Abstract:
A method for preamble detection in mobile unit to base unit wireless telephony sets a preamble detection threshold based upon a Beaulieu series computation dependent upon preamble correlation data. This preamble detection threshold adjusts for noise by assuming the noise is additive White Gaussian noise (AWGN) with a known variance. The method determines the threshold for achieving a probability of false detection of the preamble from noise input of less than 0.001.

Description:
CLAIM OF PRIORITY 
   This application claims priority under 35 U.S.C. 119(c) from U.S. Provisional Application 60/562,867 filed Apr. 16, 2004. 

   TECHNICAL FIELD OF THE INVENTION 
   The technical field of this invention is determining the detection threshold for preamble detection for mobile telephone users for connection to a UMTS base-station receiver. 
   BACKGROUND OF THE INVENTION 
   One major challenge in RACH preamble detection is the determination of the detection threshold. A power estimator (PE) can be used to estimate the input noise variance. This PE based technique assumes that the signal-to-noise ratio is low enough so that what is measured by PE is just the interference plus the noise level. This approach works well and converges in a few slots in low-to-moderate signal-to-noise ratio (SNR) environments. Dynamic range adjusters (sliders) in PE may accidentally not be set the same as those in preamble detector (PD). The software complexity increases if we need to pass PE values for use in PD and path monitor (PM) algorithms. Finally, in high SNR scenarios, the PE noise estimate will be strongly influenced by the preamble itself. 
   Another existing approach to this problem configures a preamble correlation hardware block with a scrambling code that is not used by the mobiles in the particular cell or sector of concern. This requires additional hardware that is used wastefully. Furthermore, the estimation of the detection threshold is still not resolved. 
   SUMMARY OF THE INVENTION 
   This invention uses statistics derived with Beaulieu&#39;s method to determine the detection threshold assuming input additive white Gaussian noise (AWGN) with known variance. Prior methods use the PE to estimate the input noise variance. However, it is preferable to use the PD itself to characterize the noise variance. For one, this would capture the impact of the PD hardware such as bit-widths. In high SNR scenarios, the orthogonality of the PRACH signatures may allow a more accurate estimate of the noise. 
   In a typical wireless 3G scenario, there will be free signatures. Thus for a given PRACH scrambling code not all of the 16 signatures will be assigned to users. This invention is an approach to estimate the input noise variance directly from PD outputs if signatures are free. An alternative technique based on PE may be used when no signatures are free. Simulations show that the error using this method is small even using just a few PD output observations. There is a 95% confidence that the fractional error is 2% using just 6 sets of PD outputs. 
   The PE based approach of the prior art assumes that the signal-to-noise ratio is low enough so that what is measured by PE is just the interference plus noise. This inherent assumption limits the operation range of the preamble detector. In this invention the PE and the preamble correlations are computed separately. Therefore results from PE need to be mapped to whatever configuration is used for the preamble correlations. Convergence speed of the threshold statistic computed according to this invention is much faster than the PE based solution. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     These and other aspects of this invention are illustrated in the drawings, in which: 
       FIG. 1  illustrates a block diagram of a preamble detector; 
       FIG. 2  illustrates a graph of the probability density function versus z for Ŵ=512 and {circumflex over (N)} nca =2 and several values of i; 
       FIG. 3  illustrates a graph of the fractional error percentage versus access slots to produce a 95% that the factional error is less than the ordinate; 
       FIG. 4  illustrates a graph of the probability of false detection of the preamble P FA  for several values of L={circumflex over (N)} nca  assuming 
     
       
         
           
             
               
                 W 
                 ^ 
               
               = 
               
                 
                   512 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   and 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       σ 
                       
                         n 
                         , 
                         nca 
                       
                       2 
                     
                     2 
                   
                 
                 = 
                 1 
               
             
             ; 
           
         
       
     
       FIG. 5  illustrates a graph of the probability that the estimate has fractional error of 1% versus the number of samples N; 
       FIG. 6  illustrates a graph of the probability that the estimate has fractional error of 5% versus the number of packets N; 
       FIG. 7  illustrates a graph of the base  10  logarithm of the number of samples N versus α needed to achieve 95% confidence; and 
       FIG. 8  illustrates a graph of the probability that the fractional error is less than α versus α for 4096 samples. 
   

   DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
   Preamble detection is the first step in the wireless telephone base station sorting out the signals from plural wireless telephone users. The base station must determine which received radio transmission corresponds to which user. This determination is necessary to apply the proper signal conditioning such as echo cancellation. This determination also permits the base station to route a recovered voice stream to the counterpart party to a particular wireless telephone user. 
   Setting a preamble detection threshold is crucial to proper detection. A too high threshold would prevent detection of some valid preambles corresponding to valid radio transmissions. If the threshold is set too low, then there is an increased chance of a false positive preamble detection from received noise. Setting the preamble detection threshold thus involves a compromise between missing too many proper transmissions and making too many false determinations. 
     FIG. 1  illustrates an example preamble detector (PD) that supports the detection of preambles on the PRACH. Within a predefined search window, the PD stores the 16 largest detection results and the associated offsets for each of the 16 signatures. An offset step size of ½ or 1 chip can be used. The PD passes these results to a chip rate assist (CRA) digital signal processor (DSP). The chip rate assist DSP determines if a preamble is present, acknowledges its detection and programs the finger despreader (FD) and path monitor (PM) accordingly. 
   Input buffer  101  receives the detected radio frequency signals as 8-bit I/Q signals. Input buffer  101  preferably has the capacity to store 1408 samples of each of 12 data streams of 2 times OSF 8-bit I/Q data per sample. Input buffer  101  supplies 64 chips to correlator  102 . Correlator  102  correlates the 64 chips from input buffer  101  with PRACH scrambling codes from code generator  103 . Correlator  102  preferably performs  2048  simultaneous 64-chip correlations. 
   Suppose x n  and y n  model the respective in-phase and quadrature input of additive White Gaussian noise (AWGN) samples, each having zero mean with variance 
               σ   n   2     2     .         
Then let X n,ca  and Y n,ca  model the coherent accumulation of x n  and y n  respectively over N ca  chips. X n,ca  and Y n,ca  are also Gaussian with mean zero and variance
 
   
     
       
         
           
             
               σ 
               
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           = 
           
             
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             ⁢ 
             
               
                 
                   σ 
                   n 
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   The PD forms the absolute value of the coherent accumulation via rotator  104 , coherent accumulator  105 , coherent scratch memory  106  and Hadamard transformer  107 . Rotator  104  preferably has a range of ±30 kHz in steps of 60 Hz. Coherent accumulator  105  employs coherent scratch buffer  106  to store intermediate results. Coherent scratch buffer  106  preferably has a capacity of 32,768 32-bit data words. Coherent accumulator  105  provides an output dynamic range selection to Hadamard transform  107 . The result Z n,ca =√{square root over (X n,ca   2 +Y n,ca   2 )} has Rayleigh distribution with mean 
             m     z     n   ,   ca         =           σ     n   ,   ca     2     2     ⁢       π   2         =           N   ca     ⁢     σ   n   2       2     ⁢       π   2                 
and variance
 
   
     
       
         
           
             σ 
             
               z 
               
                 n 
                 , 
                 ca 
               
             
             2 
           
           = 
           
             
               
                 ( 
                 
                   2 
                   - 
                   
                     π 
                     2 
                   
                 
                 ) 
               
               ⁢ 
               
                 
                   σ 
                   
                     n 
                     , 
                     ca 
                   
                   2 
                 
                 2 
               
             
             = 
             
               
                 ( 
                 
                   2 
                   - 
                   
                     π 
                     2 
                   
                 
                 ) 
               
               ⁢ 
               
                 
                   
                     
                       N 
                       ca 
                     
                     ⁢ 
                     
                       σ 
                       n 
                       2 
                     
                   
                   2 
                 
                 . 
               
             
           
         
       
     
   
   The PD then non-coherently combines {circumflex over (N)} nca  coherent packets Z n,ca  via non-coherent accumulator  109  and non-coherent scratch buffer  110 . As coherent scratch buffer  106 , non-coherent scratch buffer  110  preferably has a capacity of 32,768 32bit words. This forms 
             Z     n   ,   nca       =         ∑       N   ^     nca       ⁢     Z     n   ,   ca         =       ∑       N   ^     nca       ⁢           X     n   ,   ca     2     +     Y     n   ,   ca     2         .               
It is convenient to normalize the non-coherent output by
 
             σ     n   ,   ca         2           
yielding
 
               Z   ~       n   ,   nca       =           2     ⁢           ⁢     Z     n   ,   nca           σ     n   ,   ca         .           
The results of non-coherent accumulator  109  are sorted via sorter  111  and output via output buffer  112 . Output buffer  112  preferably has a capacity of 2048 32-bit words.
 
   The next section describes a Beaulieu series approach to determine the statistics of {tilde over (Z)} n,nca . Let X i , i=1 to L be independent Rayleigh variables with probability density function (pdf) given by: 
               f     x   i       ⁡     (     x   i     )       =     {               ⁢           x   i       σ   i   2       ⁢     ⅇ         -     x   i   2       /   2     ⁢     σ   i   2           ,                 ⁢     x   ≥   0                     ⁢     0   ,                 ⁢   otherwise                   
where (2−π/2)σ i   2  is the variance of the distribution.
 
   Denote the sum of the L Rayleigh variables by 
           X   =       ∑     i   =   1     L     ⁢     X   i             
the complementary distribution G x (x), cumulative distribution F x (x), and probability density function f x (x). Then:
 
                       G   x     ⁡     (     ɛ   ⁢           ⁢   L     )       =       1   2     +       2   π     ⁢       ∑       n   =   1       n   ⁢           ⁢   odd       ∞     ⁢           (     A   in     )     L     ⁢     sin   ⁡     (     L   ⁢           ⁢     θ   in       )         n             ⁢     
     ⁢   with           (   1   )                 A   in     =           [       F   1           1         ⁡     (     1   ,     1   2     ,         -     n   2       ⁢     ω   2     ⁢     σ   i   2       2       )       ]     2     +       π   2     ⁢     n   2     ⁢     ω   2     ⁢     σ   i   2     ⁢     ⅇ       -     n   2       ⁢     ω   2     ⁢     σ   i   2                       (   2   )                 θ   in     =       tan     -   1       ⁢     {                 π   2     ⁢     n   2     ⁢     ω   2     ⁢     σ   i   2     ⁢     ⅇ       -     n   2       ⁢     ω   2     ⁢     σ   i   2         ⁢     cos   ⁡     (     n   ⁢           ⁢   ωɛ     )         -                   F   1           1         ⁡     (     1   ,     1   2     ,         -     n   2       ⁢     ω   2     ⁢     σ   i   2       2       )       ⁢     sin   ⁡     (     n   ⁢           ⁢   ωɛ     )                             F   1           1         ⁡     (     1   ,     1   2     ,         -     n   2       ⁢     ω   2     ⁢     σ   i   2       2       )       ⁢     cos   ⁡     (     n   ⁢           ⁢   ωɛ     )         +                 π   2     ⁢     n   2     ⁢     ω   2     ⁢     σ   i   2     ⁢     ⅇ       -     n   2       ⁢     ω   2     ⁢     σ   i   2         ⁢     sin   ⁡     (     n   ⁢           ⁢   ωɛ     )                 }               (   3   )               
where  1 F 1 (.,.,.) is the confluent hypergeometric function and Φis a term affecting the convergence rate of G x (εL).
 
   Denote the i th  largest normalized PD output in a single antenna search window of W as ({tilde over (Z)} n,nca ) i . Ding, C-S, et al., “Statistical Estimation of the Cumulative Distribution Function for Power Dissipation in VLSI Circuits,” Proceedings of the 34 th  Design Automation Conferences, Jun. 9-13, 1997, pp. 371-376 then shows that these ordered statistics have a power dissipation function given by: 
   
     
       
         
           
             
               
                 
                   p 
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                         i 
                       
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                           z 
                           ) 
                         
                       
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                       1 
                     
                   
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                       z 
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                 4 
                 ) 
               
             
           
         
       
     
   
   p(({tilde over (Z)} n,nca ) i =z) and the expected value E(({tilde over (Z)} n,nca ) i )=ξ i,W,N     nca    can be derived numerically for any value of Ŵ, {circumflex over (N)} nca  and i by setting σ i   2 =1 in Equations 1 to 3 and using the results in Equation 4.  FIG. 2  illustrates the probability density function p(({tilde over (Z)} n,nca ) i =z) versus z for Ŵ=512 and {circumflex over (N)} nca =2 and several values of i. Table 1 provides ξ i,W,N     nca    for Ŵ=512 and Ŵ=1024 and several values of {circumflex over (N)} nca . 
   
     
       
             
             
           
             
             
             
             
             
             
             
             
             
           
             
             
             
             
             
             
             
             
             
           
         
             
               TABLE 1 
             
           
           
             
                 
             
             
               Ŵ= 512 
               Ŵ= 1024 
             
           
        
         
             
                 
               {circumflex over (N)} nca  = 1 
               {circumflex over (N)} nca  = 2 
               {circumflex over (N)} nca  = 4 
               {circumflex over (N)} nca  = 8 
               {circumflex over (N)} nca  = 1 
               {circumflex over (N)} nca  = 2 
               {circumflex over (N)} nca  = 4 
               {circumflex over (N)} nca  = 8 
             
             
                 
             
           
        
         
             
               1 
               3.667330 
               5.778324 
               9.477385 
               16.161117 
               3.864348 
               6.038004 
               9.821462 
               16.620744 
             
             
               2 
               3.376443 
               5.394726 
               8.968832 
               15.481237 
               3.585999 
               5.671316 
               9.335858 
               15.972554 
             
             
               3 
               3.221183 
               5.189702 
               8.696589 
               15.116448 
               3.438471 
               5.476691 
               9.077762 
               15.627374 
             
             
               4 
               3.113599 
               5.047540 
               9.507629 
               14.862889 
               3.336718 
               5.342364 
               8.899480 
               15.388651 
             
             
               5 
               3.030589 
               4.937802 
               8.361650 
               14.666792 
               3.258494 
               5.239050 
               8.762274 
               15.204764 
             
             
               6 
               2.962647 
               4.847953 
               8.242052 
               14.505987 
               3.194668 
               5.154723 
               8.650226 
               15.054484 
             
             
               7 
               2.904924 
               4.771599 
               8.140357 
               14.369148 
               3.140590 
               5.083256 
               8.555224 
               14.926984 
             
             
               8 
               2.854606 
               4.705025 
               8.051642 
               14.249691 
               3.093565 
               5.021097 
               8.472561 
               14.815983 
             
             
               9 
               2.809909 
               4.645877 
               7.972784 
               14.143442 
               3.051888 
               4.965997 
               8.399259 
               14.717503 
             
             
               10 
               2.769633 
               4.592568 
               7.901681 
               14.047586 
               3.014412 
               4.916442 
               8.333313 
               14.628864 
             
             
               11 
               2.732926 
               4.543978 
               7.836843 
               13.960130 
               2.980325 
               4.871362 
               8.273304 
               14.548171 
             
             
               12 
               2.699166 
               4.499281 
               7.777177 
               13.879610 
               2.949035 
               4.829974 
               8.218192 
               14.474035 
             
             
               13 
               2.667882 
               4.457857 
               7.721858 
               13.804921 
               2.920090 
               4.791685 
               8.167192 
               14.405404 
             
             
               14 
               2.638708 
               4.419221 
               7.670244 
               13.735204 
               2.893142 
               4.756035 
               8.119696 
               14.341466 
             
             
               15 
               2.611355 
               4.382992 
               7.621829 
               13.669780 
               2.867913 
               4.722662 
               8.075221 
               14.281576 
             
             
               16 
               2.585590 
               4.348863 
               7.576204 
               13.608102 
               2.844191 
               4.691273 
               8.033382 
               14.225217 
             
             
                 
             
           
        
       
     
   
   The unnormalized preamble output (Z n,nca ) has a mean 
             E   ⁡     (       (     Z     n   ,   nca       )     i     )       =         σ     n   ,   nca         2       ⁢       ξ     i   ,   W   ,     N   nca         .             
With N B  preamble detection attempts we can thus form an estimate of σ n,ca  using:
 
   
     
       
         
           
             
               
                 
                   σ 
                   
                     n 
                     , 
                     ca 
                   
                 
                 = 
                 
                   
                     
                       2 
                     
                     
                       16 
                       ⁢ 
                       
                         N 
                         B 
                       
                     
                   
                   ⁢ 
                   
                     
                       ∑ 
                       
                         j 
                         = 
                         1 
                       
                       
                         N 
                         B 
                       
                     
                     ⁢ 
                     
                       
                         ∑ 
                         
                           i 
                           = 
                           1 
                         
                         16 
                       
                       ⁢ 
                       
                         
                           
                             ( 
                             
                               Z 
                               
                                 n 
                                 , 
                                 nca 
                               
                             
                             ) 
                           
                           
                             j 
                             , 
                             i 
                           
                         
                         
                           ξ 
                           
                             i 
                             , 
                             W 
                             , 
                             
                               N 
                               nca 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 ( 
                 5 
                 ) 
               
             
           
         
       
     
   
   Preamble detector simulations show that such an estimator converges very rapidly without bias.  FIG. 3  illustrates a graph of the fractional error percentage versus access slots to produce a 95% that the factional error is less than the ordinate. For example, assuming L=2 and W=512,  FIG. 3  shows that N B =6 access slots are enough for 95% confidence that the standard deviation estimate is within 2% of the actual. N B =35 yields a 95% confidence that the estimate is within 1%. 
   The performance requirement of RACH for preamble detection is determined by the two parameters probability of false detection of the preamble P FA  and the probability of detection of preamble P D . The performance is measured by the required energy-per-chip to noise power spectral density ratio, E c /N 0  at a probability of detection, P D , of 0.99 and 0.999. P FA  is defined as a conditional probability of erroneous detection of the preamble when input is only noise and interference. P D  is defined as conditional probability of detection of the preamble when the signal is present. P FA  should be 10 −3  or less. Only one signature is used and it is known by the receiver. 
   P FA  depends on the diversity combining method. Consider “selection diversity” and “diversity combining.” With selection diversity, the length 16 sorted lists are taken from each antenna and concatenated. If any of the resulting 32 values exceeds the threshold, a detection is declared. With selection diversity, the equivalent search window is twice the search window used in each antenna. With diversity combining, the sorted length 16 lists are taken to see if particular offsets are in both lists. If so, the search values are added. A new list of length between 16 and 32 is formed with the combined results. If a particular offset is only observed in a single antenna, it is added as is to the new list. Each of the resultant samples is based on the single antenna search window. However, each sample, in the worst case assuming only noise is present, represents the non-coherent accumulation over twice the number of coherent packets used to generate the single antenna results. Table 2 generalizes the adjustments needed with multiple antennas. 
   
     
       
             
             
             
             
             
           
         
             
                 
               TABLE 2 
             
             
                 
                 
             
             
                 
                 
                 
               Equivalent 
                 
             
             
                 
                 
               Equivalent 
               Number of 
               Diversity 
             
             
                 
               Diversity 
               Search 
               Coherent 
               Method 
             
             
                 
               Method 
               Window Ŵ 
               Packets {circumflex over (N)} nca   
               Details 
             
             
                 
                 
             
           
           
             
                 
               Diversity 
               Ŵ 
               M {circumflex over (N)} nca   
               Combine 
             
             
                 
               Combining 
                 
                 
               results 
             
             
                 
                 
                 
                 
               with common 
             
             
                 
                 
                 
                 
               offsets 
             
             
                 
               Selection 
               MŴ 
               {circumflex over (N)} nca   
               Concatenate 
             
             
                 
               Diversity 
                 
                 
               lists 
             
             
                 
                 
             
             
                 
               Ŵ= single antenna search window size in offsets 
             
             
                 
               {circumflex over (N)} nca  = single antenna number of coherent packets 
             
           
        
       
     
   
   We can use Beaulieu&#39;s approach to determine P(Z n,nca ) for a given value of {circumflex over (N)} nca  and then P FA  from: 
             P   FA     =       P   ⁡     (       all   ⁢           ⁢     Z     n   ,   nca         ≤   τ     )       =       P   ⁡     (       Z     n   ,   nca       ≤   τ     )         W   ^                 FIG. 4  illustrates a graph of the probability of false detection of the preamble P FA  versus Z n,nca  for several values of L={circumflex over (N)} nca  assuming
 
             W   ^     =       512   ⁢           ⁢   and   ⁢           ⁢       σ     n   ,   nca     2       2         =   1.             FIG. 4  suggests that for P FA &lt;0.001, we&#39;d need a detection threshold of 7.95, 12.40, and 20.09 for L={circumflex over (N)} nca =2, 4 and 8, respectively. Since Z n,nca  scales directly with σ n,ca , the results of  FIG. 4  can be used for any value of σ n,ca  assuming a constant value of Ŵ. The detection threshold is simply
 
             7.95   ⁢       σ     n   ,   ca         2         ,       12.40   ⁢       σ     n   ,   ca         2       ⁢           ⁢   and   ⁢           ⁢   20.09   ⁢       σ     n   ,   ca         2       ⁢           ⁢   for   ⁢           ⁢   L     =   2     ,   4   ,     and   ⁢           ⁢   8     ,         
respectively, for instance in the  FIG. 4  scenario.
 
   In general, the detection threshold τ, is given by: 
                 τ   =       C         N   ^     nca     ,     W   ^         ⁢       σ     n   ,   ca         2                 (   6   )               
where C {circumflex over (N)}     nca     ,Ŵ  is a fixed constant depending on {circumflex over (N)} nca  and Ŵ, for example given in Table 3 for Ŵ=512 and Ŵ=1024 and {circumflex over (N)} nca  is 2, 4, 6 and 8.
 
   
     
       
             
             
           
             
             
             
             
             
             
             
             
           
         
             
               TABLE 3 
             
           
           
             
                 
             
             
               Ŵ= 512 
               Ŵ= 1024 
             
           
        
         
             
               {circumflex over (N)} nca  = 1 
               {circumflex over (N)} nca  = 2 
               {circumflex over (N)} nca  = 4 
               {circumflex over (N)} nca  = 8 
               {circumflex over (N)} nca  = 1 
               {circumflex over (N)} nca  = 2 
               {circumflex over (N)} nca  = 4 
               {circumflex over (N)} nca  = 8 
             
             
                 
             
             
               5.104674 
               7.769179 
               12.155090 
               9.756381 
               5.221810 
               7.952168 
               12.398603 
               20.086897 
             
             
                 
             
           
        
       
     
   
   Substituting Equation 5 into Equation 6 gives an estimate of the threshold: 
                 τ   =       1     16   ⁢     N   B         ⁢       ∑     j   =   1       N   B       ⁢       ∑     i   =   1     16     ⁢           C         N   ^     nca     ,     W   ^         ⁡     (     Z     n   ,   nca       )         j   ,   i         ξ     i   ,   W   ,     N   nca                         (   7   )               
The accuracy of this estimate is as accurate as the estimate σ n,ca . For PD outputs from 10 access slots, we should have a great deal of confidence that the fractional error of our estimate is very small.
 
   Roughly 10 samples (access slot, signature pairs) are required for statistical confidence. Thus, the following approach is proposed. For initialization of the first 10 samples t, form the threshold estimate using all t samples: 
           τ   =       1     16   ⁢   t       ⁢       ∑     j   =   1     t     ⁢       ∑     i   =   1     16     ⁢           C         N   ^     nca     ,     W   ^         ⁡     (     Z     n   ,   nca       )         j   ,   i         ξ     i   ,   W   ,     N   nca                       
Upon reaching equilibrium, form the instantaneous threshold estimate:
 
           τ   =       1   16     ⁢       ∑     i   =   1     16     ⁢             C         N   ^     nca     ,     W   ^         ⁡     (     Z     n   ,   nca       )         j   ,   i                   ξ     i   ,   W   ,     N   nca                     
and insert into infinite impulse response (IIR) filter:
 τ k =(1−α)τ k−1 +ατ 
where α=¼.
 
   In an alternative to this invention, the power estimate could be used to estimate the standard deviation of the input interference in the preamble detector threshold selection. Let x n  and y n  be normal variables with zero mean and variance σ 2 . It is well known that the sample mean 
             Z   =           ∑     n   =   1     N     ⁢     x   n   2       +     y   n   2       N       ,         
is the ML estimate (unbiased) of 2σ 2 .
 
   The term x n   2 +y n   2  is the chi-squared with mean 2σ 2  and variance 4σ 4 . Assuming that N is large, according to the central limit theorem Z is approximately normal with mean 2σ 2  and variance (4/N)σ 4 . The probability that Z is within a certain fractional error from the mean can then be estimated by: 
                   p   ⁡     (            Z   -     2   ⁢     σ   2              &lt;     α2σ   2       )       =     1   -     2   ⁢     p   ⁡     (       Z   -     2   ⁢     σ   2         &gt;     α2σ   2       )                       =     1   -     2   ⁢     p   ⁡     (     Z   &gt;       (     α   +   1     )     ⁢   2   ⁢     σ   2         )                       =     1   -     2   ⁢     Q   (           (     α   +   1     )     ⁢   2   ⁢     σ   2       -     2   ⁢     σ   2               4   N     ⁢     σ   4           )                     =     1   -     2   ⁢   Q   ⁢     (     ⁢     N     ⁢   α   ⁢     )                     
For instance,  FIGS. 5 and 6  show the probability that the estimate has fractional error of 1 and 5%, respectively (e.g., for α=0.01 and α=0.05).
 
     FIG. 7  illustrates the value of N on a logarithmic scale required to achieve 95% confidence, i.e., p(|Z−2σ 2 |&lt;α2σ 2 )=0.95, as a function of α. 
     FIG. 8  illustrates the probability that the fractional error is less than a as a function of α for N=4096. 
   Applying the above results to estimate the input interference level, we set a goal of a confidence of 95% that the noise variance estimate has a fractional error relative to the actual within 1 to 2%. Simulations show that for N=4096, the error is within 1.2% only 56% of the time. An N of 26677 is required for 95% confidence. An N of 4096 yields 95% confidence that error is within 3%, 99% within 4%, and 99.9% within 5%. Given N=9603, 95% of the time the error is within 2%. For N=38414, 95% of the time the error is within 1%. Averaging over about 20000 samples or about 8 slots on generation of the noise estimate yields 95% confidence that the error is within 1.4%. Employing a number of samples of this order is advisable. 
   Implementation with power estimation is similar to that described above. Initialization of the first 80 256-chip PE outputs N, includes forming sample mean from PE outputs: 
           Z   =           ∑     n   =   1     N     ⁢     x   n   2       +     y   n   2       N           
and forming the detection threshold estimate from:
 
           τ   =       C         N   ^     nca     ,     W   ^         ⁢         4096     N   nca       ⁢   Z               
Upon equilibrium take the PE output Z and feed into infinite impulse filter (IIR) filter:
 Z k =(1−α)Z k−1 +αZ 
where α= 1/16. The detection threshold is formed from:
 
   
     
       
         
           τ 
           = 
           
             
               C 
               
                 
                   
                     N 
                     ^ 
                   
                   nca 
                 
                 ⁢ 
                 
                   
                     , 
                     W 
                   
                   ^ 
                 
               
             
             ⁢ 
             
               
                 
                   4096 
                   
                     N 
                     nca 
                   
                 
                 ⁢ 
                 Z