# Patent

## Title
Fractional carrier frequency offset estimator and method of estimating fractional carrier frequency offset between transmitter and receiver

## Abstract

Fractional carrier frequency offset estimators and related methods are disclosed. The invention may be particularly applied in WiMAX systems. The disclosed method of estimating fractional carrier frequency offset between a transmitter and a receiver includes calculating a first delay correlation based on a first and a second preamble repetitions included in a received signal, processing the phase of the first delay correlation to compensate for a phase rotation of the first delay correlation, and estimating the fractional carrier frequency offset by multiplying a processed phase value, generated after the step of processing the phase of the first delay correlation, by a predetermined value.

## Background
CROSS REFERENCE TO RELATED APPLICATIONS
This application claims the benefit of U.S. Provisional Application No. 60/831,931, filed 20, Jul. 2006.


BACKGROUND OF THE INVENTION
1. Field of the Invention
The invention relates to fractional carrier frequency offset estimators and methods of estimating fractional carrier frequency offset between receiver and transmitter. The invention may be particularly applied in Orthogonal Frequency Division Multiplexing (OFDM) related systems including Orthogonal Frequency Division Multiple Access (OFDMA) system, such as WiMAX system.
2. Description of the Related Art
FIG. 1 illustrates a wireless communication system 100, comprising a transmitter 102 and a receiver 104. In the transmitter 102, a signal s(t) is mixed with a signal ej2πftxl and is then broadcast by an antenna 106. The emitted broadcast signal is received by an antenna 108 of the receiver 104. In the receiver 104, the signal received by the antenna 108 is mixed with a signal of e−j2πfrxf to generate a received signal r(t), wherein r(t)=s(t)·ej2πftxt·e−j2πfrxl=s(t)·ej2π(ftx−frx)t=s(t)·ej2πfΔf, where fΔ represents carrier frequency offset between the transmitter 102 and the receiver 104. After normalizing the carrier frequency fΔ to the subcarrier spacing of the system, the carrier frequency offset fΔ is divided into integral portion fint and fractional portion ffrac, wherein fΔ=fint+ffrac. In a conventional technique, such as that disclosed in U.S. Pat. No. 5,732,113, the fractional carrier frequency offset between the transmitter and receiver (ffrac) is estimated by evaluating a delay correlation based on the first and second preamble repetitions of the received signal r(t) and multiplying the phase of the delay correlation by a predetermined value.
The preamble of a received signal in a WLAN (Wireless Local Area Network, IEEE 802.11a) system is shown in FIG. 2. The preamble comprises a short preamble and a long preamble. The short preamble comprises ten identical preamble repetitions A1 ̃A10, and the long preamble comprises two identical preamble repetitions B1 and B2. In the described conventional technique, the fractional carrier frequency offset (ffrac) can be estimated by using any two repetitions (e.g., A1 and A2) of the short preamble of a received signal r(t). Based on the sampled data in the first and second preamble repetitions (A1 and A2), a delay correlation, z12, is obtained by the formula:




  
    
      z
      12
    
    =
    
      
        ∑
        
          n
          =
          0
        
        
          N
          -
          1
        
      
      ⁢
      
        
      
      ⁢
      
        
          r
          n
        
        ·
        
          r
          
            (
            
              n
              +
              D
            
            )
          
          *
        
      
    
  
  ,




where N represents the accumulation samples included in one preamble repetition, D represents the delay between the identical samples of the two preamble repetitions (e.g., A1 and A2), and ‘*’ represents a complex conjugate operation. The sample duration of the received signal r(t) is T. Because 0≦n≦(N−1), rn represents samples in the first preamble repetition A1, rn+D represents samples in the second preamble repetition A2. FIG. 1 shows—r(t) equals s(t)·ej2πfΔt—rn is sn·ej2πfΔnTs and rn+D is sn+D·ej2πfΔ(n+D)Ts. When noise and frequency disturbance are not present in the communication system, sn equals sn+D. The delay correlation, z12, can be further simplified as the following:





  
    
      
        
          z
          12
        
        =
        
          
            
              ∑
              
                n
                =
                0
              
              
                N
                -
                1
              
            
            ⁢
            
              
            
            ⁢
            
              
                s
                n
              
              ·
              
                e
                
                  j
                  ⁢
                  
                    
                  
                  ⁢
                  2
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  
                    f
                    Δ
                  
                  ⁢
                  
                    nT
                    s
                  
                
              
              ·
              
                
                  (
                  
                    
                      s
                      
                        n
                        +
                        D
                      
                    
                    ·
                    
                      e
                      
                        j
                        ⁢
                        
                          
                        
                        ⁢
                        2
                        ⁢
                        π
                        ⁢
                        
                          
                        
                        ⁢
                        
                          
                            f
                            Δ
                          
                          ⁡
                          
                            (
                            
                              n
                              +
                              D
                            
                            )
                          
                        
                        ⁢
                        
                          T
                          s
                        
                      
                    
                  
                  )
                
                *
              
            
          
          =
          
            
              e
              
                
                  -
                  j
                
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                
                  f
                  Δ
                
                ⁢
                
                  DT
                  s
                
              
            
            ⁢
            
              
                ∑
                
                  n
                  =
                  0
                
                
                  N
                  -
                  1
                
              
              ⁢
              
                
              
              ⁢
              
                
                  
                    
                    
                      s
                      n
                    
                    
                  
                  2
                
                .
              
            
          
        
      
    
    
      
        (
        
          eq
          .
          
            
          
          ⁢
          1
        
        )
      
    
  




Based on the phase of the delay correlation (∠z12), the fractional carrier frequency offset between the transmitter and receiver can be determined by multiplying the phase of the delay correlation (∠z12) by −1/(2πDTs), thus, when the first and second preamble repetitions are exactly identical, the fractional carrier frequency offset is estimated by the formula:





  
    
      
        
          f
          frac
        
        =
        
          
            -
            
              1
              
                2
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                
                  DT
                  s
                
              
            
          
          ⁢
          ∠
          ⁢
          
            
          
          ⁢
          
            z
            12
          
        
      
    
    
      
        (
        
          eq
          .
          
            
          
          ⁢
          2
        
        )
      
    
  




The IEEE 802.16e standard, commonly referred to as the WiMAX (Worldwide Interoperability for Microwave Access), its preamble has three-repetition property. Thus, the delay correlation operation can be applied. The 3-repetition preamble is designed for 3-sector cellular planning. FIG. 3 shows the cellular planning of a WiMAX system. Each cell is divided into three sectors; sector 1, sector 2 and sector 3. In sector 1, a delay correlation, z12, based on the first and second preamble repetitions of a received signal, is C1·e−j2πfΔDTs·ejφ1 because the second preamble repetition is not exactly identical to the first preamble repetition. Compared with equation 1, the delay correlation z12 of a signal in sector 1 has a phase rotation of φ1. Similarly, delay correlations of signals in sector 2 and sector 3 have phase rotations of φ2 and φ3, respectively. Thus, an improved method for fractional carrier frequency offset estimation is desirable.
BRIEF SUMMARY OF THE INVENTION
The invention relates to fractional carrier frequency offset estimators and methods of estimating fractional carrier frequency offset between receivers and transmitters. The invention may be applied in Orthogonal Frequency Division Multiple Access (OFDMA) systems.
An exemplary embodiment of the method for estimating fractional carrier frequency offset between a transmitter and a receiver comprises: calculating a first delay correlation based on a first and a second preamble repetitions included in a received signal; processing the phase of the first delay correlation to compensate for a phase rotation of the first delay correlation; estimating the fractional carrier frequency offset by multiplying a processed phase value, generated after the step of processing the phase of the first delay correlation, by a predetermined value. The invention further discloses fractional carrier frequency offset estimators complying with the disclosed method.
The above and other advantages will become more apparent with reference to the following description provided in conjunction with the accompanying drawings.



BRIEF DESCRIPTION OF THE DRAWINGS
The invention can be more fully understood by reading the subsequent detailed description and examples with references made to the accompanying drawings, wherein:
FIG. 1 illustrates a wireless communication system;
FIG. 2 shows the preamble of a signal of WLAN system;
FIG. 3 shows cell structure of a WiMAX system;
FIG. 4 shows a flow chart of one embodiment of the disclosed method;
FIG. 5 shows one embodiment of the disclosed fractional carrier frequency offset estimator based on equation 3; and
FIG. 6 shows one embodiment of the disclosed fractional carrier frequency offset estimator based on equation 4.



DETAILED DESCRIPTION OF THE INVENTION
The following description is made for the purpose of illustrating the general principles of the invention and should not be taken in a limiting sense. The scope of the invention is best determined by reference to the appended claims.
For WiMAX system (IEEE 802.16e), the preamble has 3-repetition property in time domain, thus, the delay-correlation operation can be applied. However, these three repetitions are not exactly identical for sampled preamble sequence, if sampling rate is not threefold signal bandwidth (In communication system, sampling rate is usually twofold or fourfold signal bandwidth). The delay correlation will induce an undesired phase term. For a mathematical expression:




  
    
      
        
          
            z
            12
          
          =
          
            
              
                ∑
                
                  n
                  =
                  0
                
                
                  N
                  -
                  1
                
              
              ⁢
              
                
              
              ⁢
              
                
                  r
                  n
                
                ⁢
                
                  r
                  
                    n
                    +
                    D
                  
                  *
                
              
            
            =
            
              
                ∑
                
                  n
                  =
                  0
                
                
                  N
                  -
                  1
                
              
              ⁢
              
                
              
              ⁢
              
                
                  s
                  n
                
                ⁢
                
                  e
                  
                    j2π
                    ⁢
                    
                      
                    
                    ⁢
                    
                      f
                      Δ
                    
                    ⁢
                    
                      nT
                      s
                    
                  
                
                ⁢
                
                  s
                  
                    n
                    +
                    D
                  
                
                ⁢
                
                  e
                  
                    
                      -
                      j2π
                    
                    ⁢
                    
                      
                    
                    ⁢
                    
                      
                        f
                        Δ
                      
                      ⁡
                      
                        (
                        
                          n
                          +
                          D
                        
                        )
                      
                    
                    ⁢
                    
                      T
                      s
                    
                  
                
              
            
          
        
      
    
    
      
        
          =
          
            
              e
              
                
                  -
                  j2π
                
                ⁢
                
                  
                
                ⁢
                
                  f
                  Δ
                
                ⁢
                
                  DT
                  s
                
              
            
            ⁢
            
              
                ∑
                
                  n
                  =
                  0
                
                
                  N
                  -
                  1
                
              
              ⁢
              
                
              
              ⁢
              
                
                  
                  
                    a
                    n
                  
                  
                
                ⁢
                
                  e
                  
                    jθ
                    n
                  
                
                ⁢
                
                  
                  
                    a
                    
                      n
                      +
                      D
                    
                  
                  
                
                ⁢
                
                  e
                  
                    -
                    
                      jθ
                      
                        n
                        +
                        D
                      
                    
                  
                
              
            
          
        
      
    
    
      
        
          =
          
            
              C
              ·
              
                e
                
                  
                    -
                    j2π
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    f
                    Δ
                  
                  ⁢
                  
                    DT
                    s
                  
                
              
            
            ⁢
            
              e
              
                j
                ⁢
                
                  
                
                ⁢
                φ
              
            
          
        
      
    
  
  ,




with sn=|an|ejθn, and





  
    C
    ×
    
      e
      
        j
        ⁢
        
          
        
        ⁢
        φ
      
    
  
  =
  
    
      ∑
      
        n
        =
        0
      
      
        N
        -
        1
      
    
    ⁢
    
      
    
    ⁢
    
      
        
        
          a
          n
        
        
      
      ⁢
      
        
        
          a
          
            n
            +
            D
          
        
        
      
      ⁢
      
        
          e
          
            j
            ⁡
            
              (
              
                
                  θ
                  n
                
                -
                
                  θ
                  
                    n
                    +
                    D
                  
                
              
              )
            
          
        
        .
      
    
  




We refer to φ an undesired phase term or phase rotation. In addition, the φ value will depend on different preambles. The WiMAX system deploys a cellular planning, three sectors per cell, and each sector has its distinct preamble pattern. Therefore, for different sector (that is different preamble), the undesired phase term has different value. Assume the undesired phase term of delay correlation for sector 1 is φ1, sector 2 is φ2, and sector 3 is φ3. The difference between any two phase terms is 120 degree. In WiMAX system, the experiment shows that the undesired phase terms approximate 3 values, 0 degree and +/−120 degrees. When a WiMAX subscriber received the WiMAX base station signal, it can not distinguish which sector the subscriber belongs to before cell searching. Thus, the delay correlation operation will have an undesired phase term, If multiplying the delay correlation function z12 by ejφ1, ejφ2, ejφ3, and generates three processed delay correlations z1, z2, and z3, respectively, wherein,

z1=z12·ejφ1=C·e−j2πfΔDTsej(φ−φ1);

z2=z12·ejφ2=C·e−j2πfΔDTsej(φ−φ2);

z3=z12·ejφ3=C·e−j2πfΔDTsej(φ−φ3) 

Take the preamble of sector 2 for example. When the received preamble belongs to sector 2 (i.e.,





  
    (
    
      
        i
        .
        e
        .
      
      ,
      
        φ
        =
        
          
            φ
            2
          
          =
          
            
              120
              °
            
            =
            
              
                2
                3
              
              ⁢
              π
              ⁢
              
                
              
              ⁢
              radian
            
          
        
      
    
    )
  
  ,




the phase terms of the processed delay correlation are:





  
    
      ∠
      ⁢
      
        
      
      ⁢
      
        z
        1
      
    
    =
    
      
        
          
            -
            2
          
          ⁢
          π
          ⁢
          
            
          
          ⁢
          
            f
            Δ
          
          ⁢
          
            DT
            s
          
        
        +
        
          (
          
            
              
                2
                3
              
              ⁢
              π
            
            -
            0
          
          )
        
      
      =
      
        
          
            -
            2
          
          ⁢
          π
          ⁢
          
            
          
          ⁢
          
            f
            Δ
          
          ⁢
          
            DT
            s
          
        
        +
        
          
            2
            3
          
          ⁢
          π
        
      
    
  
  ,
  
    

  
  ⁢
  
    
      ∠
      ⁢
      
        
      
      ⁢
      
        z
        2
      
    
    =
    
      
        
          
            -
            2
          
          ⁢
          π
          ⁢
          
            
          
          ⁢
          
            f
            Δ
          
          ⁢
          
            DT
            s
          
        
        +
        
          (
          
            
              
                2
                3
              
              ⁢
              π
            
            -
            
              
                2
                3
              
              ⁢
              π
            
          
          )
        
      
      =
      
        
          -
          2
        
        ⁢
        π
        ⁢
        
          
        
        ⁢
        
          f
          Δ
        
        ⁢
        
          DT
          s
        
      
    
  
  ,
  
    

  
  ⁢
  
    
      ∠
      ⁢
      
        
      
      ⁢
      
        z
        3
      
    
    =
    
      
        
          
            -
            2
          
          ⁢
          π
          ⁢
          
            
          
          ⁢
          
            f
            Δ
          
          ⁢
          
            DT
            s
          
        
        +
        
          (
          
            
              
                2
                3
              
              ⁢
              π
            
            +
            
              
                2
                3
              
              ⁢
              π
            
          
          )
        
      
      =
      
        
          
            -
            2
          
          ⁢
          π
          ⁢
          
            
          
          ⁢
          
            f
            Δ
          
          ⁢
          
            DT
            s
          
        
        +
        
          
            4
            3
          
          ⁢
          π
        
      
    
  
  ,




The summation of ∠z1, ∠z2 and ∠z3 is (−6πfΔDTs+2π). Fractional carrier frequency offset between transmitter and receiver (ffrac) can be estimated by converting (−6πfΔDTs+2π) to a range −π ̃π and multiplying the converted phase value by





  -
  
    
      1
      
        6
        ⁢
        π
        ⁢
        
          
        
        ⁢
        
          DT
          s
        
      
    
    .
  




When the received signal belongs to sector 1 or 3, the fractional carrier frequency offset ffrac can be similarly estimated. The fractional carrier frequency offset between transmitter and receiver (ffrac) can be estimated by the formula:





  
    f
    frac
  
  =
  
    
      -
      
        1
        
          6
          ⁢
          π
          ⁢
          
            
          
          ⁢
          
            DT
            s
          
        
      
    
    ×
    
      
        (
        
          
            ∠
            ⁢
            
              
            
            ⁢
            
              z
              1
            
          
          +
          
            ∠
            ⁢
            
              
            
            ⁢
            
              z
              2
            
          
          +
          
            ∠
            ⁢
            
              
            
            ⁢
            
              z
              3
            
          
        
        )
      
      .
    
  




Furthermore, the summation of ∠z1, ∠z2 and ∠z3 can be simplified as the following:





  
    
      
        
          
            •
            ⁢
            
              
            
            ⁢
            z
            ⁢
            
              
            
            ⁢
            1
          
          +
          
            •
            ⁢
            
              
            
            ⁢
            z
            ⁢
            
              
            
            ⁢
            2
          
          +
          
            •
            ⁢
            
              
            
            ⁢
            z
            ⁢
            
              
            
            ⁢
            3
          
        
        =
        
          
            
              
                
                  (
                  
                    
                      
                        -
                        2
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      
                        f
                        Δ
                      
                      ⁢
                      
                        DT
                        s
                      
                    
                    +
                    φ
                    -
                    
                      φ
                      1
                    
                  
                  )
                
                +
              
            
          
          
            
              
                
                  (
                  
                    
                      
                        -
                        2
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      
                        f
                        Δ
                      
                      ⁢
                      
                        DT
                        s
                      
                    
                    +
                    φ
                    -
                    
                      φ
                      2
                    
                  
                  )
                
                +
                
                  (
                  
                    
                      
                        -
                        2
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      
                        f
                        Δ
                      
                      ⁢
                      
                        DT
                        s
                      
                    
                    +
                    φ
                    -
                    
                      φ
                      3
                    
                  
                  )
                
              
            
          
        
      
    
  
  
    
      
        =
        
          
            3
            ·
            
              (
              
                
                  
                    -
                    2
                  
                  ⁢
                  π
                  ⁢
                  
                    
                  
                  ⁢
                  
                    f
                    Δ
                  
                  ⁢
                  
                    DT
                    s
                  
                
                +
                φ
              
              )
            
          
          -
          
            (
            
              
                φ
                1
              
              +
              
                φ
                2
              
              +
              
                φ
                3
              
            
            )
          
        
      
    
  
  
    
      
        =
        
          
            
              3
              ·
              •
            
            ⁢
            
              
            
            ⁢
            
              z
              12
            
          
          -
          
            (
            
              
                φ
                1
              
              +
              
                φ
                2
              
              +
              
                φ
                3
              
            
            )
          
        
      
    
  
  
    
      
        =
        
          
            [
            
              
                3
                ·
                ∠
              
              ⁢
              
                
              
              ⁢
              
                z
                12
              
            
            ]
          
          
            -
            π
          
          π
        
      
    
  




because the summation of φ1, φ2 and φ3 is 2π radian in WiMAX systems and [phase]−ππ represents a function of converting phase to a range −π ̃π. Thus, the fractional carrier frequency offset ffrac is estimated by the formula:





  
    
      
        
          f
          frac
        
        =
        
          
            -
            
              1
              
                6
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                
                  DT
                  s
                
              
            
          
          ×
          
            
              [
              
                
                  3
                  ·
                  ∠
                
                ⁢
                
                  
                
                ⁢
                
                  z
                  12
                
              
              ]
            
            
              -
              π
            
            π
          
        
      
    
    
      
        (
        
          eq
          .
          
            
          
          ⁢
          3
        
        )
      
    
  




It should be noted that equation 3 is not limited to a system in which φ1 is 0°, φ2 is 120° and φ3 is −120°. Equation 3 is applicable to estimating fractional carrier frequency offset in any system where the phase difference between φ1, φ2 and φ3 is 120°.
Although the invention performs well in fixed channel models, such as the SUI-4 channel model and the AWGN channel model, the invention does not perform very well in channel models for mobile devices, such as the ITU-R Vehicular-A channel model. In such a case, an alternative fractional carrier frequency offset estimator was invented, it requires two delay correlations z12 and z13, where z12 is obtained from the first and second preamble repetitions of the received preamble and z13 is obtained from the first and third preamble repetitions of the received preamble. z12 and z13 are obtained by the formulae:



  
    
      z
      12
    
    =
    
      
        
          ∑
          
            n
            =
            0
          
          
            N
            -
            1
          
        
        ⁢
        
          
        
        ⁢
        
          
            r
            n
          
          ·
          
            r
            
              (
              
                n
                +
                D
              
              )
            
            *
          
        
      
      =
      
        
          
            ∑
            
              n
              =
              0
            
            
              N
              -
              1
            
          
          ⁢
          
            
          
          ⁢
          
            
              s
              n
            
            ·
            
              e
              
                j
                ⁢
                
                  
                
                ⁢
                2
                ⁢
                
                  πf
                  Δ
                
                ⁢
                
                  nT
                  s
                
              
            
            ·
            
              
                (
                
                  
                    s
                    
                      n
                      +
                      D
                    
                  
                  ·
                  
                    e
                    
                      j2π
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          f
                          Δ
                        
                        ⁡
                        
                          (
                          
                            n
                            +
                            D
                          
                          )
                        
                      
                      ⁢
                      
                        T
                        s
                      
                    
                  
                
                )
              
              *
            
          
        
        =
        
          A
          ·
          
            e
            
              
                -
                j
              
              ⁢
              
                
              
              ⁢
              2
              ⁢
              π
              ⁢
              
                
              
              ⁢
              
                f
                Δ
              
              ⁢
              
                DT
                s
              
            
          
          ·
          
            e
            jφ
          
        
      
    
  




  and




  
    
      z
      13
    
    =
    
      
        
          ∑
          
            n
            =
            0
          
          
            N
            -
            1
          
        
        ⁢
        
          
        
        ⁢
        
          
            r
            n
          
          ·
          
            r
            
              (
              
                n
                +
                
                  2
                  ⁢
                  D
                
              
              )
            
            *
          
        
      
      =
      
        
          
            ∑
            
              n
              =
              0
            
            
              N
              -
              1
            
          
          ⁢
          
            
          
          ⁢
          
            
              s
              n
            
            ·
            
              e
              
                j2π
                ⁢
                
                  
                
                ⁢
                
                  f
                  Δ
                
                ⁢
                
                  nT
                  s
                
              
            
            ·
            
              
                (
                
                  
                    s
                    
                      n
                      +
                      
                        2
                        ⁢
                        D
                      
                    
                  
                  ·
                  
                    e
                    
                      j2
                      ⁢
                      
                        
                      
                      ⁢
                      π
                      ⁢
                      
                        
                      
                      ⁢
                      
                        
                          f
                          Δ
                        
                        ⁡
                        
                          (
                          
                            n
                            +
                            
                              2
                              ⁢
                              D
                            
                          
                          )
                        
                      
                      ⁢
                      
                        T
                        s
                      
                    
                  
                
                )
              
              *
            
          
        
        =
        
          B
          ·
          
            e
            
              
                -
                j2π
              
              ⁢
              
                
              
              ⁢
              
                f
                Δ
              
              ⁢
              2
              ⁢
              
                DT
                s
              
            
          
          ·
          
            e
            jθ
          
        
      
    
  




  where




  
    
      
        A
        ·
        
          e
          jφ
        
      
      =
      
        
          ∑
          
            n
            =
            0
          
          
            N
            -
            1
          
        
        ⁢
        
          
        
        ⁢
        
          
            
            
              s
              n
            
            
          
          ⁢
          
            
            
              s
              
                n
                +
                D
              
            
            
          
          ⁢
          
            e
            
              j
              ⁡
              
                (
                
                  
                    φ
                    n
                  
                  -
                  
                    φ
                    
                      n
                      +
                      D
                    
                  
                
                )
              
            
          
        
      
    
    ,
    
      

    
    ⁢
    and
  




  
    
      B
      ·
      
        e
        jθ
      
    
    =
    
      
        ∑
        
          n
          =
          0
        
        
          N
          -
          1
        
      
      ⁢
      
        
      
      ⁢
      
        
          
          
            s
            n
          
          
        
        ⁢
        
          
          
            s
            
              n
              +
              
                2
                ⁢
                D
              
            
          
          
        
        ⁢
        
          
            e
            
              j
              ⁡
              
                (
                
                  
                    φ
                    n
                  
                  -
                  
                    φ
                    
                      n
                      +
                      
                        2
                        ⁢
                        D
                      
                    
                  
                
                )
              
            
          
          .
        
      
    
  



In WiMAX system, θ=−φ. The summation of the phases of the delay correlations z12 and z13 is −6πfΔDTs+(φ+θ) and can be further simplified to −6πfΔDTs. Thus, the fractional carrier frequency offset ffrac is obtained by the formula:





  
    
      
        
          f
          frac
        
        =
        
          
            -
            
              1
              
                6
                ⁢
                π
                ⁢
                
                  
                
                ⁢
                
                  DT
                  s
                
              
            
          
          ×
          
            (
            
              
                •
                ⁢
                
                  
                
                ⁢
                
                  z
                  12
                
              
              +
              
                •
                ⁢
                
                  
                
                ⁢
                
                  z
                  13
                
              
            
            )
          
        
      
    
    
      
        (
        
          eq
          .
          
            
          
          ⁢
          4
        
        )
      
    
  




It should be noted that equation 4 is not limited to channel models for mobile devices, such as the ITU-R Vehicular-A channel model. The fractional carrier frequency offset estimator based on equation 4 still performs well in other channel models, such as the SUI-4 and AWGN channel models.
A flow chart of an embodiment of the disclosed method is shown in FIG. 4. Based on the first and second preamble repetitions included in a received signal r(t), a first delay correlation z12 is obtained in step 402, wherein




  
    
      z
      12
    
    =
    
      
        ∑
        
          n
          =
          0
        
        
          N
          -
          1
        
      
      ⁢
      
        
      
      ⁢
      
        
          r
          n
        
        ·
        
          r
          
            (
            
              n
              +
              D
            
            )
          
          *
        
      
    
  
  ,




where N represents the accumulation samples included in one preamble repetition, D represents the delay between the identical samples of the two preamble repetitions, and ‘*’ acts as complex conjugation. There is a phase rotation φ included in the first delay correlation z12. In step 404, the first delay correlation phase (∠z12) is processed to compensate for the phase rotation of the first delay correlation (φ), generating a processed phase value f(∠z12). In step 406, the processed phase value f(∠z12) is multiplied by a predetermined value A to estimate fractional carrier frequency offset between transmitter and receiver. It should be noted that z12 is not limited to be calculated by the first and second preamble repetitions in a received signal r(t), z12 could be calculated by any two adjacent preamble.

In an embodiment of the invention, the received signal r(t) may belong to a first sector, a second sector or a third sector. When the received signal belongs to the first sector, phase rotation of the first delay correlation z12 is φ1. When the received signal belongs to the second sector, phase rotation of the first delay correlation z12 is φ2. When the received signal belongs to the third sector, phase rotation of the first delay correlation z12 is φ3. Phase difference between φ1, φ2 and φ3 is 120°. The modulation scheme of the received signal in the invention is Orthogonal Frequency Division Multiplexing (OFDM), particularly applied to the IEEE 802.16e standard, commonly referred to as WiMAX.
In an embodiment of the invention based on equation 3, the process in step 404 further comprises evaluating a first value by multiplying the phase of the first delay correlation (∠z12) by 3, and converting the first value (3·∠z12) to a range −π ̃π to generate the processed phase value f(∠z12), wherein f(∠z12)=[3·∠z12]−ππ, where [phase]−ππ, acts as a function for converting phase to a range −π ̃π. The predetermined value A equals




  
    -
    
      1
      
        6
        ⁢
        π
        ⁢
        
          
        
        ⁢
        
          DT
          s
        
      
    
  
  ,




where π represents ratio of the circumference of a circle to its diameter, and Ts represents the sample duration of the received signal r(t).

In an embodiment of the invention based on equation 4, the process in step 404 further comprises calculating a second delay correlation z13 based on the first and third preamble repetitions of the received signal r(t) (according to the formula:




  
    
      
        z
        13
      
      =
      
        
          ∑
          
            n
            =
            0
          
          
            N
            -
            1
          
        
        ⁢
        
          
        
        ⁢
        
          
            r
            n
          
          ·
          
            r
            
              (
              
                n
                +
                
                  2
                  ⁢
                  D
                
              
              )
            
            *
          
        
      
    
    )
  
  ,




and adding the phases of the first and second delay correlations up to generate the processed phase value f(∠z12), wherein f(∠z12)=(∠z12+∠z13). The predetermined value A equals





  
    -
    
      1
      
        6
        ⁢
        π
        ⁢
        
          
        
        ⁢
        
          DT
          s
        
      
    
  
  ,




where π represents the ratio of the circumference of a circle to its diameter, and Ts represents the sample duration of the received signal r(t).

An embodiment of a fractional carrier frequency offset estimator of the invention based on equation 3 is shown in FIG. 5. The fractional carrier frequency offset estimator 500 comprises a first delay correlation calculator 502, a phase rotation compensator 504, and a first multiplier 506. The first delay correlation calculator 502 calculates a first delay correlation z12 based on the first and second preamble repetitions of a received signal r(t). The first delay correlation z12 is calculated according to the formula:




  
    
      z
      12
    
    =
    
      
        ∑
        
          n
          =
          0
        
        
          N
          -
          1
        
      
      ⁢
      
        
      
      ⁢
      
        
          r
          n
        
        ·
        
          r
          
            (
            
              n
              +
              D
            
            )
          
          *
        
      
    
  
  ,




where N represents the accumulation samples included in one preamble repetition, D represents the delay between the identical samples of the two preamble repetitions, and ‘*’ acts as a conjugate function. There is a phase rotation φ included in the first delay correlation z12. To compensate for the phase rotation of the first delay correlation (φ), the phase rotation compensator 504 processes the phase of the first delay correlation (∠z12) and generates a processed phase value f(∠z12). As shown in FIG. 5, the phase rotation compensator 504 further comprises a phase estimator 508, a second multiplexer 510, and a phase converter 512. The phase estimator 508 estimates the phase of the first delay correlation (∠z12). The second multiplexer 510 multiplies the phase of the first delay correlation (∠z12) by 3 to obtain a first value (3·∠z12). The phase converter 512 converts the first value to a range −π ̃π to generate the processed phase value [3·∠z12]−ππ, where [phase]−ππ, acts as a function of converting phase to a range −π ̃π. The first multiplier 506 multiplies the processed phase value [3·∠z12]−ππ by a predetermined value





  (
  
    -
    
      1
      
        6
        ⁢
        π
        ⁢
        
          
        
        ⁢
        
          DT
          s
        
      
    
  
  )




to estimate fractional carrier frequency offset between a transmitter and a receiver, where π represents ratio of the circumference of a circle to its diameter, and Ts represents the sample duration of the received signal r(t).

An embodiment of fractional carrier frequency offset estimator of the invention based on equation 4 is shown in FIG. 6. Compared to the fractional carrier frequency offset estimator 500, the fractional carrier frequency offset estimator 600 has a different phase rotation compensator 602. The phase rotation compensator 602 comprises a first phase estimator 604, a second delay correlation calculator 606, a second phase estimator 608, and an adder 610. The second delay correlation calculator 606 calculates a second delay correlation z13 based on the first and third preamble repetitions of the received signal r(t) (according to the formula:




  
    
      
        
          z
          13
        
        =
        
          
            ∑
            
              n
              =
              0
            
            
              N
              -
              1
            
          
          ⁢
          
            
          
          ⁢
          
            
              r
              n
            
            ·
            
              r
              
                (
                
                  n
                  +
                  
                    2
                    ⁢
                    D
                  
                
                )
              
              *
            
          
        
      
      )
    
    .
  
  ⁢
  
    
  




It should be noted that z13 is not limited to be calculated by the first and third preamble repetitions of the received signal r(t). The first and second phase estimators 604 and 608 respectively estimate the phases of the first and second delay correlations (∠z12 and ∠z13). The adder 610 adds the phases of the first and second delay correlations (∠z12 and ∠z13) to generate the processed phase value (∠z12+∠z13). The fractional carrier frequency offset estimator 600 performs well in channel models for mobile devices, such as the ITU-R Vehicular-A channel model.

While the invention has been described by way of example and in terms of preferred embodiment, it is to be understood that the invention is not limited thereto. To the contrary, it is intended to cover various modifications and similar arrangements (as would be apparent to those skilled in the art). Therefore, the scope of the appended claims should be accorded to the broadest interpretation so as to encompass all such modifications and similar arrangements.

## Claims

1. A method of estimating fractional carrier frequency offset between a transmitter and a receiver, comprising:
calculating a first delay correlation based on a first and a second preamble repetitions included in a received signal;
processing the phase of the first delay correlation to compensate for a phase rotation of the first delay correlation, wherein a processed phase value is generated; and
estimating the fractional carrier frequency offset between the transmitter and the receiver by multiplying the processed phase value by a predetermined value,
wherein in the step of processing the phase of the first delay correlation further comprises:
evaluating a first value by multiplying the phase of the first delay correlation by 3; and
converting the first value to a range −π ̃π to generate the processed phase value.

2. The method as claimed in claim 1, wherein the received signal can be classified as belonging to a first sector, a second sector or a third sector, wherein phase rotations of first delay correlations calculated from received signals belonging to the three different sectors are different and have a phase difference of 120 degrees therebetween.

3. The method as claimed in claim 1, wherein the modulation scheme of the received signal is Orthogonal Frequency Division Multiplexing.

4. The method as claimed in claim 3, wherein the Orthogonal Frequency Division Multiple Access is used in IEEE 802.16e standard, commonly referred to as WiMAX.

5. The method as claimed in claim 1, wherein the predetermined value is the inverse of minus 6·π·D·Ts, where π represents the ratio of the circumference of a circle to its diameter, Ts represents the sample duration of the received signal, and D represents the delay between the first and the second preamble repetitions.

6. A method of estimating fractional carrier frequency offset between a transmitter and a receiver, comprising:
calculating a first delay correlation based on a first and a second preamble repetitions included in a received signal;
processing the phase of the first delay correlation to compensate for a phase rotation of the first delay correlation, wherein a processed phase value is generated; and
estimating the fractional carrier frequency offset between the transmitter and the receiver by multiplying the processed phase value by a predetermined value,
wherein the step of processing the phase of the first delay correlation further comprises:
calculating a second delay correlation based on the first and a third preamble repetitions of the received signal; and
adding the phases of the first and second delay correlations to generate the processed phase value.

7. The method as claimed in claim 6, wherein the predetermined value is the inverse of minus 6·πD·Ts, where π represents ratio of the circumference of a circle to its diameter, Ts represents the sample duration of the received signal, and D represents the delay between the first and second preamble repetitions.

8. A fractional carrier frequency offset estimator, comprising:
a first delay correlation calculator, calculating a first delay correlation based on a first and a second preamble repetitions of a received signal;
a phase rotation compensator, compensating for a phase rotation of the first delay correlation by processing the phase of the first delay correlation, wherein a processed phase value is generated; and
a first multiplier, multiplying the processed phase value by a predetermined value to estimate fractional carrier frequency offset between a transmitter and a receiver,
wherein the phase rotation compensator further comprises:
a phase estimator, estimating the phase of the first delay correlation;
a second multiplexer, evaluating a first value by multiplying the phase of the first delay correlation by 3; and
a phase converter, converting the first value to a range to generate the processed phase value.

9. The fractional carrier frequency offset estimator as claimed in claim 8, wherein the received signal can be classified as belonging to a first sector, a second sector or a third sector, wherein phase rotations of first delay correlations calculated from received signals belonging to the three different sectors are different and have a phase difference of 120 degrees therebetween.

10. The fractional carrier frequency offset estimator as claimed in claim 8, wherein the modulation scheme of the received signal is Orthogonal Frequency Division Multiple Access.

11. The fractional carrier frequency offset estimator as claimed in claim 10, wherein the Orthogonal Frequency Division Multiple Access is used in IEEE 802.16e standard, commonly referred to as WiMAX.

12. The fractional carrier frequency offset estimator as claimed in claim 8, wherein the predetermined value is the inverse of minus 6·π·D·Ts, where π represents ratio of the circumference of a circle to its diameter, Ts represents the sample duration of the received signal, and D represents delay between the first and second preamble repetitions.

13. A fractional carrier frequency offset estimator, comprising:
a first delay correlation calculator, calculating a first delay correlation based on a first and a second preamble repetitions of a received signal;
a phase rotation compensator, compensating for a phase rotation of the first delay correlation by processing the phase of the first delay correlation, wherein a processed phase value is generated; and
a first multiplier, multiplying the processed phase value by a predetermined value to estimate fractional carrier frequency offset between a transmitter and a receiver,
wherein the phase rotation compensator further comprises:
a first phase estimator, estimating the phase of the first delay correlation;
a second delay correlation calculator, calculating a second delay correlation based on the first and a third preamble repetitions of a received signal;
a second phase estimator, estimating the phase of the second delay correlation; and
an adder, adding the phases of the first and second delay correlations to generate the processed phase value.

14. The fractional carrier frequency offset estimator as claimed in claim 13, wherein the predetermined value is the inverse of minus 6·π·D·Ts, where π represents ratio of the circumference of a circle to its diameter, Ts represents the sample duration of the received signal, and D represents delay between the first and second preamble repetitions.