Abstract:
A method and an apparatus for detecting a frequency correction burst embedded in a signal which includes ordered samples. The method comprises the following: (a) correlating K sequences of the ordered samples with a predetermined waveform to produce K correlation outputs, each of the K correlation outputs corresponding to one of the K sequences, having a magnitude and being associated with an order index, K being an integer, the predetermined waveform corresponding to the frequency correction burst; (b) storing the K correlation outputs in a buffer according to the order indices; and (c) estimating a parameter based on the K correlation outputs, the parameter indicating detection of the frequency correction burst. The estimated parameter includes a frequency error and an ending time of a frequency correction burst. A quality metric associated with each frequency error is also computed.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates generally to initial acquisition of a signal in a communication system which uses a time division multiple access (TDMA) method for transmission. More particularly, the present invention relates to a method and an apparatus for detecting a frequency correction burst, estimating the frequency error and the burst arrival time for a TDMA communication system, such as the Global System for Mobile communications (GSM). 
     2. Description of Related Art 
     The TDMA method allows multiple access by subdividing a duration T f , called the frame duration, into N non-overlapping time slots, each of duration T f /N. Each user who wishes to transmit information is assigned to a particular time slot within each frame. The Global System for Mobile Communications (GSM) is an exemplary communication system which uses the TDMA method for multiple access. 
     The Global System for Mobile Communications (GSM), originally known as Groupe Spéciale Mobile, was developed as a standard for cellular communication in Europe. GSM offers a wide range of functionality, transmission of voice and data. It has been adopted throughout Europe and the world. The original primary GSM has been expanded to encompass other versions for wider total bandwidths, such as extended GSM, DCS 1800 and PCs 1900. 
     GSM is a purely digital system. The primary GSM system, also known as GSM 900, uses the 900 MHz band, of which 890-915 MHz is for mobile transmissions, and 935-960 MHz is for base transmissions. There are 124 channels (174 channels for extended GSM, 374 channels for DCS 1800 or PCS 1900 ) and each channel is 200 kHz wide. The TDMA method is used with 8 time slots, numbered 0 to 7, per channel. GSM uses wide channels to allow high speed digital transmissions, resulting in reducing the effect of fading and minimizing production costs. Gaussian Minimum Shift Keying (GMSK) is used as the modulation process. 
     The first stage of synchronization between a receiver and a transmitter is called the initial acquisition process. In a mobile communication environment, there are always initial time and frequency errors due to uncompensated local oscillator (LO) error, range uncertainty between the transmitter and the receiver, and motion in a multi-path channel environment. Upon completion of initial acquisition, the time and frequency errors are reduced to within acceptable limits. 
     In GSM, a base station uses a Broadcast Common Control Channel (BCCH) to transmit signaling information. As shown in FIG. 1, the frame structure of the control channel includes 51 TDMA frames lasting 235.38 milliseconds (each TDMA frame is 4.615 ms in duration). Within this repetitive frame structure, each of the information data shown in FIG. 1 is transmitted in time slot 0 of a corresponding TDMA frame. Synchronization data include a Frequency correction Burst (FB) and a Synchronization Burst (SB). Within 235.38 ms, there are 5 FBs, appearing in time slots 0 of frames 0, 10, 20, 30,40. 
     The FB is a sine wave at 67.7 kHz above the carrier, lasting 546.12 μs. The FB is formed by differentially encoding a string of 148 bits of values “0” and then modulating the resulting 147 bits. The data structure of the string of 148 bits is as shown in FIG.  2 . The string consists of 142 bits (all 0s) preceded by 3 tailing bits (three 0s) and followed by three tailing bits (three 0s) and 8.25 guarding bits (all 1s). 
     In the acquisition mode, the receiver has to detect the FBs and perform timing and frequency error estimation. The receiver then uses the frequency error estimate to correct the reference frequency of the local oscillator, and the timing estimate in the subsequent demodulation of the Synchronization Burst (SB) which is sent in time slots 0 of frames 1, 11, 21, 31, 41. 
     In the acquisition mode, a receiver first scans the whole frequency band used for GSM (or extended GSM, or DCS 1800 or PCS 1900 ) and measures the relative signal strength on up to 124 channels (174 channels for extended GSM, 374 channels for DCS 1800 or PCS 1900 ), each channel having a bandwidth of 200 kHz, to obtain a list of the strongest channels. Then, the receiver will start to search for the FBs over the channels in the list. 
     According to the GSM standard, the required frequency accuracy is 0.1 ppm, which is 90 Hz for a 900 MHz carrier. Assuming that the frequency stability of the crystal oscillator of a receiver is 5 parts per million (ppm) (including aging and temperature factors, etc.), the maximum initial frequency offset can be about 5 kHz for GSM 900, and about 10 kHz for DCS 1800 and PCS 1900, when the receiver is first turned on. This initial frequency error must be rapidly detected and reduced to within the GSM requirement. 
     Accordingly, there is a need for a method and an apparatus to detect the FBs and estimate the initial frequency error in real time with high accuracy in order to meet the GSM standard. 
     SUMMARY OF THE INVENTION 
     The present invention is a method and: an apparatus for detecting a frequency correction burst embedded in a signal which includes ordered samples. The method comprises the following: (a) correlating K sequences of the ordered samples with a predetermined waveform to produce K correlation outputs, each of the K correlation outputs corresponding to one of the K sequences, having a magnitude and being associated with an order index, K being an integer, the predetermined waveform corresponding to the frequency correction burst; (b) storing the K correlation outputs in a buffer according to the order indices; and (c) estimating a parameter based on the K correlation outputs, the parameter indicating detection of the frequency correction burst. 
     The estimated parameter includes a frequency error and an ending time of a frequency correction burst. A quality metric associated with each frequency error is also computed. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a diagram of the known frame structure of the control channel used by the base station. 
     FIG. 2 is a diagram of the known data structure of a Frequency correction Burst (FB) before differential encoding and modulation. 
     FIG. 3 shows a block diagram of one embodiment of the FB detector of the present invention. 
     FIG. 4 is a graph of the output magnitude of the FB correlator versus the frequency error for the case of M=16. 
     FIG. 5 shows a diagram of the structure of the FB correlator. 
     FIG. 6 is a diagram of the general structure of a running sum module that calculates the sum of the M most recent samples. 
     FIG. 7 is a diagram of another embodiment of the FB detector which uses the modules shown in FIGS. 5 and 6. 
     FIG. 8 is a diagram of the timing estimator. 
     FIG. 9 is a graph of the sum of the squared magnitudes of two correlation outputs as a function of time, illustrating the principle of timing estimation. 
     FIG. 10 is a diagram of one embodiment of the frequency error estimator. 
     FIG. 11 is a diagram of another embodiment of the frequency error estimator, the output of which is an average of the previous 116 estimates. 
     FIG. 12 is a diagram of a first system for detecting a frequency correction burst with low probability of missed detection. 
     FIG. 13 is a diagram of a second system for detecting a frequency correction burst with low probability of missed detection. 
     FIG. 14 is a graph of the magnitude of a FB correlator output as a function of the frequency error for the two cases M=16 and M=64, M being the number of signal samples used by the FB correlator to produce one correlation output. 
    
    
     DESCRIPTION OF THE INVENTION 
     In the following description, for purposes of explanation, numerous details are set forth in order to provide a thorough understanding of the present invention. However, it will be apparent to one skilled in the art that these specific details are not required in order to practice the present invention. 
     The present invention can be practiced in any communication system which uses a frequency correction burst embedded in a transmitted signal to facilitate initial acquisition at a receiver. However, for ease of explanation, the invention is described below as applied to the primary GSM. Thus, this description is not intended to be construed in a limiting sense. 
     The FB detector of this invention detects a FB, estimates the FB&#39;s ending time (or arrival time) with an accuracy of one symbol (3.69 μs), and estimates a frequency error. A quality metric of the estimations is also provided. The quality metric is an important indicator in adverse reception condition such as multipath fading and can be used to determine the number of FBs over which a frequency error estimation or timing estimation should be averaged. The detector operates in real time, i.e., the FB detection and frequency error estimation are performed and updated at the same time for each input sample. 
     FIG. 3 shows a block diagram of one embodiment of the FB detector of the present invention. The FB detector  30  comprises a FB correlator  32 , a circular buffer  34 , a threshold module  36 , a timing estimator  38 , and a frequency error estimator  42 . The sampling rate at the input of the FB detector  30  is preferably one sample per symbol, or about 271 kHz. 
     Referring to FIG. 3, a complex, low-pass equivalent form of a received signal is stored in a sample buffer  10 . A decimator  20  is used to reduce the number of samples per symbol, for example, from 4 samples to 1 sample per symbol. The function of the decimator  20  can be equivalently implemented by appropriately addressing the sample buffer  10 . 
     The following discussion describes the underlying principle of the FB detector. In GSM, the GMSK modulator is fed with 147 consecutive 0s and generates a burst of a sinusoidal wave of angular frequency ω c +ω 1 , where ω c  is the angular carrier frequency, ω 1 =π/(2T b ) is the angular frequency deviation, and T b =1/270833≈3.69 μs is the symbol duration. 
     In the low-pass equivalent complex form, the received FB signal in the sample buffer  10  is r(nT s )=e jφ ·e j(ω     1     +Δω)nT     s   , where T s =T b /4 is the sampling period (assuming the sampling rate is 4 samples per symbol), φ is a constant and Δω is the frequency error with |Δω|&lt;2π·5000. 
     After decimation to one sample per symbol, r n =e jφ ·e j(ω     1     +Δω)(nT     b     +kT     s)   , where epoch k is a value from the set {0,1,2,3}. 
     The tap coefficients of the correlator  32  are C m =C 0 ·e jπm/2  for m=1, 2 , . . . , M−1 with |C 0 |=1. M indicates the length of the FB correlator  32 . In a preferred embodiment of the FB correlator  32 , C 0 =−j (or e −jπ/2 ) and M=16. This yields C 15 =−1. 
     The output of the FB correlator  32  is                      y   n     =         ∑     m   =   0       M   -   1                         r     n   -   m       ·     C   0     ·          j                 m                   π   /   2             =       ∑     m   =   0       M   -   1                 jφ     ·                j        (       ω   1     +   Δω     )            (     n   -   m     )          T   b       +     kT   s       )       ·     C   0     ·          j                   mx   /   2                           =            jφ     ·            j        (       ω   1     +   Δω     )            (       nT   b     +     kT   s       )                  ∑     m   =   0       M   -   1                     j        (       ω   1     +   Δω     )            (     -     mT   b       )         ·     C   0     ·          j                   mx   /   2                           =            jφ     ·            j        (       ω   1     +   Δω     )            (       nT   b     +     kT   s       )         ·     C   0              ∑     m   =   0       M   -   1                   j        (   Δω   )            (     -     mT   b       )                         =          jφ     ·            j        (       ω   1     +   Δω     )            (       nT   b     +     kT   s       )         ·     C   0     ·       (     1   -            -   j                   M                 Δω                   T   b           )     /     (     1   -            -   j                   Δ                 ω                   T   b           )                     =          jφ     ·            j        (       ω   1     +   Δω     )            (       nT   b     +     kT   s       )         ·     C   0     ·            -     j        (     M   -   1     )            Δω                     T   b     /   2         ·       sin        (     M                 Δω                     T   b     /   2       )         sin        (     Δω                     T   b     /   2       )                         (   1   )                                
     Therefore, |Y n |                 y   n          =            sin        (     M                 Δω                     T   b     /   2       )         sin        (     Δω                     T   b     /   2       )                ,                          
     which is independent of k and n. This shows that the magnitude of the output of the correlator is independent of the sampling epoch, even when the sampling rate at the input of the FB detector  30  is one sample per symbol (T b ). 
     For M=16, |y n |=16 for Δω=0, and |y n |=13.81 for Δω=2π·5000. FIG. 4 shows the magnitude of the output of the FB correlator  32  as a function of the frequency error. For M larger than 16, the magnitude of the FB correlator output may yield a higher peak when no frequency error is present, but, as the frequency error increases, the magnitude of the FB correlator output decreases and reaches zero much faster than it does in the situation where M is smaller than 16. In this exemplary embodiment, M is chosen to be 16. 
     In order to obtain the frequency error information, the real part of the complex number                    y   n     ·     y     n   -   1     *       =            C        2     ·          j                 Δω                   T   b             ,       where                 C     =       sin        (     M                 Δω                     T   b     /   2       )         sin        (     Δω                     T   b     /   2       )           ,           (   2   )                                
     is considered. The real part is equal to 
       Re ( y   n   ·y*   n−1 )=| C|   2 ·cos(Δω T   b +π/2)=−| C|   2 ·sin(Δω T   b )≈−| C|   2   ΔωT   b   (3) 
     if Δω T   b  is small, and where Re(.) denotes the real part of the argument. |C| is in fact the magnitude of the output of the FB correlator and can be compensated by an appropriate scaling. 
     
       
         Similarly, the imaginary part of  Y   n   ·Y*   n−4   =|C|   2   ·e   j4π/2   ·e   jΔω·4T     b    is  Im ( y   n   ·Y*   n−4 )=| C|   2 ·sin(Δω·4 T   b )≈|C| 2 ·Δω·4 T   b  if Δω·4 T   b  is small  (4) 
       
     
     where Im(.) denotes the imaginary part of the argument. The imaginary part in equation (4) contains the phase difference information between the two samples y n  and y n−4 , namely, Δω4T b . 
     The following numerical examples illustrate the accuracy of the above approximations in equations (3) and (4). If Δω=2π·5000, then sin(Δω·T b )=0.115737 and Δω·T b =0.115997. Thus, the relative error of the approximation in equation (3) with respect to unity is less than 2.6×10 −4 . If Δω=2π·5000, then sin(Δω·4T b )=0.44752 and Δω·4T b =0.464. Thus, the relative error of the approximation in Equation (4) with respect to unity for this case is less than 1.65×10 −2 . If Δω=2π·500, then sin(Δω·4T b )=0.046382317 and Δω·4T b =0.046398963, and the relative error of the approximation in Equation (4) with respect to unity is less than 1.665×10 −5 . This shows that when the frequency error is small (e.g., 500 Hz), the above approximations are very accurate. Such an estimation accuracy for a larger frequency error is not needed, since the frequency of the voltage controlled oscillator (VCO) in the receiver, controlled by the frequency error output of the FB detector, will be reduced to within |Δω|2π·500 after the first FB is detected. Once the frequency error is within this range, the major task of the receiver is to make residual estimation error smaller in order to maintain a tracking accuracy of 0.1 ppm (i.e., 90 Hz for GSM) in a noisy environment (which includes quantization noise and interference). 
     The output of the FB correlator  32  can be expressed as a recursive equation as follows:                      y   n     =       ∑     m   =   0       M   -   1              r     n   -   m       ·     C   0     ·          j                 m                   π   /   2                         =         r   n     ·     C   0       +       ∑     m   =   1       M   -   1              r     n   -   m       ·     C   0     ·          j                 m                   π   /   2             +       r     n   -   M       ·     C   0     ·          j                 M                   π   /   2           -       r     n   -   M       ·     C   0     ·          j                   Mπ   /   2                         =         r   n     ·     C   0       +       ∑     m   =   1     M            r     n   -   m       ·     C   0     ·          j                 m                   π   /   2             -       r     n   -   M       ·     C   0     ·          jM                   π   /   2                         =         r   n     ·     C   0       +            jπ   /   2       ·       ∑     m   =   0       M   -   1              r     n   -   1   -   m       ·     C   0     ·          j                 m                   π   /   2               -       r     n   -   M       ·     C   0     ·          j                 M                   π   /   2                         =         r   n     ·     C   0       +     j   ·     y     n   -   1         -         r     n   -   M       ·     C   0                 j                   Mπ   /   2                         =         -   j     ·     r   n       +     j   ·     y     n   -   1         +       j   ·     r     n   -   16                         (     M   =   16     )                       (   5   )                                
     The structure shown in FIG. 5 is used to implement the above recursive equation. The FB correlator  32 , as implemented in FIG. 5, has two inputs, one output, and requires no scalar multiplication since multiplications by the imaginary numbers j and −j only require rotations. Thus, the FB correlator 32 can be implemented such that the number of arithmetic operations is reduced. 
     It is noted that the structure of the FB correlator  32  as shown in FIG. 5 can be used for any M which is equal to 2 L  where L is an integer greater than 1. For other values of M, the multipliers used in the FB correlator  32  will be different than j and −j. 
     Similarly, a summation of M samples can be implemented with a structure having a small number of arithmetic operations. FIG. 6 shows such a structure. A running sum module  60  has two inputs and one output. The running sum module  60  performs a running sum of M consecutive samples, where the number M indicates the number of samples that separate the two inputs. 
     When the memories needed for the delay elements in FIGS. 5 and 6 are implemented in conjunction with the sample buffer  10 , they can be part of the sample buffer  10 , and the function of the decimator  20  can be realized by appropriately addressing the sample buffer. FIG. 7 shows the structure of such an implementation. When the receiver is in the acquisition mode, the samples in the sample buffer  10  are preferably stored in a circular way, i.e., the sample with the ending address is stored right before the sample with the starting address of the sample buffer  10 . The sample buffer  10  may also be used for other modes after the acquisition process. 
     The output y n  of the FB correlator  32  are stored in a circular buffer  34  at a rate of one sample per symbol. The magnitude of y n  reaches its maximum during a frequency correction burst. A threshold is provided in order to determine whether a frequency correction burst is received. In consideration of the variations of the signal level, the threshold is calculated relative to the average power level of the contributing input samples of the FB correlator  32 . The threshold calculation module  36  computes the average power of a number input samples and scales it to an appropriate value. In the exemplary implementation shown in FIG. 7, the number of input samples used for this computation is  16 . 
     The timing estimator  38  determines whether a frequency correction burst has occurred and outputs the ending time of the detected frequency correction burst. Equivalently, the arrival time of the detected frequency correction burst can also be outputted instead. In a GSM system, the TDMA frame timing  39  which is the start time of a TDMA frame can be easily computed from the starting or ending times of the frequency correction bursts, since the frequency correction bursts occur within a known repeating pattern within the signaling multi-frame structure of GSM as shown in FIG.  1 . 
     FIG. 8 is a block diagram of the timing estimator  38 . The timing estimator  38  comprises a combining module  381 , a peak detector  396  and a counter module  387 . 
     The combining module  381  comprises two squared magnitude computing modules  382 ,  384  and an adder  386 . The counter module comprises an adder  388 , two comparators  390 ,  394  and a counter  392 . 
     The two squared magnitude computing modules  382 ,  384  compute the squared magnitudes of y n−131  and y n3 , respectively. The adder  388  subtracts the threshold  37  from the value |y n | 2 . If the output of the adder  388  is strictly positive, i.e.; if |y n | 2  is greater than the threshold  37 , then the comparator  390  outputs a 1, otherwise it outputs a 0. The counter  392  counts consecutive Is that are outputted from the comparator  390 . Any 0 outputted from the comparator  390  will reset the counter  392 . This resetting occurs often for FSM when the received signal is a randomly modulated GMSK signal, instead of a FB. If the count value reaches  131 , then the comparator  394  outputs a control signal value of 1 which enables the peak detector  396 . 
     The peak detector 396 is enabled by the control signal to receive a number of values of |y n | 2 +|y n−131 | 2 , for example 32 values. The number  131 , which is equal to the length of a frequency correction burst minus the length of the FB correlator, is chosen in order to obtain an accurate timing estimation and reduce the false detection probability of a frequency correction burst. The enabled peak detector  396  examines a set of values of |Y n | 2 +|y n−131 | 2  and determines the sample index nmax which corresponds to the maximal value in the set. This peak index nmax  40  is the estimate of the ending time of the frequency correction burst. The TDMA frame timing  39  which is the start time of a TDMA frame is computed from the peak index  40 . 
     FIG. 9 illustrates the principle of the timing estimation. The FB correlator output y nmax  represents the correlation of 16 samples over the end portion of the frequency correction burst, while Y nmax−131  represents the correlation of 16 samples over the beginning portion of the burst. Since the two correlations are spaced  131  symbols apart in time, and since a frequency correction burst in the GSM is a pure sine wave of  147  symbols, the peak detector  396  will not achieve its maximum in |Y n | 2 +|y n−131|   2  with any index value n that is smaller or larger than nmax. 
     FIG. 10 is a block diagram of the frequency error estimator  42  which comprises a second combining module  421  and a frequency module  435 . 
     The second combining module  421  includes a complex conjugate module  422  and a multiplier module  424 . The frequency module  435  includes an imaginary part module  426 , a squared magnitude module  428 , a scale calculation module  432 , and a multiplier module  434 . 
     The complex conjugate module  422  computes the complex conjugate of a correlation output y n−4  stored in the circular buffer  34 . The multiplier module  424  multiplies the output of the complex conjugate module  422  with the current correlation output y n  and outputs the result to the imaginary part module  426  which extracts the imaginary part of the complex number y n y* n−4 . The squared magnitude module  428  computes the squared magnitude of the current correlation output Y n  and outputs this value as a quality metric  44 . The scale calculation module  432  receives |y| 2  and computes a scale value corresponding to this value of |y n | 2 . The multiplier module  434  multiplies Im(y n ·y* n−4 ) by the scale value to compensate for the factor |y n | 2  in Im(y n ·y* n−4 ), and outputs the result as a frequency error. Due to the scale value, the frequency error  41  has an appropriate unit such as Hz. 
     The frequency error estimator  42  also includes an optional data selector  436  which continuously receives the frequency errors outputted from the multiplier module  434 , and, upon reception of the peak index  40 , outputs a frequency error  43  which corresponds to the peak index  40 . A multiplexer (MUX) can be used to implement the data selector  436 . 
     In a second embodiment of the frequency estimator  42 , to reduce the effects of noise and interference on the estimation accuracy, the frequency error  41  and quality metric  44  are averages of a number of estimates. FIG. 11 shows an implementation of this embodiment. 
     Referring to FIG. 11, the frequency error estimator  42  comprises a second combining module  421 ′, a frequency module  435 ′ and an optional data selector  436 . 
     The second combining module  421 ′ includes two complex conjugate modules  422  and  423  and two multiplier modules  424  and  425 . The frequency module  435 ′ includes two imaginary part modules  426  and  427 , two squared magnitude modules  428  and  429 , two running sum modules  430  and  431 , a scale calculation module  432 , and a multiplier module  434 . 
     The complex conjugate modules  422 ,  423  compute the complex conjugates of correlation outputs y n−4  and y n−132 , respectively, which are stored in the circular buffer  34 . The multiplier modules  424 ,  425  multiply the outputs of the complex conjugate modules  422 ,  423  with the correlation outputs y n  and y n−128 , respectively, and output the results to the imaginary part modules  426 ,  427  which extract the imaginary parts of the complex numbers Y n y* n−4  and y n−128 y* n−132 , respectively. 
     The squared magnitude modules  428 ,  429  computes the squared magnitudes of the correlation outputs y n  and y n−128 . These two values are inputted to the running sum module  430  which outputs a quality metric  44 . Each of the running sum modules  430 ,  431  is as shown in FIG. 6 where the order indices of input 1  and input 2  differ by  128 . The scale calculation module  432  receives the output of the running sum module  430 , which is a sum of 128 estimates of |y n   2 , and computes a scale value corresponding to this value. 
     The running sum module  431  receives as inputs the outputs of the imaginary part modules  426 ,  427 , and outputs a sum of  128  estimates of Im(y n ·y* n−4)    
     The multiplier module  434  multiplies the output of the running sum module  431  by the output of the scale calculation module  432  to compensate for the magnitude factor in the sum of 128 estimates of Im(y n ·y n−4 ), and outputs the resulting value as a frequency error  41 . Due to the appropriate scale value, the frequency error  41  has an appropriate unit such as Hz. 
     The optional data selector  436  continuously receives frequency errors from the multiplier module  434 , and, upon reception of the peak index  40 , outputs a frequency error  43  which corresponds, to the peak index  40 . 
     In the frequency error estimator  42  of FIG. 11, running sums of  128  estimates for both the frequency error and the quality metric, namely,            ∑     k   =   0     127                       Im        (       y     n   -   k       ·     y     n   -   k   -   4     *       )                     and                             ∑       k   =   0     127                            y     n   -   k            2         ,                          
     are calculated. Since the order indices of each pair of correlation outputs used in the calculation of an estimate for a frequency error differ by 4, the total number of estimates used in a running sum is chosen to be 128=137−4+1 in order to include in the summation all the estimates that are within a frequency correction burst when the latest estimate in the summation is at the end of the frequency correction burst. The frequency error estimator  42  operates continuously for each new correlation output y n  coming in the circular buffer  34 . When the timing estimator  38  ouputs the ending time of a detected frequency correction burst, a frequency error  43  and a corresponding quality metric  44  are outputted almost instantly. 
     FIG. 12 is a diagram of a system  120  which includes three frequency error estimators  124 ,  126 ,  128 . The system  120  can detect a frequency correction burst in a received signal with low probability of missed detection. The system  120  also includes two mixers  121 ,  123 . Each of the mixers translates the carrier frequency of the received signal by a predetermined amount of frequency. 
     Referring to FIG. 12, the system  120  processes the received signal r(i) through three processing paths to produce three sets of outputs. Each set of outputs includes a TDMA frame timing estimate, a frequency error estimate, and an associated quality metric. Each of the TDMA frame timing estimates can be derived from a corresponding peak index. 
     The advantage of the system  120  over the single path system of FIG. 3 (or FIG. 7) is that the system  120  provides a lower probability of missed detection when the frequency error to be estimated is large. A large initial frequency error may be due to the use of an inexpensive crystal oscillator having low accuracy. The preferred embodiment of the system of FIG. 3 can perform satisfactorily when the frequency error is within the range±5 kHz. If the frequency error due to an inaccurate crystal oscillator is outside this range, then the system of FIG. 3 may miss the detection of a FB. In contrast, the system  120  of FIG. 12 has a lower probability of missed detection due to the two additional paths. The mixers  121 ,  123  in these two paths translate the center frequency of the received signal by frequency amounts f 1 , f 2 , respectively. This allows the FB detectors  124 ,  128  of these two paths to search for a FB at two different frequencies than the one at which the FB detector  126  is searching. Thus, the system  120  can estimate a larger frequency error than the system of FIG.  3 . If the frequency amount f 1  is the negative of f 2 , then the frequency search range is symmetrical with respect to the frequency at which the FB detector  126  is operating. If f 1 =−f 2 , then the implementation of the mixers  121  and  123  is also simplified. 
     In addition to being capable of detecting a FB despite a large initial frequency error due to the crystal oscillator of the receiver, the system  120  also has better resilience to carrier phase changes. It can be shown analytically, and verified by simulation, that the system  120  can detect a FB correctly even when the digital amplitude gain control (AGC) amplifiers in the receiver cause a 180 degrees phase change in the received signal. 
     It is clear to one skilled in the art that, for very large frequency errors, for example, for an error larger than±10 kHz, more than three paths may be needed. The system  120  of FIG. 12 can be easily expanded to operate in such a case. 
     One embodiment of another system which also provides a lower probability of missed detection than the system of FIG. 3 is illustrated in FIG.  13 . This system  130  requires less hardware logic and cycle complexity than the system  120  of FIG. 12, but may waste one or more FB time in order to figure out whether a frequency error is very large. 
     Referring to FIG. 13, the system  130  includes the FB detector  30  of the present invention, six mixers  131 ,  132 ,  133 ,  134 ,  135 ,  136 , six FB correlators  141 ,  142 ,  143 ,  144 ,  145 ,  146 , and six peak modules  151 ,  152 ,  153 ,  154 ,  155 ,  156 . This is only an exemplary implementation of the system  130 . The system  130  can be implemented with any number of mixers, FB correlators and peak modules. 
     Referring to FIG. 13, the system  130  has  7  signal paths including a path marked “NOMIX path”. Except for the NOMIX path, each of the remaining paths includes a mixer  131 ,  132 ,  133 ,  134 ,  135 , or  136 , a FB correlator  141 ,  142 ,  143 ,  144 ,  145 , or  146 , and a peak module.  151 ,  152 ,  153 ,  154 ,  155 , or  156 . Each of the mixers is used to translate the center frequency of the received signal r(i) by a predetermined frequency amount. The predetermined amounts of frequency which are used as inputs to the mixers  131 ,  132 ,  133 ,  134 ,  135 ,  136  are f 1 , f 2 , f 3 , f 4 , f 5 , f 6 , respectively. For ease of implementation, f 1 , f 2 , f 3 , f 4 , f 5 , f 6  can be chosen such that f 1 =−f 4 , f 2 =−f 5 , f 3 =−f 6 . Due to their simple relationships, sampled values of these 6 frequencies can be retrieved from only three look-up tables, instead of six look-up tables. The three look-up tables are preferably stored in an on-chip random access memory (RAM). 
     Each of the FB correlators operates to search for a FB in a different translated signal. Thus, the FB correlators are searching for a FB over different ranges of frequencies. Appropriate choices of the frequencies f 1 , f 2 , f 3 , f 4 , f 5 , f 6  can make these ranges of frequencies mutually exclusive. The outputs of each of the FB correlators are correlation outputs, which are complex numbers, as described above. 
     Each of the peak modules  151 ,  152 ,  153 ,  154 ,  155 ,  156  computes a maximum of the magnitudes of the correlation outputs received from a respective FB correlator. Each of the peak modules also includes a register to store a threshold, compares the respective maximum with the threshold and only outputs the maximum if it is greater than the threshold. Each of the peak modules can also include a buffer to store a set of maxima. All of the maxima produced by all the peak modules  151 ,  152 ,  153 ,  154 ,  155 ,  156 , along with the outputs from the FB detector  30  will be examined by a decision module  160  at certain time, e.g., at a start of a TDMA frame. Either one of the maxima or the outputs of the FB detector will be chosen by the decision module  160 . If one of the maxima is chosen, then the corresponding mixer input frequency is selected to bias the reference oscillator of the receiver. For example, if the output. MAX 1  of the peak module  151  is chosen, then the frequency f 1  is selected to bias the oscillator. If the outputs of the FB detector is selected, then the selected frequency error  43  is used to biased the reference oscillator. 
     Each of the six FB correlators is the FB correlator of the present invention. In the embodiment of the system  130  as shown in FIG. 13, each of the FB correlators  141 ,  142 ,  143 ,  144 ,  145 ,  146  uses 64 samples of input signal to produce one correlation output, while the FB correlator  32  inside the FB detector  30  uses only 16 samples. The choice of  64  samples for the six FB correlators  141 ,  142 ,  143 ,  144 ,  145 ,  146  is to provide better performance in signal selectivity. By signal selectivity, it is meant that each of the six FB correlators operates within a very narrow frequency range and at least one of the six FB correlators will produce a high magnitude output. 
     FIG. 14 illustrates the magnitude of a FB correlator output as a function of frequency error for the two cases M=16 samples and M=64 samples. Referring to FIG. 14, the main lobe of the output magnitude for the case M=64 rolls off a lot faster the one for the case M=16. Since the main lobe  160  is narrow, the distance between the frequencies of the sidelobes and the frequency of the received signal is larger than the one for the case where M=16. Thus, each of the six FB correlators  141 ,  142 ,  143 ,  144 ,  145 ,  146  operates in a very small range of frequency. This requires that the paths of the six FB correlators be separated from each other in frequency by small amounts, in order to cover a full interval of search. This separation can be done by selecting suitable mixer frequency values f 1 , f 2 , f 3 , f 4 , f 5 , f 6 . 
     The frequency error due to an inexpensive crystal oscillator typically has a Gaussian probability distribution function with low variance. Thus, with high probability, the initial frequency error (for example, at start-up, when the cellular phone is turned on) is within±5 kHz, and can be detected with the FB detector  30  of the path marked “NOMIX path” in FIG.  13 . For the few times when the initial frequency error is greater than±5 kHz, one of the other paths in FIG. 13 will detect this error. This error estimate will then be used to bias the crystal oscillator of the receiver appropriately to reduce the error to within the operating range (±5 kHz) of the NOMIX path. The main purpose of the paths containing the FB correlators  141 ,  142 ,  143 ,  144 ,  145 ,  146  is to detect when the initial error is out of operating range of the FB detector  30 . For example, if the frequency error is near. −f 1  kHz, the output of the FB correlator  141  which succeeds the +f 1  kHz mixer  131  will have the highest magnitude output among all the FB correlator magnitude outputs. Thus, the output of the peak module  151  will be selected. The corresponding frequency f 1  will then be used to bias the oscillator. Subsequently, future FBs will arrive at the FB detector  30  with almost no frequency error. 
     The systems  120  of FIG. 12 and 130 of FIG. 13 find particular applications in the systems DCS 1800 and PCS 1900, where the initial frequency errors can be as large as twice the one in the GSM system when the same LO is used to generate the reference carrier frequency. 
     While this invention has been described with reference to illustrative embodiments, this description is not intended to be construed in a limiting sense. Various modifications of the illustrative embodiments, as well as other embodiments of the invention, which are apparent to persons skilled in the art to which the invention pertains are deemed to lie within the spirit and scope of the invention.