Patent Publication Number: US-8116691-B2

Title: Systems and methods for improving reference signals for spatially multiplexed cellular systems

Description:
RELATED REFERENCES 
     This application is the national phase under 35 U.S.C. §371 of PCT International Application No. PCT/JP2008/055157 with an International Filing Date of Mar. 13, 2008, which claims the benefit of U.S. patent application Ser. No. 11/686,251, entitled “SYSTEMS AND METHODS FOR IMPROVING REFERENCE SIGNALS FOR SPATIALLY MULTIPLEXED CELLULAR SYSTEMS” filed on Mar. 14, 2007. 
     TECHNICAL FIELD 
     The present invention relates generally to computers and computer-related technology. More specifically, the present invention relates to systems and methods that improve the design of reference signals for spatially multiplexed cellular systems. 
     BACKGROUND ART 
     A wireless communication system typically includes a base station in wireless communication with a plurality of user devices (which may also be referred to as mobile stations, subscriber units, access terminals, etc.). The base station transmits data to the user devices over a radio frequency (RF) communication channel. The term “downlink” refers to transmission from a base station to a user device, while the term “uplink” refers to transmission from a user device to a base station. 
     Orthogonal frequency division multiplexing (OFDM) is a modulation and multiple-access technique whereby the transmission band of a communication channel is divided into a number of equally spaced sub-bands. A sub-carrier carrying a portion of the user information is transmitted in each sub-band, and every sub-carrier is orthogonal with every other sub-carrier. Sub-carriers are sometimes referred to as “tones.” OFDM enables the creation of a very flexible system architecture that can be used efficiently for a wide range of services, including voice and data. OFDM is sometimes referred to as discrete multi-tone transmission (DMT). 
     The 3rd Generation Partnership Project (3GPP) is a collaboration of standards organizations throughout the world. The goal of 3GPP is to make a globally applicable third generation (3G) mobile phone system specification within the scope of the IMT-2000 (International Mobile Telecommunications-2000) standard as defined by the International Telecommunication Union. The 3GPP Long Term Evolution (“LTE”) Committee is considering OFDM as well as OFDM/OQAM (Orthogonal Frequency Division Multiplexing/Offset Quadrature Amplitude Modulation), as a method for downlink transmission, as well as OFDM transmission on the uplink. 
     Wireless communications systems (e.g., Time Division Multiple Access (TDMA), Orthogonal Frequency-Division Multiplexing (OFDM)) usually calculate an estimation of a channel impulse response between the antennas of a user device and the antennas of a base station for coherent receiving. Channel estimation may involve transmitting known reference signals that are multiplexed with the data. Reference signals may include a single frequency and are transmitted over the communication systems for supervisory, control, equalization, continuity, synchronization, etc. Wireless communication systems may include one or more mobile stations and one or more base stations that each transmit a reference signal. Reference signals may be designed such that a mobile station may re-use a reference signal that was previously used by a different mobile station. As such, benefits may be realized from systems and methods that improve the design of reference signals for spatially multiplexed cellular systems. 
     DISCLOSURE OF INVENTION 
     A method to generate sequences used for wireless communication systems is described. Initial sequences are set to a first set of sequences. A second set of sequences is obtained by projecting the first set of sequences to a space of tight frames. A third set of sequences is obtained by projecting each of subsets of the second set of sequences to a space of orthogonal matrices. A forth set of sequences is obtained by projecting the third set of sequences to a space of circulant matrices. A fifth set of sequences is obtained by projecting the forth set of sequences to a space of matrices with preferable PAPR. Said steps are iterate at least once by setting the fifth set of sequences to the first set of sequences. The fifth set of sequences after the iterating step has been executed is outputted. 
     A method for using a numerical method to design reference signals for multiple input multiple output (MIMO) systems is described. An input multiple input multiple output signal is determined. A nearest tight frame to one or more given structured vectors is obtained. One or more structured vectors from the nearest tight frame is obtained. The one or more structured vectors are projected onto the space of circulant matrices. One or more classes of matrices associated with reference signals are outputted. 
     A transmitter that is configured to use a numerical method to design reference signals for multiple input multiple output (MIMO) systems is also described. The transmitted includes a processor and memory in electronic communication with the processor. Instructions are stored in the memory. An input multiple input multiple output signal is determined. A nearest tight frame to one or more given structured vectors is obtained. One or more structured vectors from the nearest tight frame is obtained. The one or more structured vectors is projected onto the space of circulant matrices. One or more classes of matrices associated with reference signals are outputted. 
     A computer-readable medium comprising executable instructions is also described. An input multiple input multiple output signal is determined. A nearest tight frame to one or more given structured vectors is obtained. One or more structured vectors from the nearest tight frame is obtained. The one or more structured vectors is projected onto the space of circulant matrices. One or more classes of matrices associated with reference signals are outputted. 
     Various embodiments of the invention are now described with reference to the Figures, where like reference numbers indicate identical or functionally similar elements. The embodiments of the present invention, as generally described and illustrated in the Figures herein, could be arranged and designed in a wide variety of different configurations. Thus, the following more detailed description of several exemplary embodiments of the present invention, as represented in the Figures, is not intended to limit the scope of the invention, as claimed, but is merely representative of the embodiments of the invention. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
       Exemplary embodiments of the invention will become more fully apparent from the following description and appended claims, taken in conjunction with the accompanying drawings. Understanding that these drawings depict only exemplary embodiments and are, therefore, not to be considered limiting of the invention&#39;s scope, the exemplary embodiments of the invention will be described with additional specificity and detail through use of the accompanying drawings in which: 
         FIG. 1  illustrates an exemplary wireless communication system in which embodiments may be practiced; 
         FIG. 2  illustrates some characteristics of a transmission band of an RF communication channel in accordance with an OFDM-based system; 
         FIG. 3  illustrates communication channels that may exist between an OFDM transmitter and an OFDM receiver according to an embodiment; 
         FIG. 4  illustrates one embodiment of a MIMO system that may be implemented with the present systems and methods; 
         FIG. 5  illustrates a block diagram of certain components in an embodiment of a transmitter; 
         FIG. 6  is a block diagram illustrating one embodiment of components used to design a reference signal to be transmitted in a MIMO system; 
         FIG. 7  is a flow diagram illustrating one embodiment of a method for designing a reference signal in a MIMO system; 
         FIG. 8  is a flow diagram illustrating a further embodiment of an algorithm that may be utilized to design a reference signal; 
         FIG. 9  is a flow diagram illustrating a method of an algorithm that may be utilized to design a reference signal in a MIMO system; and 
         FIG. 10  illustrates various components that may be utilized in a communications device. 
     
    
    
     BEST MODE FOR CARRYING OUT THE INVENTION 
     The word “exemplary” is used exclusively herein to mean “serving as an example, instance, or illustration.” Any embodiment described herein as “exemplary” is not necessarily to be construed as preferred or advantageous over other embodiments. 
     Many features of the embodiments disclosed herein may be implemented as computer software, electronic hardware, or combinations of both. To clearly illustrate this interchangeability of hardware and software, various components will be described generally in terms of their functionality. Whether such functionality is implemented as hardware or software depends upon the particular application and design constraints imposed on the overall system. Skilled artisans may implement the described functionality in varying ways for each particular application, but such implementation decisions should not be interpreted as causing a departure from the scope of the present invention. 
     Where the described functionality is implemented as computer software, such software may include any type of computer instruction or computer executable code located within a memory device and/or transmitted as electronic signals over a system bus or network. Software that implements the functionality associated with components described herein may comprise a single instruction, or many instructions, and may be distributed over several different code segments, among different programs, and across several memory devices. 
     As used herein, the terms “an embodiment”, “embodiment”, “embodiments”, “the embodiment”, “the embodiments”, “one or more embodiments”, “some embodiments”, “certain embodiments”, “one embodiment”, “another embodiment” and the like mean “one or more (but not necessarily all) embodiments of the disclosed invention(s)”, unless expressly specified otherwise. 
     The term “determining” (and grammatical variants thereof) is used in an extremely broad sense. The term “determining” encompasses a wide variety of actions and therefore “determining” can include calculating, computing, processing, deriving, investigating, looking up (e.g., looking up in a table, a database or another data structure), ascertaining and the like. Also, “determining” can include receiving (e.g., receiving information), accessing (e.g., accessing data in a memory) and the like. Also, “determining” can include resolving, selecting, choosing, establishing and the like. 
     The phrase “based on” does not mean “based only on,” unless expressly specified otherwise. In other words, the phrase “based on” describes both “based only on” and “based at least on.” 
     Reference signals may be used in communication systems. Reference signals may include a single frequency and are transmitted over the communication systems for supervisory, control, equalization, continuity, synchronization, etc. Communication systems may include one or more mobile stations and one or more base stations that each transmit a reference signal. Reference signals may be designed such that a mobile station may re-use a reference signal that was previously used by a different mobile station or is used at the same time at another mobile station in another cell far enough apart so as to negligibly interfere. Truncation or cyclic extension of a particular set of Zadoff-Chu sequences has been utilized to design reference signals for re-use. However, truncation or cyclic extension may result in a tedious integer programming problem for a sequence assignment. In addition, a guarantee of minimal correlation does not exist when truncation or cyclic extension of a particular set of Zadoff-Chu sequences are implemented. Further, because of variable correlation properties of candidate proposed reference signals, detailed planning regarding the mobile station may be done which may be particularly vexing if adjacent networks exist in the same band that are operated by different operators. 
     The present systems and methods design reference signals for multiple input multiple output (MIMO) systems in which reference signals are allocated amongst one or more mobile stations, for use in single user or multiple user MIMO systems. In one embodiment, the present systems and methods design uplink reference signals in a cellular system. Communications from mobile stations to base stations may be classified as “uplink” communications. Conversely, communications from base stations to mobile stations may be classified as “downlink” communications. Transmitting uplink reference signals in a cellular system may pose stringent requirements on time and frequency resources on the mobile station. These stringent requirements may impede an optimum design of the reference signals for the mobile station, which may desire to implement a single or multiple carrier modulation with cyclic prefix, where there is synchronization between the transmission of multiple uplink signals and their respective base stations and where sectorization amongst cells of mobile stations is employed to maximize the capacity per cell. In addition, the present systems and methods employ multiple bandwidth allocations simultaneously to multiple base stations. In one embodiment, each bandwidth segment allocated to a mobile station is an integer amount of some basic unit. 
     In designing a set of reference signals, certain design considerations may be implemented. For example, the set may be large enough to cover at least three sectors per cell, with at least two reference signals per sector. In one embodiment, four reference signals per sector are present. A further design consideration may be that the set of reference signals may be orthogonal in each sector of a given cell. The set of reference signals may also be orthogonal in all sectors adjacent to a given sector. If the reference signals are orthogonal and the reference signals are known to adjacent sectors, a best minimum mean square receiver may be designed and implemented. 
     For those reference signals that are not in adjacent sectors, or which are not orthogonal, another design consideration may be that these reference signal are minimally correlated, with approximately the same correlation, and approach (if not meet) the Welch Bound. Sets of sequences that approach or meet the Welch Bound may denote a tight frame, where each vector possesses a unit norm, i.e. ∥ ∥ 2 =1. A further design consideration is the set of reference signals may also have a Peak to Average Power Ratio (PAPR) that approaches (if not equal) to 1. The PAPR may be defined as, for a sequence vector c as: 
     
       
         
           
             
               
                 
                   
                     = 
                     
                       
                         
                            
                           c 
                            
                         
                         ∞ 
                         2 
                       
                       
                         
                           c 
                           H 
                         
                         ⁢ 
                         c 
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                   ) 
                 
               
             
           
         
       
     
     where ∥c∥ ∞   2  denotes the square maximum modulus component of c and where ( ) H  denotes a conjugate transpose. 
     Another example of a design consideration may be that amongst subsets of sequences with orthogonal elements, each element may be a cyclic shift of another element. This property may be useful to provide robust performance if a transmission system which transmits a cyclic prefix for multipath elimination encounters multipath components with a delay spread greater than the cyclic prefix length. An additional design consideration is that in a system where multiple bandwidths are employed simultaneously, the set of reference signal sequences may be recursively generated from a base sequence. 
     In one embodiment, the amount of reference signal space (time and frequency resources) may be exactly large enough. For example, the basic unit of bandwidth allocation may allow for 19 or any larger prime number of reference signals available for two reference signals per sector. In a further example, the basic unit of bandwidth allocation may allow for 37 or any larger prime number of reference signals for four reference signals per sector. As in this case, if the amount of reference signal space is exactly large enough, Zadoff-Chu sequences may be taken as the reference sequences as they meet the design considerations previously described. However, such resource availability or sequence numerology may not be plausible. The present systems and methods provide an algorithm for designing reference signals based on alternating projections when such resources or sequence numerology are not available. 
       FIG. 1  illustrates an exemplary wireless communication system  100  in which embodiments may be practiced. A base station  102  is in wireless communication with a plurality of user devices  104  (which may also be referred to as mobile stations, subscriber units, access terminals, etc.). A first user device  104   a , a second user device  104   b , and an Nth user device  104   n  are shown in  FIG. 1 . The base station  102  transmits data to the user devices  104  over a radio frequency (RF) communication channel  106 . 
     As used herein, the term “OFDM transmitter” refers to any component or device that transmits OFDM signals. An OFDM transmitter may be implemented in a base station  102  that transmits OFDM signals to one or more user devices  104 . Alternatively, an OFDM transmitter may be implemented in a user device  104  that transmits OFDM signals to one or more base stations  102 . 
     The term “OFDM receiver” refers to any component or device that receives OFDM signals. An OFDM receiver may be implemented in a user device  104  that receives OFDM signals from one or more base stations  102 . Alternatively, an OFDM receiver may be implemented in a base station  102  that receives OFDM signals from one or more user devices  104 . 
       FIG. 2  illustrates some characteristics of a transmission band  208  of an RF communication channel  206  in accordance with an OFDM-based system. As shown, the transmission band  208  may be divided into a number of equally spaced sub-bands  210 . As mentioned above, a sub-carrier carrying a portion of the user information is transmitted in each sub-band  210 , and every sub-carrier is orthogonal with every other sub-carrier. 
       FIG. 3  illustrates communication channels  306  that may exist between an OFDM transmitter  312  and an OFDM receiver  314  according to an embodiment. As shown, communication from the OFDM transmitter  312  to the OFDM receiver  314  may occur over a first communication channel  306   a . Communication from the OFDM receiver  314  to the OFDM transmitter  312  may occur over a second communication channel  306   b.    
     The first communication channel  306   a  and the second communication channel  306   b  may be separate communication channels  306 . For example, there may be no overlap between the transmission band of the first communication channel  306   a  and the transmission band of the second communication channel  306   b.    
     In addition, the present systems and methods may be implemented with any modulation that utilizes multiple antennas/MIMO transmissions. For example, the present systems and methods may be implemented for MIMO Code Division Multiple Access (CDMA) systems or Time Division Multiple Access (TDMA) systems. 
       FIG. 4  illustrates one embodiment of a MIMO system  400  that may be implemented with the present systems and methods. The illustrated MIMO system  400  includes a first transmit antenna (Tx 1 )  402 A and a second transmit antenna (Tx 2 )  402 B. The system  400  also includes a first receive antenna (Rx 1 )  404 A and a second receive antenna (Rx 2 )  404 B. The transmit antennas  402 A,  402 B may be used to transmit a signal  406 ,  408 ,  410 ,  412  to the receive antennas  404 A,  404 B. 
     In single antenna systems, multi-path propagation may be detrimental to the performance of the system. The multiple propagation paths may cause “copies” of a signal to arrive at a receiver at slightly different times. These time delayed signals may then become interference when trying to recover the signal of interest. The MIMO system  400  is designed to exploit the multi-path propagation to obtain a performance improvement. For example, the first receive antenna (Rx 1 )  404 A may receive a mixture of a first signal  406  and a third signal  410  which are sent from the first transmit antenna (Tx 1 )  402 A and the second transmit antenna (Tx 2 )  402 B. The first and third signals  406 ,  410  may be sent over a first channel h 1,1  and a second third channel h 2,1 . The proportion of the first and third signals that is received at the first receive antenna (Rx 1 )  404 A depends on the transmission channels h 1,1 , h 2,1 . A simplified equation for the signal received at the first receive antenna (Rx 1 )  404 A may be:
 
 Rx   1 =( h   1,1   ×Tx   1 )+( h   2,1   ×Tx   2 )   (Equation 2)
 
     The first receive antenna (Rx 1  )  404 A receives a combination of what was transmitted from the first and second transmit antennas  402 A,  402 B. The MIMO system  400  may implement various coding schemes that define which signals  406 ,  408 ,  410 ,  412  should be transmitted, and at what times, to enable an original signal to be recovered when it is received in combination with another signal. These coding schemes may be known as “space-time” codes because they define a code across space (antennas) and time (symbols). 
       FIG. 5  illustrates a block diagram  500  of certain components in an embodiment of a transmitter  504 . Other components that are typically included in the transmitter  504  may not be illustrated for the purpose of focusing on the novel features of the embodiments herein. 
     Data symbols may be modulated by a modulation component  514 . The modulated data symbols may be analyzed by other subsystems  518 . The analyzed data symbols  516  may be provided to a reference processing component  510 . The reference processing component  510  may generate a reference signal that may be transmitted with the data symbols. The modulated data symbols  512  and the reference signal  508  may be communicated to an end processing component  506 . The end processing component  506  may combine the reference signal  508  and the modulated data symbols  512  into a signal. The transmitter  504  may receive the signal and transmit the signal to a receiver through an antenna  502 . 
       FIG. 6  is a block diagram  600  illustrating one embodiment of components used to design a reference signal to be transmitted in a MIMO system. In one embodiment, an initial sequence retriever  602  may obtain initial sequences (first set of sequences). A first sequence projection component  604  may project an obtained sequence set (first set of sequences) to a nearest tight frame. A subsets projection component  606  may be implemented to project subsets of the nearest tight frame (second set of sequences) to one or more orthogonal matrices. In one embodiment, a matrices projection component  608  may project the one or more orthogonal matrices (third set of sequences)to a nearest circulant matrix. In one embodiment, a second sequence projection component  610  may project each of the obtained sequence sets (forth set of sequences) onto a minimum Peak to Average Power Ratio (PAPR) vector. An iterator  612  may be utilized to iterate the steps performed by the first sequence projection component  604 , the subsets projection component  606 , the matrics projection component  608  and the second sequence projection component  610 . The iterator  612  may iterate these steps T times. A sequence output component  614  may output the sequences (fifth set of sequences) after T iterations have been executed. 
       FIG. 7  is a flow diagram illustrating one embodiment of a method  700  for designing a reference signal in a MIMO system. The method  700  may be implemented by the components discussed previously in regards to  FIG. 6 . In one embodiment, the existence of a fixed point of a MIMO signal is verified  702 . For example, for a set of Zadoff-Chu sequences of lengths 19 or 37 (as previously described), the Zadoff-Chu sequences may be returned as used as an input to design the reference signal. A nearest tight frame to one or more structured vectors may be obtained  704 . One or more structured vectors may then be obtained  706  from the previously computed nearest tight frame. The one or more structured vectors may be projected  708  onto the space of circulant matrices and one or more classes of matrices may be outputted  710 . The outputted matrices may indicate the design of the reference signal transmitted in a MIMO system. 
       FIG. 8  is a flow diagram  800  illustrating a further embodiment of an algorithm that may be utilized to design a reference signal. In one embodiment, a first matrix is provided  802 . The first matrix may be on the unit hyper-sphere. Sequences may be on the unit hyper-sphere to ensure a satisfactory constant envelope property initially. The first matrix may include zero components if the starting sequence is on the unit hyper-sphere. A second matrix may be computed  804 . The second matrix may be a nearest tight frame to the first matrix. The nearest tight frame may include an estimation of the first matrix. 
     In one embodiment, a third matrix may be computed  806 . The third matrix may be the closest matrix with a minimum peak to average power ratio to the second matrix. The third matrix may also be expanded and a fourth matrix may be computed  808  from the expansion. In one embodiment, a fifth matrix is computed  810  that is a nearest circulant matrix to the fourth matrix. The first matrix may be set  812  to the fifth matrix. In other words, the first matrix may be assigned the included in the fifth matrix. The fourth matrix and the fifth matrix may be outputted  814 . In addition, a maximum inner product of the fourth and fifth matrices may also be outputted  814 . 
     The following may represent steps taken to compute a correlated set of matrices that is the closest matrix with a minimum peak to average power ratio. A sequence of N column vectors {x n } N   n=1 , x n  εe d , d≦N, may be assigned as columns of a matrix X=[x 1  x 2  . . . x N ]. The matrix may be referred to as a frame. Each vector may have unit length, without any loss in generality. Block of K of these vectors may be grouped into a set of matrices, {X i } K   i=1  so that (with MK=N) X=[X 1  X 2  . . . X M ]. The correlation between vectors may be represented as &lt;x k , x n &gt; which is the standard inner product in complex Euclidean d-space. 
     The Welch Bound is, for any frame, for k≠n: 
     
       
         
           
             
               
                 
                   
                     
                       max 
                       
                         k 
                         ≠ 
                         n 
                       
                     
                     ⁢ 
                     
                       &lt; 
                       
                         x 
                         k 
                       
                     
                   
                   , 
                   
                     
                       x 
                       n 
                     
                     &gt; 
                     ≥ 
                     
                       
                         
                           N 
                           - 
                           d 
                         
                         
                           d 
                           ⁡ 
                           
                             ( 
                             
                               N 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     3 
                   
                   ) 
                 
               
             
           
         
       
     
     A frame that meets or approaches the Welch Bound may be referred to as a tight frame. The design considerations previously mentioned imply that for any &lt;x k , x n &gt; not in the same X i , &lt;x k , x n &gt;≦α, where α is a constant determined by the Welch Bound provided above. If any matrix Z ε e d×N , is provided, the matrix that comes closest in distance (as measured in element-wise or Frobenius norm) may be given by α (ZZ H )/ 1/2 Z. This condition may also enforce an orthnormality condition between rows of X, if an optimal X exists. 
     The design considerations previously mentioned also imply that X i *X i =I K ; (with K≦d). In other words, each column in any X i  may be orthogonal to any other column in X i . The above may be repeated with the role of X above being assumed by X i   H . Further, if as few as two sequences are required per cell (i.e., per matrix X i ), a “phase parity check” may be implemented to provide orthogonality between column vectors in X i  when there are zero entries in any column of X i . In other words, the phase of the zero components are chosen such that orthogonality if maintained once each column vector has minimal Peak to Average Power Ratio. 
     The following may illustrate steps taken to obtain the circulant matrix nearest to a given matrix. A matrix Z=[z 1  . . . z N ], may be provided, where each z i  is a column vector ε e N . A circulant matrix C=[c 0  . . . c N−1 ], may be obtained that is closest in Frobenius (element-wise) norm to Z. In one embodiment, F may be given as the Discrete Fourier Transform (DFT) matrix: 
     
       
         
           
             
               
                 
                   F 
                   = 
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           1 
                         
                         
                           … 
                         
                         
                           1 
                         
                       
                       
                         
                           1 
                         
                         
                           
                             ⅇ 
                             
                               
                                 - 
                                 j2π 
                               
                               / 
                               N 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             ⅇ 
                             
                               
                                 - 
                                 
                                   j2π 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       N 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                               / 
                               N 
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           ⋱ 
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           1 
                         
                         
                           
                             ⅇ 
                             
                               
                                 - 
                                 
                                   j2π 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       N 
                                       - 
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                                     ) 
                                   
                                 
                               
                               / 
                               N 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             ⅇ 
                             
                               
                                 - 
                                 
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                                   ⁡ 
                                   
                                     ( 
                                     
                                       N 
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                               ⁢ 
                               
                                 
                                   ( 
                                   
                                     N 
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                                   ) 
                                 
                                 / 
                                 N 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   
                     Equation 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     4 
                   
                   ) 
                 
               
             
           
         
       
     
     A diagonal “delay” matrix D may be defined as D=diag(1 e −j2π   /N  e −j2π   2/N  . . . e −j2π   (N−1)/N ). For any ciculant matrix C, C=F H ΛF, where Λ is the DFT of the sequence/vector c 0 . In addition, it may be shown that c i+1 mod N =F H DF c i =(F H DF) (i+1) mod N  c 0 . 
     Then 
     
       
         
           
             
               
                  
                 
                   Z 
                   - 
                   C 
                 
                  
               
               F 
               2 
             
             = 
             
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   N 
                 
                 ⁢ 
                 
                   
                      
                     
                       
                         Z 
                         i 
                       
                       - 
                       
                         c 
                         
                           i 
                           - 
                           1 
                         
                       
                     
                      
                   
                   2 
                 
               
               = 
               
                 
                   ∑ 
                   
                     i 
                     = 
                     1 
                   
                   N 
                 
                 ⁢ 
                 
                   
                     
                        
                       
                         
                           Z 
                           i 
                         
                         - 
                         
                           
                             
                               ( 
                               
                                 
                                   F 
                                   H 
                                 
                                 ⁢ 
                                 DF 
                               
                               ) 
                             
                             
                               ( 
                               
                                 i 
                                 - 
                                 1 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             c 
                             0 
                           
                         
                       
                        
                     
                     2 
                   
                   . 
                 
               
             
           
         
       
     
     In one embodiment, 
               ζ   =     [           z   1               z   2             ⋮             z   N           ]       ,     andB   =     [           I   N                 F   H     ⁢   DF             ⋮               (       F   H     ⁢   DF     )       N   -   1             ]             
to minimize c 0 , which uniquely determines C,
     c 0  is given by c 0 =B + ζ,   where B +  is the Moore-Penrose pseudo-inverse of B. In other words, B + =(B H  B) −1  B H .   

     Matrices where the number of column vectors are not equal to the number of row vectors may be referred to as reduced rank matrices (Z has fewer than N columns). Modifications may be implemented to the recurrence relation c i+1 mod N =F H DF c i  and the forming of the appropriate matrix B. If only two vectors were required that were cyclic shifted three elements apart, then c 1 =(F H DF) 3 c 0  and B may include the matrix elements I N  and (F H DF) 2 . 
       FIG. 9  is a flow diagram  900  illustrating a method of an algorithm that may be utilized to design a reference signal in a MIMO system. A matrix Z 0  ε e d×N , may be provided  902 . In one embodiment, the matrix Z 0  is on the unit hyper-sphere with all non-zero components. The following may occur for t=1 to T. 
     In one embodiment, α(ZZ H ) 1/2 Z may be computed  904  and assigned to the matrix Y. This may result in the tight frame nearest to Z. The following constraints may be implemented. If zero entries exist in column vectors of Y, phases to their related components in Y may be added so that orthogonality is maintained. For m=1 to M, (α (Wm H Wm) 1/2 Wm H ) H  may be computed  906  and assigned to a vector Vm. The matrix V=[V 1  V 2  . . . V M ] may be assembled. 
     In one embodiment, the max  k≠n &lt;v k , v n &gt; may be computed. Further, a Q matrix may be computed  908  that is a nearest circulant matrix to V and max  k≠n &lt;q k , q n &gt; may also be computed. A W matrix may be computed  910 . The W matrix may be the closest matrix with minimum PAPR to Y. The W matrix may be expressed as W=[W 1  W 2  . . . W M ]. The Z matrix may be assigned  912  as the Q matrix. If a circulant matrix is not desired, the Z matrix may be assigned as the V matrix. In one embodiment, t is updated as t+1. The V matrix and the Q matrix may be outputted  914 . In addition, max  k≠n &lt;v k , v n &gt; and max  k≠n &lt;q k , q n &gt; may also be outputted  914 . 
       FIG. 10  illustrates various components that may be utilized in a communications device  1002 . The communications device  1002  may include any type of communications device such as a mobile station, a cell phone, an access terminal, user equipment, a base station transceiver, a base station controller, etc. The communications device  1002  includes a processor  1006  which controls operation of the communications device  1002 . The processor  1006  may also be referred to as a CPU. Memory  1008 , which may include both read-only memory (ROM) and random access memory (RAM), provides instructions and data to the processor  1006 . The instructions contain the method of designing the reference signals above-mentioned. The instructions are carried out by the processor  1006 . A portion of the memory  1008  may also include non-volatile random access memory (NVRAM). 
     The communications device  1002  may also include a housing  1022  that contains a transmitter  1012  and a receiver  1014  to allow transmission and reception of data. The transmitter  1012  and receiver  1014  may be combined into a transceiver  1024 . An antenna  1026  is attached to the housing  1022  and electrically coupled to the transceiver  1024 . Additional antennas (not shown) may also be used. 
     The communications device  1002  may also include a signal detector  1010  used to detect and quantify the level of signals received by the transceiver  1024 . The signal detector  1010  detects such signals as total energy, pilot energy, power spectral density, and other signals. 
     A state changer  1016  controls the state of the communications device  1002  based on a current state and additional signals received by the transceiver  1024  and detected by the signal detector  1010 . The communications device  1002  may be capable of operating in any one of a number of states. 
     The various components of the communications device  1002  are coupled together by a bus system  1020  which may include a power bus, a control signal bus, and a status signal bus in addition to a data bus. However, for the sake of clarity, the various buses are illustrated in  FIG. 10  as the bus system  1020 . The communications device  1002  may also include a digital signal processor (DSP)  1018  for use in processing signals. The communications device  1002  illustrated in  FIG. 10  is a functional block diagram rather than a listing of specific components. 
     Information and signals may be represented using any of a variety of different technologies and techniques. For example, data, instructions, commands, information, signals, bits, symbols, and chips that may be referenced throughout the above description may be represented by voltages, currents, electromagnetic waves, magnetic fields or particles, optical fields or particles, or any combination thereof. 
     The various illustrative logical blocks, modules, circuits, and algorithm steps described in connection with the embodiments disclosed herein may be implemented as electronic hardware, computer software, or combinations of both. To clearly illustrate this interchangeability of hardware and software, various illustrative components, blocks, modules, circuits, and steps have been described above generally in terms of their functionality. Whether such functionality is implemented as hardware or software depends upon the particular application and design constraints imposed on the overall system. Skilled artisans may implement the described functionality in varying ways for each particular application, but such implementation decisions should not be interpreted as causing a departure from the scope of the present invention. 
     The various illustrative logical blocks, modules, and circuits described in connection with the embodiments disclosed herein may be implemented or performed with a general purpose processor, a digital signal processor (DSP), an application specific integrated circuit (ASIC), a field programmable gate array signal (FPGA) or other programmable logic device, discrete gate or transistor logic, discrete hardware components, or any combination thereof designed to perform the functions described herein. A general purpose processor may be a microprocessor, but in the alternative, the processor may be any conventional processor, controller, microcontroller, or state machine. A processor may also be implemented as a combination of computing devices, e.g., a combination of a DSP and a microprocessor, a plurality of microprocessors, one or more microprocessors in conjunction with a DSP core, or any other such configuration. 
     The steps of a method or algorithm described in connection with the embodiments disclosed herein may be embodied directly in hardware, in a software module executed by a processor, or in a combination of the two. A software module may reside in RAM memory, flash memory, ROM memory, EPROM memory, EEPROM memory, registers, hard disk, a removable disk, a CD-ROM, or any other form of storage medium known in the art. An exemplary storage medium is coupled to the processor such that the processor can read information from, and write information to, the storage medium. In the alternative, the storage medium may be integral to the processor. The processor and the storage medium may reside in an ASIC. The ASIC may reside in a user terminal. In the alternative, the processor and the storage medium may reside as discrete components in a user terminal. 
     The methods disclosed herein comprise one or more steps or actions for achieving the described method. The method steps and/or actions may be interchanged with one another without departing from the scope of the present invention. In other words, unless a specific order of steps or actions is required for proper operation of the embodiment, the order and/or use of specific steps and/or actions may be modified without departing from the scope of the present invention. 
     While specific embodiments and applications of the present invention have been illustrated and described, it is to be understood that the invention is not limited to the precise configuration and components disclosed herein. Various modifications, changes, and variations which will be apparent to those skilled in the art may be made in the arrangement, operation, and details of the methods and systems of the present invention disclosed herein without departing from the spirit and scope of the invention. 
     In one embodiment, a set of reference signals is provided to cover three sectors of a cell. At least two reference signals per sector may be provided. The set of reference signals may be orthogonal in each sector of a given cell. The set of reference signals may be orthogonal in sectors adjacent to a given cell. Reference signals not in adjacent sectors may be minimally correlated. The set of reference signals may comprise a Peak to Average Power Ratio that approximates the value of one. 
     Multiple bandwidths may be employed simultaneously. A set of sequences may be recursively generated from a base sequence. A matrix may be provided on the unit hyper-sphere with non-zero components. The correlation between each of the one or more structured vectors may be outputted. 
     A sequence set may be projected to a nearest tight frame. Subsets of a nearest tight frame may be projected to one or more orthogonal matrices. One or more orthogonal matrices may be projected to a nearest circulant matrix. Each sequence may be projected onto a minimum Peak to Average Power Ratio vector. 
     In one embodiment, a matrix may be provided on the unit hyper-sphere with non-zero components. The correlation between each of the one or more structured vectors may be outputted. A sequence set may be projected to a nearest tight frame. Subsets of a nearest tight frame may be projected to one or more orthogonal matrices. One or more orthogonal matrices may be projected to a nearest circulant matrix. Each sequence may be projected onto a minimum Peak to Average Power Ratio vector.