (MIMO) system is configured to perform a method for generating channel quality feedback information. The method includes receiving, from a MIMO transmitter, pilot signals in each MIMO layer. The method also includes selecting an optimal precoder matrix for each MIMO layer using a first detection metric. The method further includes determining a signal-to-noise ratio (SNR) for each MIMO layer using a second detection metric and the optimal precoder.

Description:
CROSS-REFERENCE TO RELATED APPLICATION(S) AND CLAIM OF PRIORITY 
       [0001]    The present application claims priority to U.S. Provisional Patent Application Ser. No. 61/543,200 filed Oct. 4, 2011, entitled “METHOD AND APPARATUS FOR LOW COMPLEXITY FEEDBACK”. The content of the above-identified patent documents is incorporated herein by reference. 
     
    
     TECHNICAL FIELD 
       [0002]    The present application relates generally to determination of channel quality feedback in wireless mobile communication systems and, more specifically, to an improved low-complexity feedback algorithm that is suitable for use in a multiple input, multiple output (MIMO) system. 
       BACKGROUND 
       [0003]    Detection of signals and providing periodic channel quality feedback in multiple input, multiple output (MIMO) wireless transmission systems presents a challenging problem involving complex and extensive computations. For mobile handsets, the number of computations that must be performed to provide feedback for each transmitted symbol can require substantial power consumption, decreasing battery life. 
       SUMMARY 
       [0004]    For use in a receiver in a multiple input, multiple output (MIMO) system, a method for generating channel quality feedback information is provided. The method includes receiving, from a MIMO transmitter, pilot signals in each MIMO layer. The method also includes selecting an optimal precoder matrix for each MIMO layer using a first detection metric. The method further includes determining a signal-to-noise ratio (SNR) for each MIMO layer using a second detection metric and the optimal precoder. 
         [0005]    For use in a receiver in a MIMO system, an apparatus configured to generate channel quality feedback information is provided. The apparatus includes a processor. The processor is configured to receive, from a MIMO transmitter, pilot signals in each MIMO layer. The processor is also configured to select an optimal precoder matrix for each MIMO layer using a first detection metric. The processor is further configured to determine a SNR for each MIMO layer using a second detection metric and the optimal precoder. 
         [0006]    A receiver configured for use in a MIMO system and capable of generating channel quality feedback information is provided. The receiver includes a plurality of antenna elements and a processor coupled to the plurality of antenna elements. The processor is configured to receive, from a MIMO transmitter, pilot signals in each MIMO layer. The processor is also configured to select an optimal precoder matrix for each MIMO layer using a first detection metric. The processor is further configured to determine a SNR for each MIMO layer using a second detection metric and the optimal precoder. 
         [0007]    Before undertaking the DETAILED DESCRIPTION below, it may be advantageous to set forth definitions of certain words and phrases used throughout this patent document: the terms “include” and “comprise,” as well as derivatives thereof, mean inclusion without limitation; the term “or,” is inclusive, meaning and/or; the phrases “associated with” and “associated therewith,” as well as derivatives thereof, may mean to include, be included within, interconnect with, contain, be contained within, connect to or with, couple to or with, be communicable with, cooperate with, interleave, juxtapose, be proximate to, be bound to or with, have, have a property of, or the like; and the term “controller” means any device, system or part thereof that controls at least one operation, such a device may be implemented in hardware, firmware or software, or some combination of at least two of the same. It should be noted that the functionality associated with any particular controller may be centralized or distributed, whether locally or remotely. Definitions for certain words and phrases are provided throughout this patent document, those of ordinary skill in the art should understand that in many, if not most instances, such definitions apply to prior, as well as future uses of such defined words and phrases. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0008]    For a more complete understanding of the present disclosure and its advantages, reference is now made to the following description taken in conjunction with the accompanying drawings, in which like reference numerals represent like parts: 
           [0009]      FIG. 1  illustrates an exemplary wireless network capable of implementing MIMO techniques, according to one or more embodiments of this disclosure; 
           [0010]      FIGS. 2A and 2B  illustrate components and signals within a transmitter and a receiver for a MIMO signal transmission system in a wireless network, according to one or more embodiments of this disclosure; 
           [0011]      FIG. 3  depicts a representative MIMO system having nTx transmitter antennas and nRx receiver antennas; 
           [0012]      FIG. 4  depicts a plot illustrating a number of different functions that have the same optimization over a sparse set; and 
           [0013]      FIGS. 5A through 5F  illustrate comparative performance plots in a number of simulations for a minimum mean square error (MMSE) based feedback metric and a detector using the maximum ratio combining (MRC) metric described herein, according to an embodiment of this disclosure. 
       
    
    
     DETAILED DESCRIPTION 
       [0014]      FIGS. 1 through 5F , discussed below, and the various embodiments used to describe the principles of the present disclosure in this patent document are by way of illustration only and should not be construed in any way to limit the scope of the disclosure. Those skilled in the art will understand that the principles of the present disclosure may be implemented in any suitably arranged wireless communication system. 
         [0015]    The following documents and standards descriptions are hereby incorporated into the present disclosure as if fully set forth herein: 
         [0016]    “Evolved Universal Terrestrial Radio Access (E-UTRA); User Equipment (UE) Radio Access Capabilities (Release 10)”, 3GPP Technical Specification No. 36.211, version 10.2.0, June 2011 (hereinafter “REF1”); and “Evolved Universal Terrestrial Radio Access (E-UTRA); User Equipment (UE) radio transmission and reception (Release 10)”, 3GPP Technical Specification No. 36.101, version 10.3.0, June 2011 (hereinafter “REF2”). 
         [0017]    MIMO antenna systems are an integral part of fourth generation communications systems such as Long Term Evolution (LTE), LTE Advanced (LTE-A) and Worldwide Interoperability for Microwave Access (WiMAX). To achieve high spectral efficiency, as many as eight antennas are supported at both the receiver and transmitter in LTE Release 10. In addition, higher order modulations such as Quadrature Amplitude Modulation with 64 constellation points (64-QAM) are used in a high signal-to-noise ratio (SNR) scenario. 
         [0018]      FIG. 1  is a high level diagram illustrating an exemplary wireless network implementing MIMO techniques, according to one or more embodiments of this disclosure. The wireless network  100  illustrated in  FIG. 1  is provided solely for purposes of explaining the subject matter of the present disclosure, and is not intended to suggest any limitation regarding the applicability of that subject matter. Other wireless networks may employ the subject matter depicted in the drawings and described herein without departing from the scope of the present disclosure. In addition, those skilled in the art will recognize that the complete structure and operation of a wireless network and the components thereof are depicted in the drawings and described therein. Instead, for simplicity and clarity, only so much of the structure and operation of the wireless network and the components thereof as are unique to the present disclosure or necessary for an understanding of the present disclosure are depicted and described. 
         [0019]    In the illustrated embodiment, wireless network  100  includes a base station (BS)  101 , BS  102 , and BS  103 . Depending on the network type, other well-known terms may be used instead of “base station,” such as “Evolved Node B” (eNB) or “access point” (AP). For simplicity and clarity, the term “base station” will be used herein to refer to the network infrastructure components that provide or facilitate wireless communications network access to remote (mobile or fixed) terminals. 
         [0020]    The BS  101  communicates with BS  102  and BS  103  via network  130  operating according to a standardized protocol (e.g., X2 protocol), via a proprietary protocol, or preferably via Internet protocol (IP). IP network  130  may include any IP-based network or a combination thereof, such as the Internet, a proprietary IP network, or another data network. 
         [0021]    The BS  102  provides wireless broadband access to a first plurality of mobile stations (MSs) within coverage area  120  of BS  102 . In the example illustrated, the first plurality of MSs includes MS  111 , which may be located in a small business; MS  112 , which may be located in an enterprise; MS  113 , which may be located in a wireless fidelity (WiFi) hotspot; MS  114 , which may be located in a first residence; MS  115 , which may be located in a second residence; and MS  116 , which may be a mobile device, such as a cell phone, a wireless laptop, a wireless-enabled tablet, or the like. For simplicity and clarity, the term “mobile station” or “MS” is used herein to designate any remote wireless equipment that wirelessly accesses or communicates with a BS, whether the MS is a mobile device (e.g., cell phone, wireless-enabled tablet or laptop, etc.) or is normally considered a stationary device (e.g., desktop personal computer, wireless television receiver, etc.). In other systems, other well-known terms may be used instead of “mobile station,” such as “user equipment” (UE), “subscriber station” (SS), “remote terminal” (RT), “wireless terminal” (WT), and the like. 
         [0022]    The BS  103  provides wireless broadband access to a second plurality of MSs within coverage area  125  of BS  103 . The second plurality of MSs includes MS  115  and MS  116 . In an exemplary embodiment, BSs  101 - 103  communicate with each other and with MSs  111 - 116  using MIMO techniques. While only six MSs are depicted in  FIG. 1 , it will be understood that wireless network  100  may provide wireless broadband access to additional MSs. 
         [0023]      FIGS. 2A and 2B  are diagrams of components and signals within a transmitter and a receiver for a MIMO signal transmission system in a wireless network, according to one or more embodiments of this disclosure. 
         [0024]    As shown in  FIG. 2A , the MIMO signal transmission system  200  includes a transmitter  201  coupled to an array of L antenna or antenna elements  203 - 1  to  203 -L and a receiver  202  coupled to an array of L antenna or antenna elements  204 - 1  to  204 -L, with the transmitter  201  forming part of one of BSs  101 - 103  and the receiver  202  forming part of one of the MSs  111 - 116  in the embodiment. As understood by those skilled in the art, each BS  101 - 103  and each MS  111 - 116  includes both a transmitter and a receiver each separately coupled to the respective antenna array to transmit or receive radio frequency signals over channel H, such that the transmitter  201  may alternatively be disposed within one of the MSs  111 - 116  and the receiver  202  may alternatively be disposed within one of the BSs  101 - 103 . 
         [0025]    In the example depicted, the transmitter  201  includes encoding and modulation circuitry comprising a channel encoder  205  receiving and encoding data for transmission, an interleaver  206  coupled to the channel encoder  205 , a modulator  207  coupled to the interleaver  206 , and a de-multiplexer  208  coupled to the modulator  207  and antenna elements  203 - 1  to  203 -L. In the example depicted, the receiver  202  includes a MIMO demodulator  209  coupled to the antenna elements  204 - 1  to  204 -L, a de-interleaver  210  coupled to the MIMO demodulator  209  and a channel decoder  211  coupled to the de-interleaver  210 . In addition, transmitter  201  and receiver  202  may each include a programmable processor or controller including and/or connected to memory and coupled to the respective transmitter and receiver chains for controlling operation of the respective BS or MS. Using such components, synchronization signals are transmitted by a BS and received by an MS in the manner described in further detail below. 
         [0026]      FIG. 2B  illustrates an example of MIMO signal transmission. As discussed above, MIMO signal transmission utilizes multiple antenna elements at both the transmitter and receiver. The MIMO transmission arrangement  250  illustrated by  FIG. 2B  includes transmitter antenna elements  203 - 1  and  203 - 2  located within one of BSs  101 ,  102  and  103 , and receiver antenna elements  204 - 1  and  204 - 2  located within one of MSs  111 ,  112 ,  113 ,  114 ,  115  and  116 . For simplicity, a 2×2 MIMO system (i.e., two transmit antennas and two receive antennas) MIMO system is illustrated, although those skilled in the art will understand how the mathematics discussed below extends to larger systems. The mathematical equation governing signals transmitted over channel H between antenna elements  203 - 1  and  203 - 2  and antenna elements  204 - 1  and  204 - 2  is given as: 
         [0000]    
       
         
           
             
               [ 
               
                 
                   
                     
                       Y 
                       1 
                     
                   
                 
                 
                   
                     
                       Y 
                       2 
                     
                   
                 
               
               ] 
             
             = 
             
               
                 
                   [ 
                   
                     
                       
                         
                           h 
                           11 
                         
                       
                       
                         
                           h 
                           12 
                         
                       
                     
                     
                       
                         
                           h 
                           21 
                         
                       
                       
                         
                           h 
                           22 
                         
                       
                     
                   
                   ] 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           X 
                           1 
                         
                       
                     
                     
                       
                         
                           X 
                           2 
                         
                       
                     
                   
                   ] 
                 
               
               + 
               
                 [ 
                 
                   
                     
                       
                         n 
                         1 
                       
                     
                   
                   
                     
                       
                         n 
                         2 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
         [0000]    where Y 1  and Y 2  are the received signal at antenna elements  204 - 1  and  204 - 2 , respectively, X 1  and X 2  are the symbols transmitted by antenna elements  203 - 1  and  203 - 2 , respectively, h 11  and h 12  represent characteristics of channel H between antenna element  203 - 1  and antenna elements  204 - 1  and  204 - 2 , respectively, h 21  and h 22  represent channel characteristics between antenna element  203 - 2  and antenna elements  204 - 1  and  204 - 2 , respectively, and n 1  and n 2  are independent identically distributed Gaussian noise signals with variance σ 2 . 
         [0027]    MIMO detection is used to recover estimates of the bits in X 1  and X 2 . Since the system is coded, interest is focused on the soft estimates (i.e., log-likelihood ratios or “LLRs”) instead of the actual bits themselves, where the soft estimates are then fed to the turbo decoder. The performance of any detector is finally evaluated according to the resulting block error rate (BLER) (sometimes also referred to as frame error rate or “FER”) performance as a function of the SNR. 
         [0028]    In LTE Release 8, periodic feedback of the channel state is sent from the UE to the ENodeB. This feedback includes three components:
       Channel Quality Indicator (CQI): Indicates the spectral efficiency of each layer in the channel;   Precoding Matrix Indicator (PMI): Precoder matrix to be used with the channel;   Rank Indicator (RI): The number of layers that are feasible for the channel.       
 
         [0032]    To illustrate these concepts,  FIG. 3  depicts a representative MIMO system having nTx transmitter antennas and nRx receiver antennas. The MIMO system  300  may represent one or more of the MIMO systems depicted in  FIGS. 1 ,  2 A, and  2 B. The number of layers that are transmitted on the MIMO system  300  is L. The channel matrix, H, has dimensions (nRx×nTx), while the precoder matrix P has dimensions (nTx×L). The equation governing this system is given as: 
         [0000]        Y=HPX+n   [Eqn. 1]
 
         [0000]    where Y is a vector of length nRx, P is the precoder matrix, and X is a vector of length L, and n is a noise vector. 
         [0033]    The rank of this system is equal to L. The precoder is restricted to a finite set of choices. For example, for an example 4×4 MIMO system, the precoder can only be selected from Table 1 below, where 
         [0000]    
       
         
           
             
               W 
               n 
             
             = 
             
               I 
               - 
               
                 
                   
                     2 
                      
                     
                       u 
                       n 
                     
                      
                     
                       u 
                       n 
                       H 
                     
                   
                   
                     
                       u 
                       n 
                       H 
                     
                      
                     
                       u 
                       n 
                     
                   
                 
                 . 
               
             
           
         
       
     
         [0034]    One approach to selecting the rank and precoder for feedback is to work backwards from the detection algorithm used in the receiver. For example, a MIMO system is considered that includes a receiver that uses a MMSE detector. A MMSE filter for the system in Equation 1 is given as: 
         [0000]        F   MMSE =( HP ) H ( HPP   H   H   H +σ 2   I ) −1   [Eqn. 2]
 
         [0000]    where σ 2  is a variance of the noise. The MMSE filter is dependent on the MIMO channel H, which can be estimated at the receiver using pilot symbols transmitted from the base station. 
         [0035]    The MMSE filter can be applied to Equation 1 to get: 
         [0000]        F   MMSE   Y=F   MMSE   HPX+F   MMSE   n   [Eqn. 3]
 
         [0036]    Equation 3 can be decomposed as L separate equations, each of the L equations having the form: 
         [0000]        {circumflex over (X)}   L   ={tilde over (H)}   ll   X   l +Σ {i=1,i≠1}   L   {tilde over (H)}   li   X   i   +ñ   l   [Eqn. 4]
 
         [0000]    where {tilde over (Y)} l  and ñ l  are the components of the vectors F MMSE Y and F MMSE n respectively, and H ij  are the components of the L×L matrix F MMSE HP. 
         [0037]    Using Equation 4, the signal-to-noise ratio (SNR) for layer l can be determined as: 
         [0000]    
       
         
           
             
               
                 
                   
                     SNR 
                     l 
                   
                   = 
                   
                     
                       
                          
                         
                           
                             H 
                             ~ 
                           
                           ll 
                         
                          
                       
                       2 
                     
                     
                       
                         
                           ∑ 
                           
                             
                               i 
                               = 
                               1 
                             
                             , 
                             
                               i 
                               ≠ 
                               l 
                             
                           
                           L 
                         
                          
                         
                           
                              
                             
                               H 
                               li 
                             
                              
                           
                           2 
                         
                       
                       + 
                       
                         
                           σ 
                           ~ 
                         
                         l 
                         2 
                       
                     
                   
                 
               
               
                 
                   [ 
                   
                     Eqn 
                     . 
                     
                         
                     
                      
                     5 
                   
                   ] 
                 
               
             
           
         
       
     
         [0000]    where {tilde over (σ)} l   2  represents the noise variance experienced on layer l, as per Equation 4. 
         [0038]    The SNR per layer may be mapped to determine effective spectral efficiency per layer. Using the spectral efficiency per layer, it is possible to determine how much throughput can be achieved with various combinations of precoder matrices and rank. The combination that provides the best throughput can be indicated. The CQI may then be reported based on that rank and precoder. 
         [0039]    The MMSE matrix in Equation 2 may need to be reevaluated for each precoder matrix. This results in a large number of mathematical operations at the receiver. For example, in a 4×4 MIMO system, there are 16 precoder choices for each selection of rank ε{1, 2, 3, 4}, as shown in Table 1. The number of multiplications required to compute the feedback using this technique for each resource block is 9584. 
         [0040]    In accordance with an embodiment of this disclosure, a new feedback algorithm uses a new, low-complexity metric to determine the optimal precoder. Using the new metric, the optimal precoder is quickly determined without the need to evaluate the MMSE matrix for each precoder candidate. The new metric is based on the following observation. Suppose there is a function f(x) that is to be maximized over a finite and sparse set. Then there is an infinite number of functions f i (X) indexed by i, such that argmax xεS (f(x))=argmax xεS (f i (x)). An example is shown in  FIG. 4 , where three very different functions have the same maximizing value x 2  over the sparse set {x 1 , x 2 , x 3 , x 4 }. Therefore, to find the optimizer of a given function, it may be advantageous to compute the optimizer of another function that is computationally easier. 
         [0041]    In one embodiment of this disclosure, the new metric is based upon the maximum ratio combining (MRC) detection metric, which has substantially lower complexity than the original MMSE metric. Use of the MRC detection metric instead of the MMSE metric is based on the fact that, over a sparse set, many different functions have the same optimizing argument. Simulation results are provided below that establish that the low complexity MRC metric is a good substitute for the MMSE metric. 
         [0042]    To explain the MRC metric, it is helpful to describe the MRC detector as follows. Considering Equation 1 (shown above). To detect the i th  layer, Equation 1 is multiplied by (HP i ) H . The result of the multiplication is the following: 
         [0000]      ( HP   i ) H   Y=|HP   i | 2   X   i +Σ j=1,j≠i   L   P   i   H   H   H   HP   j   X   j +( HP   i ) H   n   [Eqn. 6]
 
         [0043]    Thus, the SNR using the MRC detector is given as: 
         [0000]    
       
         
           
             
               
                 
                   
                     SNR 
                     i 
                   
                   = 
                   
                     
                       
                         
                            
                           
                             HP 
                             i 
                           
                            
                         
                         4 
                       
                       
                         
                           
                             ∑ 
                             
                               
                                 j 
                                 = 
                                 1 
                               
                               , 
                               
                                 j 
                                 ≠ 
                                 i 
                               
                             
                             L 
                           
                            
                           
                             
                                
                               
                                 
                                   P 
                                   i 
                                   H 
                                 
                                  
                                 
                                   H 
                                   H 
                                 
                                  
                                 
                                   HP 
                                   j 
                                 
                               
                                
                             
                             2 
                           
                         
                         + 
                         
                           
                              
                             
                               HP 
                               i 
                             
                              
                           
                           2 
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   [ 
                   
                     Eqn 
                     . 
                     
                         
                     
                      
                     7 
                   
                   ] 
                 
               
             
           
         
       
     
         [0044]    Since the objective of the new metric is reduced computational complexity, the denominator of the right side of Equation 7 can be ignored. This eliminates the need to calculate the cross terms P i   H H H HP j . Equation 7 is thus reduced to the following: 
         [0000]        SNR   i   ∝|HP   i | 4   [Eqn. 8]
 
         [0045]    Finally we choose a precoder that maximizes the quantity in Equation 8, using the following equation: 
         [0000]        f ( SNR   i=1   L )=Σ i=1   L   SNR   i   [Eqn. 9]
 
         [0046]    It will be understood by those skilled in the art that the use of Equations 8 and 9 is just one of many possible choices for the SNR, and f(SNR i=1   L ) that could be made while choosing a feedback algorithm metric. 
         [0047]    In summary, the feedback algorithm proceeds as follows: 
         [0048]    Stage 1) Select the optimal precoder based upon the MRC metric, using Equation 9. 
         [0049]    Stage 2) Use the optimal precoder for each rank to report the best rank and PMI based upon the aforementioned MMSE metric. 
         [0050]    Use of the MRC metric in the feedback algorithm described herein provides a significant savings in complex computations. For example, in a 4×4 MIMO system, the original MMSE based algorithm requires 9584 multiplications. In contrast, use of the MRC metric described herein results in a feedback algorithm that requires only 625 multiplications. Thus, the number of multiplications is reduced by a factor of 15.33. 
         [0051]      FIGS. 5A through 5F  illustrate comparative performance plots in a number of simulations from REF2 for the original MMSE metric and a detector using the MRC metric described herein, according to an embodiment of this disclosure. The performance is evaluated for low, medium, and high correlation to gauge the effect of neglecting the interference term in Equation 8.  FIG. 5A  depicts results from a scenario from REF2 with a low correlation Doppler 5 Hz EPA channel.  FIG. 5B  depicts results from a scenario with a medium correlation Doppler 5 Hz EPA channel.  FIG. 5C  depicts results from a scenario with a high correlation Doppler 5 Hz EPA channel.  FIG. 5D  depicts results from a scenario from REF2 with a low correlation Doppler 5 Hz ETU channel.  FIG. 5E  depicts results from a scenario with a medium correlation Doppler ETU EPA channel.  FIG. 5F  depicts results from a scenario with a high correlation Doppler 5 Hz ETU channel. 
         [0052]    As shown in  FIGS. 5A through 5F , the MRC metric provides results that are nearly as good as the original MMSE metric in certain scenarios, and even better than the MMSE metric in other scenarios, while computational complexity is 15 times less than the original MMSE metric. It is observed that neglecting interference from other layers does not harm the MRC-based reduced complexity approach for high and medium correlations. In fact, the MRC metric outperforms the original MMSE metric in these scenarios. 
         [0053]    The embodiments described above provide a very low complexity MIMO detection algorithm that is suitable for many applications, including use in LTE-Advanced modem chips. The disclosed embodiments of the detection algorithm provide increased throughput, improved cellular reception, and improved battery power conservation. 
         [0054]    Although the present disclosure has been described with an exemplary embodiment, various changes and modifications may be suggested to one skilled in the art. It is intended that the present disclosure encompass such changes and modifications as fall within the scope of the appended claims. 
         [0000]    
       
         
               
             
               
               
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 4 × 4 MIMO Precoders 
               
             
          
           
               
                 Codebook 
                   
                 Number of layers ν 
               
             
          
           
               
                 index 
                 u n   
                 1 
                 2 
                 3 
                 4 
               
               
                   
               
             
          
           
               
                 0 
                 u 0  = [1 −1 −1 −1] T   
                 W 0   {1}   
                 W 0   {14} /{square root over (2)} 
                 W 0   {124} /{square root over (3)} 
                 W 0   {1234} /2 
               
               
                 1 
                 u 1  = [1 −j 1 j] T   
                 W 1   {1}   
                 W 1   {12} /{square root over (2)} 
                 W 1   {123} /{square root over (3)} 
                 W 1   {1234} /2 
               
               
                 2 
                 u 2  = [1 1 −1 1] T   
                 W 2   {1}   
                 W 2   {12} /{square root over (2)} 
                 W 2   {3214} /2 
                 W 2   {3214} /2 
               
               
                 3 
                 u 3  = [1 j 1 −j] T   
                 W 3   {1}   
                 W 3   {12} /{square root over (2)} 
                 W 3   {123} /{square root over (3)} 
                 W 3   {3214} /2 
               
               
                 4 
                 u 4  = [1 (−1 − j)/{square root over (2)} −j (1 − j)/{square root over (2)}] T   
                 W 4   {1}   
                 W 4   {14} /{square root over (2)} 
                 W 4   {124} /{square root over (3)} 
                 W 4   {1234} /2 
               
               
                 5 
                 u 5  = [1 (1 − j)/{square root over (2)} j (−1 − j)/{square root over (2)}] T   
                 W 5   {1}   
                 W 5   {14} /{square root over (2)} 
                 W 5   {1234} /2 
                 W 5   {1234} /2 
               
               
                 6 
                 u 6  = [1 (1 + j)/{square root over (2)} −j (−1 + j)/{square root over (2)}] T   
                 W 6   {1}   
                 W 6   {13} /{square root over (2)} 
                 W 6   {134} /{square root over (3)} 
                 W 6   {1324} /2 
               
               
                 7 
                 u 7  = [1 (−1 + j)/{square root over (2)} j (1 + j)/{square root over (2)}] T   
                 W 7   {1}   
                 W 7   {13} /{square root over (2)} 
                 W 7   {1324} /2 
                 W 7   {1324} /2 
               
               
                 8 
                 u 8  = [1 −1 1 1] T   
                 W 8   {1}   
                 W 8   {12} /{square root over (2)} 
                 W 8   {124} /{square root over (3)} 
                 W 8   {1234} /2 
               
               
                 9 
                 u 9  = [1 −j −1 −j] T   
                 W 9   {1}   
                 W 9   {14} /{square root over (2)} 
                 W 9   {134} /{square root over (3)} 
                 W 9   {1234} /2 
               
               
                 10 
                 u 10  = [1 1 1 −1] T   
                 W 10   {1}   
                 W 10   {13} /{square root over (2)} 
                 W 10   {1324} /2 
                 W 10   {1324} /2 
               
               
                 11 
                 u 11  = [1 j −1 j] T   
                 W 11   {1}   
                 W 11   {13} /{square root over (2)} 
                 W 11   {134} /{square root over (3)} 
                 W 11   {1324} /2 
               
               
                 12 
                 u 12  = [1 −1 −1 1] T   
                 W 12   {1}   
                 W 12   {12} /{square root over (2)} 
                 W 12   {1234} /2 
                 W 12   {1234} /2 
               
               
                 13 
                 u 13  = [1 −1 1 −1] T   
                 W 13   {1}   
                 W 13   {1324} /2 
                 W 13   {1324} /2 
                 W 13   {1324} /2 
               
               
                 14 
                 u 14  = [1 1 −1 −1] T   
                 W 14   {1}   
                 W 14   {13} /{square root over (2)} 
                 W 14   {123} /{square root over (3)} 
                 W 14   {3214} /2 
               
               
                 15 
                 u 15  = [1 1 1 1] T   
                 W 15   {1}   
                 W 15   {12} /{square root over (2)} 
                 W 15   {1234} /2 
                 W 15   {1234} /2