Abstract:
A pseudonise (PN) code acquisition scheme employs all elements of a smart antenna array and an adaptive threshold. The basic structure is the combination of a conventional PN correlation searcher, an adaptive beamformer, and an adaptive threshold setting circuit. During each observation interval, the adaptive beamformer adaptively updates the weight vector for the smart antenna elements using the accumulated received signal despread with trial PN code phase error as input, preferably at the chip rate. A spatially correlated signal is then formed by weighting an accumulated value of the signal received by each antenna in the array over the observation period with the corresponding final weight of smart antenna weight vector as calculated by the adaptive beamformer. This spatially correlated signal is then compared to a threshold to determine whether PN code acquisition has occurred. In preferred embodiments, an adaptive threshold setting algorithm is employed to adapt to varying environment for efficient PN code acquisition. The adaptive threshold setting circuit accumulates the signal received by each element over the observation period, multiplies the accumulated values by the updated weight vector and combines the weighted values to an average estimated power. The average estimated power is then employed to scale a fixed reference threshold to create the adaptive threshold.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates to wireless telecommunications generally, and more particularly to the application of a smart antenna to PN code acquisition in a code division multiple access (CDMA) wireless communications system. 
     2. Description of the Related Art 
     In third generation (3G) wireless communications systems, particularly wide band (W)-CDMA systems such as those described in the 3rd Generation Partnership Project, “Physical Channels and Mapping of Transport Channels onto Physical Channels (FDD),” 3GPP Technical Specification, TS25.211, v3.2.0, March 2000; 3rd Generation Partnership Project, “Spreading and Modulation (FDD),” 3GPP Technical Specification, TS25.213, v3.2.0, March 2000; and 3rd Generation Partnership Project, “FDD: Physical Layer Procedures,” 3GPP Technical Specification, TS25.214, v3.2.0, March 2000 (collectively “3GPP”), or in the CDMA2000 standard in TIA, Interim V&amp;V Text for CDMA2000 Physical Layer (Revision 8.3), Mar. 16, 1999 (“TIA”), an option is to employ smart antenna technology in the base station. A smart antenna can suppress interfering signals from different direction of arrival angles (DOAs) from the desired users by using spatial diversity. Smart antenna technologies attract much attention these days as they support more users with a high quality of service and high data rates, up to 1.92 mega bits per second (Mbps). Examples of smart antennas and system architectures employing smart antennas may be found in commonly-owned co-pending U.S. application Ser. No. 09/610,470, filed Jul. 5, 2000, entitled “Smart Antenna with Adaptive Convergence Parameter;” Ser. No. 09/661,155, filed Sep. 13, 2000, entitled “Smart Antenna with No Phase Calibration for CDMA Reverse Link;” and Ser. No. 09/669,633, filed Sep. 26, 2000, entitled “New Cellular Architecture.” The contents of all of these applications are hereby incorporated by reference herein. 
     Despite the interest in smart antenna technology generally, little attention has been paid to pseudonoise (PN) code acquisition in CDMA systems that employ a smart antenna at a base station. As used herein, PN code acquisition refers to a portion of a process referred to in the art as synchronization. Synchronization is generally regarded as encompassing two processes: PN code acquisition (in which a phase error for a known PN code is resolved to within a specified boundary—that is, a coarse PN phase code error correction), and PN code tracking, in which fine differences in PN phase code errors are detected and corrected. Thus, PN code acquisition, despite its misleading moniker, refers to a coarse correction for a PN phase error between a receiver and a transmitter (e.g., a base station and a mobile unit, or vice-versa), and does not refer to a process by which a PN code itself (as opposed to a PN code phase error) is detected. 
     PN code acquisition may be difficult when the smart antenna weight vector does not correspond to the desired signal&#39;s DOA (because the smart antenna will suppress signals from other DOAs). Because of this potential problem, existing systems use only one (omnidirectional) element output out of the M array elements for PN acquisition purposes. See F. Adachi, M. Sawahashi, and H. Suda, “Wideband DS-CDMA for Next-Generation Mobile Communications Systems,” IEEE Communications Magazine, pp. 56-69, September 1998 (“Adachi et al.”). Thus, the benefit of a smart antenna (e.g., the potential gain of the smart antenna) has not been used for PN code acquisition. This causes another problem as the mobile units in the system may transmit signals with low power because of the expected high smart antenna gain at a base station, with the result that the received signal-to-interference-plus-noise-ratio (SINR) at a base station may not be sufficient for PN code acquisition when only one element is employed. 
     BRIEF SUMMARY OF THE INVENTION 
     The present invention is an efficient PN code acquisition scheme which employs multiple elements of an antenna array and an adaptive threshold. The invention is particularly useful for CDMA wireless communications, especially for Direct Sequence (DS-)CDMA wireless communications. The basic structure of preferred embodiments is the combination of a conventional PN correlation searcher, an adaptive beamformer, and an adaptive threshold setting circuit. During each observation interval, which consists of multiple chips, the adaptive beamformer adaptively updates the weight vector for the smart antenna elements (as used herein, a smart antenna element refers to a single antenna, such as an omnidirectional antenna, in an array of antennas that collectively form the smart antenna) using the accumulated received signal despread with trial PN code phase error as input, preferably at the chip rate. The adaptive beam former may use any one of a number of algorithms for this purpose. In a preferred embodiment, a normalized least mean square algorithm is used. A spatially correlated signal is then formed by weighting an accumulated value of the signal received by each antenna in the array over the observation period with the corresponding final weight of smart antenna weight vector as calculated by the adaptive beamformer algorithm. This spatially correlated signal is then compared to a threshold to determine whether PN code acquisition has occurred. If the threshold is exceeded, PN code acquisition is declared. Otherwise, a new trial PN code phase error is selected and the process is repeated. 
     In conventional serial search algorithms, the above-mentioned threshold for PN code acquisition has been fixed and can be calculated from a given false alarm probability P f  and a given bit-energy-to-interference power spectral density ratio E b /I o . But in a real mobile fading environment, E b /I o  often varies. In preferred embodiments, an adaptive threshold setting algorithm is employed to adapt to varying environment for efficient PN code acquisition. An adaptive threshold setting algorithm has been analyzed for a receiver with a single antenna element, in Kwonhue Choi, Kyungwhoon Cheun, and Kwanggeog Lee, “Adaptive PN code Acquisition Using Instantaneous Power-Scaled Detection Threshold Under Rayleigh Fading and Gaussian Pulsed Jamming,” The 4 th  CDMA International Conference, Proceedings Vol. II pp. 162-169, Seoul, Korea, Sep. 8-11, 1999 (“Choi et al.”) , the contents of which are hereby incorporated herein by reference. The present invention develops an adaptive threshold setting algorithm for a receiver with multiple array elements. The adaptive threshold setting circuit is actually an average power estimator in preferred embodiments. While the adaptive beamformer updates the weight vector  w (i) adaptively, the power estimator estimates the instantaneous received signal and interference power prior to PN code despreading for each observation interval of NT c  seconds. The average estimated power is then employed to scale a fixed reference threshold to create the above-mentioned adaptive threshold used to determine whether or not PN code acquisition has been achieved. The PN code acquisition time with the proposed PN code acquisition scheme with M=5 elements, by way of example, can be 210% shorter at a given SINR than the PN code acquisition time for a system, such as the system described in Adachi et al., in which only a single element is used for PN code acquisition. 
     In one aspect of the present invention, the PN code acquisition system may be applied to a base station, wherein the antennas are in the base station. According to another aspect of the present invention, the system is applied to a mobile station, wherein the smart antennas are in the mobile station. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The features, objects, and advantages of the present invention will become more apparent from the detailed description set forth below when taken in conjunction with the drawings in which like reference characters identify correspondingly throughout and wherein: 
     FIG. 1 shows a block diagram of a proposed PN code acquisition scheme for a DS-CDMA system configured in accordance with one embodiment of the present invention; 
     FIG. 2 shows a flow chart of the proposed PN code acquisition scheme for a DS-CDMA system configured in accordance with one embodiment of the present invention; 
     FIG. 3 shows the theoretical and simulation false alarm probabilities P f  versus bit-energy-to-interference power spectral density ratio E b /I o  in dB for M=1, 3, and 5 elements cases under a fading environment, indicating that the PN code acquisition scheme configured in accordance with one embodiment of the present invention is robust against the received signal power variations; 
     FIG. 4 shows the theoretical and simulation detection probabilities P d  versus bit-energy-to-interference power spectral density ratio E b /I o  in dB for M=1, 3, and 5 elements under fading environment, indicating that the PN code acquisition scheme configured in accordance with one embodiment of the present invention improves detection probability P d  significantly; and 
     FIG. 5 shows the theoretical and simulation average acquisition time T acq  versus bit-energy-to-interference power spectral density ratio E b /I o  in dB for M=1, 3, and 5 elements under fading environment, indicating that the PN code acquisition scheme configured in accordance with one embodiment of the present invention improves average acquisition time T acq  significantly. 
    
    
     DETAILED DESCRIPTION 
     In the following detailed description, many specific details, such as chip rates and numbers of elements in smart antenna arrays, are provided by way of example in order to provide a thorough understanding of the present invention. The details discussed in connection with the preferred embodiments should not be understood to limit the present invention. 
     The present invention is described below in the context of a reverse link from a mobile to a base station in a CDMA wireless communications system. However, the invention is equally applicable to forward link PN code acquisition. The invention is also believed to be particularly applicable to Direct Sequence CDMA (DS-CDMA) systems, and thus will be discussed in the context of the same herein, but should not be understood to be limited to DS-CDMA systems. 
     FIG. 1 shows a block diagram of the proposed PN code acquisition scheme for a DS-CDMA system configured in accordance with one embodiment of the present invention. The input signal r′(t) received from the each element  101  of the smart antenna array is passed through a demodulator  102  and a matched filter  103 . The output of each matched filter  103  is sampled at the chip rate by a chip sampler  104 , and correlated (despread) with the trial PN code phase error by a correlator  105 . The output of the correlators  105  is stored by accumulators  106 . The output of the accumulators  106  is used by an update process for the smart antenna weight vector. 
     In preferred embodiments, the weight vector of the smart antenna is updated with an LMS algorithm. In highly preferred embodiments, the normalized least mean square (LMS) algorithm described in S. Haykin,  Adaptive filter Theory,  3rd Edition, Prentice Hall, Upper Saddle River, N.J., 1996 (“Haykin”), is employed to update the weight vector. Other vector update algorithms are also possible, including those algorithms discussed in commonly assigned co-pending U.S. application Ser. 09/610,470, filed Jul. 5, 2000, entitled “Smart Antenna with Adaptive Convergence Parameter.” It is not necessary for the vector update to be an LMS algorithm. During each observation interval of NT c  (where T c  is the chip period and N is the number of chip periods in an observation interval) seconds, an adaptive beamformer  107  updates the weight vector  w (i) adaptively, preferably at the chip rate. The input to adaptive beamformer  107  is the accumulator  106  output after despreading using the trial PN code phase error as described above. 
     In preferred embodiments, the vector update process starts after the observation interval begins. This is done in order to conserve processing power. The sample time index i=N s  (where i is the chip or sample time index) on which the update starts is chosen to reflect the time required for the adaptive algorithm to converge. For example, in preferred embodiments using the normalized LMS algorithm described above and in which the number of chips N in the observation period is equal to 256, N, is chosen as 192, which provides for 256−192=64 iterations of the algorithm. This reflects that fact that the algorithm typically takes approximately 40 iterations to converge and includes a safety factor. 
     The update process ends at i=N−2 for the first observation interval. The last weight vector  w (N−1) calculated by the update algorithm in an observation interval, which can be regarded as a best weight vector  w   opt , is used to generate an antenna array spatial correlation output {overscore (z)} with the last accumulation output at i=N−1. The square of {overscore (z)} is then compared to an adaptive threshold (discussed in further detail below) to determine whether PN code acquisition has been achieved. If the threshold is exceeded, the PN code has been acquired and the process ends. Otherwise, a new trial PN code phase error is selected and the algorithm is repeated. The weight vector adaptation is performed at each successive observation interval of NT c  seconds until PN code acquisition is achieved. 
     The preferred LMS algorithm takes only 5M computations per chip period, where M is the number of the antenna array elements  101 . The update rate can be sufficiently small such as a chip rate of 1.2288 Mcps. The present invention employs preferably all array elements  101  of an array for a PN correlation searcher while only one element is employed in the existing literatures (“Adachi et al.”). 
     The following discussion explains the above-described process in greater detail. The array antenna elements  101  are assumed to be substantially identical and have the same response to any direction and the antenna spacing is one half of the carrier wavelength. The array response vector can be written as 
     
       
             a     o (θ)=[1 e   −jπ sin θ    . . . e   −jπ(M−1)sin θ ] T   (1) 
       
     
     where θ is the DOA from the desired user signal, and T denotes the transpose. The received signal at the m-th element can be written as 
     
       
           r′   m ( t )={square root over (2 P +L )}α( t ) b ( t−τT   c ) c ( t−τT   c ) e   j(w     c     t+ψ     m     )   e   jφ(t)   e   −jπ(m−1)sin θ   +n   m,BPF ( t )  (2) 
       
     
     where P is the received signal power, α(t) and φ(t) are the fading amplitude and phase, respectively, b(t) is information data, c(t) is the PN spreading sequence, w c  and ψ m  are the angular carrier frequency and phase, respectively, T c  is a chip interval, and τ is a PN code phase offset from a reference, which is assumed to be a random integer uniformly distributed between 0 and PN sequence period L. The n m,BPF (t) in equation (2) represents an additive white Gaussian noise (AWGN) bandpass interference plus noise signal. A fading channel with a single path is assumed for simplicity. The present invention can be applied for a multipath channel also. A simplified search, which shifts the trial PN code sequence by T c , is considered here for simplicity. The phase of the PN code can be shifted in a fraction of chip interval in a complete search and it takes long simulation time. A single dwell is assumed for simplicity. The invention can be applied for a complete and multi-dwell PN code searches also. The PN spreading sequence c(t) is written as                c        (   t   )       =       ∑   i            c        (   i   )              ∏     T   c            (     t   -     iT   c       )                   (   3   )                                
     where c(i)ε{1, −1} with equal probability, Π Tc  is a rectangular shape pulse with unit amplitude and duration T c . The carrier synchronization is not assumed because noncoherent PN code single dwell serial search is employed. Also, assume a pilot channel is used instead of pilot symbols, i.e., b(t)=1 for all time as the pilot channel in cdma2000. Then, the equivalent baseband signal r m (t) at the m-th element after down-conversion with carrier frequency can be written as 
     
       
           r   m ( t )={square root over ( P +L )}α( t ) c ( t−τT   c ) e   j[Φ     m     +ψ(t)−π(m−1)sin θ]   +n   m ( t )  (4) 
       
     
     where n m (t)=n m   1 (t)+jn m   Q (t) is a complex AWGN with mean zero and variance I 0 . The output of matched filter H*(f)  103  is sampled at chip rate and the samples r m (i) are written as 
     
       
           r   m ( i )={square root over ( E   c +L )}α( i ) c ( i− τ) e   j[ψ     m     +φ(i)−π(m−1)sin θ]   +n   m ( i )  (5) 
       
     
     where i and E c  denote the chip index and chip energy, respectively. After correlating with the local PN code at  105 , r m (i) becomes 
     
       
           x   m ( i )=[{square root over ( E   c +L )}α( i ) c ( i− τ) e   jφ     m     (i)   +n   m ( i )] c ( i− {circumflex over (τ)})  (6) 
       
     
     where {circumflex over (τ)} is the estimated PN code phase and 
     
       
         φ m ( i )=ψ m +φ( i )−π( m− 1)sin θ  (7) 
       
     
     All antenna element outputs can be written in a vector form as 
     
       
             x   ( i )=[{square root over ( E   c +L )}α( i ) c ( i −τ)   a   (φ)+   n   ( i )] c ( i −{circumflex over (τ)})  (8) 
       
     
     where 
     
       
             a (φ)=[   e   jφ     1     (i)   e   jφ     2     (i)    . . . e   jφ     M     (i) ] T   (9) 
       
     
     can be regarded as a modified array response vector, compared with  a   0 (θ) in equation (1). Assume an observation interval of NT c  for a PN code phase shift. Energy is accumulated every chip at each branch after PN despreading  105 . The accumulator  106  output at iT c  for the m-th branch can be written as                  y   m          (   i   )       =       ∑       i   ′     =   0     i            x   m          (     i   ′     )                 (   10   )                                
     where i=0, 1, . . . , N−1 for the first observation interval, and m=1, 2, . . . , M. The adaptive beamformer  107  preferably starts updating weight vector  w (i) adaptively at i=N s  and ends updating at i=N−2 for the first observation interval. (Those of skill in the art will recognize that it is also possible to begin updating earlier or later, even as early as the start of the observation interval.) The same process is applied for the other observation intervals until PN code is acquired. The normalized LMS algorithm is employed to update the weight vector. This weight vector adaptation is performed every observation interval of NT c  seconds. The input to the adaptive beamformer  107  is written as 
     
       
             y   ( i )=[ y   1 ( i ) y   2 ( i ) . . .  y   M ( i )]T.  (11) 
       
     
     The last weight vector  w (N−1) in an observation interval, which can be regarded as a best weight vector  w   opt , is used to generate a spatial correlation output {overscore (z)} with the last accumulation output  y (N−1). The antenna array spatial correlation output {overscore (z)}  109  can be written as                      z   _     =           w   _     H          (     N   -   1     )                y   _          (     N   -   1     )       /     (   MN   )                     =       1   MN            ∑     m   =   1     M            ∑     i   =   0       N   -   1            [       w   optm            x   m          (   i   )         ]                         =       1   MN            ∑     j   =   1     MN          z   j           ,                 (   12   )                                
     where H stands for Hermitian, i.e., conjugate and transpose, 
     
       
             w   ( N− 1)=   w     opt   =[w   opt1   w   opt2    . . . w   optM ] H ,  (13) 
       
     
     and 
     
       
           z   j   =w   optm   x   m ( i ),  j =( m− 1) N+i+ 1.  (14) 
       
     
     The final decision variable Z  110  is normalized and written as              Z   =              z   _          2     =                1   MN            ∑     j   =   1     MN          z   j              2     .               (   15   )                                
     As discussed above, this final decision variable Z is then compared to a threshold, preferably an adaptive threshold as calculated by an adaptive threshold setting circuit  108 , in order to determine whether PN code acquisition has been achieved. In preferred embodiments, the adaptive threshold setting circuit  108  estimates the average power during each observation interval NT c . Referring to FIG.  1  and considering |r m (i)| 2 =x m (i)| 2 , we can get the average power {overscore (P)} as                      P   _     =       ∑     m   =   1     M          {       [       1   M                 w   optm          2       ]     ·     [       1   N            ∑     i   =   0       N   -   1                     r   m          (   i   )            2         ]       }                   =       1   MN            ∑     m   =   1     M            ∑     i   =   0       N   -   1                     w   optm            x   m          (   i   )              2                       =       1   MN            ∑     j   =   1     MN                   z   j          2     .                       (   16   )                                
     If Z is larger than a scaled threshold k·{overscore (P)}, a PN code acquisition is declared and the tracking loop is triggered for a single dwell serial search, where k is the fixed reference threshold. Otherwise, the acquisition scheme shifts the locally generated PN code phase (i.e., creates a new trial PN code phase error), and the searcher continues until a correct PN code phase is claimed (i.e., PN code acquisition is achieved). 
     The probability of Z being greater than k·{overscore (P)} is given as                  P   r          {     Z   &gt;     k   ·     P   _         }       =       P   r            {                1   MN            ∑     j   =   1     MN          z   j              2     &gt;       k   MN            ∑     j   =   1     MN                 z   j          2           }     .               (   17   )                                
     To ease the derivation of the false alarm probability P f  and detection probability P d  later, we rewrite equation (17) using new decision variables U and V as 
     
       
           P   r   {Z&gt;k·{overscore (P)}}=P   r   {U&gt;V}   (18) 
       
     
     with 
     
       
           U =(1 −k ) MN |{overscore (z)}| 2 ,  (19) 
       
     
     
       
           V=k ( MN −1) {overscore (ν)}   z ,  (20) 
       
     
     where {overscore (z)} is the sample mean of z j  given in equation (12), and {overscore (ν)} z  denotes the sample variance of z j  given as                  v   _     z     =       1     MN   -   1              ∑     j   =   1     MN                     z   j     -     z   _            2     .                 (   21   )                                
     Under hypothesis H 1  when the received PN code and the local PN code are aligned with the same phase, equation (18) gives the detection probability P d , and under hypothesis H 0  when PN codes are out of phase, equation (18) gives the false alarm probability P f  as follows: 
     
       
           P   d   =P   r   {U&gt;V|H   1 },  (22) 
       
     
       P   f   =P   r   {U&gt;V|H   0 }.  (23) 
     From equations (19) and (20), we observe that both P d  and P f  are equal to 1 when k&lt;0 and both equal to 0 when k&gt;1. So we focus on the range of k between 0 and 1 in this invention. 
     The following is the theoretical analysis. Assume a slow Rayleigh fading channel so that the amplitude α(i) as well as the phase φ(i) remain constant over each observation time NT c  but are independent between the observation intervals. Then α(i) and φ(i) can be written as 
     
       
         α( i )=α, 0 ≦i≦N− 1,  (24) 
       
     
     
       
         φ( i )=φ, 0 ≦i≦N− 1,  (25) 
       
     
     for the first observation interval, where α and φ are constants. Therefore, (7) becomes 
     
       
         φ m ( i )=ψ m +φ−π( m− 1)sin θ=φ m , 0≦ i≦N −1,  (26) 
       
     
     for the first observation interval, where φ m  is constant. 
     (1) When PN Code is Synchronized 
     This case means 
     
       
           c ( i −τ)= c ( i −{circumflex over (τ)}).  (27) 
       
     
     For a theoretical analysis, we apply perfect weight vector  w   pft  equal to the modified array vector in equation (9) to obtain an ideal performance when PN code is synchronized. In other words, we assume that the smart antenna can track the DOA of the desired user signal perfectly as 
     
       
             w     pft   = a   (φ)  (28) 
       
     
     during the end period of an observation interval, N s ≦i≦N−1, when PN code is synchronized. In practice, a smart antenna cannot track perfectly the DOA of the desired user. The ideal case in equation (28) provides an upper and a lower bound of detection probability P d  and average acquisition time T acq , respectively. Equation (28) implies 
     
       
           w   opt,m   =w   pft,m   *   =e   −jφ     m   , 1≦ m≦M.   (29) 
       
     
     Substituting equations (6), (27) and (29) into (14), we have 
     
       
           z   j   ={square root over (E c +L )}α+   n   j   , j= ( m− 1) N+i+ 1,  (30) 
       
     
     where n j =e −jφ     m   n m (i)c(i−{circumflex over (τ)}), which can be easily shown to be a complex AWGN variable with zero mean and variance equal to I 0 . Substituting equation (30) into equation (12), we get the sample mean 
     
       
           {overscore (z)}={square root over (E   c +L )}α+ {overscore (n)},   (31) 
       
     
     where          n   _     =       1   MN            ∑     j   =   1     MN          n   j                                
     is a complex AWGN variable with zero mean and variance equal to I 0 /MN. Substituting equations (30) and (31) into equations (19) and (20), we have 
     
       
           U =(1 −k ) MN|{square root over (E c +L )}α+   {overscore (n)} | 2 ,  (32) 
       
     
     
       
           V=k ( MN −1) {overscore (ν)}   n ,  (33) 
       
     
     where              v   _     n     =       1     MN   -   1              ∑     j   =   1     MN                 n   j     -     n   _                  ,                          
     is the sample variance of n j . From equation (32), we can show the decision variable U has the non-central chi-square distribution 20 with its conditional probability density function as                    p   U1          (     u   |   α     )       =         exp        {     -     [     u   +       (     1   -   k     )          MNE   c          α   2          ]   /     [       (     1   -   k     )          I   0       ]           }               (     1   -   k     )          I   0                I   0          (       2            (     1   -   k     )          MNE   c          α   2        u             (     1   -   k     )          I   0         )           ,     u   ≥   0     ,           (   34   )                                
     where I 0 (x) is the zero-th order modified Bessel function of the first kind, and α is Rayleigh-distributed with its probability density function as                p        (   α   )       =         2      α       σ   f   2                   -     α   2       /     σ   f   2                   (   35   )                                
     where σ f   2  is the average fading channel power. The probability density function (pdf) of U under synchronization is                    p   U1          (   u   )       =         ∫   0   ∞              p   U1          (     u   |   α     )            p        (   α   )               α         =       exp        (       -   u     /   β     )       β         ,     u   ≥   0     ,           (   36   )                                
     where 
     
       
         β=(1 −k )( MNE   c σ f   2   +I   0 ).  (37) 
       
     
     From equation (33), we can show the decision variable V has the central chi-square distribution with 2(MN-1) degrees of freedom with its probability density function as below:                    p   V1          (   v   )       =         v     (     MN   -   2     )            exp        [       -   v     /     (     2        σ   2       )       ]             σ     2        (     MN   -   1     )              2     (     MN   -   1     )            Γ        (     MN   -   1     )             ,     v   ≥   0     ,           (   38   )                                
     where 
     
       
         σ 2   =kI   0 /2.  (39) 
       
     
     From equations (32) and (33), we know U and V are statistically independent because sample mean and sample variance are statistically independent when sampled from a Gaussian distribution. Then the detection probability can be derived to be                      P   d     =         P   r          {       U   &gt;   V     |     H   1       }       =       [     β     β   +     2        σ   2           ]       MN   -   1                     =         [         (     1   -   k     )          (         MNE   c          σ   f   2       +     I   0       )             (     1   -   k     )          MNE   c          σ   f   2       +     I   0         ]       MN   -   1       .                   (   40   )                                
     (2) When PN Code is not Synchronized 
     This case means 
     
       
           c ( i −τ)≠ c ( i −{circumflex over (τ)}).  (41) 
       
     
     Assume the beamforming direction of smart antenna is different from the DOA of the desired user signal. This is true in general when the PN code is not synchronized. Assume that the weight vector can be written as 
     
       
             w     opt   = w   ( N −1)=   a   (η)=[ e   jη     1     e   jη     2      . . . e   jη     M   ] T   ≠ a   (φ)  (42) 
       
     
     without loss of generality. Equation (40) implies 
     
       
           w   optm   =e   −jη     m     ≠e   −jφ     m   , 1 ≦m≦M.   (43) 
       
     
     Substituting equations (6), (41) and (43) into equation (14), we have 
     
       
           z   j   ={square root over (E c +L )}α   c ( i −τ) c ( i −{circumflex over (τ)}) e   j (φ   m     −η     m)     +e   −jη     m     n   m ( i ) c ( i −{circumflex over (τ)}).  (44) 
       
     
     We can show z j  is a complex AWGN variable n 0  with zero mean and variance equal to (E c σ f   2 +I 0 ). Equation (44) can be rewritten as 
     
       
           z   j   =n   0 .  (45) 
       
     
     Substituting equation (45) into equation (12), we have                  z   _     =         1   MN            ∑     j   =   1     MN          n   0         =       n   _     0         ,           (   46   )                                
     which is also a complex AWGN variable with zero mean and variance equal to (E cσ   f   2 +I 0 )/(MN). Substituting equations (45) and (46) into equations (19) and (20), we have 
     
       
           U =(1 −k ) MN|{overscore (n)}   0 | 2 ,   (47) 
       
     
     
       
           V=k ( MN −1) {overscore (ν)}   n     0   ,  (48) 
       
     
     where            v   _       n   0       =       1     MN   -   1              ∑     j   =   1     MN                 n   0     -       n   _     0                                       
     is the sample variance of n 0 . Similarly as subsection (1), we can show the decision variables U and V follow central chi-square distribution with 2 and 2(MN−1) degrees of freedom, respectively, and their probability density function are given as below                    p   U0          (   u   )       =       exp        (       -   u     /   β     )       β       ,                u   ≥   0     ,           (   49   )                     p   V0          (   v   )       =         v     (     MN   -   2     )            exp        [       -   v     /     (     2        σ   2       )       ]             σ     2        (     MN   -   1     )              2     (     MN   -   1     )            Γ        (     MN   -   1     )             ,     v   ≥   0     ,           (   50   )                                
     where 
     
       
         β=(1 −k )( E   c σ f   2   +I   0 ),   (51) 
       
     
     
       
         σ 2   =k ( E   c σ f   2   +I   0 )/2.   (52) 
       
     
     Similarly as subsection (1), we also know U and V are statistically independent from equations (47) and (48), due to the fact that sample mean and sample variance are statistically independent when sampled from a Gaussian distribution. Then the false alarm probability can be derived to be                      P   f     =         P   r          {       U   &gt;   V     |     H   0       }       =       [     β     β   +     2        σ   2           ]       MN   -   1                     =         (     1   -   k     )       MN   -   1       .                   (   53   )                                
     We observe that the false alarm probability P f  is a function of k, M and N only and does not depend on E c  and I 0 . Thus the adaptive threshold algorithm can approximately achieve a constant false alarm rate (CFAR). 
     FIG. 2 shows the corresponding flow chart of the proposed PN code acquisition system with the present invention. All the vectors in FIG. 2 have M elements, where M is the number of antenna elements. First, we start from the received sequence  r (i) in block  201 , which is formed by the output of the chip samplers  104  in FIG. 1, where i is the time index. Next,  x (i) (the despread signal) in block  202  is formed by the output of the correlators  105 . Next, the  y (i) in block  203  is formed from the accumulator  106  output. The output of block  203  is then used in block  204  and block  208 . At i=N s  the adaptive beamformer  107  starts updating the weight vector  w (i).  W   0 =(1 1 . . . 1) H  in block  205  is the initial weight vector at i=N s . In block  204 , the input to the adaptive beamformer  107  is  ŷ (i), which is the normalized  y (i). The output of block  204  is used in updating the weight vector block  206 . Block  206  shows the LMS algorithm for the adaptive beamformer  107 . As shown in block  206 , an intermediate value  w (i) of the weight vector is formed at each update, which is then used at the subsequent update. The updated weight vector from block  206  is output to block  207 . At i=N−1, the final weight vector shown in block  207  is multiplied with the final accumulator output  y (N−1) shown in block  208  to generate the spatial correlation output  109  {overscore (z)} in block  209 . Block  210  calculates the magnitude square of {overscore (z)} using the output of block  209  to get the decision variable Z  110 , which is output to blocks  212 . The power is estimated at block  211  utilizing the outputs of block  201  and  207 . In block  212 , Z is compared with the threshold scaled by the adaptive threshold setting circuit  108  output shown in block  211  and then the decision as to whether to declare PN code acquisition is made. 
     To verify the theoretical results, simulation has been done with the PN code acquisition scheme configured in accordance with one embodiment of the present invention. The simulation parameters are described as the following. P f =0.01, the number of antenna elements M=1, 3, and 5, N s =192, N=256, bit rate=9600 bits/second, chip rate =1.2288×10 6  chips/second, and penalty factor K p =256 were assumed. The fixed reference threshold k was chosen to guarantee P f  to be equal to 0.01 for the adaptive threshold case. k=0.0179 was chosen for one element case; k=0.0108 for three element case; and k=0.0089 for five elements case. The fixed reference I 0  was chosen to make P f  in equation (59) to be equal to 0.01 at E b /I o =−15 dB for the nonadaptive threshold case. One sample per chip was taken for simulation. Jakes&#39; fading model was used for fading environment. The average fading channel power σ f   2 =1, velocity ν=80 miles/hour and carrier frequency f c =900 MHz were chosen for fading case. Spreading factor 128 and E b /I o =128E c /I o  were used where E b  is information bit energy. The simulation parameters are summarized in Table 1. 
     
       
         
               
             
               
               
               
             
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Simulation Parameters 
               
             
          
           
               
                   
                 Description 
                 Parameter Value 
               
               
                   
                   
               
               
                   
                 False Alarm Probability 
                 P f  = 0.01 
               
               
                   
                 Observation Interval 
                 N = 256 
               
               
                   
                 Starting Time Index of 
                 N s  = 192 
               
               
                   
                 Updating Weight Vector 
               
               
                   
                 Number of Samples per Chip 
                 1 sample/chip 
               
               
                   
                 Penalty Factor 
                 K p  = 256 
               
               
                   
                 Bit Rate 
                 9600 bits/second 
               
               
                   
                 Chip Rate 
                 1.2288 × 10 6  chips/second 
               
               
                   
                 Spreading Factor 
                 128 
               
               
                   
                 Bit-Energy-to-Interference Power 
                 E b /I o  = 128 E c /I o   
               
               
                   
                 Spectral Density Ratio 
               
               
                   
                 Carrier Frequency 
                 f c  = 900 MHz 
               
               
                   
                 Mobile User Velocity 
                 v = 80 miles/hour 
               
               
                   
                 Average Fading Channel Power 
                 σ f   2  = 1 
               
             
          
           
               
                   
                 Fixed Reference 
                 M = 1 
                 k = 0.0179 
               
               
                   
                 Threshold k 1   
                 M = 3 
                 k = 0.0108 
               
               
                   
                   
                 M = 5 
                 k = 0.089 
               
               
                   
                 Interference Power 
                 M = 1 
                 I 0  = 0.9947 
               
               
                   
                 Spectral Density Ratio 
                 M = 3 
                 I 0  = 0.9981 
               
               
                   
                   
                 M = 5 
                 I 0  = 0.9987 
               
               
                   
                   
               
               
                   
                   1 k is chosen such that the PN code acquisition yield a given P f  = 0.01 for the adaptive threshold case. When {overscore (P)} = 1, k becomes the fixed threshold for the nonadaptive threshold case. That is why k is called the fixed reference threshold.  
               
             
          
         
       
     
     FIG. 3 shows the theoretical and simulation false alarm probabilities P f  versus bit-energy-to-interference power spectral density ratio E b /I o  in dB for M=1, 3, and 5 cases under fading environment with a PN code acquisition scheme configured in accordance with one embodiment of the present invention. The simulation P f  for M=1, 3, and 5 cases fluctuate very slightly around the theoretical P f =0.01. And the simulation P f &#39;s do not increase as E b /I o  increases while P f  does for a nonadaptive threshold scheme. This means the present invention can approximately achieve CFAR and is robust against the received signal power variations. 
     FIG. 4 shows the theoretical and simulation detection probabilities P d  versus bit-energy-to-interference power spectral density ratio E b /I o  in dB for M=1, 3, and 5 under fading environment, with a PN code acquisition scheme configured in accordance with one embodiment of the present invention. At P d =0.8, M=3 and 5 case simulation results show 3.5 dB and 4.9 dB improvement in SINR, respectively, compared with M=1 case. 
     FIG. 5 shows the theoretical and simulation average acquisition time T acq  versus bit-energy-to-interference power spectral density ratio E b /I o  in dB for M=1, 3, and 5 under fading environment with a PN code acquisition scheme configured in accordance with one embodiment of the present invention. At T acq =10 4 NT c , M=3 and 5 case can improve 2.7 dB and 4.2 dB in SINR, respectively, compared with M=1 case. Therefore, when a base station employs the present invention, a random access user may transmit signal with smaller power than the single antenna case. Interference can be reduced and capacity of the system can be increased. Or at E b /I 0 =−10 dB, the PN code acquisition time with the proposed PN code acquisition scheme of M=5 elements can be significantly shortened, e.g., 210%, compared to the existing PN code acquisition scheme of a single element in “Adachi et al.” We observe that simulation results are different from the theoretical results when M≧2. This is because the smart antenna weight vector does not perfectly match with the array response vector of the desired user signal&#39;s DOA while perfect weight vector was assumed for the theoretical results. 
     In conclusion, a PN code acquisition scheme using a smart antenna and an adaptive threshold setting of the present invention is robust against the received signal power variations and could improve performance 210% in PN code acquisition time or at least 4.5 dB in SINR to achieve the same detection probability when M=5 elements are used. 
     Those of skill in the art will readily appreciate that various alternative designs and embodiments for practicing the invention are possible. For example, while a single dwell has been described above, the present invention is equally applicable to other PN searches, e.g., double dwell, serial and complete searches. Additionally, those of skill in the art will recognize that the various devices described herein (e.g., accumulators) can be implemented in hardware and/or software. It is therefore to be understood that within the scope of the appended claims, the invention may be practiced otherwise than as specifically described herein.