In telecommunication and wireless transceivers, the oscillators used to generate the digital to analog converter (DAC) and analog to digital convertor (ADC) sampling instants at the transmitter and receiver cannot have exactly the same period. Consequently, the sampling instants slowly shift, relatively to each other.
This phenomenon is usually known and referred to as SFO (Sampling Frequency Offset), and has been addressed, including regarding OFDM communication systems. This sampling clock error, or SFO, has two main effects:                A slow shift of the symbol timing point, which rotates subcarriers, and,        A loss of SNR (Signal to Noise Ratio) due to the inter-carrier interference (ICI) generated by the slightly incorrect sampling instants, which in turn causes loss of the orthogonality of the subcarriers.        
The FIGS. 1a and 1b depict more clearly undesirable effects due to synchronization issues between receiver and transmitter.
The FIG. 1a shows a phase difference between the DAC and ADC sampling frequencies of, respectively, the transmitter Tx and the receiver Rx.
The period Ts between 2 subsequent samples remains constant and equal at transmitter Tx side and receiver Rx side, but a phase difference ε shifts forward the samples at receiver side. This leads to STO (Sample Time Offset), which can be easily compensated.
The FIG. 1b shows the problem caused by SFO (Sample Frequency Offset), mentioned above. This time, the period Ts′ between 2 received samples at receiver side is different (here, longer) than the period Ts of the transmitted samples. Therefore, at each sample, the received sample is shifted by an increasing time ε.
After a certain number of samples, a sample will be missed. This would lead to the impossibility to decode the corresponding OFDM symbol.
Also, after a certain number of samples less than the total number of samples per OFDM symbol, the ADC will fall on a sample of the next OFDM symbol. This situation leads to Inter-Symbol Interference (ISI), which turn leads to the loss of orthogonality of the subcarriers and dramatically decrease of the signal to noise ratio (SNR).
Similar problems happen when the ADC sampling frequency is greater than the DAC sampling frequency: samples will be repeated instead of being missed, but the consequence remains impossibility to decode some OFDM symbols and a decrease of the signal to noise ratio (SNR).
This SFO-related problem is partially solved by using pilot signal.
Pilot-based techniques are employed in the frequency domain, i.e. after the DFT (Discrete Fourier Transform) block of the OFDM receiver chain.
The SFO is measured by the normalized sampling error given as follows:
      t    Δ    =                    T        ′            -      T        T  
Where T and T′ are, respectively, the transmitter and receiver sampling periods.
Then, the overall effect after DFT on the received subcarriers Rl,k is as follows:
                              R                      l            ,            k                          =                                            ⅇ                                                j                  ⁢                                                                          ⁢                  2                  ⁢                                                                          ⁢                  π                  ⁢                                                                          ⁢                                                            k                      ⁢                                                                                                                                  t                      Δ                                                        ⁢                  l                                ⁢                                                                                        ⁢                                          T                s                                            T                u                                      ⁢                                          X                                  l                  ,                  k                                            ·                              sinc                ⁡                                  (                                      π                    ⁢                                                                                  ⁢                    kt                                    )                                            ·                              H                                  l                  ,                  k                                                              +                      W                          l              ,              k                                +                                    N                              t                Δ                                      ⁡                          (                              l                ,                k                            )                                                          (        1        )            
Where                l is the OFDM symbol index,        k is the subcarrier index,        Ts and Tu are the durations of the total OFDM symbol and of the useful data portion, respectively,        Hl,k is the transfer function associated with the transmission channel between the emitter and the receiver,        Wl,k is the additive white noise        And NtΔ(l,k) is the additional interference due to the sampling frequency offset. The power of this last term can be approximated by:        
      P          t      Δ        ≈                    π        2            3        ⁢                  (                  t          Δ                                            k                )            2      
Therefore, the degradation grows as the square of the product of the offset tΔ and the subcarrier index k. This means that the outermost subcarriers are most severely impacted. The degradation can also be expressed directly as SNR loss in decibels. The following approximation is derived:
      D    n    ≈          ⁢            log      10        ⁡          (              1        +                                            π              2                        3                    ⁢                                    E              s                                      N              0                                ⁢                                    (                              k                                  t                  Δ                                            )                        2                              )      
WLAN OFDM systems typically have relatively small number of subcarriers and quite small offset tΔ. Hence, ktΔ<<1, so that the interference caused by sampling frequency offset can usually be ignored.
The equation (1) also shows the most significant problem caused by the offset, namely the term
            ⅇ                        j          ⁢                                          ⁢          2          ⁢                                          ⁢          π          ⁢                                          ⁢                                    k              ⁢                                                                                  t              Δ                                ⁢          l                ⁢                                        ⁢                  T        s                    T        u              ,which represents the phase error due to SFO.
As discussed above, in WLAN, we only care about correcting the phase error induced by the SFO. This is done by relying on the pilots to estimate their phase difference according to the following:
The sampling frequency offset (SFO) is estimated by using the knowledge of the linear relationship between the phase rotation caused by the offset and the pilot subcarrier index. The received pilot subcarriers, in a simplified form, are:
                              R                      l            ,            k                          =                              H            k                    ⁢                      P                          l              ,              k                                ⁢                      ⅇ                          j              ⁢                                                          ⁢              2              ⁢                                                          ⁢              π              ⁢                                                          ⁢                              k                                                                                          ⁢                                      t                    Δ                                                              ⁢              l              ⁢                                                T                  s                                                  T                  u                                                                                        (        2        )            
Let Zl,k=Rl,kR*l-1,k 
This implies:
                                          Z                          l              ,              k                                =                                    H              k                        ⁢                          P                              l                ,                k                                      ⁢                                                            ⅇ                                      j                    ⁢                                                                                  ⁢                    2                    ⁢                                                                                  ⁢                    π                    ⁢                                                                                  ⁢                                          k                                                                                                                        ⁢                                                  t                          Δ                                                                                      ⁢                    l                    ⁢                                                                  T                        s                                                                    T                        u                                                                                            ⁡                                  (                                                            H                      k                                        ⁢                                          P                                                                        l                          -                          1                                                ,                        k                                                              ⁢                                          ⅇ                                              j                        ⁢                                                                                                  ⁢                        2                        ⁢                                                                                                  ⁢                        π                        ⁢                                                                                                  ⁢                                                  k                                                                                                                                            ⁢                                                          t                              Δ                                                                                                      ⁢                        l                        ⁢                                                                              T                            s                                                                                T                            u                                                                                                                                )                                            *                                      ⁢                                  ⁢                              Z                          l              ,              k                                =                                                                                      H                  k                                                            2                        ⁢                                                                            P                                      l                    ,                    k                                                                              2                        ⁢                          ⅇ                              j                ⁢                                                                  ⁢                2                ⁢                                                                  ⁢                π                ⁢                                                                  ⁢                                  k                                                                                                    ⁢                                          t                      Δ                                                                      ⁢                l                ⁢                                                      T                    s                                                        T                    u                                                                        ⁢                          ⅇ                                                -                  j                                ⁢                                                                  ⁢                2                ⁢                                                                  ⁢                π                ⁢                                                                  ⁢                                                      k                                                                                                              ⁢                                              t                        Δ                                                                              ⁡                                      (                                          l                      -                      1                                        )                                                  ⁢                                                      T                    s                                                        T                    u                                                                                      ⁢                                  ⁢                              Z                          l              ,              k                                =                                                                                      H                  k                                                            2                        ⁢                                                                            P                                      l                    ,                    k                                                                              2                        ⁢                          ⅇ                              j                ⁢                                                                  ⁢                2                ⁢                                                                  ⁢                π                ⁢                                                                  ⁢                                  k                                                                                                    ⁢                                          t                      Δ                                                                      ⁢                l                ⁢                                                      T                    s                                                        T                    u                                                                                                          (        3        )            
According to the above equation, we can see that Zl,k is the multiplication of a pilot of the lth received OFDM symbol (after the DFT) by the complex conjugate of the same pilot of the previous OFDM symbol.
Finding the phase of Zl,k is equivalent to finding the phase error caused by the SFO (which is the error that we care about in WLAN, as mentioned before).
From the above question (3), an estimation {circumflex over (t)}Δ of the SFO tΔ can be found by the following equation:
            t      ^        Δ    =                    T        u                    2        ⁢                                  ⁢        π        ⁢                                  ⁢        kl        ⁢                                  ⁢                  T          s                      ⁢          arg      ⁡              (                  Z                      l            ,            k                          )            
For better estimation accuracy, since we have more than 1 pilot per OFDM symbol, the above estimation can be applied between the negative and positive pilots of the same received OFDM.
                    t        ^            Δ        =                  1                  2          ⁢                                          ⁢          π                    ⁢                        T          u                          T          s                    ⁢              1                                            min                              k                ∈                                  C                  2                                                      ⁢                          (              k              )                                +                                    max                              k                ∈                                  C                  2                                                      ⁢                          (              k              )                                          ⁢              (                              ϕ                          2              ,              l                                -                      ϕ                          1              ,              l                                      )                  Where    ⁢          :                  ϕ              1        ,        l              =          ∡      [                        ∑                      k            ∈                          C              1                                      ⁢                  Z                      l            ,            k                              ]        And            ϕ              2        ,        l              =          ∡      [                        ∑                      k            ∈                          C              ⁢                                                          ⁢              2                                      ⁢                  Z                      l            ,            k                              ]      
And where C1 and C2 are the sets of pilots at negative and positive subcarriers respectively. In the case or WLAN 802.11 a/g, C1=[−21, −7], C2=[7, 21], minkC2=7, and maxkC2=21.
The correction can then be done by different approaches.
A first approach consists in correcting the clock frequency of the ADC like depicted in FIGS. 2a and 2b. 
The decision block of FIG. 2a or the timing error detector block (TED) of FIG. 2b estimates the SFO tΔ as explained above.
A second approach is illustrated by FIGS. 3a and 3b, and consists in performing an inverse rotation of the subcarriers after the DFT. This is done by the ROTOR block in FIG. 3a, which is piloted by the Decision block which estimates the SFO tΔ as explained above.
FIG. 3b shows a non-synchronous sampling system where the sampling rate is fixed and the sampling time offset is compensated by using digital devices such as an FIR interpolating filter. Since it does not require a feedback signal for adjusting the sampling frequency (at ADC), it is simpler to implement then the synchronous sampling systems. However, the nonsynchronous sampling scheme is more vulnerable to SFO if it is not compensated properly. Since a sample can be inserted or lost in one OFDM symbol when SFO is present, the nonsynchronous sampling scheme performs the operations of skip/duplication/interpolation before the FFT operation and compensates for the effect of phase rotation by using FEQ (Frequency-domain Equalizer).
As these approaches are not fully satisfactory, there is a need to improve the situation and to propose a new and alternative approach.