Abstract:
A pseudonoise code teaching loop (PNCTL) measures the difference in energy between a smart antenna array output that is despread using an early pseudonoise code and a late pseudonoise code and contains a chip time shifting current based on this difference. Using multiple elements of the array significantly improves performance.

Description:
BACKGROUND OF THE INVENTION  
     Filed of the Invention  
       [0001]     The present invention relates to spread spectrum communications generally, and more particularly to a method and apparatus for maintaining pseudo-noise code synchronization.  
         [0002]     Smart antennas, consisting of multiple antenna elements and an adaptive algorithm, have received much attention recently because a smart antenna can suppress interfering signals if the direction of arrival angle (DOA) from the desired signal is different from that of interference, (Ayman F. Naguib, “Adaptive Antennas for CDMA Wireless Networks,” Ph.D. Dissertation Stanford University, August 1996 and Y. S. Song, H. M. Kwon, and B. J. Min, “Computationally Efficient Smart Antennas for CDMA Wireless Comm.”, IEEE Trans. on Vehicular Technology, Vol. 50, No. 6, November 2001.) Some encouraging field experiment results for smart antennas have been reported (M. Sawahashi, et al., “Enhanced Wireless Access Technologies and Experiments for W-CDMA Communications”, IEEE Personal Comm., pp. 6-16, December 2000). In a code division multiple access (CDMA) system, pseudonoise (PN) code synchronization is a difficult and critical process before data demodulation. The locally generated PN code should be synchronized with the received PN code for correct data demodulation. In general, PN code synchronization consists of two steps: PN code acquisition and PN code tracking. PN code acquisition is a coarse PN code synchronization to within a chip interval, and PN code tracking is a fine PN code synchronization (Andrew J. Viterbi, “CDMA Principle of Spread Spectrum Communication”, Addison Wesley, N.Y., 1995 and H. Meyr and G. Ascheid, “Synch. in Digital Communications”, John Wiley and Sons, 1990.) Most smart antenna studies have focused mainly on improving the bit error rate (BER) and capacity of a CDMA wireless communications system. Also, most of the time, perfect PN code synchronization has been assumed, or only one of the multiple antenna elements has been employed for PN code synchronization (Ayman F. Naguib, “Adaptive Antennas for CDMA Wireless Networks”, Ph.D. Dissertation Stanford University, August 1996; Y. S. Song, H. M. Kwon, and B. J. Min, “Computationally Efficient Smart Antennas for CDMA Wireless Comm.”, IEEE Trans. on Vehicular Technology, Vol. 50, No. 6, November 2001; Yoo S. Song, Hyuck M Kwon, and Byung J. Min, “Smart Antennas for 3G and Future Generation CDMA”, IEEE International Conference on Phased Array Systems and Technology, Dana Point, Calif., pp. 69-72, May 21-25, 2000) and M. Sawahashi, et al., “Enhanced Wireless Access Technologies and Experiments for W-CDMA Communications”, IEEE Personal Comm., pp. 6-16, December 2000.) This may be due to existing conventional antenna diversity schemes where antenna element spacing is large, e.g., ten times a wavelength, and arrived signals at different antenna elements are independently faded. This independent fading may make it difficult to coordinate the elements for the purpose of PN code synchronization. However, antenna element spacing in a smart antenna design for a CDMA system is usually assumed to be a half wavelength or less, and received signals at different antenna elements are the same except for phase differences due to the small spatial separation when the element response is identical. Calibration may be required, depending on the smart antenna algorithms, if the element response is not identical. If a normalized least mean square (NLMS) algorithm is used, then phase calibration is not necessary even though radio frequency (RF) mixers for each element have different phase responses (T. Chulajata, H. M. Kwon, and K. Y. Min, “Adaptive Antenna Array with No Phase Calibration for CDMA Reverse Link”, IEEE VTC&#39;2000 Fall, Boston, Mass., Sep. 24-28, 2000). Also, mobiles may transmit signals with low power because of expected high smart antenna gains at a base station, and if only one element is employed for PN code synchronization, then the received signal-to-interference-plus-noise power ratio (SINR) at a base station may not be sufficient. As a result, a call drop may occur due to weak PN code tracking at a base station. It is more desirable for mobile users not to experience frequent call drops than to have high voice quality and fast PN code acquisition. These are among the motivations for a PN code chip time tracking loop (PNCTL) which can maintain its lock status for a longer time by utilizing all antenna elements instead of one element.  
         [0003]     PN code acquisition using an antenna array has been addressed in other studies R. K. Madyastha and B. Aazhang, “Synch. and Detection of Spread Spectrum Signals in Multipath Channels Using Antenna Arrays”, IEEE MILCOM, pp. 1170-1174, 1995 and S. Kim and S. L. Miller, “Subspace-based Synch. Using Antenna Array for DS-CDMA Signals in Multipath Time Varying Fading Channels”, IEEE VTC Spring, pp. 547-551, Tokyo, May 2000) based on signal subspace techniques. However, many of these are difficult to implement because they require enormous computational loads, and also they assume short PN codes. The current and near future CDMA mobile applications e.g., the second generation (2G) and third (3G) CDMA systems (Andrew J. Viterbi, “CDMA Principle of Spread Spectrum Communication”, Addison Wesley, N.Y., 1995 and Third Generation Partnership Project; “Spreading and Modulation (FDD),” 3GPP Technical Specification, TS25.213, v3.2.0, March 2000) have employed long PN codes. Also, a practical PN code acquisition scheme employing all smart antenna elements has been studied in Bing Wang and Hyuck M. Kwon, “PN Code Acquisition with Adaptive Antenna Array and Adaptive Threshold for DS-CDMA Wireless Communications”, IEEE GLOBECOM, San Francisco, Calif., Nov. 27-Dec. 1, 2000. However, to the inventors&#39; knowledge, practical PNCTLs using all smart antenna elements are not available yet. The invention employs all smart antenna elements for PN code tracking. Verification that tracking performance can be improved significantly when all smart antenna element outputs are added and fed into a typical noncoherent digital PNCTL is set forth herein.  
         [0004]     Root-mean-square-chip-time-jitter (RMS) and mean-time-to-lose-lock (MTLL) are typical performance criteria for a PNCTL. Once the PN code is locked, longer MTLL is better for mobile users. Longer MTLL can be achieved as RMS gets smaller. Much analysis and design have appeared for analog and digital phase lock loop (DPLL) and PNCTLs, (e.g., Andrew J. Viterbi, “CDMA Principle of Spread Spectrum Communication”, Addison Wesley, N.Y., 1995; H. Meyr and G. Ascheid, “Synch. in Digital Communications”, John Wiley and Sons, 1990; R. De Gaudenzi, M. Luise, and R. Viola, “A Digital Chip Timing Recovery Loop for Band Limited Direct-Sequence Spread-Spectrum Signals”, IEEE Trans. on Comm., Vol. 41, no. 11, pp. 1760-1769. November 1993; R. De Gaudenzi and M. Luise, “Decision Directed Coherent Delay Lock Tracking Loop for DS-Spread Spectrum Signals”, IEEE Trans. on Comm., Vol. 39, No. 5, pp. 758-765, May 1991; R. De Gaudenzi, “Direct Sequence Spread Spectrum Chip Tracking in the Presence of Unresolvable Multipath Components”, IEEE Trans. on Vehicular Technology, Vol. 48, No. 5, pp. 1573-1583, September 1999; and A. Weinberg and B. Liu, “Discrete Time Analysis of Nonuniform Sampling First and Second Order Digital Phase Lock Loops”, IEEE Trans. on Comm., Vol. Com-22, No. 2, pp. 123-137, February 1974), under additive white Gaussian noise (AWGN) and fading environments. The analyses in these studies suggest that decreasing loop bandwidth or increasing desired signal transmitting power is the way to improve MTLL. However, the present invention, without changing loop bandwidth and desired signal power, improves MTLL by using all smart antenna elements since a smart antenna can improve the SINR output.  
         [0005]     A typical noncoherent early-late gate PNCTL of the first and second order are considered herein (Andrew J. Viterbi, “CDMA Principle of Spread Spectrum Communication”, Addison Wesley, N.Y., 1995.) Then, the analysis in R. De Gaudenzi, M. Luise, and R. Viola, “A Digital Chip Timing Recovery Loop for Band Limited Direct-Sequence Spread-Spectrum Signals”, IEEE Trans. on Comm., Vol. 41, No. 11, pp. 1760-1769, November 1993 and R. De Gaudenzi and M. Luise, “Decision Directed Coherent Delay Lock Tracking Loop for DS-Spread Spectrum Signals”, IEEE Trans. on Comm, Vol. 39, No. 5, pp. 758-765, May 1991, to the second order early-late gate PNCTL, while only the first order was studied in the first of these 2 references. Also, the analysis is extended (A. Weinberg and B. Liu, “Discrete Time Analysis of Nonuniform Sampling First and Second Order Digital Phase Lock Loops”, IEEE Trans. on Comm., Vol. Com-22, No. 2, pp. 123-137, February 1974) to a fading environment and PNCTL because a typical mobile channel is under fading while Weinberg and Liu (in A. Weinberg and B. Liu, “Discrete Time Analysis of Nonuniform Sampling First and Second Order Digital Phase Lock Loops”, IEEE Trans. on Comm., Vol. Com-22, No. 2, pp. 123-137, February 1974) presented the steady state phase error probability density function and variance of the first and second order DPLL only under an AWGN environment.  
         [0006]     The proposed PNCTL of smart antennas in a 3G W-CDMA uplink dedicated physical control channel (DPCCH) has been simulated. To do that, the on-time array output was used as input to NLMS adaptive algorithm whose reference value is set to the number of elements, M. The NLMS algorithm requires only (3M+1) complex multiplication and 2M additions and does not require any phase calibration. Array outputs with early and late PN codes are used as inputs to the noncoherent PNCTL. In addition, loop bandwidth and RMS are expressed in terms of poles of the overall PNCTL transfer function for systematic design instead of using a trial and error method. Optimum pole locations to minimize RMS for the first and second order PNCTLs are found. Finally, through both analysis and simulation, significant improvements in RMS and MTLL are demonstrated. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0007]     A more complete appreciation of the invention and many of the attendant advantages and features thereof will be readily understood by reference to the following detailed description when considered in connection with the accompanying drawings wherein:  
         [0008]      FIG. 1  is a block diagram of a noncoherent PN code chip time tracking loop (PNCTL) with a smart antenna according to an embodiment of the present invention.  
         [0009]      FIG. 2  is an equivalent PN code chip time tracking loop (PNCTL) using a linear model.  
         [0010]      FIG. 3  is a plot of the magnitude |{tilde over (f)}(p 1 ,p 2 )| in e.g. (17) versus pole locations.  
         [0011]      FIG. 4  is a plot of a normalized loop bandwidth B L T c  for the first and second order PN code tracking loops (PNCTLs) using equations (25) and (15), when chip rate is 3.84 Mcps and window size N t  is 256 chips for energy accumulation, as a function of pole location.  
         [0012]      FIG. 5  is a plot of a root mean square (RMS) chip time jitter of the first order PN code tracking loop under fading and AWGN environments with the number of antenna elements M as a parameter.  
         [0013]      FIG. 6  is a plot a root mean square (RMS) chip time jitter of the second order PN code tracking loop under fading and AWGN environments with the number of antenna elements M as a parameter.  
         [0014]      FIG. 7  is a plot of the mean time to lose lock (MTLL) of a first order PN code tracking loop under fading and AWGN environments with the number of antenna elements M as a parameter.  
         [0015]      FIG. 8  is a plot of the mean time to lose lock (MTLL) of the second order PN code tracking loop under fading and AWGN environments with the number of antenna elements M as a parameter. 
     
    
     DETAILED DESCRIPTION  
     1. Introduction  
       [0016]     The present invention will be discussed with reference to preferred embodiments of methods and devices for psuedonoise code tracking loops. Specific details, such as numbers of antenna elements and the order of loop filters, are set forth in order to provide a through understanding of the present invention. The preferred embodiments discussed herein should not be understood to limit the invention. Furthermore, for ease of understanding, certain method steps are delineated as separate steps; however, these steps should not be construed as necessarily distinct nor order dependent in their performance.  
         [0017]     Section 2 describes a noncoherent PNCTL. Section 3 presents analysis: (1) the overall transfer function of PNCTL, and (2) RMS and MTLL of the first and second orders under AWGN and fading environments. Section 4 shows both analysis and simulation results for RMS and MTLL. Section 5 makes conclusions.  
       2. System Modeling  
       [0018]      FIG. 1  shows a block diagram of the proposed noncoherent PNCTL with smart antennas. A single dwell serial PN code search is considered for demonstration. Other search can be employed. When the received and local PN codes are synchronized within a fraction of a chip time, i.e., when PN code acquisition is achieved, the PNCTL in  FIG. 1  is triggered. The DPCCH in the W-CDMA, or called 3GPP system (Third Generation Partnership Project, “Spreading and Modulation (FDD)”, 3GPP Technical Specification, TS25.213, v3.2.0, March 2000), is used for the PN code chip time tracking demonstration. The received signal is down-converted and passed through matched filters. In the 3G W-CDMA system, a square root raised cosine (RRC) filter with frequency response H(f) is employed at both the transmitter and receiver with roll-off factor β=0.22. The impulse time response of the filter response H(f) can be written as  
               RRC   ⁡     (   t   )       =         sin   ⁡     (     π   ⁢     t     T   c       ⁢     (     1   -   β     )       )       +     4   ⁢   β   ⁢     t     T   c       ⁢     cos   ⁡     (     π   ⁢     t     T   c       ⁢     (     1   -   β     )       )             π   ⁢     t     T   c       ⁢     (     1   -       (     4   ⁢   β   ⁢     t     T   c         )     2       )                 (   1   )               
 where T c  is a chip time interval. The received signal is multiplied by three local PN codes. One code is advanced by Δ=T c /2 seconds for the early gate, another is delayed by Δ=T c /2 seconds for the late gate, and the third is on-time and used for signal demodulation. The outputs multiplied by the local PN codes are accumulated over N t  chips to achieve processing gain, and denoted by an M×1 vector y(k) for M element antenna where k denotes the update index, i.e., kN t T c . An inner product y H (k)w(k) is taken at every k to obtain an array output where superscripts H and T in the paper denote the Hermitian and transpose, respectively. The same weight vector w(k) is multiplied to the three de-spread outputs to get array outputs. The on-time array output is fed into an adaptive algorithm to update the weight vector. The NLMS adaptive algorithm is used for weight vector updating (Simon Haykin, “Adaptive Filter Theory”, pp. 437, Prentice Hall 1996). The reference value is set to the number of antenna elements M because ideal weight vector w(k) should be matched with array response vector a(θ(k))=(1, e −jπsin(θ(k)) , . . . , e −j(M−1)π sin(θ(k)) ) T  where θ(k) is the DOA from the desired signal, and ideal array output w H (k)a(θ(k)) would be equal to M. Array outputs for early and late gates are fed into the square devices in the PNCTL,  FIG. 1 , to measure energy. Measured energy is denoted by Z −  and Z +  for the advanced and delayed branches, respectively. Energy difference Z Δ  is passed through loop filter F(z) and multiplied with loop filter gain γ. The voltage controlled oscillator (VCO) takes the loop filter output and controls the chip time shifting amount for the local PN code. The transfer function of VCO is modeled by z −1 /(1−z −1 ) (Andrew J. Viterbi, “CDMA Principle of Spread Spectrum Communication”, Addison Wesley, N.Y., 1995), where z denotes a sample delay. Four samples are taken per chip. The transfer function of loop filter is denoted by F(z) and  
         F   ⁡     (   z   )       =     1   +       az     -   1         1   -     z     -   1                   
 for a second order PNCTL (Andrew J. Viterbi, “CDMA Principle of Spread Spectrum Communication”, Addison Wesley, N.Y., 1995, pp. 66). The first order PNCTL is a special case of the second order with a=0, where a is for the Doppler rate compensation. The loop parameters a and γ are determined for given loop bandwidths or pole locations. The Jakes Rayleigh fading model is employed for multipath simulation. 
 
       3. Analysis  
       [0019]     Inverse Fourier transform of |H(f)| 2  can be written as  
               g   ⁡     (   τ   )       =         cos   ⁡     (     πβτ   /     T   c       )         1   -       (     2   ⁢     βτ   /     T   c         )     2         ⁢       sin   ⁡     (     πτ   /     T   c       )         πτ   /     T   c                   (   2   )             
 
 where H(f) is the RRC frequency response of time response in (1). The output of the receiver matched filter in  FIG. 1  is proportional to g(τ) because the transmitter also employs a square root raised cosine filter H(f). If samples of g(τ) are taken at integer times T c , then there would be no degradation in desired signal power when the PN code is synchronized. Let ε≡τ/T c  denote the normalized chip timing error. Then, g(τ) at τ=εT c  is  
               g   ⁡     (     ɛ   ⁢           ⁢     T   c       )       =         cos   ⁡     (   πβɛ   )         1   -       (     2   ⁢   βɛ     )     2         ⁢         sin   ⁡     (   πɛ   )       πɛ     .               (   3   )             
 
 The loop error characteristic of early and late gate PNCTL (Andrew J. Viterbi, “CDMA Principle of Spread Spectrum Communication”, Addison Wesley, N.Y., 1995) can be written as  
               η   ⁡     (   ɛ   )       =         g   2     ⁡     [       (     ɛ   -     1   2       )     ⁢     T   c       ]       -       g   2     ⁡     [       (     ɛ   +     1   2       )     ⁢     T   c       ]                 (   4   )             
 
 by using (3) where early and late gates employ a half chip delayed and advanced version of received samples. The PNCTL characteristic can be assumed to be linear when chip time error ε is small. The slope at ε=0 is obtained by taking a derivative of (4) as  
                     κ   ≡       ⁢       ⅆ           ⁢     η   ⁡     (   ɛ   )           ⅆ           ⁢   ɛ         ⁢     ❘     ɛ   =   0                   =       ⁢     16   ⁢           ⁢     cos   ⁡     (     βπ   2     )       ⁢           βπ   ⁡     (     1   -     β   2       )       ⁢     sin   ⁡     (     πβ   /   2     )         +     2   ⁢     (     1   -     3   ⁢     β   2         )     ⁢     cos   ⁡     (     πβ   /   2     )                 π   2     ⁡     (     1   -     β   2       )       3                     ≅       ⁢   3.2416                 (   5   )             
 
 where β=0.22 is used. Let Z Δ  denote the energy difference between the early and late branches in  FIG. 1 . Then, average energy difference can be written as 
 
 E[Z   Δ   ]=N   t   2   E   c η(ε)≅ N   t   2   E   c κ  (6) 
 
 by using the loop error characteristic in (4) (Andrew J. Viterbi, “CDMA Principle of Spread Spectrum Communication”, Addison Wesley, N.Y., 1995). 
 
         [0020]      FIG. 2  shows an equivalent PNCTL model using (6) where V 0  is an upper bound of total interference variance in the tracking loop when a smart antenna is employed and can be written as 
   V   0 =2 N   t   2 ( I   0,in   /M ) 2 (1+2N t   E   c /( I   0,in   /M ))  (7)  
 where I 0,in /2 is variance of input interference fed into each antenna element (Y. S. Song, H. M. Kwon, and B. J. Min, “Computationally Efficient Smart Antennas for CDMA Wireless Comm.”, IEEE Trans. On Vehicular Technology, Vol. 50, No. 6, November 2001; Andrew J. Viterbi, “CDMA Principle of Spread Spectrum Communication”, Addison Wesley, N.Y., 1995; Yoo S. Song, Hyuck M. Kwon, and Byung J. Min, “Smart Antennas for 3G and Future Generation CDMA”, IEEE International Conference on Phased Array Systems and Technology, Dana Point, Calif., pp. 69-72, May 21-25, 2000). The overall loop transfer function can be found from the equivalent PNCTL as  
               H   ⁡     (   z   )       =           N   t   2     ⁢     E   c     ⁢   κγ   ⁢           ⁢     F   ⁡     (   z   )       ⁢       z     -   1         1   -     z     -   1               1   +       N   t   2     ⁢     E   c     ⁢   κγ   ⁢           ⁢     F   ⁡     (   z   )       ⁢       z     -   1         1   -     z     -   1                 =         N   t   2     ⁢     E   c     ⁢   κγ   ⁢           ⁢     F   ⁡     (   z   )       ⁢     z     -   1           1   -     z     -   1       +       N   t   2     ⁢     E   c     ⁢   κγ   ⁢           ⁢     F   ⁡     (   z   )       ⁢     z     -   1                       (   8   )               
 which is equal to that of the single antenna element case (Andrew J. Viterbi, “CDMA Principle of Spread Spectrum Communication”, Addison Wesley, N.Y., 1995). The major difference between a smart antenna and a single antenna element is interference suppressed by a factor M as (7). 
 
 3.1. Root Mean Square Chip Time Jitter (RMS) Analysis 
 
         [0021]     By using  
         F   ⁡     (   z   )       =     1   +       az     -   1         1   -     z     -   1                 
 
 for the second order PNCTL, (8) can be rewritten as  
               H   ⁡     (   z   )       =           N   t   2     ⁢     E   c     ⁢   κγ   ⁢           ⁢     (       z     -   1       -       (     1   -   a     )     ⁢     z     -   2           )         1   +       (       -   2     +       N   t   2     ⁢     E   c     ⁢   κγ       ⁢           )     ⁢     z     -   1         +       (     1   -       N   t   2     ⁢     E   c     ⁢   κγ   ⁢           ⁢     (     1   -   a     )         )     ⁢     z     -   2             .             (   9   )             
 
 Let p 1  and p 2  denote two poles of H(z) in (9). Then, the two poles satisfy the following: 
 
 p   1   +p   2 =2 −N   t   2   E   c κγ  (10-a) 
 
and 
 
 p   1   p   2 =1− N   t   2   E   c κγ(1− a ).  (10-b) 
 
 Therefore, loop filter gain γ and Doppler rate compensation coefficient a can be expressed in terms of poles as  
               γ   =       2   -     p   1     -     p   2           N   t   2     ⁢     E   c     ⁢   κ         ⁢     
     ⁢   and           (     11   ⁢     -     ⁢   a     )               a   =         1   -     p   1     -     p   1     +       p   1     ⁢     p   2           2   -     p   1     -     p   2         .             (     11   ⁢     -     ⁢   b     )             
 
 Variance of chip time error for the second order PNCTL is derived in Appendix in detail and can be written as  
                     Var   (     τ     T   c       )     =         V   0         (       N   t   2     ⁢     E   c     ⁢   κ     )     2       ⁢     ∮       H   ⁡     (   z   )       ⁢     H   ⁡     (     z     -   1       )       ⁢       ⅆ   z       2   ⁢   πⅈz                         =           V   0     ⁢     γ   2             (       N   t   2     ⁢     E   c     ⁢   κγ     )     2     ⁢     (     a   -   1     )         ⁢         2   ⁢   a     +     2   ⁢     N   t   2     ⁢     E   c     ⁢   κγ     -     3   ⁢     aN   t   2     ⁢     E   c     ⁢   κγ     +       a   2     ⁢     N   t   2     ⁢     E   c     ⁢   κγ         (       2   ⁢     N   t   2     ⁢     E   c     ⁢   κγ     -       aN   t   2     ⁢     E   c     ⁢   κγ     -   4     )                       (   12   )             
 
 while variance of the first order PNCTL is shown in (Andrew J. Viterbi, “CDMA Principle of Spread Spectrum Communication”, pp. 66, Addison Wesley, N.Y., 1995) which is a special case of (12) with a=0. One-side loop bandwidth in Hertz is given as  
               B   L     =       ∫   0     π   /     T   c         ⁢       [       H   ⁡     (     ⅇ     jω   ⁢           ⁢     T   c         )       ⁢     H   ⁡     (     ⅇ       -   jω     ⁢           ⁢     T   c         )         ]     ⁢                 ⁢     ⅆ   ω         2   ⁢   π       .                 (   13   )             
 
 Thus, (12) can be expressed in terms of loop bandwidth by using (13) as  
               Var   (     τ     T   c       )     =         2   ⁢     V   0     ⁢     B   L     ⁢     T   c           (       N   t   2     ⁢     E   c     ⁢   κ     )     2       .             (   14   )             
 
 Normalized loop bandwidth of the second order PNCTL can be obtained from (12) and (14) and expressed in terms of poles with (10) as  
                       B   L     ⁢     T   c       =         2   ⁢   a     +     2   ⁢     N   t   2     ⁢     E   c     ⁢   κγ     -     3   ⁢     aN   t   2     ⁢     E   c     ⁢   κγ     +       a   2     ⁢     N   t   2     ⁢     E   c     ⁢   κγ         2   ⁢     (     a   -   1     )     ⁢     (       2   ⁢     N   t   2     ⁢     E   c     ⁢   κγ     -       aN   t   2     ⁢     E   c     ⁢   κγ     -   4     )                     =           2   ⁢     (     2   -     p   1     -     p   2       )       +     a   ⁡     (         p   1     ⁢     p   2       +     2   ⁢     p   1       +     2   ⁢     p   2       -   3     )             -   2     ⁢     (     a   -   1     )     ⁢     (         p   1     ⁢     p   2       +     p   1     +     p   2     +   1     )         .                   (   15   )             
 
 RMS of the second order PNCTL can be obtained by taking the square root of (12) or (14). However, it is more useful to represent RMS in terms of loop bandwidth as  
               σ   ɛ     =           2   ⁢     V   0     ⁢     B   L     ⁢     T   c             N   t   2     ⁢     E   c     ⁢   κ       .             (   16   )             
 
 Equation (16) shows that RMS is proportional to loop bandwidth B L T c . 
 
         [0022]     To find the optimum poles that minimize variance of chip time error, we take a derivative of (12) with respect to p 1  and p 2 , after substituting (11) into (12) as  
                             ∂   2         ∂     p   1       ⁢     ∂     p   2           ⁢     Var   (     τ     T   c       )       ⁢     ❘       γ   =       2   -     p   1     -     p   2           N   t   2     ⁢     E   c     ⁢   κ         ,     a   =       1   -     p   1     -     p   2     +       p   1     ⁢     p   2           2   -     p   1     -     p   2                 =         V   0         (       N   t   2     ⁢     E   c     ⁢   κ     )     2       ⁢       f   ~     ⁡     (       p   1     ,     p   2       )                     =   0           ⁢     
     ⁢   where           (   17   )                   f   ~     ⁡     (       p   1     ,     p   2       )       =       Num     f   ~         Den     f   ~                 (   18   )             
 
 Num   {tilde over (f)}   =p   1   3   p   2   3 +2 p   1   2   p   2   3   +p   1   p   2   3 +2 p   1   p   2   3 +2 p   1   3   p   2   2 −11 p   1   2   p   2   2 −4 p   1   p   2   2   +p   2   2   +p   2   2   +p   1   3   p   2 −4 p   1   2   p   2 +13 p   1   p   2   +p   1   2 +2 p   1 −7  (19) 
 
 Den   {tilde over (f)} =0.5(1 +p   1 )(1+ p   2 )( p   1   p   2 −1) 3 (1+ p   1   +p   2   +p   1   p   2 ).  (20) 
 
  FIG. 3  shows magnitude |{tilde over (f)}(p 1 ,p 2 )| by varying two pole locations, p 1  and p 2 . If two poles p 1  and p 2  are located in the right hand side of unit circle z-domain, it is observed that |{tilde over (f)}(p 1 ,p 2 )| approaches zero as poles are close to 1 on the real axis. This implies that optimum pole locations are equal to 1. Therefore, variance of chip time error will be minimum as poles approach 1 on the real axis. However, pole locations are not allowed to exceed, nor to equal, 1 because of tracking loop stability. 
 
         [0023]     Under fading environment, (7) can be modified into conditional variance of interference V 0|α  for given fading amplitude a as 
 
 V   0|α 2 N   t   2 ( I   0,in   /M ) 2 +4α 2   N   t   3 ( I   0,in   /M ) E   c .  (21) 
 
 Then, the average variance of chip timing error under fading environment can be obtained with a probability density function (pdf) of fading amplitude, p R (α), as  
                 Var   (     τ     T   c       )     =       ∫   0   ∞     ⁢       Var   (       τ     T   c       ❘   α     )     ⁢       p   R     ⁡     (   α   )       ⁢           ⁢     ⅆ   α           ⁢     
     ⁢   where           (   22   )                       Var   (       τ     T   c       ❘   α     )     =       ⁢         V     0   |   α             (       N   t   2     ⁢     α   2     ⁢     E   c       )     2     ⁢     κ   2         ⁢     ∮       H   ⁡     (     z   |   α     )       ⁢     H   ⁡     (       z     -   1       |   α     )       ⁢       ⅆ   z       2   ⁢   π   ⁢           ⁢   ⅈz                         =       ⁢       V     0   |   α             (       N   t   2     ⁢     α   2     ⁢     E   c     ⁢   κ     )     2     ⁢     (     a   -   1     )                         ⁢         2   ⁢   a     +     2   ⁢     N   t   2     ⁢     α   2     ⁢     E   c     ⁢   κγ     -     3   ⁢     aN   t   2     ⁢     α   2     ⁢     E   c     ⁢   κγ     +       a   2     ⁢     N   t   2     ⁢     α   2     ⁢     E   c     ⁢   κγ           2   ⁢     N   t   2     ⁢     α   2     ⁢     E   c     ⁢   κγ     -       aN   t   2     ⁢     α   2     ⁢     E   c     ⁢   κγ     -   4                     (   23   )             
 
 For a Rayleigh fading model,  
                 p   R     ⁡     (   α   )       =       α     σ   α   2       ⁢     ⅇ       -     α   2         2   ⁢     σ   α   2                     (   24   )             
 
 where σ α   2 =1. RMS under fading channel can be obtained by taking a square root of (22). 
 
         [0024]     Similarly, analysis for the first order PNCTL can be done as a special case of the second order with a=0 in (9)-(23). The first order PNCTL transfer function has a single pole p at 1−N t   2 E c κγ, and loop filter gain γ can be expressed in terms of the pole as γ=(1−p)/N t   2 E c κ. And normalized loop bandwidth of the first order PNCTL can be written as  
                 B   L     ⁢     T   c       =           N   t   2     ⁢     E   c     ⁢   κγ       2   ⁢     (     2   -       N   t   2     ⁢     E   c     ⁢   κγ       )         =         1   -   p       2   ⁢     (     1   +   p     )         .               (   25   )             
 
 RMS of the first order PNCTL can be obtained by replacing B L T c  in (14) with (25) and taking a square root of (14). The optimum pole location that minimizes variance (or RMS) can be found by replacing γ with (1−p)/N t   2 E c κ and taking a derivative of variance with respect to p as  
                 ⅆ     ⅆ   p       ⁢     (           V   0     ⁢   γ         N   t   2     ⁢     E   c     ⁢     κ   ⁡     (     2   -       N   t   2     ⁢     E   c     ⁢   κγ       )           ⁢     |     γ   =       1   -   p         N   t   2     ⁢     E   c     ⁢   κ             )       =           V   0     ⁡     (     -   2     )             (       N   t   2     ⁢     E   c     ⁢   κ     )     2     ⁢       (     1   +   p     )     2         &lt;   0.             (   26   )             
 
 Note from (26) that the derivative is negative, which implies that variance decreases asp increases. Therefore, RMS will be minimum when p is 1. RMS of the first order PNCTL under fading environment can be obtained similar to (22). 
 
 3.2. Mean Time to Lose Lock (MTLL) Analysis 
 
         [0025]     Loop noise Ar(k) of variance V 0  in  FIG. 2  is the difference between Z Δ (k) and its average E[Z Δ (k)] and can be written as 
 
 N ( k )= Z   Δ ( k )− E[Z   Δ ( k )]= Z   Δ ( k )− N   t   2   E   c η{ε( k )}  (27) 
 
 from (6) where k denotes the PNCTL update index. Since loop filter F(z) can be represented as 1+az −1 /(1-z −1 ) for the second order PNCTL and ε(k) is the VCO output in  FIG. 1 , the z-transform of ε(k) can be written as  
               ɛ   ⁡     (   z   )       =       γ   ⁡     (     -       z     -   1         1   -     z     -   1             )       ⁢     (     1   +       a   ⁢           ⁢     z     -   1           1   -     z     -   1             )     ⁢       Z   Δ     ⁡     (   z   )                 (   28   )             
 
 where the negative sign is used in (28) because the VCO output is subtracted from the input timing in  FIG. 2 . Z Δ (z) in (28) denotes the forward z-transform for Z Δ (k). By taking the inverse z-transform, (28) can be rewritten as  
                       ɛ   ⁡     (   k   )       -     2   ⁢     ɛ   ⁡     (     k   -   1     )         +     ɛ   ⁡     (     k   -   2     )         =       ⁢     -     γ   ⁡     [         Z   Δ     ⁡     (     k   -   1     )       +       (     a   -   1     )     ⁢       Z   Δ     ⁡     (     k   -   2     )           ]                     =       ⁢         -   γ     ⁢           ⁢     N   ι   2     ⁢     E   c     ⁢   η   ⁢     {     ɛ   ⁡     (     k   -   1     )       }       -                     ⁢       γ   ⁢           ⁢   N   ⁢     (     k   -   1     )       -     (     a   -   1     )                       ⁢       [       γ   ⁢           ⁢     N   ι   2     ⁢     E   c     ⁢   η   ⁢     {     ɛ   ⁡     (     k   -   2     )       }       +     γ   ⁢           ⁢   N   ⁢     (     k   -   2     )         ]     .                   (   29   )             
 
 Let ε(k)≡s(k+1)+(a−1)s(k). Then (29) becomes  
                 [       s   ⁢     (     k   +   1     )       -     2   ⁢     s   ⁢     (   k   )         +     s   ⁢     (     k   -   1     )         ]     +       (     a   -   1     )     ⁡     [       s   ⁡     (   k   )       -     2   ⁢     s   ⁡     (     k   -   1     )         +     s   ⁡     (     k   -   2     )         ]         =           ⁢         -   γ     ⁢           ⁢     N   ι   2     ⁢     E   c     ⁢   η   ⁢     {       s   ⁢     (   k   )       +       (     a   -   1     )     ⁢     s   ⁡     (     k   -   1     )           }       -     γ   ⁢           ⁢     N   ⁡     (     k   -   1     )         +         (     a   -   1     )     ⁡     [         -   γ     ⁢           ⁢     N   ι   2     ⁢     E   c     ⁢   η   ⁢     {       s   ⁡     (     k   -   1     )       +       (     a   -   1     )     ⁢     s   ⁡     (     k   -   2     )           }       -     γ   ⁢           ⁢     N   ⁡     (     k   -   2     )           ]       .               (   30   )             
 
 Equation (30j can be broken into two equivalent equations. Let y 1 (k)=s(k−2) and y 2 (k)=s(k−1). Then, (30) can be rewritten as the following state equations: 
 
 y   1 ( k+ 1)= y   2 ( k ) 
 
 y   2 ( k+ 1)=2 y   2 ( k )− y   1 ( k )−γN t   2   E   c η( y   2 ( k )+( a− 1) y   1 ( k ))−γ N ( k− 2)  (31) 
 
 where 
 
ε( k )= y   2 ( k+ 2)+( a− 1) y   1 ( k+ 2).  (32) 
 
 The y 1 (k) and y 2 (k) are discrete time continuous variable Markov processes. The joint pdf of y 1 (k) and y 2 (k) satisfies the Chapman-Kolmogorov equation (A. Weinberg and B. Liu, “Discrete Time Analysis of Nonuniform Sampling First and Second Order Digital Phase Lock Loops,” IEEE Trans. on Comm., Vol. Com-22, No. 2, pp. 123-137, February 1974) as  
                 p     k   +   1       ⁡     (       y   1     ,       y   2     |     y     1   ,   0         ,     y     2   ,   0         )       =       ∫               -   ∞       ∞     ⁢       ∫     -   ∞     ∞     ⁢         q   k     ⁡     (       y   1     ,       y   2     |     x   1       ,     x   2       )       ⁢       p   k     ⁡     (       x   1     ,       x   2     |     y     1   ,   0         ,     y     2   ,   0         )       ⁢     ⅆ     x   1       ⁢     ⅆ     x   2                     (   33   )             
 
 where y 1,0 =y 1 (0) and y 2,0 =y 2 (0) are the initial chip timing errors, p k (·,·|y 1,0 , y 2,0 ) is the joint pdf of y 1 (k) and y 2 (k) for given y 1,0  and y 2,0 , and q k (y 1 , y 2 |x 1 , x 2 ) is the joint transition pdf of y 1 (k+1) and y 2 (k+1) for given y 1 (k)=x 1  and y 2 (k)=x 2 . Noting that noise N(k) is independent of y 1 (k) and y 2 (k), we can observe from (31) that y 2 (k+1) for given (y 1 (k)=x 1 , y 2 (k)=x 2 ) is Gaussian with mean and variance equal to E k (y 1 , y 2 |x 1 , x 2 )=2x 2 −x 1 −γN t   2 E c η(x 2 +(a−1)x 1 ) and γ 2 V 0 , respectively. Thus, the conditional density of y 2 (k+1), given (y 1 (k)=x 1 , y 2 (k)=x 2 ), is independent of k since the channel is assumed to be stationary. Therefore, from (31), one obtains  
                 q   k     ⁡     (       y   1     ,       y   2     |     x   1       ,     x   2       )       =       δ   ⁡     (       y   1     -     x   2       )       ⁢     1       2   ⁢     πγ   2     ⁢     V   0           ⁢       exp   ⁡     [     -         (       y   2     -     2   ⁢     x   2       +     x   1     +     γ   ⁢           ⁢     N   ι   2     ⁢     E   c     ⁢     η   ⁡     (       x   2     +       (     a   -   1     )     ⁢     x   1         )           )     2       2   ⁢     γ   2     ⁢     V   0           ]       .               (   34   )             
 
 Then (33) becomes  
                         p     k   +   1       ⁡     (       y   1     ,     y   2       )       =       ⁢       ∫     -   ∞     ∞     ⁢       1       2   ⁢     πγ   2     ⁢     V   0           ⁢   exp                       ⁢     [     -         (       y   2     -     2   ⁢     y   1       +     x   1     +     γ   ⁢           ⁢     N   ι   2     ⁢     E   c     ⁢     η   ⁡     (       y   1     +       (     a   -   1     )     ⁢     x   1         )           )     2       2   ⁢     γ   2     ⁢     V   0           ]                     ⁢         p   k     ⁡     (       x   1     ,     y   1       )       ⁢     ⅆ     x   1                     =       ⁢       ∫     -   ∞     ∞     ⁢       K   ⁡     (       y   1     ,     y   2     ,     x   1       )       ⁢       p   k     ⁡     (       x   1     ,     y   1       )       ⁢     ⅆ     x   1                   ⁢     
     ⁢   where           (   35   )                 K   ⁢           ⁢     (       y             ⁢   1       ,     y             ⁢   2       ,     x             ⁢   1         )       =       1       2   ⁢     πγ   2     ⁢     V   0           ⁢       exp   ⁡     [     -         (       y   2     -     2   ⁢     y   1       +     x   1     +     γ   ⁢           ⁢     N   ι   2     ⁢     E   c     ⁢     η   ⁡     (       y   1     +       (     a   -   1     )     ⁢     x   1         )           )     2       2   ⁢     γ   2     ⁢     V   0           ]       .               (   36   )             
 
 From (31), and since N(k−1) is Gaussian with zero mean and variance V 0 , y 1 (k+1) and y 2 (k+1) will be also jointly Gaussian for all k with zero means. The jointly Gaussian density function is written as  
                 p     k   +   1       ⁡     (       y   1     ,     y   2       )       =       1     2   ⁢       πσ   2     ⁡     (     k   +   1     )       ⁢       1   -       ρ   2     ⁡     (     k   +   1     )               ⁢   exp   ⁢     {     -         y   1   2     -     2   ⁢     ρ   ⁡     (     k   +   1     )       ⁢     y   1     ⁢     y   2       +     y   2   2           2   ⁡     [     1   -       ρ   2     ⁡     (     k   +   1     )         ]       ⁢       σ   2     ⁡     (     k   +   1     )             }               (   37   )             
 
 where σ 2 (k+1) and ρ(k+1) denote variances of y 1 (k+1) and y 2 (k+1) and their correlation coefficients, respectively. The ρ(k+1) and σ(k+1) are two unknown variables and can be obtained from (35) and (37) by computing two pair points (y 1 , y 2 ) recursively for each k and assuming that the digital loop is initially locked, i.e., y 1,0 =0 and y 2,0 =0, i.e., p 0 (y 1 ,y 2 )=δ(y 1 ,y 2 ), where δ(x) is the Dirac delta function. 
 
 For example,  
                         p     k   +   1       ⁡     (     0   ,   0     )       =       ⁢     1     2   ⁢       πσ   2     ⁡     (     k   +   1     )       ⁢       1   -       ρ   2     ⁡     (     k   +   1     )                           =       ⁢       1     2   ⁢       πσ   2     ⁡     (   k   )       ⁢       2   ⁢     πγ   2     ⁢     V   0         ⁢       1   -       ρ   2     ⁡     (   k   )               ⁢       ∫     -   ∞     ∞     ⁢   exp                       ⁢       [       -         (       x   1     +     γ   ⁢           ⁢     N   ι   2     ⁢     E   c     ⁢   η   ⁢     {       (     a   -   1     )     ⁢     x   1       }         )     2       2   ⁢     γ   2     ⁢     V   0           -       x   1   2         2   ⁡     [     1   -       ρ   2     ⁡     (   k   )         ]       ⁢       σ   2     ⁡     (   k   )             ]     ⁢     ⅆ     x   1                 ⁢     
     ⁢   and           (   38   )                         p     k   +   1       ⁡     (     n   ,   0     )       =       ⁢       1     2   ⁢       πσ   2     ⁡     (     k   +   1     )       ⁢       1   -       ρ   2     ⁡     (     k   +   1     )               ⁢   exp                     ⁢     {     -       n   2         2   ⁡     [     1   -       ρ   2     ⁡     (     k   +   1     )         ]       ⁢       σ   2     ⁡     (     k   +   1     )             }                 =       ⁢       1     2   ⁢       πσ   2     ⁡     (   k   )       ⁢       2   ⁢     πγ   2     ⁢     V   0         ⁢       1   -       ρ   2     ⁡     (   k   )               ⁢       ∫     -   ∞     ∞     ⁢   exp                       ⁢       [             -         (         -   2     ⁢   n     +     x   1     +     γ   ⁢           ⁢     N   ι   2     ⁢     E   c     ⁢     η   ⁢     (     n   +       (     a   -   1     )     ⁢     x   1         )           )     2       2   ⁢     γ   2     ⁢     V   0           -                   x   1   2     -     2   ⁢     ρ   ⁢     (   k   )       ⁢     x   1     ⁢   n     +     n   2         2   ⁢     (     1   -       ρ   2     ⁢     (   k   )         )     ⁢       σ   2     ⁢     (   k   )                 ]     ⁢     ⅆ     x   1                       (   39   )             
 
 where n is any nonzero real number and set to 0.01. We compute the rightmost terms in (38) and (39) numerically and recursively for each k to obtain p k+1 (0,0) and p k+1 (n,0). Thus, we have  
                 σ   2     ⁡     (     k   +   1     )       =     1     2   ⁢   π   ⁢       1   -       ρ   2     ⁡     (     k   +   1     )           ⁢       p             ⁢     k   ⁢           +           ⁢   1         ⁡     (     0   ,   0     )                   (   40   )                   p     k   +   1       ⁡     (     n   ,   0     )       =         p     k   +   1       ⁡     (     0   ,   0     )       ⁢   exp   ⁢     {     -         n   2     ⁢   π   ⁢           ⁢       p     k   +   1       ⁡     (     0   ,   0     )             1   -       ρ   2     ⁡     (     k   +   1     )               }               (   41   )             
 
 Using (40) and (41), we can solve ρ(k+1) as  
                 ρ   ⁡     (     k   +   1     )       =           -   4     ⁢   AC         2   ⁢   A         ,            ρ   ⁡     (     k   +   1     )            &lt;   1.             (   42   )             
 
 where  
         A   =         [     ln   ⁡     (         p   k     ⁡     (     n   ,   0     )           p     k   +   1       ⁡     (     0   ,   0     )         )       ]     2     ⁢           ⁢   and       ⁢               
       C   =           (     n   2     )     2     ⁢     π   2     ⁢       p     k   +   1     2     ⁡     (     0   ,   0     )         -         [     ln   ⁡     (         p   k     ⁡     (     n   ,   0     )           p     k   +   1       ⁡     (     0   ,   0     )         )       ]     2     .           
 
 Also, we can have σ 2 (k+1) by substituting ρ(k+1)in (42) into (40). 
 
         [0026]     The mean and variance of timing error ε(k) can be obtained from (32) as  
                 μ     ɛ   ,   k       =         ∫     -   ∞     ∞     ⁢       ∫     -   ∞     ∞     ⁢       [       y   2     +       (     a   -   1     )     ⁢     y   1         ]     ⁢       p     k   +   2       ⁡     (       y   1     ,     y   2       )       ⁢           ⁢     ⅆ     y   1       ⁢           ⁢     ⅆ     y   2             =   0       ⁢     
     ⁢   and           (   43   )                       σ     ɛ   ,   k     2     =       ⁢       ∫     -   ∞     ∞     ⁢       ∫     -   ∞     ∞     ⁢         [       y   2     +       (     a   -   1     )     ⁢     y   1       -     μ     ɛ   ,   k         ]     2     ⁢       p     k   +   2       ⁡     (       y   1     ,     y   2       )       ⁢           ⁢     ⅆ     y   1       ⁢           ⁢     ⅆ     y   2                         =       ⁢         σ   2     ⁡     (     k   +   2     )       +     2   ⁢     (     a   -   1     )     ⁢       σ   2     ⁡     (     k   +   2     )       ⁢     ρ   ⁡     (     k   +   2     )         +                     ⁢         (     a   ⁢           -           ⁢   1     )     2     ⁢         σ             ⁢   2       ⁡     (     k   +   2     )       .                     (   44   )             
 
 Therefore, the pdf of chip timing error ε(k) can be written as  
                 p   k     ⁡     (   ɛ   )       =       1       2   ⁢           ⁢   π   ⁢           ⁢     σ     ɛ   ,   k     2           ⁢       exp   ⁡     [     -         (     ɛ   -     μ     ɛ   ,   k         )     2       2   ⁢           ⁢     σ     ɛ   ,   k     2           ]       .               (   45   )             
 
 The probability that the timing error is out of the limit±ε L  for the first time at the k-th iteration is denoted by P k  and can be written as 
 
 P   k   =Pr {# of iterations= k  for which |ε k |≧ε L   }=Pr {# of iterations≦k for which|ε k |≧ε L   }−Pr {of iterations&lt; k  for which |ε k |≧ε L }
 
                     P   k     =       ⁢     Pr   ⁢     {       #   ⁢           ⁢   of   ⁢           ⁢   iterations     =       k   ⁢           ⁢   for   ⁢           ⁢   which   ⁢           ⁢          ɛ   k            ≥     ɛ   L         }                   =       ⁢       Pr   ⁢     {       #   ⁢           ⁢   of   ⁢           ⁢   iterations     ≤     k   ⁢           ⁢   for   ⁢           ⁢   which   ⁢           ⁢          ɛ   k            ≥     ɛ   L       }       -                     ⁢     Pr   ⁢     {       #   ⁢           ⁢   of   ⁢           ⁢   iterations     &lt;     k   ⁢           ⁢   for   ⁢           ⁢   which   ⁢           ⁢          ɛ   k            ≥     ɛ   L       }                   =       ⁢       (     1   -       ∫     -     ɛ   L         ɛ   L       ⁢         p   k     ⁡     (   ɛ   )       ⁢           ⁢     ⅆ   ɛ           )     -     (     1   -       ∫     -     ɛ   L         ɛ   L       ⁢         p     k   -   1       ⁢     (   ɛ   )       ⁢           ⁢     ⅆ   ɛ           )                   =       ⁢         ∫     -     ɛ   L         ɛ   L       ⁢         p     k   -   1       ⁢     (   ɛ   )       ⁢           ⁢     ⅆ   ɛ         -       ∫     -     ɛ   L         ɛ   L       ⁢         p   k     ⁡     (   ɛ   )       ⁢           ⁢       ⅆ   ɛ     .                         (   46   )             
 
 Therefore, MTLL will be  
                   MTLL   =       ∑     k   =   1     ∞     ⁢     kP   k                   =       ∑     k   =   1     ∞     ⁢     k   (       ∫     -     ɛ   L         ɛ   L       ⁢       [         p     k   -   1       ⁡     (   ɛ   )       -       p   k     ⁡     (   ɛ   )         ⁢           ]     ⁢     ⅆ   ɛ         )                   =     1   +       ∑     k   =   1     ∞     ⁢       ∫     -     ɛ   L         ɛ   L       ⁢         p   k     ⁡     (   ɛ   )       ⁢           ⁢       ⅆ   ɛ     .                           (   47   )             
 
         [0027]     Under fading environment, first we obtain the conditional probability density of the timing error ε(k) for a given fading amplitude α similarly to (45), and then take an average of the conditional pdf. Thus, the MTLL can be obtained under fading environment. For example, under fading environment, the steady state pdf of ε can be written as  
                 p   ⁡     (   ɛ   )       =       ∫   0   ∞     ⁢       α   (       1       2   ⁢           ⁢   π   ⁢           ⁢     σ     ɛ   ⁢        α       2           ⁢     exp   ⁡     [     -       ɛ   2       2   ⁢           ⁢     σ     ɛ   ⁢        α       2           ]         )     ⁢     ⅇ       -     α   2       2       ⁢           ⁢     ⅆ   α           ⁢     
     ⁢   where           (   48   )                 σ     ɛ   ⁢        α       2     =       ∫     -   ∞     ∞     ⁢       ∫     -   ∞     ∞     ⁢         [       y   2     +       (     a   -   1     )     ⁢     y   1         ]     2     ⁢     p   (       y   1     ,       y   2     ⁢        α   )     ⁢           ⁢     ⅆ     y   1       ⁢           ⁢     ⅆ     y   2                           (   49   )                       p   ⁢     (     ⁢     y   1       ,         y   2     ⁢        α   )       =       ⁢       ∫     -   ∞     ∞     ⁢       1       2   ⁢           ⁢   π   ⁢           ⁢     γ   2     ⁢     V     0   ⁢        α               ⁢   exp                         ⁢       [     -             (       y   2     -     2   ⁢     y   1       +   x   +     γ   ⁢           ⁢     α   2     ⁢     N   t   2                           E   c     ⁢     η   ⁡     (       y   1     +       (     a   -   1     )     ⁢   x       )         )     2             2   ⁢           ⁢     γ   2     ⁢     V     0   ⁢        α               ]     ⁢     p   (     x   ,     y   1       )     ⁢     ⅆ   x                     (   50   )             
 
 and V 0|a  is given in (21). 
 
         [0028]     The MTLL for the first order PNCTL in (R. DeGaudenzi, M. Luise, and R. Viola, “A Digital Chip Timing Recovery Loop for Band Limited Direct-Sequence Spread-Spectrum Signals,” IEEE Trans. on Comm., Vol. 41, No. 11, pp. 1760-1769, November 1993) is a special case of the second order PNCTL described herein, with a=0.  
       4. Numerical and Simulation Results  
       [0029]     For simulation, the DOA of the desired signal was chosen arbitrarily between −90° and 90° with a zero angle spread. Interference was added at each antenna element with a mean of zero and a variance of I 0,in /2. A linear array of M=4 antenna elements was assumed. The constant parameter and convergence parameter in the NLMS were chosen as 0.1 and 1.5, respectively. The number of chip intervals N t  for energy accumulation was set to spreading factor 256 of DPCCH. The antenna weight vector and PNCTL were updated at every control symbol time=0.006 ms. Normalized bandwidth B L T c  was chosen as 0.0556. The corresponding pole for the first order PNCTL was 0.8. The corresponding pole locations for the second order PNCTLs were 0.9 and 0.9364. Also the RRC filter specified in W-CDMA was used, and so κ=3.2.  
         [0030]     Jakes fading model was used with mobile speeds in the range of 10 km/h and 1000 km/h. Although results are not shown in this paper, it was observed that RMS chip time jitter and MTLL are not sensitive to mobile speeds. This may be due to noncoherent energy detection in the tracking loop.  
         [0031]      FIG. 4  shows normalized bandwidth B L T c  versus the second pole location 1  with the first pole location as a parameter using (25) and (15) for the second order PNCTL, respectively. As the pole approaches 1, the bandwidth gets smaller, which means RMS decreases according to (16) and MTLL increases as indicated in (R. DeGaudenzi, M. Luise, and R. Viola, “A Digital Chip Timing Recovery Loop for Band Limited Direct-Sequence Spread-Spectrum Signals,” IEEE Trans. on Comm., Vol. 41, No. 11, pp. 1760-1769, November 1993).    1  For the first order PNCTL, the pole location in the horizontal axis of  FIG. 4  represents its single pole location.    
         [0032]      FIGS. 5 and 6  show RMS under AWGN and fading environments for the first and second order PNCTL, respectively, with the number of smart antenna elements M as a parameter. The solid and dashed lines represent analysis results under AWGN and fading environments, respectively. The triangles and circles show corresponding simulation results for AWGN and fading environments, respectively. Results show significant improvement by employing smart antennas, e.g., 6 dB improvement in E c /I 0  with M=4 elements to achieve the same RMS, compared to M=1. Performance under fading environment is slightly degraded, compared to AWGN, especially at low E c /I 0 . It is also observed that simulation results agree well with analysis.  
         [0033]      FIGS. 7 and 8  show MTLL under AWGN and fading environments for the first and second order PNCTLs, respectively, with M as a parameter. Again, performance is significantly improved, e.g., by 6 dB with a smart antenna of M=4 elements. Fading channel degrades performance by 0.5 dB, compared to AWGN. The second order PNCTL is about 1 dB better than the first order for the same bandwidth.  
       5. Conclusions  
       [0034]     The invention employs a smart antenna to improve a noncoherent PN code tracking loop. RMS and MTLL were analyzed for the first and second order PNCTLs under fading and AWGN environments. Simulation results agree well with the theoretical results. The proposed PN tracking scheme was simulated with a 3G W-CDMA DPCCH channel. Chip time tracking performance can be improved significantly, e.g., by 6 dB with four antenna elements, compared to a tracking scheme with a single antenna element. For a given bandwidth, the chip time tracking loop with the second order PNCTL can achieve a better MTLL than that achieves with the first order. RMS and MTLL performance under fading environment are slightly worse than that under AWGN. In addition, relations between poles of the PNCTL transfer function with normalized bandwidth and RMS, allow the use of a systematic PNCTL design instead of the tedious trial and error method. Optimum pole locations are 1 on the real axis to minimize RMS for both the first and second order PNCTLs. In conclusion, a smart antenna can significantly improve chip time tracking performance as well as bit error rate.  
         [0035]     Obviously, numerous other modifications and variations of the present invention are possible in light of the above teachings. 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.  
       APPENDIX  
       [0036]     Variance of Chip Timing Error in Second Order PN Code Tracking Loop.  
                       Var   ⁢           ⁢     (           ⁢     τ             ⁢     T             ⁢   c           )       =         V   0         (       N   t   2     ⁢     E   c     ⁢   κ     )     2       ⁢     ∮       H   ⁡     (   z   )       ⁢     H   ⁡     (     z     -   1       )       ⁢       ⅆ   z       2   ⁢           ⁢   π   ⁢           ⁢   ⅈ   ⁢           ⁢   z                         =           V   0     ⁢     γ   2         2   ⁢           ⁢   π   ⁢           ⁢   ⅈ       ⁢     ∮       q   ⁡     (   z   )       ⁢     ⅆ   z                   ⁢     
     ⁢   where           (   A1   )                       q   ⁡     (   z   )       =         [     z   +     (       -   1     +   a     )       ]     ⁡     [     1   +       (       -   1     +   a     )     ⁢   z       ]           [       z   2     +   Az   +   B     ]     ⁡     [     1   +   Az   +     Bz   2       ]                     =         [     z   +     (       -   1     +   a     )       ]     ⁡     [     1   +       (       -   1     +   a     )     ⁢   z       ]           (     z   -     z   1       )     ⁢     (     z   -     z   2       )     ⁢     (     1   -       z   1     ⁢   z       )     ⁢     (     1   -       z   2     ⁢   z       )                       (   A2   )               A   =       -   2     +       N   t   2     ⁢     E   c     ⁢   κ   ⁢           ⁢   γ               (   A3   )               B   =     1   -       N   t   2     ⁢     E   c     ⁢   κ   ⁢           ⁢       γ   ⁡     (     1   -   a     )       .                 (   A4   )             
 
 Let z 1  and z 2  denote roots of z 2 +Az+B=0 inside the unit circle. Then 1/z 1  and 1/z 2  are roots of 1+Az+Bz 2 =0, which are outside the unit circle. We know z 1 +z 2 =−A and z 1 z 2 =B. Then,  
                     Var   ⁢           ⁢     (     τ             ⁢     T             ⁢   c           )       =       ⁢                 ⁢       V   0     ⁢     γ   2                   ⁢     2   ⁢           ⁢   π   ⁢           ⁢   ⅈ         ⁢           ⁢     ∮       q   ⁡     (   z   )       ⁢           ⁢     ⅆ   z                               ⁢     =        ⁢                 ⁢       V   0     ⁢           ⁢     γ   2                   ⁢     2   ⁢           ⁢   π   ⁢           ⁢   ⅈ         [           ⁢     2   ⁢           ⁢   π   ⁢           ⁢   ⅈ   ⁢           ⁢     (           ⁢       Residue   ⁡     [       q   ⁢           ⁢     (   z   )       ,     z             ⁢   1         ]       +     ⁢                                     ⁢       Residue   ⁢           [       q   ⁢           ⁢     (   z   )       ,     z   2       ]     +   0   +   0     )     ]               =       ⁢       V   0     ⁢       γ   2     (         (         [     z   +     (       -   1     +   a     )       ]     ⁡     [     1   +       (       -   1     +   a     )     ⁢   z       ]           (     z   -     z   2       )     ⁢     (     1   -       z   1     ⁢   z       )     ⁢     (     1   -       z   2     ⁢   z       )         )         |   z     =     z   1         +                         ⁢           [     z   +     (       -   1     +   a     )       ]     ⁡     [     1   +       (       -   1     +   a     )     ⁢   z       ]           (     z   -     z   1       )     ⁢     (     1   -       z   1     ⁢   z       )     ⁢     (     1   -       z   2     ⁢   z       )         ⁢            z   =     z   2         )                   =       ⁢           V   0     ⁢     γ   2         1   -       z   1     ⁢     z   2           ⁢             1   +     2   ⁢     (       z   1     +     z   2       )     ⁢     (       -   1     +   a     )       +                   (       -   1     +   a     )     2     +       z   1     ⁢     z   2       +       z   1     ⁢         z   2     ⁡     (       -   1     +   a     )       2                   -       (       z   1     +     z   2       )     2       +     2   ⁢     z   1     ⁢     z   2       +       z   1   2     ⁢     z   2   2       +   1                     =       ⁢         V   0     ⁢     γ   2             (       N   t   2     ⁢     E   c     ⁢   κ   ⁢           ⁢   γ     )     2     ⁢     (     a   -   1     )                         ⁢         (       2   ⁢   a     +     2   ⁢     N   t   2     ⁢     E   c     ⁢   κ   ⁢           ⁢   γ     -     3   ⁢     aN   t   2     ⁢     E   c     ⁢   κ   ⁢           ⁢   γ     +       a   2     ⁢     N   t   2     ⁢     E   c     ⁢   κ   ⁢           ⁢   γ       )       (       2   ⁢     N   t   2     ⁢     E   c     ⁢   κ   ⁢           ⁢   γ     -       aN   t   2     ⁢     E   c     ⁢   κ   ⁢           ⁢   γ     -   4     )       .                   (   A5   )