Abstract:
Efficient algorithms for estimating LSFCs with no aid of SSFCs by taking advantage of the channel hardening effect and large spatial samples available to a massive MIMO base station (BS) are proposed. The LSFC estimates are of low computational complexity and require relatively small training overhead. In the uplink direction, mobile stations (MSs) transmit orthogonal uplink pilots for the serving BS to estimate LSFCs. In the downlink direction, the BS transmits either pilot signal or data signal intended to the MSs that have already established time and frequency synchronization. The proposed uplink and downlink LSFC estimators are unbiased and asymptotically optimal as the number of BS antennas tends to infinity.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims priority under 35 U.S.C. §119 from U.S. Provisional Application No. 61/904,076, entitled “Large-Scale Fading Coefficient Estimation in Wireless Massive MIMO Systems,” filed on Nov. 14, 2013, the subject matter of which is incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     The disclosed embodiments relate generally to wireless network communications, and, more particularly, to large-scale fading coefficient estimation for wireless massive multi-user multiple-input multiple-output (MU-MIMO) systems. 
     BACKGROUND 
     A cellular mobile communication network in which each serving base station (BS) is equipped with an M-antenna array, is referred to as a large-scale multiuser multiple-input multiple-output (MIMO) system or a massive MIMO system if M&gt;&gt;1 and M&gt;&gt;K, where K is the number of active user antennas within its serving area. A massive MIMO system has the potential of achieving transmission rate much higher than those offered by current cellular systems with enhanced reliability and drastically improved power efficiency. It takes advantage of the so-called channel-hardening effect that implies that the channel vectors seen by different users tend to be mutually orthogonal and frequency-independent. As a result, linear receiver is almost optimal in the uplink and simple multiuser pre-coders are sufficient to guarantee satisfactory downlink performance. 
     To achieve such performance, channel state information (CSI) is needed for a variety of link adaptation applications such as precoder, modulation and coding scheme (SCM) selection. CSI in general includes large-scale fading coefficients (LSFCs) and small-scale fading coefficients (SSFCs). LFCSs summarize the pathloss and shadowing effects, which are proportional to the average received-signal-strength (RSS) and are useful in power control, location estimation, handover protocol, and other application. SSFCs, on the other hand, characterize the rapid amplitude fluctuations of the received signal. While all existing MIMO channel estimation focus on the estimation of the SSFCs and either ignore or assume perfect known LFCSs, it is desirable to know SSFCs and LSFCs separately. This is because LSFCs can not only be used for the aforementioned applications, but also be used for the accurate estimation of SSFCs. 
     LSFCs are long-term statistics whose estimation is often more time-consuming than SSFCs estimation. Conventional MIMO CSI estimation usually assume perfect LSFC information and deal solely with SSFCs. For co-located MIMO systems, it is reasonable to assume that the corresponding LSFCs remain constant across all spatial sub-channels and the SSFC estimation can sometime be obtained without the LSFC information. Such assumption is no longer valid in a multiuser MIMO system, where the user-BS distances spread over a large range and the SSFCs cannot be derived without the knowledge of LSFCs. 
     In the past, the estimation of LSFC has been largely neglected, assuming somehow perfectly known prior to SSFC estimation. When one needs to obtain a joint LSFC and SSFC estimate, the minimum mean square error (MMSE) or least squares (LS) criterion is not directly applicable. The expectation-maximization (EM) approach is a feasible alternate but it requires high computational complexity and convergence is not guaranteed. A solution for efficiently estimating LSFCs with no aid of SSFCs is sought in a massive multiuser MIMO system. 
     SUMMARY 
     Efficient algorithms for estimating LSFCs with no aid of SSFCs by taking advantage of the channel hardening effect and large spatial samples available to a massive MIMO base station (BS) are proposed. The LSFC estimates are of low computational complexity and require relatively small training overhead. In the uplink direction, mobile stations (MSs) transmit orthogonal uplink pilots for the serving BS to estimate LSFCs. In the downlink direction, the BS transmits either pilot signal or data signal intended to the MSs that have already established time domain and frequency domain synchronization. The proposed uplink and downlink LSFC estimators are unbiased and asymptotically optimal as the number of BS antennas tends to infinity. 
     In one embodiment, a base station (BS) receives radio signals transmitted from K mobile stations (MSs) in a massive MIMO uplink channel where M&gt;&gt;K. The BS vectorizes the received radio signals denoted as a matrix Yε             M×T . The transmitted radio signals are orthogonal pilot signals denoted as a matrix Pε           K×T  transmitted from the K MSs, and T≧K is the pilot signal length. The BS derives an estimator of large-scale fading coefficients (LSFCs) of the uplink channel without knowing small-scale fading coefficients (SSFCs) of the uplink channel. The BS may also receive pilot signals that are transmitted for J times over coherent radio resource blocks from the K MS. The BS then derives a more accurate estimator of the LSFCs of the uplink channel based on the multiple pilot transmissions. In addition, the BS calculates element-wise expression of the LSFCs for each of the kth uplink channel based on the LSFCs estimator.
     In another embodiment, a mobile station (MS) receives radio signals transmitted from a base station (BS) having M antennas in a massive MIMO system. The transmitted radio signals are denoted as a matrix Q transmitted from the BS to K MS and M&gt;&gt;K. The MS determines a received radio signal denoted as a vector x k  received by the MS that is the kth MS associated with a k th  downlink channel. The k th  MS derives an estimator of a large-scale fading coefficient (LSFC) of the k th  downlink channel without knowing a small-scale fading coefficient (SSFC) of the kth downlink channel. In a semi-blind LSFC estimation, matrix Q is a semi-unitary matrix consisting of orthogonal pilot signals, and the LSFC of the k th  downlink channel is derived based on x k  and the transmitting power of the pilot signals. In a blind LSFC estimation, matrix Q represents pre-coded data signals transmitted to K′ MS that are different from the K MS. The LSFC of the k th  downlink channel is derived based on x k  and the transmitting power of the data signals with unknown data information and unknown beamforming or precoding information. 
     Other embodiments and advantages are described in the detailed description below. This summary does not purport to define the invention. The invention is defined by the claims. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates simplified block diagrams of a base station and a plurality of mobile stations in a single-cell massive MU-MIMO system in accordance with one novel aspect. 
         FIG. 2  is an exemplary diagram illustrating an uplink MIMO system in accordance with one novel aspect. 
         FIG. 3  is a flow chart of an uplink LSFC estimator that estimates all accessing MSs&#39; LSFCs simultaneously in a massive MIMO system. 
         FIG. 4A  is a flow chart of an uplink LSFC estimator that estimates the LSFC for each accessing MS individually in a massive MIMO system. 
         FIG. 4B  is a flow chart of an uplink LSFC estimator that estimates the LSFC for each accessing MS individually using a row of a diagonal matrix as pilot in a massive MIMO system. 
         FIG. 5  shows an exemplary schematic view of an uplink LSFC estimator in a massive MIMO system. 
         FIG. 6  is a flow chart of an uplink LSFC estimator that estimates all accessing MSs&#39; LSFCs simultaneously with multiple pilot transmissions in a massive MIMO system. 
         FIG. 7A  is a flow chart of an uplink LSFC estimator that individually estimates the LSFC for each MS with multiple pilot transmissions in a massive MIMO system. 
         FIG. 7B  is a flow chart of an uplink LSFC estimator that individually estimates the LSFC for each MS using a row of a diagonal matrix as pilot with multiple pilot transmissions in a massive MIMO system. 
         FIG. 8  shows an exemplary schematic view of an uplink LSFC estimator with multiple pilot transmissions in a massive MIMO system. 
         FIG. 9  shows the MSE performance with respect to BS antenna number and SNR of the proposed uplink LSFC estimator without SSFC knowledge using only one training block. 
         FIG. 10  shows the MSE performance with respect to BS antenna number and SNR of the conventional single-block uplink LSFC estimator with perfect SSFC knowledge and the proposed uplink LSFC estimator without SSFC knowledge using multiple training blocks. 
         FIG. 11  is a flow chart of a method of estimating uplink LSFC in accordance with one novel aspect. 
         FIG. 12  is an exemplary diagram illustrating a downlink MIMO system in accordance with one novel aspect. 
         FIG. 13  is a flow chart and schematic view of a downlink semi-blind LSFC estimator that resides at each MS and estimates the LSFC of the MS using a semi-unitary matrix as pilot in a massive MIMO system. 
         FIG. 14  is a flow chart and schematic view of a downlink blind LSFC estimator that resides at each MS and estimates the LSFC of the MS exploiting only the statistics of the unknown broadcast signal in a massive MIMO system. 
         FIG. 15A-15B  show the MSE performance with respect to SNR and training period of the proposed downlink LSFC estimators without SSFC knowledge. 
         FIG. 16  is a flow chart of a method of estimating downlink LSFC in accordance with one novel aspect. 
     
    
    
     DETAILED DESCRIPTION 
     Reference will now be made in detail to some embodiments of the invention, examples of which are illustrated in the accompanying drawings. 
     Notation: (.) T , (.) H , (.)* represent the transpose, conjugate transpose, and conjugate of the enclosed items, respectively, vec(.) is the operator that forms one tall vector by stacking columns of the enclosed matrix, whereas Diag(.) translate a vector into a diagonal matrix with the vector entries being the diagonal terms. While E{.}, ∥.∥, and ∥.∥ F , denote the expectation, vector l 2 -norm, and Frobenius norm of the enclosed items, respectively,             and ⊙ respectively denote the Kronecker and Hadamard product operator. Denoted by I L , 1 L , and 0 L  respectively, are the (L×L) identity matrix, L-dimensional all-one and all-zero column vectors, whereas 1 L×S  and 0 L×S  are the matrix counterparts of the latter two. Almost surely convergence is denoted by
             ⟶     a   .   s   .           
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       FIG. 1  illustrates simplified block diagrams of a base station and a plurality of mobile stations in a single-cell massive MU-MIMO system  100  in accordance with one novel aspect. Massive MU-MIMO system  100  comprises a base station BS  101  having an M-antenna array and K single-antenna mobile stations MS # 1  to MS #K, wherein M&gt;&gt;K. For a multi-cell uplink system, pilot contamination may become a serious design concern in the worst case when the same pilot sequences (i.e., the same pilot symbols are place at the same time-frequency locations) happen to be used simultaneously in several neighboring cells and are perfectly synchronized in both carrier and time. In practice, there are frequency, phase, and timing offset between any pair of pilot signals and the number of orthogonal pilots is often sufficient to serve mobile users in multiple cells. Moreover, neighbor cells may use the same pilot sequence but the pilot symbols are located in non-overlapping time-frequency units, hence a pilot sequence is more likely be interfered by uncorrelated asynchronous data sequence whose impact is not as serious as the worst case and can be mitigated by proper inter-cell coordination, frequency planning and some interference suppression techniques. Throughout the present application, the discussion will be focused on the single-cell narrowband scenario. The proposed method, however, is not limited thereto. 
     In the example of  FIG. 1 , BS  101  comprises memory  102 , a processor  103 , a scheduler  104 , a MIMO codec  105 , a precoder/beamformer  106 , a channel estimator  107 , and a plurality of transceivers coupled to a plurality of antennas. Similarly, each MS comprises memory, a processor, a MIMO codec, a precoder/beamformer, a channel estimator, and a transceiver couple to an antenna. Each wireless device receives RF signals from the antenna, converts them to baseband signals and sends them to the processor. Each RF transceiver also converts received baseband signals from the processor, converts them to RF signals, and sends out to the antenna. For example, processor  103  processes the received baseband signals and invokes different functional modules to perform features in the device. Memory  102  stores program instructions and data to control the operations of the device. The functional modules carry out embodiments of the current invention. The functional modules may be configured and implemented by hardware, firmware, software, or any combination thereof. 
       FIG. 2  is an exemplary diagram illustrating an uplink MIMO system  200  in accordance with one novel aspect. MIMO system  200  comprises a base station BS  201  having M antennas, and K mobile stations MS  1  to MS K. In the uplink direction, each k th  MS transmits pilot training signals p k  to be received by BS  201  via M antennas. We assume a narrowband communication environment in which a transmitted signal suffers from both large-scale and small-scale fading. The large-scale fading coefficients (LSFCs) for each uplink channel is denoted as β k &#39;s, while the small-scale fading coefficients (SSFCs) for each uplink channel is denoted as h k &#39;s. The K uplink packets place their pilot of length T at the same time-frequency location so that, without loss of generality, the corresponding received signals, arranged in matrix form, Yε             M×T  at the BS can be expressed as:
             Y   =           ∑     k   =   1     K     ⁢           ⁢         β   k       ⁢     h   k     ⁢     p   k   H         +   N     =         HD   β     1   /   2       ⁢   P     +   N             
where
         H=[h 1 , . . . , h K ]ε           M×K  contains the SSFCs that characterize the K uplink channels, h k =Φ k   1/2 {tilde over (h)} k , {tilde over (h)} k ˜CN(0 M ,I M ), where Φ k  is the spatial correlation matrix at the BS side with respect to the k th  user   D β =Diag(β 1 , . . . , β K ) contains the LSFCs that characterize the K uplink channels, vector β=[(β 1 , . . . , β K ) T ] whose elements β k =s k d k   −α  describes the shadowing effect, parameterized by independent identically distributed (i.i.d) s k &#39;s with log 10 (s k )˜N(0,σ s   2 ), and the pathloss which depends on the distance between the BS and MS d k , with α&gt;0   P=[p 1 , . . . , p K ] H ε           K×T  is the K×T matrix where M&gt;&gt;T≧K and p k  is the pilot sequence sent by MS k and p j   H p k =0, ∀j≠k (orthogonal pilot sequences)   N=[n ij ], n ij ˜CN(0,1) is the noise matrix whose entries are distributed according to CN(0,1).       

     We invoke the assumption that independent users are relatively far apart (with respect to the wavelength) and the k th  uplink channel vector is independent of the l th  vector, ∀l≠k. We assume that {tilde over (h)} k  are i.i.d. and the SSFC H remains constant during a pilot sequence period, i.e., the channel&#39;s coherence time is greater than T, while the LSFC β varies much slower. 
     Unlike most of the existing works that focus on the estimation of the composite channel matrix HD β   1/2 , or equivalently, ignore the LSFC, it is beneficial for system performance to know H and D β   1/2  separately. Even though the decoupled treatment of LSFCs and SSFCs has been seen recently, the assumption that the former is well known is usually made. In ordinary MIMO systems, MMSE or LS criterion cannot be used directly to jointly estimate LSFC and SSFC owing to their coupling, and EM algorithm is a feasible alternative. However, EM has high computational complexity and convergence is not guaranteed. In accordance with one novel aspect, a timely accurate LSFC estimator for uplink massive MIMO without the prior knowledge of SSFC is proposed. 
       FIG. 3  is a flow chart of an uplink LSFC estimator that estimates all accessing MSs&#39; LSFCs simultaneously in a massive MIMO system  200  of  FIG. 2 . In step  311 , each MS k transmits assigned UL pilot p k . In step  312 , the BS receives pilot signals transmitted from all K MSs, which becomes the received signals denoted as Y. In step  313 , the BS vectorizes 
                 1   M     ⁢     Y   H     ⁢   Y     -       I   T     .           
Finally, in step  314 , the BS derives an estimator of LSFC {circumflex over (β)} by multiplying it with Diag(∥p 1 ∥ −4 , . . . , ∥p k ∥ −4 )·((1 T   T             P)⊙(P*         1 T   T ). The derivation of LSFC {circumflex over (β)} is as follows:

                           1   M     ⁢     Y   H     ⁢   Y     -     I   T       =           2   M     ⁢   ℜ   ⁢     {       P   H     ⁢     D   β     1   /   2       ⁢     H   H     ⁢   N     }       +       P   H     ⁢     D   β     ⁢   P     +       P   H     ⁢       D   β     1   2       ⁡     (         1   M     ⁢     H   H     ⁢   H     -     I   K       )       ⁢     D   β     1   2       ⁢   P     +       1   M     ⁢     N   H     ⁢   N     -     I   T       ⁢     
     ⁢           ⁢     →     a   .   s       ⁢       P   H     ⁢     D   β     ⁢   P         ⁢     
     ⁢           ⁢       vec   ⁡     (         1   M     ⁢     Y   H     ⁢   Y     -     I   T       )       ⁢     →     a   .   s       ⁢       (       (       1   T     ⊗     P   H       )     ⊙     (       P   T     ⊗     1   T       )       )     ⁢   β       ⁢     
     ⁢       β   ^     =       Diag   ⁡     (              p   1            -   4       ,   …   ⁢           ,            p   K            -   4         )       ·     (       (       1   T   T     ⊗   P     )     ⊙     (       P   *     ⊗     1   T   T       )       )     ·     vec   ⁡     (         1   M     ⁢     Y   H     ⁢   Y     -     I   T       )                   (   1   )               
where due to the large number of BS antennas M, the large sample size of the receive signal shows the following convergence:
 
     
       
         
           
             
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     By exploiting the properties of massive MIMO, the proposed LFSC estimator has low computational complexity while outperform the one derived from EM algorithm. The proposed LFSC estimator is of low complexity, as no matrix inversion is needed when orthogonal pilots are used and does not require any knowledge of SSFCs. Furthermore, the configuration of massive MIMO makes the estimator robust against noise. 
       FIG. 4A  is a flow chart of an uplink LSFC estimator that estimates the LSFC for each accessing MS individually in a massive MIMO system  200  of  FIG. 2 . In step  411 , each MS k transmits assigned UL pilot p k . In step  412 , the BS receives pilot signals from all K MSs, denoted as Y. In step  413 , {circumflex over (β)}=[{circumflex over (β)} 1 , . . . , {circumflex over (β)} K ] is decoupled as, ∀k, and the BS derives an estimator of each LSFC {circumflex over (β)} k  for each uplink channel to be: 
     
       
         
           
             
               
                 
                   
                     
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       FIG. 4B  is a flow chart of an uplink LSFC estimator that estimates the LSFC for each accessing MS individually using a row of a diagonal matrix as pilot in a massive MIMO system  200  of  FIG. 2 . In step  421 , each MS k transmits assigned UL pilot p k . In the example of  FIG. 4B , the pilot is chosen to be a row of diagonal matrix, i.e., P=Diag(s 1 , . . . . , s K ) and T=K, where each pilot sequence p k =[0, . . . , 0, s k , 0, . . . 0]. In step  422 , the BS receives pilot signals from all K MSs, denoted as Y, where Y=[y 1 , . . . , y K ]. In step  423 , the BS decouples the LSFC estimator {circumflex over (β)} k  for each uplink channel to be: 
     
       
         
           
             
               
                 
                   
                     
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     This estimator coincides with our prediction that the instantaneous received signal strength minus the noise power, ∥y k ∥ 2 −M, is approximately equal to the strength of the desired signal and thus fairly reflects the gain provided by large-scale fading if it is divided by M s k   2 , the total power emitted by user k (s k   2 ) times the number of copies received at the BS (M). 
       FIG. 5  shows an exemplary schematic view of an uplink LSFC estimator  501  in a massive MIMO system  200  of  FIG. 2 . In the example of  FIG. 5 , a base station having M antennas receives radio signal Y from K mobile stations MSs, each MS k transmits a pilot sequence p k , and the noise variance is σ 2 . The UL LSFC estimator  501  is able to derive the LSFC {circumflex over (β)} k  for each uplink channel with low computational complexity and without prior knowledge of the small-scale fading coefficients. 
       FIG. 6  is a flow chart of an uplink LSFC estimator that estimates all accessing MSs&#39; LSFCs simultaneously with multiple pilot transmissions in a massive MIMO system  200  of  FIG. 2 . In step  611 , each MS k transmits assigned UL pilot p k  J times. The J-time pilot transmissions can be achieved in different ways. In one example, the MS may transmit the pilot p k  by repeating the transmission J times in time domain. In another example, the MS may transmit the pilot p k  by repeating the transmission J times in frequency domain. Note that, in our example, although the ip k  remains the same during the J transmissions, different p k  can be used for each of the different transmissions. In step  612 , the BS receives pilot signals from all K MSs, denoted as Y 1 , . . . , Y J , where Y i  is the i th  received signal block at the BS. In step  613 , the BS vectorizes 
                 1   MJ     ⁢       ∑     i   =   1     J     ⁢           ⁢       Y   i   H     ⁢     Y   i           -       1   J     ⁢       I   T     .             
Finally, in step  614 , the BS derives an estimator of LSFC {circumflex over (B)} by multiplying it with Diag(∥p 1 ∥ −4 , . . . , ∥p K ∥ −4 )·((1 T   T             P)⊙(P*         1 T   T )). If the J coherent resource blocks on time-frequency domain in which the LSFCs remain constant are available, then we have:

     
       
         
           
             
               
                 
                   
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                                
                             
                             
                               - 
                               4 
                             
                           
                         
                         ) 
                       
                     
                     · 
                     
                       ( 
                       
                         
                           ( 
                           
                             
                               1 
                               T 
                               T 
                             
                             ⊗ 
                             P 
                           
                           ) 
                         
                         ⊙ 
                         
                           ( 
                           
                             
                               P 
                               * 
                             
                             ⊗ 
                             
                               1 
                               T 
                               T 
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     · 
                     
                       vec 
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               1 
                               MJ 
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 J 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 
                                   Y 
                                   i 
                                   H 
                                 
                                 ⁢ 
                                 
                                   Y 
                                   i 
                                 
                               
                             
                           
                           - 
                           
                             
                               1 
                               J 
                             
                             ⁢ 
                             
                               I 
                               T 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
       FIG. 7A  is a flow chart of an uplink LSFC estimator that individually estimates the LSFC for each MS with multiple pilot transmissions in a massive MIMO system  200  of  FIG. 2 . In step  711 , each MS k transmits assigned UL pilot p k  J times. In step  712 , the BS receives pilot signals from all K MSs, denoted as Y 1 , . . . , Y J . In step  713 , {circumflex over (β)} k  is decoupled from {circumflex over (β)}=[{circumflex over (β)} 1 , . . . , {circumflex over (β)} K ], ∀k, and the BS derives an estimator of each LSFC {circumflex over (β)} k  for each uplink channel to be: 
     
       
         
           
             
               
                 
                   
                     
                       β 
                       ^ 
                     
                     k 
                   
                   = 
                   
                     
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           J 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             p 
                             k 
                             H 
                           
                           ⁢ 
                           
                             Y 
                             i 
                             H 
                           
                           ⁢ 
                           
                             Y 
                             i 
                           
                           ⁢ 
                           
                             p 
                             k 
                           
                         
                       
                       - 
                       
                         MJ 
                         ⁢ 
                         
                           
                              
                             
                               p 
                               k 
                             
                              
                           
                           2 
                         
                       
                     
                     
                       MJ 
                       ⁢ 
                       
                         
                            
                           
                             p 
                             k 
                           
                            
                         
                         4 
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
       FIG. 7B  is a flow chart of an uplink LSFC estimator that individually estimates the LSFC for each MS using a row of a diagonal matrix as pilot with multiple pilot transmissions in a massive MIMO system  200  of  FIG. 2 . In step  721 , each MS k transmits assigned UL pilot p k  J times. In the example of  FIG. 7B , the pilot is chosen to be a row of diagonal matrix, i.e., P=Diag(s 1 , . . . , s K ), where each pilot sequence p k =[0, . . . , 0, s k , 0, . . . 0]. In step  722 , for each of the i th  pilot transmission, the BS receives pilot signals from all K MSs, denoted as Y i , where Y i =[y 1   (i) , . . . , y K   (i) ]. In step  723 , the BS decouples the LSFC estimator {circumflex over (β)} k  for each uplink channel to be: 
     
       
         
           
             
               
                 
                   
                     
                       β 
                       ^ 
                     
                     k 
                   
                   = 
                   
                     
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           J 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                              
                             
                               y 
                               k 
                               
                                 ( 
                                 i 
                                 ) 
                               
                             
                              
                           
                           2 
                         
                       
                       - 
                       MJ 
                     
                     
                       MJs 
                       k 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     While the diagonal pilots give lower computational burden, the requirement that an MS needs to transmit all pilot power in a time slot to achieve the same performance shows a risk of disobeying the maximum user output power constraint. The decision of a suitable uplink pilot pattern is a trade-off between the computational complexity and maximum user output power. In one alternative example, a Hadamard matrix is adopted as the pilot pattern. A Hadamard matrix is a square matrix whose rows or columns are mutually orthogonal and of ±1 entries. It is conjectured that a Hadamard matrix or rows of it as the pilot matrix P, the computation effort can be reduced significantly due to the fact that the calculation of Yp k  in equation (2) or Y i p k  in equation (5) involves only column additions and subtractions of Y/Y i . 
       FIG. 8  shows an exemplary schematic view of an uplink LSFC estimator  801  with multiple pilot transmissions in a massive MIMO system  200  of  FIG. 2 . In the example of  FIG. 8 , a base station BS having M antennas receives radio signal Y 1 , . . . , Y J  from K mobile stations MSs, each MS k transmits a pilot sequence p k  J times, and the noise variance is σ 2 . The UL LSFC estimator  801  is able to derive the LSFC {circumflex over (β)} k  for each uplink channel with low computational complexity and without prior knowledge of the small-scale fading coefficients. 
       FIG. 9  shows the MSE performance with respect to BS antenna number and SNR of the proposed uplink LSFC estimator without SSFC knowledge using only one training block (J=1). As shown in  FIG. 9 , the MSE performance improves as the number of antenna M increases, and as the SNR increases. 
       FIG. 10  shows the MSE performance with respect to BS antenna number and SNR of the conventional single-block uplink LSFC estimator with perfect SSFC knowledge and the proposed uplink LSFC estimator without SSFC knowledge using multiple training blocks. As shown in  FIG. 10 , the MSE performance of the conventional single-block uplink LSFC estimator with perfect SSFC knowledge is the best, as depicted by the dashed-line. However, the MSE performance of the proposed uplink LSFC estimator without SSFC knowledge improves as the number of antenna M increases, and as the number of training blocks J increases. Furthermore, the configuration of massive MIMO makes the estimator robust against noise. 
       FIG. 11  is a flow chart of a method of estimating uplink LSFC in accordance with one novel aspect. In step  1101 , a base station (BS) receives radio signals transmitted from K mobile stations (MSs) in a massive MIMO uplink channel where M&gt;&gt;K. In step  1102 , the BS vectorizes the received radio signals denoted as a matrix Yε             M×T , the transmitted radio signals are orthogonal pilot signals denoted as a matrix Pε           K×T  transmitted from the K MSs, and T≧K is the pilot signal length. In step  1103 , the BS derives an estimator of large-scale fading coefficients (LSFCs) of the uplink channel without knowing small-scale fading coefficients (SSFCs) of the uplink channel. In step  1104 , the BS receives pilot signals that are transmitted for J times over coherent radio resource blocks from the K MS. In step  1105 , the BS derives a more accurate estimator of the LSFCs of the uplink channel based on the multiple pilot transmissions. In step  1106 , the BS calculates element-wise expression of the LSFCs for each of the kth uplink channel based on the LSFCs estimator.
       FIG. 12  is an exemplary diagram illustrating a downlink MIMO system  1200  in accordance with one novel aspect. MIMO system  1200  comprises a base station BS  1201  having M antennas, and K mobile stations MS  1  to MS K. In the downlink direction, BS  1201  transmits downlink packets via some or all of its M antennas to be received by some or all of the K MSs. We assume a narrowband communication environment in which a transmitted signal suffers from both large-scale and small-scale fading. The large-scale fading coefficients (LSFCs) for each downlink channel is denoted as β k &#39;s, while the small-scale fading coefficients (SSFCs) for each downlink channel is denoted as g k &#39;s. The length-T downlink packets of different BS antennas are placed at the same time-frequency locations so that, without loss of generality, the corresponding received samples, arranged in matrix form, X H =[x 1 , . . . , x K ] H  at MSs can be expressed as
 
 X   H   =[x   1   , . . . ,x   K ] H   =D   β   1/2   G   H   Q+Z   H  
 
where
         G=[g 1 , . . . , g K ]ε           M×K  contains the SSFCs that characterize the K downlink channels, g k =Φ k   1/2 {tilde over (g)} k , {tilde over (g)} k ˜CN(0 M ,I M ), where Φ k  is the spatial correlation matrix at the BS side with respect to the k th  user   D β =Diag(β 1 , . . . , β K ) contains the LSFCs that characterize the K downlink channels, vector β=[β 1 , . . . , β K ] T  whose elements β k =s k d k   −α  describes the shadowing effect, parameterized by independent identically distributed (i.i.d) s k &#39;s with log 10 (s k )˜N(0,σ s   2 ), and the pathloss which depends on the distance between the BS and MS d k , with α&gt;0   Q=[q 1 , . . . , q T ]ε           M×T  is a M×T matrix where T≦M, which can be a pilot matrix containing orthogonal columns q i   H q j =0, ∀i≠j or a data matrix intended to serving different MSs   Z=[z ij ], z ij ˜CN(0,1) is the noise matrix whose entries are distributed according to CN(0,1).       

     We invoke the assumption that independent users are relatively far apart (with respect to the wavelength) and the k th  downlink channel vector is independent of the l th  vector, ∀l≠k. We assume that {tilde over (g)} k  are i.i.d. and the SSFC G remains constant during a pilot/data sequence period, i.e., the channel&#39;s coherence time is greater than T, while the LSFC β varies much slower. In accordance with one novel aspect, several accurate LSFC estimators for downlink massive MIMO without the prior knowledge of SSFC are proposed. By exploiting the properties of massive MIMO, it has low computational complexity. 
       FIG. 13  is a flow chart and schematic view of a downlink semi-blind LSFC estimator  1311  that resides at each MS and estimates the LSFC of the MS using a semi-unitary matrix as pilot in a massive MIMO system  1200  of  FIG. 12 . In step  1301 , the BS transmits downlink pilot Q to the K MSs. In step  1302 , the k th  MS receives signal x k   H . In step  1303 , the k th  MS recovers the LSFC {circumflex over (β)} k  for each downlink channel to be: 
                       β   ^     k     =                x   k          2     -   T            Q        F   2               (   7   )               
where Q is a semi-unitary matrix and MS knows nothing but pilot power ∥Q∥ F   2 .
 
     In the embodiment of  FIG. 13 , let Q be a pilot matrix of the following form with 
     
       
         
           
             
               q 
               t 
             
             = 
             
               
                 
                   P 
                 
                 ⁡ 
                 
                   [ 
                   
                     
                       0 
                       
                         1 
                         × 
                         
                           ( 
                           
                             M 
                             - 
                             TR 
                             + 
                             r 
                           
                           ) 
                         
                       
                     
                     , 
                     
                       u 
                       
                         1 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         t 
                       
                     
                     , 
                     
                       0 
                       
                         1 
                         × 
                         
                           ( 
                           
                             R 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                     , 
                     
                       u 
                       
                         2 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         t 
                       
                     
                     , 
                     
                       0 
                       
                         1 
                         × 
                         
                           ( 
                           
                             R 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                     , 
                     … 
                     ⁢ 
                     
                         
                     
                     , 
                     
                       u 
                       Tt 
                     
                     , 
                     
                       0 
                       
                         1 
                         × 
                         
                           ( 
                           
                             R 
                             - 
                             r 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                   
                   ] 
                 
               
               T 
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             where 
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               
                 R 
                 = 
                 
                   ⌊ 
                   
                     M 
                     / 
                     T 
                   
                   ⌋ 
                 
               
               , 
               
                 r 
                 ≤ 
                 
                   R 
                   - 
                   1 
                 
               
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               U 
               = 
               
                 
                   [ 
                   
                     u 
                     ij 
                   
                   ] 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 is 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 unitary 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 matrix 
               
             
           
         
       
       
         
           
             
               QQ 
               H 
             
             = 
             
               P 
               · 
               
                 Diag 
                 ⁡ 
                 
                   ( 
                   
                     
                       0 
                       
                         1 
                         × 
                         
                           ( 
                           
                             M 
                             - 
                             TR 
                             + 
                             r 
                           
                           ) 
                         
                       
                     
                     , 
                     1 
                     , 
                     
                       0 
                       
                         1 
                         × 
                         
                           ( 
                           
                             R 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                     , 
                     1 
                     , 
                     
                       0 
                       
                         1 
                         × 
                         
                           ( 
                           
                             R 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                     , 
                     … 
                     ⁢ 
                     
                         
                     
                     , 
                     1 
                     , 
                     
                       0 
                       
                         1 
                         × 
                         
                           ( 
                           
                             R 
                             - 
                             r 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               
                 
                   Q 
                   H 
                 
                 ⁢ 
                 Q 
               
               = 
               
                 P 
                 · 
                 
                   I 
                   T 
                 
               
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               
                 
                   
                      
                     
                       x 
                       k 
                     
                      
                   
                   2 
                 
                 ≈ 
                 
                   
                     
                       β 
                       k 
                     
                     ⁢ 
                     
                       g 
                       k 
                       H 
                     
                     ⁢ 
                     
                       QQ 
                       H 
                     
                     ⁢ 
                     
                       g 
                       k 
                     
                   
                   + 
                   
                     
                        
                       
                         z 
                         k 
                       
                        
                     
                     2 
                   
                 
               
               ⁢ 
               
                 → 
                 
                   a 
                   . 
                   s 
                   . 
                 
               
               ⁢ 
               
                 
                   
                     
                       β 
                       k 
                     
                     ⁢ 
                     
                       P 
                       · 
                       T 
                     
                   
                   + 
                   
                     T 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     if 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     T 
                   
                 
                 ⪢ 
                 1. 
               
             
           
         
       
     
     In the example of  FIG. 13 , each k th  MS receives radio signal x k  from a BS having M antennas. The BS transmits a pilot signal denoted by matrix Q, which is a semi-unitary matrix. The pilot power is ∥Q∥ F   2 , and the noise variance is σ 2 . The DL LSFC estimator  1311  is able to derive an estimate of the LSFC {circumflex over (β)} k  for each downlink channel with low computational complexity and without prior knowledge of the small-scale fading coefficients. Because in a massive MIMO system, M≧T&gt;&gt;1 gives 
                   z   k   H     ⁢     Q   H     ⁢     g   k       ⁢     →     a   .   s   .       ⁢   0     ,         
if T→∞, and ∥x k ∥ 2 ≈β k g k   H QQ H g k +∥z k ∥ 2 . In addition,
 
                 β   k     ⁢     g   k   H     ⁢     QQ   H     ⁢     g   k       +                z   k          2     ⁢     ⟶     a   .   s   .       ⁢     β   k       ⁢     P   ·   T       +   T         
is because
 
                 g   k   H     ⁢     QQ   H     ⁢     g   k       =     P   ⁢           ⁢       ∑     i   =   1     T     ⁢                g       M   -       (     T   -   i   +   1     )     ⁢   R     +   r   +   1     ,   k            2     ⁢     ⟶     a   .   s   .       ⁢   PT     .               
E{|g ik | 2 }=PT if T→∞.
 
       FIG. 14  is a flow chart and schematic view of a downlink blind LSFC estimator  1411  that resides at each MS and estimates the LSFC of the MS exploiting only the statistics of the unknown broadcast signal in a massive MIMO system  1200  of  FIG. 12 . In step  1401 , the BS transmits downlink data signal Q=WD to a plurality of K′ MSs excluding MS  1  to K. In step  1402 , the k th  MS receives signal x k   H . In step  1403 , the k th  MS recovers the LSFC {circumflex over (β)} k  for each downlink channel to be: 
                       β   ^     k     =                x   k          2     -   T     PT             (   8   )               
where each MS using only statistics of unknown broadcast signal to estimate {circumflex over (β)} k .
 
     In the embodiment of  FIG. 14 ,
 
Q=WD
 
where
         D=[d 1 , . . . , d T ]ε           K′×T : Data entries of D are unknown i.i.d. information intended to K′ serving MSs excluding MS  1  to K. The power of D entries is P/K′   W=[w 1 , . . . , w K′ ]ε           M×K′ : Unknown beamforming or precoding matrix, having unit-norm columns, to those K′ serving MSs, and w i   H w j =δ ij         

     
       
         
           
             
               
                 
                   
                     
                        
                       
                         x 
                         k 
                       
                        
                     
                     2 
                   
                   ≈ 
                     
                   ⁢ 
                   
                     
                       
                         β 
                         k 
                       
                       ⁢ 
                       
                         g 
                         k 
                         H 
                       
                       ⁢ 
                       
                         QQ 
                         H 
                       
                       ⁢ 
                       
                         g 
                         k 
                       
                     
                     + 
                     
                       
                          
                         
                           z 
                           k 
                         
                          
                       
                       2 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         β 
                         k 
                       
                       ⁢ 
                       
                         
                           g 
                           ~ 
                         
                         k 
                         H 
                       
                       ⁢ 
                       
                         Φ 
                         k 
                         
                           1 
                           2 
                         
                       
                       ⁢ 
                       
                         QQ 
                         H 
                       
                       ⁢ 
                       
                         Φ 
                         k 
                         
                           1 
                           2 
                         
                       
                       ⁢ 
                       
                         
                           g 
                           ~ 
                         
                         k 
                       
                     
                     + 
                     
                       
                         
                           
                              
                             
                               z 
                               k 
                             
                              
                           
                           2 
                         
                         ⁢ 
                         
                           ⟶ 
                           
                             a 
                             . 
                             s 
                             . 
                           
                         
                         ⁢ 
                         
                           β 
                           k 
                         
                       
                       ⁢ 
                       
                         tr 
                         ( 
                         
                           
                             Φ 
                             k 
                             
                               1 
                               2 
                             
                           
                           ⁢ 
                           
                             QQ 
                             H 
                           
                           ⁢ 
                           
                             Φ 
                             k 
                             
                               1 
                               2 
                             
                           
                         
                         ) 
                       
                     
                     + 
                     T 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         β 
                         k 
                       
                       ⁢ 
                       
                         tr 
                         ⁡ 
                         
                           ( 
                           
                             
                               Φ 
                               k 
                               
                                 1 
                                 / 
                                 2 
                               
                             
                             ⁢ 
                             
                               WDD 
                               H 
                             
                             ⁢ 
                             
                               W 
                               H 
                             
                             ⁢ 
                             
                               Φ 
                               k 
                               
                                 1 
                                 / 
                                 2 
                               
                             
                           
                           ) 
                         
                       
                     
                     ⁢ 
                     
                         
                     
                     + 
                     T 
                     - 
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     tr 
                     ⁡ 
                     
                       ( 
                       
                         
                           Φ 
                           k 
                           
                             1 
                             / 
                             2 
                           
                         
                         ⁢ 
                         
                           WDD 
                           H 
                         
                         ⁢ 
                         
                           W 
                           H 
                         
                         ⁢ 
                         
                           Φ 
                           k 
                           
                             1 
                             / 
                             2 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     tr 
                     ⁡ 
                     
                       ( 
                       
                         
                           D 
                           H 
                         
                         ⁢ 
                         
                           W 
                           H 
                         
                         ⁢ 
                         
                           Φ 
                           k 
                         
                         ⁢ 
                         WD 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                     def 
                   
                   ⁢ 
                     
                   ⁢ 
                   
                     tr 
                     ⁡ 
                     
                       ( 
                       
                         
                           D 
                           H 
                         
                         ⁢ 
                         AD 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       
                         tr 
                         ⁡ 
                         
                           ( 
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       d 
                                       1 
                                       H 
                                     
                                     ⁢ 
                                     
                                       Ad 
                                       1 
                                     
                                   
                                 
                                 
                                   … 
                                 
                                 
                                   
                                     
                                       d 
                                       1 
                                       H 
                                     
                                     ⁢ 
                                     
                                       Ad 
                                       T 
                                     
                                   
                                 
                               
                               
                                 
                                   ⋮ 
                                 
                                 
                                   ⋱ 
                                 
                                 
                                   ⋮ 
                                 
                               
                               
                                 
                                   
                                     
                                       d 
                                       T 
                                       H 
                                     
                                     ⁢ 
                                     
                                       Ad 
                                       1 
                                     
                                   
                                 
                                 
                                   … 
                                 
                                 
                                   
                                     
                                       d 
                                       T 
                                       H 
                                     
                                     ⁢ 
                                     
                                       Ad 
                                       T 
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ⟶ 
                         
                           a 
                           . 
                           s 
                           . 
                         
                       
                       ⁢ 
                       
                         P 
                         
                           K 
                           ′ 
                         
                       
                     
                     ⁢ 
                     
                       tr 
                       ⁡ 
                       
                         ( 
                         
                           [ 
                           
                             
                               
                                 trA 
                               
                               
                                 … 
                               
                               
                                 0 
                               
                             
                             
                               
                                 ⋮ 
                               
                               
                                 ⋱ 
                               
                               
                                 ⋮ 
                               
                             
                             
                               
                                 0 
                               
                               
                                 … 
                               
                               
                                 trA 
                               
                             
                           
                           ] 
                         
                         ) 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       PT 
                       
                         K 
                         ′ 
                       
                     
                     ⁢ 
                     trA 
                   
                 
               
             
             
               
                 
                   = 
                     
                   ⁢ 
                   
                     
                       PT 
                       
                         K 
                         ′ 
                       
                     
                     ⁢ 
                     
                       
                         tr 
                         ⁡ 
                         
                           ( 
                           
                             [ 
                             
                               
                                 
                                   
                                     
                                       w 
                                       1 
                                       H 
                                     
                                     ⁢ 
                                     
                                       Φ 
                                       k 
                                     
                                     ⁢ 
                                     
                                       w 
                                       1 
                                     
                                   
                                 
                                 
                                   … 
                                 
                                 
                                   
                                     
                                       w 
                                       1 
                                       H 
                                     
                                     ⁢ 
                                     
                                       Φ 
                                       k 
                                     
                                     ⁢ 
                                     
                                       w 
                                       
                                         K 
                                         ′ 
                                       
                                     
                                   
                                 
                               
                               
                                 
                                   ⋮ 
                                 
                                 
                                   ⋱ 
                                 
                                 
                                   ⋮ 
                                 
                               
                               
                                 
                                   
                                     
                                       w 
                                       
                                         K 
                                         ′ 
                                       
                                       H 
                                     
                                     ⁢ 
                                     
                                       Φ 
                                       k 
                                     
                                     ⁢ 
                                     
                                       w 
                                       1 
                                     
                                   
                                 
                                 
                                   … 
                                 
                                 
                                   
                                     
                                       w 
                                       
                                         K 
                                         ′ 
                                       
                                       H 
                                     
                                     ⁢ 
                                     
                                       Φ 
                                       k 
                                     
                                     ⁢ 
                                     
                                       w 
                                       
                                         K 
                                         ′ 
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ⟶ 
                         
                           a 
                           . 
                           s 
                           . 
                         
                       
                     
                   
                 
               
             
             
               
                 
                     
                   ⁢ 
                   
                     
                       PT 
                       
                         
                           K 
                           ′ 
                         
                         ⁢ 
                         M 
                       
                     
                     ⁢ 
                     
                       tr 
                       ⁡ 
                       
                         ( 
                         
                           [ 
                           
                             
                               
                                 
                                   tr 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     Φ 
                                     k 
                                   
                                 
                               
                               
                                 … 
                               
                               
                                 0 
                               
                             
                             
                               
                                 ⋮ 
                               
                               
                                 ⋱ 
                               
                               
                                 ⋮ 
                               
                             
                             
                               
                                 0 
                               
                               
                                 … 
                               
                               
                                 
                                   tr 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
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                         ) 
                       
                     
                   
                 
               
             
             
               
                 
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     In the example of  FIG. 14 , each k th  MS receives radio signal x k  from a BS having M antennas. The BS transmits a pilot signal denoted by matrix Q=WD, the power is P, and the noise variance is σ 2 . The DL LSFC estimator  1411  is able to derive the LSFC {circumflex over (β)} k  for each downlink channel with low computational complexity and without prior knowledge of the small-scale fading coefficients. Note that when N&gt;&gt;1, two independent random vectors u,vε             N×1  has two properties: i)
                 u   H     ⁢   Au     -       tr   ⁡     (   A   )       ⁢     ⟶     a   .   s   .       ⁢   0           
and ii)
 
               1   M     ⁢     u   H     ⁢     Av   ⁢     ⟶     a   .   s   .       ⁢   0.           
Thus, with large dimensions of {tilde over (g)} k &#39;s, d i &#39;s,
 
                 w     i   ′       ⁢   s     ,         g   ~     k   H     ⁢     Φ   k     1   2       ⁢     QQ   H     ⁢     Φ   k     1   2       ⁢         g   ~     k     ⁢     ⟶     a   .   s   .       ⁢     tr   (       Φ   k     1   2       ⁢     QQ   H     ⁢     Φ   k     1   2         )         ,         K   ′     P     ⁢     d   i   H     ⁢         Ad   j     ⁢     ⟶     a   .   s   .       ⁢   trA     ·     δ   ij         ,         
and
 
     
       
         
           
             
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                   . 
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                   . 
                 
               
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       FIG. 15A  shows the MSE performance with respect to SNR and training period of the proposed downlink semi-blind LSFC estimators without SSFC knowledge. As shown in  FIG. 15A , the MSE performance improves as the SNR increases, and as the training period T increases. 
       FIG. 15B  shows the MSE performance with respect to SNR and training period of the proposed downlink blind LSFC estimators without SSFC knowledge. As shown in  FIG. 15B , the MSE performance improves as the SNR increases, and as the training period T increases. 
       FIG. 16  is a flow chart of a method of estimating downlink LSFC in accordance with one novel aspect. In step  1601 , a mobile station (MS) receives radio signals transmitted from a base station (BS) having M antennas in a massive MIMO system. The transmitted radio signals are denoted as a matrix Q transmitted from the BS to K MS. In step  1602 , the MS determines a received radio signal denoted as a vector x k  received by the MS that is the kth MS associated with a k th  downlink channel. In step  1603 , the k th  MS derives an estimator of a large-scale fading coefficient (LSFC) of the k th  downlink channel without knowing a small-scale fading coefficient (SSFC) of the kth downlink channel. In step  1604 , in a semi-blind LSFC estimation, matrix Q is a semi-unitary matrix consisting of orthogonal pilot signals, and the LSFC of the k th  downlink channel is derived based on x k  and the transmitting power of the pilot signals. In step  1605 , in a blind LSFC estimation, matrix Q represents pre-coded data signals transmitted to K′ MS that are different from the K MS. The LSFC of the k th  downlink channel is derived based on x k  and the transmitting power of the data signals with unknown data information and unknown beamforming or precoding information. 
     Although the present invention has been described in connection with certain specific embodiments for instructional purposes, the present invention is not limited thereto. Accordingly, various modifications, adaptations, and combinations of various features of the described embodiments can be practiced without departing from the scope of the invention as set forth in the claims.