1. Field of the Invention
The present invention relates to a method for data communication of signals in units of a frame using an orthogonal frequency division modulation (OFDM) algorithm, and in particular, to a method to find and synchronize the initial frame using a globally optimal maximum likelihood (ML) estimate of joint carrier frequency offset and symbol timing error at the receiver.
2. Description of the Prior Art
OFDM system is a viable modulation scheme for data transmission over time varying dynamic channels. However, it is known that performance of such system is highly susceptible to non-ideal synchronization parameters. Specifically, symbol timing and carrier frequency offset become an increasingly important issue in implementation of OFDM systems for practical applications. It is known that carrier frequency offset deteriorates performance of OFDM systems by introducing interference among the sub-channels. To overcome this imperfection, various compensation methods for estimation and correction of synchronization parameters are known in the prior art.
These prior art methods for synchronization of OFDM systems can be classified into two main subclasses, namely minimum mean square error (MMSE) and ML estimators. In MMSE approach, the estimator uses the information provided by the reference signal (pilot tones) in order to minimize a cost function associated with the synchronization parameters. A salient feature of this approach is that no probabilistic assumptions are made with regard to the data. Although MMSE estimators usually result in a tractable (globally stable) and easy to implement realization, no optimal criteria (probabilistic) is associated with these estimators. Also, since part of the transmitted information is allocated to the reference pilots, the bandwidth efficiency of these methods is lower in comparison to the non-pilot schemes.
On the other hand, ML estimators provide the estimate of the unknown parameter subject to minimum probability of error criteria. Although not perfectly efficient, ML estimators are asymptotically minimum variance unbiased (MVU), i.e., their variance attains that of MVU estimator as the length of data record goes to infinity. However, due to the physical constraints, systems with infinitely long data records are not feasible for implementation purposes.
P. H. Moose, in xe2x80x9cA Technique for Orthogonal Frequency Division Multiplexing Frequency Offset Correction,xe2x80x9d IEEE Trans. on Communications, Vol. 42, No. 10, pp. 2098-2913, October 1994, describes the use of a retransmission technique in order to reveal the frequency offset parameter in the likelihood function of the received signal. Due to the redundancy introduced by repeating the data block, the data rate efficiency is decreased by a factor of two. To avoid this imperfection, a ML estimator based on cyclic prefix (CP) is described by J. van de Beek, M. Standel and P. O. Borjesson, in xe2x80x9cML Estimation of Timing and Frequency Offset in OFDM Systems,xe2x80x9d IEEE Trans. on Signal Processing, Vol. 45, No. 3, pp. 1800-1805, July 1997. In this approach, the side information provided by the CP is used to obtain the likelihood function for joint estimation of symbol timing error and frequency offset in an OFDM system.
The likelihood function described in the Van de Beek reference does not globally characterize the observation vector over the entire range of the timing offset. The ML estimator of the prior art results in a considerable performance loss over finite range of timing offset parameter. More specifically, for a given symbol length, the probability of false ML estimation approaches one as the length of cyclic prefix increases.
Currently, there is increasing interest in multi-carrier modulation (MCM) for dividing a communication channel into several subchannels and transmitting many subcarriers through a single band using frequency division multiplexing (FDM) techniques. In the MCM method, however, because several subcarriers occupying a narrow frequency domain are transmitted at one time, a relatively longer symbol period results compared with a single carrier modulation method. The MCM method has, owing to such characteristics, the advantages that equalization is easily performed and that it has immunity to impulse noise. OFDM is a type of the MCM designed to maximize the working frequency efficiency by securing orthogonality among the multiplexed subcarriers. OFDM is applied to mobile radio channels to attenuate multipath fading.
In an OFDM transmitting/receiving system, modulation and demodulation of parallel data are carried out using the Fast Fourier Transform (FFT). It is required that the sampled data be sent in predetermined frames, having passed through a FFT routine, been time-division multiplexed, and transmitted, then restored at the receiving end. However, if the synchronization is in error in the course of restoring the frame, the signals demodulated after the FFT will be influenced by interchannel and intersymbol interference. Accordingly, the problem of synchronization in reforming the frame, especially any joint carrier frequency offset or symbol timing error, must be addressed as a matter of importance.
Conventional synchronization methods as above-described encounter problems in that the process of synchronization is not only very complex, but the synchronization is not realized rapidly.
Accordingly, the present invention is directed at a synchronization method that substantially obviates on or more of the problems due to limitations and disadvantages of the prior art.
Additional features and advantages of the invention will be set forth in the description which follows, and in part will be apparent from the description, or may be learned by practice of the invention. The objectives and other advantages of the invention will be realized and attained by the methods particularly pointed out in the written description and claims hereof, as well as the appended drawings.
To achieve these and other advantages, and in accordance with the purpose of the invention as embodied and broadly described, there is provided a method of estimating joint carrier frequency and symbol timing errors in a received sample bit stream including an observation vector (OV), having an observed carrier frequency timing offset xcex5, and a plurality of data-symbol frames, having an observed symbol timing error xcex8. The method comprises the steps of generating a probability density function (PDF) based on the OV, and generating from the PDF a joint maximum likelihood (ML) estimate of a joint carrier frequency offset and a symbol timing error, the maximization performed globally over the entire span of the estimation vector [xcex5, xcex8]. In another aspect of the invention, wherein the OV comprises an L-bit cyclic extension portion and a first and a second N-bit synchronization frame, the method calls for the PDF to comprise a first term, p1, based on the observed timing error xcex8 being within the span 1 to N and a second term, p2, based on the observed timing error xcex8 being within the span N+1 to N+L.
In yet another aspect of the method wherein the received bit stream has uncorrelated independent identically distributed random signal and noise sequence variables with power of "sgr"s2 and "sgr"n2, respectively, and wherein the OV is denoted x, the PDF is given by
p(x,xcex5,xcex8)=p1(x,xcex5,xcex8)(U[xcex8xe2x88x921]xe2x88x92U[xcex8xe2x88x92Nxe2x88x921])+p2(x,xcex5,xcex8)(U[xcex8xe2x88x92Nxe2x88x921]xe2x88x92U[xcex8xe2x88x92Nxe2x88x92L+1])
wherein U[n] is the discrete time unit step function, and wherein
if (1xe2x89xa6xcex8xe2x89xa6N)                     R        =                  xe2x80x83                ⁢                  [                                                                                                                r                      xx                                        ⁡                                          [                      0                      ]                                                        ⁢                                      I                                          (                      ϑ                      )                                                                                                  0                                            0                                                                    0                                                              T                                      (                                          N                      +                      L                                        )                                                                              0                                                                    0                                            0                                                                                                        r                      xx                                        ⁡                                          [                      0                      ]                                                        ⁢                                      I                                          (                                              N                        -                        ϑ                                            )                                                                                                    ]                                                  R                      -            1                          =                  xe2x80x83                ⁢                  [                                                                                          I                                          (                      ϑ                      )                                                                                                  r                      xx                                        ⁡                                          [                      0                      ]                                                                                                  0                                            0                                                                    0                                                              T                                      (                                          N                      +                      L                                        )                                                        -                    1                                                                              0                                                                    0                                            0                                                                                  I                                          (                                              N                        -                        ϑ                                            )                                                                                                  r                      xx                                        ⁡                                          [                      0                      ]                                                                                                    ]                    
TM=xcex94Toeplitz (rxx[0] rxx[1] . . . rxx[Mxe2x88x921]
rxx[m]=xcex94("sgr"s2+"sgr"n2)xcex4[m]+"sgr"s2exe2x88x92j2xcfx80xcex5xcex4[mxe2x88x92N]
                                          T                          (              M              )                                      -              1                                ⁡                      [                          i              ,              j                        ]                          =                  xe2x80x83                ⁢                  {                                                                      1                                                                                    r                        xx                                            ⁡                                              [                        0                        ]                                                              ⁢                                          (                                              1                        -                                                                              [                            a                            ]                                                    2                                                                    )                                                                                                                                        i                    =                                                                  j                        and                                            ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          ≤                          i                          ≤                                                      M                            -                                                          N                              ⁢                                                              xe2x80x83                                                            ⁢                              or                              ⁢                                                              xe2x80x83                                                            ⁢                              N                                                        +                            1                                                    ≤                          i                          ≤                          M                                                )                                                                              ⁢                                      xe2x80x83                                                                                                                                            -                                          a                      *                                                                                                                          r                        xx                                            ⁡                                              [                        0                        ]                                                              ⁢                                          (                                              1                        -                                                                              "LeftBracketingBar"                            a                            "RightBracketingBar"                                                    2                                                                    )                                                                                                                                        i                    -                    j                                    =                  N                                                                                                                          -                    a                                                                                                      r                        xx                                            ⁡                                              [                        0                        ]                                                              ⁢                                          (                                              1                        -                                                                              "LeftBracketingBar"                            a                            "RightBracketingBar"                                                    2                                                                    )                                                                                                                                        j                    -                    i                                    =                  N                                                                                                      1                                                            r                      xx                                        ⁡                                          [                      0                      ]                                                                                                                                                                                                                i                          -                          j                                                and                                            ⁢                                              xe2x80x83                                            ⁢                      M                                        -                    N                    +                    1                                    ≤                  i                  ≤                  N                                                                                    0                                            otherwise                                                                                      where          ⁢                      xe2x80x83                    ⁢          a                ⁢                  =          △                ⁢                              xe2x80x83                    ⁢                      xe2x80x83                          ⁢                                            ⅇ                                                -                  j                                ⁢                                  xe2x80x83                                ⁢                2                ⁢                ω                ⁢                                  xe2x80x83                                ⁢                π                                      ⁢                          σ              j              2                                                          σ              s              2                        +                          σ              n              2                                                                                    P            1                    ⁡                      (                          x              ,              ϵ              ,              ϑ                        )                          =                  xe2x80x83                ⁢                              1                                                            (                                      2                    ⁢                    π                                    )                                                                      2                    ⁢                    N                                    +                  L                                            ⁢                              det                (                R                )                                              ⁢                      xe2x80x83                    ⁢                      exp            [                                                            -                  1                                                  2                  ⁢                                      (                                                                  σ                        s                        2                                            +                                              σ                        n                        2                                                              )                                                              ⁢                              (                                                      ∑                                          k                      =                      0                                                                                      2                        ⁢                        N                                            +                      L                      -                      1                                                        ⁢                                      ❘                                                                                            x                          ⁡                                                      [                            k                            ]                                                                          2                                            +                                                                                                                                            xe2x80x83                ⁢                                            ∑                              k                =                0                                            L                +                ϑ                -                1                                      ⁢                                          (                                                      "LeftBracketingBar"                                                                                            x                          ⁡                                                      [                            k                            ]                                                                          2                                            +                                        "RightBracketingBar"                                    ⁢                                                            x                      ⁡                                              [                                                  k                          +                          N                                                ]                                                              2                                                  )                            ⁢                              xe2x80x83                            ⁢                                                                    [                    a                    ]                                    2                                                  1                  -                                                            [                      a                      ]                                        2                                                                                -                                                                            xe2x80x83                        ⁢                          2              ⁢              ℛ              ⁢                              {                                                      ∑                                          k                      =                      ϑ                                                              ϑ                      +                      L                      -                      1                                                        ⁢                                      xe2x80x83                                    ⁢                                                                                    x                        *                                            ⁡                                              [                        k                        ]                                                              ⁢                                          x                      ⁡                                              [                                                  k                          +                          N                                                ]                                                              ⁢                                                                  a                        *                                                                    1                        -                                                                              "LeftBracketingBar"                            a                            "RightBracketingBar"                                                    2                                                                                                                    }                                      )                    ]                ,                  xe2x80x83                ⁢        or            
if (N+1xe2x89xa6xcex8xe2x89xa6N+L).                     R        =                  xe2x80x83                ⁢                  [                                                                      T                                      (                    s                    )                                                                              0                                                                    0                                                              T                                      (                                                                  2                        ⁢                        N                                            +                      L                      -                      ϑ                                        )                                                                                ]                                                              P            2                    ⁡                      (                          x              ,              ϵ              ,              ϑ                        )                          =                  xe2x80x83                ⁢                              1                                                            (                                      2                    ⁢                    π                                    )                                                                      2                    ⁢                    N                                    +                  L                                            ⁢                              det                (                R                )                                              ⁢                      xe2x80x83                    ⁢          exp          ⁢                      {                                                            -                  1                                                  2                  ⁢                                      (                                                                  σ                        s                        2                                            +                                              σ                        n                        2                                                              )                                                              ⁢                              (                                                      ∑                                          k                      =                      0                                                                                      2                        ⁢                        N                                            +                      L                      -                      1                                                        ⁢                                      xe2x80x83                                    ⁢                                      ❘                                                                                            x                          ⁡                                                      [                            k                            ]                                                                          2                                            +                                                                                                                                            xe2x80x83                ⁢                                            (                                                                    ∑                                          k                      =                      0                                                              ϑ                      -                      N                      -                      1                                                        ⁢                                                            "LeftBracketingBar"                                                                                                    x                            ⁡                                                          [                              k                              ]                                                                                2                                                +                                            "RightBracketingBar"                                        ⁢                                                                  x                        ⁡                                                  [                                                      k                            +                            N                                                    ]                                                                    2                                                                      +                                                      ∑                                          k                      =                      ϑ                                                              N                      +                      L                      -                      1                                                        ⁢                                                            "LeftBracketingBar"                                                                                                    x                            ⁡                                                          [                              k                              ]                                                                                2                                                +                                            "RightBracketingBar"                                        ⁢                                                                  x                        ⁡                                                  [                                                      k                            +                            N                                                    ]                                                                    2                                                                                  )                        ⁢                          xe2x80x83                        ⁢                                          a                *                                            1                -                                                      "LeftBracketingBar"                    a                    "RightBracketingBar"                                    2                                                              -                                                  xe2x80x83                ⁢                              2            ⁢            ℛ            ⁢                          {                                                ∑                                      k                    =                    0                                                        ϑ                    -                    L                    -                    1                                                  ⁢                                                                            x                      *                                        ⁡                                          [                      k                      ]                                                        ⁢                                      x                    ⁡                                          [                                              k                        +                        N                                            ]                                                        ⁢                                                            a                      *                                                              1                      -                                                                        "LeftBracketingBar"                          a                          "RightBracketingBar"                                                2                                                                                                        }                                -                                                                            xe2x80x83                        ⁢                          2              ⁢              ℛ              ⁢                              {                                                      ∑                                          k                      =                      ϑ                                                              N                      +                      L                      -                      1                                                        ⁢                                                                                    x                        *                                            ⁡                                              [                        k                        ]                                                              ⁢                                          x                      ⁡                                              [                                                  k                          +                          N                                                ]                                                              ⁢                                                                  a                        *                                                                    1                        -                                                                              "LeftBracketingBar"                            a                            "RightBracketingBar"                                                    2                                                                                                                    }                                      )                    ]                ,                  xe2x80x83                ⁢        and            
wherein the ML estimate of symbol timing error is given by       ϑ    MI    =                              arg          ⁢                      xe2x80x83                    ⁢          max                ϑ            ⁢                        T          2                ⁡                  (          ϑ          )                    ⁢              "LeftBracketingBar"        a        "RightBracketingBar"              +          2      ⁢              "LeftBracketingBar"                              T            1                    ⁡                      (            ϑ            )                          "RightBracketingBar"            
and wherein the ML estimate of carrier frequency offset is given by             ϵ      MI        =                            -          1                          2          ⁢          π                    ⁢      ∠      ⁢              xe2x80x83            ⁢                        T          1                ⁡                  (                      x            ,            ϑ                    )                      ,
where                               T                                    1              ⁢                              (                                  χ                  ,                  ϑ                                }                                      )                          =                  xe2x80x83                ⁢                  {                                                                                                                ∑                                              k                        =                        ϑ                                                                    L                        +                        ϑ                        -                        1                                                              ⁢                                                                  x                        ⁡                                                  [                          k                          ]                                                                    ⁢                                                                        x                          *                                                ⁡                                                  [                                                      k                            +                            N                                                    ]                                                                                                                                                          1                    ≤                    ϑ                    ≤                    N                                                                                                                                                                                                                                                      ∑                                                              k                                =                                0                                                                                            ϑ                                -                                N                                -                                1                                                                                      ⁢                                                                                          x                                ⁡                                                                  [                                  k                                  ]                                                                                            ⁢                                                                                                x                                  *                                                                ⁡                                                                  [                                                                      k                                    +                                    N                                                                    ]                                                                                                                                              +                                                                                                                                                                                          ∑                                                          k                              =                              ϑ                                                                                      N                              +                              L                              -                              1                                                                                ⁢                                                                                    x                              ⁡                                                              [                                k                                ]                                                                                      ⁢                                                                                          x                                *                                                            ⁡                                                              [                                                                  k                                  +                                  N                                                                ]                                                                                                                                                                                                                                                                    N                      +                      1                                        ≤                    ϑ                    ≤                                          N                      +                      L                                                                                            ⁢                          
                        ⁢            and                                                                        T            2                    ⁡                      (                          χ              ,              ϑ                        )                          =                  xe2x80x83                ⁢                  {                                                                                                                                        ∑                                                  k                          =                          ϑ                                                                          L                          +                          ϑ                          -                          1                                                                    ⁢                                              xe2x80x83                                            ⁢                                                                                                    "LeftBracketingBar"                            x                            "RightBracketingBar"                                                    ⁡                                                      [                            k                            ]                                                                          2                                                              +                                    ❘                                                            x                      ⁡                                              [                                                  k                          +                          N                                                ]                                                              2                                                                                                1                  ≤                  ϑ                  ≤                  N                                                                                                                                                                                                                              ∑                                                          k                              =                              0                                                                                      ϑ                              -                              N                              -                              1                                                                                ⁢                                                                                    "LeftBracketingBar"                                                                                                                                    x                                    ⁡                                                                          [                                      k                                      ]                                                                                                        2                                                                +                                                            "RightBracketingBar"                                                        ⁢                                                                                                                            x                                  *                                                                ⁡                                                                  [                                                                      k                                    +                                    N                                                                    ]                                                                                            2                                                                                                      +                                                                                                                                                                          ∑                                                      k                            =                            ϑ                                                                                N                            +                            L                            -                            1                                                                          ⁢                                                                              "LeftBracketingBar"                                                                                                                            x                                  ⁡                                                                      [                                    k                                    ]                                                                                                  2                                                            +                                                        "RightBracketingBar"                                                    ⁢                                                                                    x                              ⁡                                                              [                                                                  k                                  +                                  N                                                                ]                                                                                      2                                                                                                                                                                                                                N                    +                    1                                    ≤                  ϑ                  ≤                                      N                    +                    L                                                                                          
A method of synchronizing a received sample bit stream is also provided by the invention, comprising the steps of transmitting at a transmitter the bit stream including an observation vector (OV), receiving and sampling at a receiver the bit stream, which includes the OV with an observed carrier frequency timing offset xcex5, and a plurality of data-symbol frames, having an observed symbol timing error xcex8, generating a probability density function (PDF) based on the OV, generating from the PDF a joint maximum likelihood (ML) estimate of a joint carrier frequency offset and a symbol timing error, the maximization performed globally over the entire span of the estimation vector [xcex5,xcex8], and synchronizing the received bit stream by the ML estimates of carrier frequency offset and symbol timing error.
In another aspect of the synchronization method provided by the invention wherein the OV comprises an L-bit cyclic extension portion and a first and a second N-bit synchronization frame, the PDF comprises a first term, p1, based on the observed timing error xcex8 being within the span 1 to N and a second term, p2, based on the observed timing error xcex8 being within the span N+1 to N+L.
In yet another aspect of the synchronization method provided by the invention, wherein the received bit stream has uncorrelated independent identically distributed random signal and noise sequence variables with power of "sgr"s2 and "sgr"n2, respectively, and wherein the OV is denoted x, the PDF is given by
p(x,xcex5,xcex8)=p1(x,xcex5,xcex8)(U[xcex8xe2x88x921]xe2x88x92U[xcex8xe2x88x92Nxe2x88x921])+p2(x,xcex5,xcex8)(U[xcex8xe2x88x92Nxe2x88x921]xe2x88x92U[xcex8xe2x88x92Nxe2x88x92L+1])
wherein U[n] is the discrete time unit step function, and wherein
if (1xe2x89xa6xcex8xe2x89xa6N)                     R        =                  xe2x80x83                ⁢                  [                                                                                                                r                      xx                                        ⁡                                          [                      0                      ]                                                        ⁢                                      I                                          (                      ϑ                      )                                                                                                  0                                            0                                                                    0                                                              T                                      (                                          N                      +                      L                                        )                                                                              0                                                                    0                                            0                                                                                                        r                      xx                                        ⁡                                          [                      0                      ]                                                        ⁢                                      1                                          (                                              N                        -                        ϑ                                            )                                                                                                    ]                                                  R                      -            1                          =                  xe2x80x83                ⁢                  [                                                                                          I                                          (                      ϑ                      )                                                                                                  r                      xx                                        ⁡                                          [                      0                      ]                                                                                                  0                                            0                                                                    0                                                              T                                      (                                          N                      +                      L                                        )                                                        -                    1                                                                              0                                                                    0                                            0                                                                                  I                                          (                                              N                        -                        ϑ                                            )                                                                                                  r                      xx                                        ⁡                                          [                      0                      ]                                                                                                    ]                    
TM=xcex94Toeplitz (rxx[0] rxx[1] . . . rxx[Mxe2x88x921]
rxx[m]=xcex94("sgr"s2+"sgr"n2)xcex4[m]+"sgr"s2exe2x88x92j2xcfx80xcex5xcex4[mxe2x88x92N]
                    T                  (          M          )                          -          1                    ⁡              [                  i          ,          j                ]              =          {                                                                                                        1                                                                                            r                          xx                                                ⁡                                                  [                          0                          ]                                                                    ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                                                    [                              a                              ]                                                        2                                                                          )                                                                                                                                  i                    =                                                                  j                        and                                            ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          ≤                          i                          ≤                                                      M                            -                                                          N                              ⁢                                                              xe2x80x83                                                            ⁢                              or                              ⁢                                                              xe2x80x83                                                            ⁢                              N                                                        +                            1                                                    ≤                          i                          ≤                          M                                                )                                                                                                                                                                                    -                                              a                        *                                                                                                                                      r                          xx                                                ⁡                                                  [                          0                          ]                                                                    ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                                                    "LeftBracketingBar"                              a                              "RightBracketingBar"                                                        2                                                                          )                                                                                                                                                        i                      -                      j                                        =                    N                                                                                                                                          -                      a                                                                                                                r                          xx                                                ⁡                                                  [                          0                          ]                                                                    ⁢                                              xe2x80x83                                            ⁢                                              (                                                  1                          -                                                                                    "LeftBracketingBar"                              a                              "RightBracketingBar"                                                        2                                                                          )                                                                                                                                                        j                      -                      i                                        =                    N                                                                                                                    1                                                                                            r                          xx                                                ⁡                                                  [                          0                          ]                                                                    ⁢                                              xe2x80x83                                                                                                                                                                                                                                      i                            -                            j                                                    and                                                ⁢                                                  xe2x80x83                                                ⁢                        M                                            -                      N                      +                      1                                        ≤                    i                    ≤                    N                                                                                                0                                                  otherwise                                                      ⁢                          
                        ⁢            where            ⁢                          xe2x80x83                        ⁢            a                    ⁢                      =            Δ                    ⁢                      xe2x80x83                    ⁢                                                                                          ⅇ                                                                  -                        f                                            ⁢                                              xe2x80x83                                            ⁢                      2                      ⁢                      xx                                                        ⁢                                      σ                    x                    2                                                                                        σ                    s                    2                                    +                                      σ                    n                    2                                                              ⁢                              
                            ⁢                                                P                  1                                ⁡                                  (                                      x                    ,                    ϵ                    ,                    ϑ                                    )                                                      =                                          1                                                                            (                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        π                                            )                                                                                      2                        ⁢                        N                                            +                      L                                                        ⁢                                      xe2x80x83                                    ⁢                  det                  ⁢                                      xe2x80x83                                    ⁢                                      (                    R                    )                                                              ⁢                              xe2x80x83                            ⁢                              exp                ⁡                                  [                                                                                    -                        1                                                                    2                        ⁢                                                  (                                                                                    σ                              s                              2                                                        +                                                          σ                              n                              2                                                                                )                                                                                      ⁢                                          xe2x80x83                                        ⁢                                          (                                                                                                    ∑                                                          k                              =                              0                                                                                                                      2                                ⁢                                N                                                            +                              L                              -                              1                                                                                ⁢                                                      xe2x80x83                                                    ⁢                                                                                    "LeftBracketingBar"                                                              x                                ⁡                                                                  [                                  k                                  ]                                                                                            "RightBracketingBar"                                                        2                                                                          +                                                  
                                                ⁢                                                  xe2x80x83                                                ⁢                                                                              ∑                                                          k                              =                              ϑ                                                                                      L                              +                              ϑ                              -                              1                                                                                ⁢                                                      xe2x80x83                                                    ⁢                                                                                    (                                                                                                                                    "LeftBracketingBar"                                                                          x                                      ⁡                                                                              [                                        k                                        ]                                                                                                              "RightBracketingBar"                                                                    2                                                                +                                                                                                      "LeftBracketingBar"                                                                          x                                      ⁡                                                                              [                                                                                  k                                          +                                          N                                                                                ]                                                                                                              "RightBracketingBar"                                                                    2                                                                                            )                                                        ⁢                                                          xe2x80x83                                                        ⁢                                                                                                                            [                                  a                                  ]                                                                2                                                                                            1                                -                                                                                                      [                                    a                                    ]                                                                    2                                                                                                                                                                    -                                                  
                                                ⁢                                                  xe2x80x83                                                ⁢                                                  2                          ⁢                                                      xe2x80x83                                                    ⁢                          ℛ                          ⁢                                                      {                                                                                          ∑                                                                  k                                  =                                  ϑ                                                                                                  ϑ                                  +                                  L                                  -                                  1                                                                                            ⁢                                                              xe2x80x83                                                            ⁢                                                                                                                                    x                                    *                                                                    ⁡                                                                      [                                    k                                    ]                                                                                                  ⁢                                                                  xe2x80x83                                                                ⁢                                                                  x                                  ⁡                                                                      [                                                                          k                                      +                                      N                                                                        ]                                                                                                  ⁢                                                                  xe2x80x83                                                                ⁢                                                                                                      a                                    *                                                                                                        1                                    -                                                                                                                  "LeftBracketingBar"                                        a                                        "RightBracketingBar"                                                                            2                                                                                                                                                                                            }                                                                                              )                                                        ]                                                                    ,                  
                ⁢                              or            ⁢                          xe2x80x83                        ⁢            if            ⁢                          xe2x80x83                        ⁢                          (                                                N                  +                  1                                ≤                ϑ                ≤                                  N                  +                  L                                            )                        ⁢                          
                        ⁢            R                    =                                                    [                                                                                                    T                                                  (                          ϑ                          )                                                                                                            0                                                                                                  0                                                                                      T                                                  (                                                                                    2                              ⁢                              N                                                        +                            L                            -                            ϑ                                                    )                                                                                                                    ]                            ⁢                              
                            ⁢                                                P                  2                                ⁡                                  (                                      x                    ,                    ϵ                    ,                    ϑ                                    )                                                      =                                          1                                                                            (                                              2                        ⁢                                                  xe2x80x83                                                ⁢                        π                                            )                                                                                      2                        ⁢                        N                                            +                      L                                                        ⁢                                      xe2x80x83                                    ⁢                  det                  ⁢                                      xe2x80x83                                    ⁢                                      (                    R                    )                                                              ⁢                              xe2x80x83                            ⁢                              exp                [                                                                                                    -                        1                                                                    2                        ⁢                                                  (                                                                                    σ                              s                              2                                                        +                                                          σ                              n                              2                                                                                )                                                                                      ⁢                                          xe2x80x83                                        ⁢                                          (                                                                                                    ∑                                                          k                              =                              0                                                                                                                      2                                ⁢                                N                                                            +                              L                              -                              1                                                                                ⁢                                                      xe2x80x83                                                    ⁢                                                                                    "LeftBracketingBar"                                                              x                                ⁡                                                                  [                                  k                                  ]                                                                                            "RightBracketingBar"                                                        2                                                                          +                                                  
                                                ⁢                                                  xe2x80x83                                                ⁢                                                                              ∑                                                          k                              =                              ϑ                                                                                      ϑ                              -                              N                              -                              1                                                                                ⁢                                                      xe2x80x83                                                    ⁢                                                                                    "LeftBracketingBar"                                                              x                                ⁡                                                                  [                                  k                                  ]                                                                                            "RightBracketingBar"                                                        2                                                                          +                                                                              "LeftBracketingBar"                                                          x                              ⁡                                                              [                                                                  k                                  +                                  N                                                                ]                                                                                      "RightBracketingBar"                                                    2                                                +                                                  
                                                ⁢                                                                              ∑                                                          k                              =                              ϑ                                                                                      N                              +                              L                              -                              1                                                                                ⁢                                                      xe2x80x83                                                    ⁢                                                                                    "LeftBracketingBar"                                                              x                                ⁡                                                                  [                                  k                                  ]                                                                                            "RightBracketingBar"                                                        2                                                                          +                                                                              "LeftBracketingBar"                                                          x                              ⁡                                                              [                                                                  k                                  +                                  N                                                                ]                                                                                      "RightBracketingBar"                                                    2                                                                    )                                        ⁢                                          xe2x80x83                                        ⁢                                                                                            [                          a                          ]                                                2                                                                    1                        -                                                                              [                            a                            ]                                                    2                                                                                                      -                                      2                    ⁢                                          xe2x80x83                                        ⁢                    ℛ                    ⁢                                          {                                                                        ∑                                                      k                            =                            0                                                                                ϑ                            +                            N                            -                            1                                                                          ⁢                                                  xe2x80x83                                                ⁢                                                                                                            x                              *                                                        ⁡                                                          [                              k                              ]                                                                                ⁢                                                      xe2x80x83                                                    ⁢                                                      x                            ⁡                                                          [                                                              k                                +                                N                                                            ]                                                                                ⁢                                                      xe2x80x83                                                    ⁢                                                                                    a                              *                                                                                      1                              -                                                                                                "LeftBracketingBar"                                  a                                  "RightBracketingBar"                                                                2                                                                                                                                                        )                                                        -                                      
                                    ⁢                                      xe2x80x83                                    ⁢                                      2                    ⁢                                          xe2x80x83                                        ⁢                    ℛ                    ⁢                                          {                                                                        ∑                                                      k                            =                            ϑ                                                                                N                            +                            L                            -                            1                                                                          ⁢                                                  xe2x80x83                                                ⁢                                                                                                            x                              *                                                        ⁡                                                          [                              k                              ]                                                                                ⁢                                                      xe2x80x83                                                    ⁢                                                      x                            ⁡                                                          [                                                              k                                +                                N                                                            ]                                                                                ⁢                                                      xe2x80x83                                                    ⁢                                                                                    a                              *                                                                                      1                              -                                                                                                "LeftBracketingBar"                                  a                                  "RightBracketingBar"                                                                2                                                                                                                                                        }                                                                      )                                                        ]        ,      
    ⁢  and
wherein the ML estimate of symbol timing error is given by       ϑ    ML    =            arg      ⁢              xe2x80x83            ⁢      max      ⁢              xe2x80x83            ⁢                        T          x                ⁡                  (          ϑ          )                    ⁢              xe2x80x83            ⁢              "LeftBracketingBar"        a        "RightBracketingBar"              +          2      ⁢              "LeftBracketingBar"                              T            1                    ⁡                      (            ϑ            )                          "RightBracketingBar"            
and wherein the ML estimate of carrier frequency offset is given by             ϵ      ML        =                            -          1                          2          ⁢                      xe2x80x83                    ⁢          π                    ⁢              xe2x80x83            ⁢                        ∠T          1                ⁡                  (                      x            ,            ϑ                    )                      ,
where       T          1      ⁢              (                  χ          ,          ϑ                )              =      {                                                                                        ∑                                      k                    =                    ϑ                                                        L                    +                    ϑ                    -                    1                                                  ⁢                                  xe2x80x83                                ⁢                                                      x                    ⁡                                          [                      k                      ]                                                        ⁢                                      xe2x80x83                                    ⁢                                                            x                      *                                        ⁡                                          [                                              k                        +                        N                                            ]                                                                                                                          1                ≤                ϑ                ≤                N                                                                                                                              ∑                                          k                      =                      0                                                              ϑ                      -                      N                      -                      1                                                        ⁢                                      xe2x80x83                                    ⁢                                                            x                      ⁡                                              [                        k                        ]                                                              ⁢                                          xe2x80x83                                        ⁢                                                                  x                        *                                            ⁡                                              [                                                  k                          +                          N                                                ]                                                                                            +                                                      ∑                                          k                      =                      ϑ                                                              N                      +                      L                      -                      1                                                        ⁢                                      xe2x80x83                                    ⁢                                                            x                      ⁡                                              [                        k                        ]                                                              ⁢                                          xe2x80x83                                        ⁢                                                                  x                        *                                            ⁡                                              [                                                  k                          +                          N                                                ]                                                                                                                                                                  N                  +                  1                                ≤                ϑ                ≤                                  N                  +                  L                                                                    ⁢                  xe2x80x83                ⁢        and        ⁢                  xe2x80x83                ⁢                  
                ⁢                  T                      2            ⁢                          (                              χ                ,                ϑ                            )                                          =              {                                                                              ∑                                      k                    =                    ϑ                                                        L                    +                    ϑ                    -                    1                                                  ⁢                                  xe2x80x83                                ⁢                                                      "LeftBracketingBar"                                          x                      ⁢                                              "LeftBracketingBar"                                                                                                                                            [                                k                                ]                                                            ⁢                                                              xe2x80x83                                                                                      2                                                    +                                                "RightBracketingBar"                                            ⁢                                              x                        ⁡                                                  [                                                      k                            +                            N                                                    ]                                                                                      "RightBracketingBar"                                    2                                                                                    1                ≤                ϑ                ≤                N                                                                                                                              ∑                                          k                      =                      0                                                              ϑ                      -                      N                      -                      1                                                        ⁢                                      xe2x80x83                                    ⁢                                                                                    "LeftBracketingBar"                                                  x                          ⁡                                                      [                            k                            ]                                                                          "RightBracketingBar"                                            ⁢                                              xe2x80x83                                                              2                                                  +                                                      "LeftBracketingBar"                                          x                      ⁡                                              [                                                  k                          +                          N                                                ]                                                              "RightBracketingBar"                                    2                                +                                                      ∑                                          k                      =                      ϑ                                                              N                      +                      L                      -                      1                                                        ⁢                                      xe2x80x83                                    ⁢                                                                                    "LeftBracketingBar"                                                  x                          ⁡                                                      [                            k                            ]                                                                          "RightBracketingBar"                                            ⁢                                              xe2x80x83                                                              2                                                  +                                                      "LeftBracketingBar"                                          x                      ⁡                                              [                                                  k                          +                          N                                                ]                                                              "RightBracketingBar"                                    2                                                                                                      N                  +                  1                                ≤                ϑ                ≤                                  N                  +                  L                                                                        