# Patent

## Title
Method for the blind estimation of OFDM modulation parameters according to a maximum likelihood criterion

## Abstract

The present invention relates to a method for blind estimation of at least one parameter of an OFDM signal according to a maximum likelihood (ML) criterion. In one embodiment, the estimation method uses a deterministic maximum likelihood criterion. In another embodiment, the estimation method assumes that the modulation symbols modulating the sub-carriers of the OFDM multiplex are i.i.d. and Gaussian random variables. The estimation of said parameter is carried out while minimizing a cost function depending on the chosen criterion.

## Background
TECHNICAL FIELD
The present invention relates to a method for the blind estimation of OFDM (Orthogonal Frequency Division Multiplex) modulation parameters.
BACKGROUND OF THE INVENTION
OFDM modulation is well known from the prior art and is used in many telecommunications systems, such as DVB-T, ADSL, Wi-Fi (IEEE 802 a/g), WiMax (IEEE 802.16). It has the advantage of good spectral responsivity and good protection from selective frequency fading.
It will be recalled that in an OFDM system, the information symbols to be transmitted are grouped by blocks of N symbols, where N is generally a power of 2, the OFDM symbols being obtained by performing an IFFT (Inverse Fast Fourier Transform) on said blocks of information symbols. Generally, a cyclicprefix is added to the beginning of each OFDM signal to absorb the inter-symbol interference (ISI) and facilitate equalization upon reception. The OFDM signal formed by these OFDM symbols may possibly be translated in frequency.
Generally, the signal transmitted by an OFDM system can be represented in baseband by:




  
    
      
        
          
            s
            a
          
          ⁡
          
            (
            t
            )
          
        
        =
        
          
            
              E
            
            N
          
          ⁢
          
            
              ∑
              k
            
            ⁢
            
              
                g
                ⁡
                
                  (
                  
                    t
                    -
                    
                      
                        k
                        ⁡
                        
                          (
                          
                            N
                            +
                            D
                          
                          )
                        
                      
                      ⁢
                      
                        T
                        c
                      
                    
                  
                  )
                
              
              ·
              
                
                  ∑
                  
                    n
                    =
                    0
                  
                  
                    N
                    -
                    1
                  
                
                ⁢
                
                  
                    a
                    
                      k
                      ,
                      n
                    
                  
                  ⁢
                  
                    e
                    
                      
                        -
                        2
                      
                      ⁢
                      
                        
                      
                      ⁢
                      iπ
                      ⁢
                      
                        n
                        
                          NT
                          c
                        
                      
                      ⁢
                      
                        (
                        
                          t
                          -
                          
                            DT
                            c
                          
                          -
                          
                            
                              k
                              ⁡
                              
                                (
                                
                                  N
                                  +
                                  D
                                
                                )
                              
                            
                            ⁢
                            
                              T
                              c
                            
                          
                        
                        )
                      
                    
                  
                
              
            
          
        
      
    
    
      
        (
        1
        )
      
    
  




where E is the power of the signal, N is the number of carriers of the OFDM multiplex, ak,n, are the information symbols relative to the block k, belonging to a modulation alphabet M-ary, typically BPSK, QPSK or QAM, 1/Tc is the throughput of the information symbols where Tc is the “chip” time, D is the size of the cyclic prefix expressed in number of chips, g(t) is a forming impulsion for the OFDM signals having a temporal support [0,(N+D)Tc] intended to apodize the spectrum of the signal.

We have diagrammatically illustrated an OFDM signal in FIG. 1. It is made up of a sequence of OFDM symbols, each symbol having a total duration (N+D)Tc including a useful duration NTc and a guard interval Tprefix=DTc, in which the cyclic prefix is located. It will be recalled that, traditionally, the cyclic prefix is a copy of the end of the OFDM symbol inside the guard interval. In certain OFDM systems, the cyclic prefixes are simply omitted, in other words the useful portions of the symbols are separated by “empty” guard intervals. This transmission technique also makes it possible to eliminate inter-symbol interference, but does not make equalization of the signal easy.
After propagation in the transmission channel, the OFDM signal received by the receiver can be expressed by:

y(t)=hsa(t)+b(t)  (2)

where hsa is the convolution between the OFDM signal transmitted, sa(t) is the impulse response from the transmission channel h(t), and b(t) is a random function describing the noise. It will be assumed that the length of the impulse response is shorter than the length of the guard interval, so that it will be possible to neglect the inter-symbol interference (ISI).

FIG. 2 diagrammatically illustrates the structure of an OFDM receiver.
After any baseband demodulation, the received signal is sampled at 210 at the chip frequency, then the samples are subjected to a serial/parallel conversion at 220 to form blocks of N+D samples. The first D samples corresponding to the guard interval are rejected and the block of the N remaining samples corresponding to the useful portion of the OFDM symbol is subjected to a FFT at 230. The demodulated symbols obtained are then subjected to a serial conversion at 240.
Eventually, assuming that the receiver is well synchronized in time and frequency, the demodulated symbols can be expressed by:

âk,n=hnak,n+bk,n  (3)

where hn is a complex coefficient that depends on the impulse response of the transmission channel, and bk,n is a random variable representing a noise sample.

The proper operation of this receiver requires precise synchronization in time and frequency. Indeed, it is understood that a poor time synchronization will cause gradual temporal slipping of the truncation window, and a poor frequency synchronization will cause a phase rotation of the samples, which can be represented by a multiplicative factor e2iπΔfnTc, where Δf is the frequency offset between the demodulation frequency of the receiver and the carrier frequency of the OFDM multiplex.
The temporal and frequency synchronization of the receiver is generally done owing to the acquisition of a learning sequence.
The operation of this detector assumes, of course, that the parameters of the transmitted OFDM signal are known (in other words, the parameters of the OFDM symbols). “Parameters of the OFDM signal” refers here to the number N of sub-carriers, the useful duration NTc of a symbol or, equivalently, the frequency spacing 1/NTc between sub-carriers, the duration of the guard interval DTc and/or the repetition period (N+D)Tc of said symbols.
Quite often, the receiver does not know a priori the parameters of the OFDM signal and it is necessary to estimate them blindly, before any demodulation.
Several methods have been proposed to that end. They exploit the presence of the cyclic prefix in the OFDM signal and the resulting cyclostationarity properties. The estimators for the parameters are based on the auto-correlation function of the OFDM signal. An example of such an estimating method can be found in the article by P. Liu et al. entitled “A blind time-parameters estimation scheme for OFDM in multi-path channel”, published in Proc. 2005 Int'l Conference on Information, Communications and Signal Processing, vol. 1, pp. 242-247, 23-26 Sep. 2005.
These estimating methods, however, have the drawback of requiring the acquisition of a large number of OFDM symbols to calculate the auto-correlation function. Moreover, these methods do not work in the case, mentioned above, where the OFDM signal does not have cyclic prefixes. They do not work well or at all when the ratio between the prefix duration and the OFDM symbol duration, D/(D+N), is low. Indeed, in this case, the secondary peak of the auto-correlation function, due to the cyclostationarity of the signal, becomes blurred or even disappears in the noise. It is then impossible to precisely determine the gap between the main peak and the secondary peak that makes it possible to estimate the parameter NTc.
The aim of the present invention is therefore to propose a method for the blind estimation of parameters of an OFDM signal that does not have the aforementioned drawbacks.
A secondary aim of the present invention is to allow temporal and frequency synchronization of the OFDM receiver that is fast and does not require a learning sequence.
BRIEF DESCRIPTION OF THE INVENTION
The present invention is defined, according to a first embodiment, by a method for blind estimation of at least one parameter of an OFDM signal comprising a plurality of OFDM symbols, wherein the OFDM signal is baseband sampled for a given observation period to obtain a plurality of samples and in that one calculates, for a plurality of values of said parameter and a block of modulation symbols of a modulation constellation, a cost function depending on a distance between a vector y made up of the plurality of samples and a vector Hθa where a is made up of the symbols of said block and Hθ is a matrix representing the modulation of the sub-carriers of the OFDM multiplex by said modulation symbols, said matrix depending on the parameter to be estimated, said cost function being averaged for each value of the parameter on a set of blocks of possible symbols to obtain an averaged value of the cost function, the estimation of the parameter being obtained as the value of the parameter corresponding to the lowest averaged value.
According to one example, the parameter(s) of the OFDM signal is (are) the number of sub-carriers of that signal and/or the duration of the prefix of an OFDM symbol and/or the useful duration of an OFDM symbol.
According to a second embodiment, the invention also relates to a method for the blind estimation of at least one parameter of an OFDM signal comprising a plurality of OFDM symbols, characterized in that the OFDM signal is baseband sampled for a given observation period to obtain a plurality M of samples and in that one calculates, for a plurality of values of said parameter, a cost function depending on a distance between a vector y made up of said plurality of samples and a vector HθHθ†y where Hθ is a matrix representing the modulation of the sub-carriers of the OFDM multiplex by a block of modulation symbols, said matrix depending on the parameter to be estimated, and where Hθ†y is the pseudo-inverse matrix of Hθ, the estimation of the parameter being obtained as the value of the parameter corresponding to the lowest value of the cost function on said plurality of values of the parameter.
Advantageously, the pseudo-inverse matrix Hθ†y is calculated as




  
    H
    θ
    H
  
  
    q
    ⁡
    
      (
      
        1
        +
        
          D
          /
          N
        
      
      )
    
  




where HθH, q is the conjugated transpose of the matrix Hθ, q is a test value of the sampling rate of the OFDM signal, D and N are respectively a prefix length test value and a useful length test value expressed in number of chips.

A third embodiment of the invention concerns a method for the blind estimation of at least one parameter θ of an OFDM signal comprising a plurality of OFDM symbols, characterized in that the OFDM signal is baseband sampled for a given observation period to obtain a plurality of samples and in that one calculates, for a plurality of values of said parameter, a cost function Ω, decreasing function of the Gaussian conditional probability pg(y|θ), where y is a vector made up of said plurality of samples and pg(y|θ) is obtained from a guassian distribution of the modulation symbols modulating the various sub-carriers of the OFDM multiplex for said plurality of OFDM symbols, the estimation of the parameter being obtained as the value of the parameter corresponding to the lowest value of the cost function on said plurality of values of the parameter.
Advantageously, said cost function is a function of −log pg(y|θ).
According to a first example, said cost function can be calculated using the expression:




  
    Ω
    ⁡
    
      (
      
        
          -
          log
        
        ⁢
        
          
        
        ⁢
        
          
            p
            g
          
          ⁡
          
            (
            
              y
              |
              θ
            
            )
          
        
      
      )
    
  
  =
  
    
      det
      (
      
        
          2
          ⁢
          
            σ
            a
            2
          
          ⁢
          
            H
            θ
            H
          
          ⁢
          
            H
            θ
          
        
        +
        
          2
          ⁢
          
            N
            0
          
          ⁢
          
            I
            KN
          
        
      
      )
    
    -
    
      
        
          σ
          a
          2
        
        
          2
          ⁢
          
            
          
          ⁢
          
            N
            0
          
        
      
      ⁢
      
        y
        H
      
      ⁢
      
        
          
            H
            θ
          
          ⁡
          
            (
            
              
                
                  σ
                  a
                  2
                
                ⁢
                
                  H
                  θ
                  H
                
                ⁢
                
                  H
                  θ
                
              
              +
              
                
                  N
                  0
                
                ⁢
                
                  I
                  KN
                
              
            
            )
          
        
        
          -
          1
        
      
      ⁢
      
        H
        θ
        H
      
      ⁢
      y
    
  




where Hθ is a matrix representing the modulation of the sub-carriers of the OFDM multiplex by said modulation symbols, said matrix depending on the parameter to be estimated, HθH is the conjugated transpose of the matrix Hθ, IKN is the unit matrix, σa2 is an estimation of the variance of the modulation symbols, N0 is an estimation of the noise density, and det(.) is the determinant.

According to a second example, the cost function is obtained from the expression:



  
    
      Ω
      ⁡
      
        (
        
          
            -
            log
          
          ⁢
          
            
          
          ⁢
          
            
              p
              g
            
            ⁡
            
              (
              
                y
                |
                θ
              
              )
            
          
        
        )
      
    
    =
    
      
        KN
        ⁢
        
          
        
        ⁢
        
          log
          ⁡
          
            (
            
              ω
              a
              2
            
            )
          
        
      
      -
      
        
          
            σ
            a
            2
          
          
            
              N
              0
            
            ⁢
            
              ω
              a
              2
            
          
        
        ⁢
        
          y
          H
        
        ⁢
        
          H
          θ
        
        ⁢
        
          H
          θ
          H
        
        ⁢
        y
      
    
  




  where




  
    
      ω
      a
      2
    
    =
    
      
        2
        ⁢
        
          σ
          a
          2
        
        ⁢
        
          q
          ⁡
          
            (
            
              1
              +
              
                D
                N
              
            
            )
          
        
      
      +
      
        2
        ⁢
        
          N
          0
        
      
    
  



where Hθ is a matrix representing the modulation of the sub-carriers of an OFDM multiplex by said modulation symbols, said matrix depending on the parameter to be estimated, HθH is the conjugated transpose of the matrix Hθ, IKN is the unit matrix, σa2 is an estimate of the variance of the modulation symbols, N0 is the noise density, q is a test value of the sampling rate of the OFDM signal, D and N are respectively a prefix length test value and a useful length test value expressed in number of chips.




BRIEF DESCRIPTION OF THE DRAWINGS
Other features and advantages of the invention will appear upon reading one preferred embodiment of the invention done in reference to the appended figures, in which:
FIG. 1 diagrammatically illustrates an OFDM signal;
FIG. 2 diagrammatically illustrates an OFDM receiver known from the prior art;
FIG. 3 is a flowchart of the method for estimating parameters of an OFDM signal according to a first embodiment of the invention;
FIG. 4 is a flowchart of the method for estimating parameters of an OFDM signal according to a second embodiment of the invention;
FIG. 5 shows the relative performance of different methods of estimating parameters of an OFDM signal, according to the invention and according to the prior art.



DETAILED DESCRIPTION OF THE INVENTION
Below we will consider the case of a telecommunications system using an OFDM modulation. We will consider that the OFDM signal transmitted has the form given by expression (1) and that the OFDM symbols may or may not contain a prefix. For simplification, we will consider that they contain a prefix, but that it can be empty (D=0).
We will situate ourselves in the context of a blind estimation of at least one parameter of the OFDM signal, in particular an OFDM modulation parameter, such as the number of sub-carriers, the frequency spacing between these sub-carriers or, equivalently, the useful duration of the OFDM symbols. The set of parameters to be estimated is represented by a vector θ, for example θ=(N,DTc,NTc). In certain cases, this assembly can be reduced to a singleton. For example, the number N of carriers and the duration of the OFDM symbol Ts=(N+D)Tc can be known, the parameter to be blindly estimated being the useful duration NTc.
The signal received by the receiver can be expressed, after baseband demodulation and sampling at a frequency 1/Te greater than the Nyquist frequency 1/Tc, by:




  
    
      
        
          y
          ⁡
          
            (
            m
            )
          
        
        =
        
          
            
              ∑
              
                l
                =
                0
              
              L
            
            ⁢
            
              
                λ
                l
              
              ⁢
              
                
                  s
                  a
                
                ⁡
                
                  (
                  
                    m
                    -
                    
                      τ
                      l
                    
                  
                  )
                
              
            
          
          +
          
            b
            ⁡
            
              (
              m
              )
            
          
        
      
    
    
      
        (
        4
        )
      
    
  




where the λl and τl are respectively the complex attenuation coefficients and the delays associated with the different paths of the channel expressed in numbers of samples, L is the total number of paths, and b(m)=b(mTe) where b(t) is an additive white Gaussian noise.

Without loss of generality, but for the sake of simplification, we will hereinafter limit ourselves to the case of a single path channel. In that case, it can be shown that equation (4) can be put in matricial form:

y=Hθa+b  (5)

where y=(y(0), y(1), . . . , y(M−1))T, (.)T represents the transposition operation, M=└T0/Te┘ where T0 is the length of the observation window of the OFDM signal, and └x┘ is the largest integer less than or equal to x, ak=(ak,0, ak,1, . . . , ak,M−1)T and a=a0T, a1T, . . . , ak−1T)T with K=┌T0/Ts┐ and ┌x┐ is the smallest integer greater than or equal to x, b=b(0), b(1), . . . , b(M−1))T. In other words, y is the size vector M made up of samples received in baseband, a is a size vector KN representative of the sequence of the KN modulation symbols a modulating the sub-carriers n=0, . . . , N−1 of the OFDM symbols k=0, . . . , K−1 in the observation window, and b is the vector made up of the noise samples.

Generally, the matrix Hθ is representative of the channel and the OFDM modulation, this modulation being parameterized by a set θ of parameters, for example θ is the triplet θ=(N, DTc/NTc). We will use the case of a flat fading channel. The matrix Hθ is then only representative of the OFDM modulation. More precisely, if the forming function g(t) can be approximated by a window function, the elements of the matrix Hθ are null except for those defined to within a multiplicative factor by:




  
    
      
        
          
            H
            θ
          
          ⁡
          
            (
            
              m
              ,
              
                
                  
                    k
                    m
                  
                  ⁢
                  N
                
                +
                n
              
            
            )
          
        
        =
        
          
            1
            
              N
            
          
          ⁢
          
            e
            
              2
              ⁢
              
                iπ
                ⁢
                nm
              
              ⁢
              
                
                  T
                  e
                
                
                  NT
                  c
                
              
            
          
          ⁢
          
            e
            
              2
              ⁢
              iπ
              ⁢
              
                
              
              ⁢
              
                n
                ⁡
                
                  (
                  
                    
                      k
                      m
                    
                    +
                    1
                  
                  )
                
              
              ⁢
              
                
                  DT
                  c
                
                
                  NT
                  c
                
              
            
          
        
      
    
    
      
        (
        6
        )
      
    
  




where m=0, . . . , M−1, n=0, . . . , N−1 and where km is the only integer verifying:





  
    
      
        
          
            m
            ⁢
            
              
                T
                e
              
              
                
                  (
                  
                    N
                    +
                    D
                  
                  )
                
                ⁢
                
                  T
                  c
                
              
            
          
          -
          1
        
        <
        
          k
          m
        
        ≤
        
          m
          ⁢
          
            
          
          ⁢
          
            
              T
              e
            
            
              
                (
                
                  N
                  +
                  D
                
                )
              
              ⁢
              
                T
                c
              
            
          
        
      
    
    
      
        (
        7
        )
      
    
  




Generally, it is proposed according to the invention to look for the values of the parameters {circumflex over (θ)} of the OFDM signal according to a maximum likelihood (ML) criterion. According to that criterion, one looks for {circumflex over (θ)} that maximizes the conditional probability p(y|θ), i.e. the conditional probability p(y|θ,a) taken as an average over the values of a:

p(y|θ)=Ea(p(y|θ,a))  (8)

The components of the noise vector b being assumed to be independent, identically distributed and Gaussian, this amounts to looking for {circumflex over (θ)} that minimizes the quadratic distance between the vector of the received signal y and the vector Hθa, Hθ being a matrix representative of the transfer function of the channel and the parameters θ of the OFDM signal, as for example yielded by (6), the distance being taken as an average value over the vector of the modulation symbols a, i.e.:




  
    
      
        
          θ
          ^
        
        =
        
          
            argmin
            θ
          
          ⁢
          
            
              E
              a
            
            ⁡
            
              (
              
                
                  
                  
                    y
                    -
                    
                      
                        H
                        θ
                      
                      ⁢
                      a
                    
                  
                  
                
                2
              
              )
            
          
        
      
    
    
      
        (
        9
        )
      
    
  




and more generally:





  
    
      
        
          θ
          ^
        
        =
        
          
            argmin
            θ
          
          ⁢
          
            
              E
              a
            
            ⁡
            
              (
              
                Ω
                ⁡
                
                  (
                  
                    
                    
                      y
                      -
                      
                        
                          H
                          θ
                        
                        ⁢
                        a
                      
                    
                    
                  
                  )
                
              
              )
            
          
        
      
    
    
      
        (
        
          9
          ′
        
        )
      
    
  




where Ω is a cost function that increases with the distance ∥y−Hθa∥.

According to a first embodiment of the invention, the estimation of the parameters of the OFDM signal is done from expressions (9) or (9′). This estimation can for example be done using a Monte Carlo-type method when the receiver knows the law of probability of the symbols ak,n. 
According to a second embodiment of the invention, the estimation of the parameters of the OFDM signal, {circumflex over (θ)}, can be simplified by considering that the vector a is made up of deterministic symbols. In other words, the maximum likelihood criterion thus simplified, also called deterministic maximum likelihood (DML) criterion, does not take the probability distribution of a into account, but uses a first estimation of a:

â0=Hθ0†y  (10)

where θ0 is an arbitrary value, preferably chosen in the order of magnitude of the actual value, Hθ0†=(Hθ0HHθ0)−1Hθ0H is the pseudo-inverse matrix of Hθ0. 

The estimated values of the OFDM signal parameters, synthetically denoted in the form of a vector {circumflex over (θ)}, for example {circumflex over (θ)}=({circumflex over (N)},Tc,Tc) are then determined using:




  
    
      
        
          θ
          ^
        
        =
        
          
            argmax
            θ
          
          ⁢
          
            p
            ⁡
            
              (
              
                
                  y
                  |
                  θ
                
                ,
                
                  a
                  =
                  
                    
                      a
                      ^
                    
                    0
                  
                
              
              )
            
          
        
      
    
    
      
        (
        11
        )
      
    
  




If one assumes that the components of b are decorrelated and correspond to a white noise, hypothesis verified in practice if the sampling frequency 1/Te is high enough relative to the Nyquist frequency 1/Tc, the expression (11) can be written equivalently:




  
    
      
        
          θ
          ^
        
        =
        
          
            argmin
            θ
          
          ⁡
          
            (
            
              
                
                
                  y
                  -
                  
                    
                      H
                      θ
                    
                    ⁢
                    
                      
                        a
                        ^
                      
                      0
                    
                  
                
                
              
              2
            
            )
          
        
      
    
    
      
        (
        12
        )
      
    
  




or, by replacing â0 with its expression (10):





  
    
      
        
          θ
          ^
        
        =
        
          
            argmin
            θ
          
          ⁡
          
            (
            
              
                
                
                  
                    (
                    
                      
                        I
                        M
                      
                      -
                      
                        
                          
                            
                              H
                              θ
                            
                            ⁡
                            
                              (
                              
                                
                                  H
                                  
                                    θ
                                    0
                                  
                                  H
                                
                                ⁢
                                
                                  H
                                  
                                    θ
                                    0
                                  
                                
                              
                              )
                            
                          
                          
                            -
                            1
                          
                        
                        ⁢
                        
                          H
                          
                            θ
                            0
                          
                          H
                        
                      
                    
                    )
                  
                  ⁢
                  y
                
                
              
              2
            
            )
          
        
      
    
    
      
        (
        13
        )
      
    
  




and, more generally:





  
    
      
        
          θ
          ^
        
        =
        
          
            
              arg
              ⁢
              
                
              
              ⁢
              min
            
            θ
          
          ⁢
          
            (
            
              Ω
              ⁡
              
                (
                
                  
                  
                    
                      (
                      
                        
                          I
                          M
                        
                        -
                        
                          
                            
                              
                                H
                                θ
                              
                              ⁡
                              
                                (
                                
                                  
                                    H
                                    
                                      θ
                                      0
                                    
                                    H
                                  
                                  ⁢
                                  
                                    H
                                    
                                      θ
                                      0
                                    
                                  
                                
                                )
                              
                            
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            H
                            
                              θ
                              0
                            
                            H
                          
                        
                      
                      )
                    
                    ⁢
                    y
                  
                  
                
                )
              
            
            )
          
        
      
    
    
      
        (
        
          13
          '
        
        )
      
    
  




where Ω is a cost function that increases with the distance ∥(IM−Hθ(Hθ0HHθ0)−1Hθ0H)y∥

It can be shown that if T0 is a whole multiple of Ts, in other words if the length of the observation window is a whole multiple of the OFDM symbol time, the matrix Hθ0HHθ0 then reduces to:




  
    
      
        
          
            H
            
              θ
              0
            
            H
          
          ⁢
          
            H
            
              θ
              0
            
          
        
        =
        
          
            q
            ⁡
            
              (
              
                1
                +
                
                  D
                  N
                
              
              )
            
          
          ⁢
          
            I
            KN
          
        
      
    
    
      
        (
        14
        )
      
    
  




and therefore the pseudo-inverse matrix Hθ0†:





  
    
      
        
          H
          
            θ
            0
          
          †
        
        =
        
          
            H
            
              θ
              0
            
            H
          
          
            q
            ⁡
            
              (
              
                1
                +
                
                  D
                  /
                  N
                
              
              )
            
          
        
      
    
    
      
        (
        
          14
          '
        
        )
      
    
  




where q=Te/Tc is the sampling rate of the OFDM signal used for the test values of the parameters θ0 and IKN is the unit matrix with size KN×KN.

In practice, expression (14) is a satisfactory approximation of Hθ0HHθ0, even if the length of the observation window is not a whole multiple of the duration of an OFDM symbol. In the context of this approximation, the criterion (13′) is simplified as follows:




  
    
      
        
          θ
          ^
        
        =
        
          
            
              arg
              ⁢
              
                
              
              ⁢
              min
            
            θ
          
          ⁢
          
            (
            
              Ω
              (
              
                
                
                  
                    (
                    
                      
                        I
                        M
                      
                      -
                      
                        
                          
                            H
                            θ
                          
                          ⁢
                          
                            H
                            
                              θ
                              0
                            
                            H
                          
                        
                        
                          q
                          ⁡
                          
                            (
                            
                              1
                              +
                              
                                D
                                /
                                N
                              
                            
                            )
                          
                        
                      
                    
                    )
                  
                  ⁢
                  y
                
                
              
              )
            
            )
          
        
      
    
    
      
        (
        15
        )
      
    
  




where Ω is the aforementioned cost function. The values q,D,N appearing in the right portion of the expression (15) depend on the test values of the parameters θ0.

It should be noted that the process of estimating the set of parameters θ can be iterated to improve the precision thereof. Indeed, the estimated value {circumflex over (θ)} given by (13), (13′) or (15) can then be used as a new test value θ1={circumflex over (θ)}, in other words this first estimation of the parameters of the OFDM signal can be used to calculate a second matrix Hθ1, for example using expression (6), and a second estimation {circumflex over (θ)}(2) from (13), (13′) or (15) in which the matrix Hθ0 has been replaced by the matrix Hθ1. This second estimation can be used again as test value, and so on.
The preceding iterative method works well if the test value is close to the actual value of the parameters. It can, however, lead to an erroneous estimation in the case where θ0 is too far from the actual parameters, for example if the number of carriers, the prefix length and the useful length of the OFDM symbol, taken as test values, are too far from the actual values of the parameters of the OFDM signal. In that case, the estimation method may or may not converge towards a local minimum of the cost function.
In order to resolve this drawback, an alternative of the estimation method according to the second embodiment of the invention consists of using a “closed” expression of the distance appearing in (13), (13′) and (15), or respectively:




  
    
      
        
          θ
          ^
        
        =
        
          
            
              
                arg
                ⁢
                
                  
                
                ⁢
                min
              
              θ
            
            ⁢
            
              (
              
                
                  
                  
                    
                      (
                      
                        
                          I
                          M
                        
                        -
                        
                          
                            
                              
                                H
                                θ
                              
                              ⁡
                              
                                (
                                
                                  
                                    H
                                    θ
                                    H
                                  
                                  ⁢
                                  
                                    H
                                    θ
                                  
                                
                                )
                              
                            
                            
                              -
                              1
                            
                          
                          ⁢
                          
                            H
                            θ
                            H
                          
                        
                      
                      )
                    
                    ⁢
                    y
                  
                  
                
                2
              
              )
            
          
          =
          
            
              
                arg
                ⁢
                
                  
                
                ⁢
                min
              
              θ
            
            ⁢
            
              (
              
                
                  
                  
                    
                      (
                      
                        
                          I
                          M
                        
                        -
                        
                          
                            H
                            θ
                          
                          ⁢
                          
                            H
                            θ
                            †
                          
                        
                      
                      )
                    
                    ⁢
                    y
                  
                  
                
                2
              
              )
            
          
        
      
    
    
      
        (
        16
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            θ
            ^
          
          =
          
            
              
                arg
                ⁢
                
                  
                
                ⁢
                min
              
              θ
            
            ⁢
            
              (
              
                Ω
                ⁡
                
                  (
                  
                    
                    
                      
                        (
                        
                          
                            I
                            M
                          
                          -
                          
                            
                              H
                              θ
                            
                            ⁢
                            
                              H
                              θ
                              †
                            
                          
                        
                        )
                      
                      ⁢
                      y
                    
                    
                  
                  )
                
              
              )
            
          
        
      
    
    
      
        (
        
          16
          '
        
        )
      
    
  
  
    
      
        
          
        
        ⁢
        
          
            θ
            ^
          
          =
          
            
              
                arg
                ⁢
                
                  
                
                ⁢
                min
              
              θ
            
            ⁢
            
              (
              
                Ω
                ⁡
                
                  (
                  
                    
                    
                      
                        (
                        
                          
                            I
                            M
                          
                          -
                          
                            
                              
                                H
                                θ
                              
                              ⁢
                              
                                H
                                θ
                                H
                              
                            
                            
                              q
                              ⁡
                              
                                (
                                
                                  1
                                  +
                                  
                                    D
                                    /
                                    N
                                  
                                
                                )
                              
                            
                          
                        
                        )
                      
                      ⁢
                      y
                    
                    
                  
                  )
                
              
              )
            
          
        
      
    
    
      
        (
        17
        )
      
    
  




FIG. 3 illustrates a method for blind estimation of an OFDM signal parameter according to said alternative of the second embodiment of the invention.
In step 310, the OFDM signal is baseband demodulated, then sampled at 320 at a frequency 1/Te greater than the Nyquist frequency. The order of magnitude of 1/Tc is generally known, in other words the bandwidth of the OFDM signal, which makes it possible to choose the sampling frequency accordingly.
In step 330, a block of M complex samples is chosen corresponding to an observation window with a given width T0. If the duration Ts of the OFDM symbol is known, this width is advantageously chosen as a multiple of Ts.
In 340, the vector y is built from M complex samples, in 350 the values of the parameters of θ are initialized, as well as the minimum of the cost function at a high value Ωmin, then one enters an iterative loop that goes through a range of values of said parameters.
At each iteration, in 360 one calculates the matrix Hθ and depending on the case, the pseudo-inverse matrix Hθ† (expressions 16, 16′) or the matrix HθHθH (expression 17) then, in 365, the value η taken from the function by the cost function Ω, for the current vector θ.
In 370, one tests whether this value is below Ωmin If it is, Ωmin, is updated by η in 375 and the corresponding values of the parameters θ are stored in memory. One then tests, in 380, whether the range of parameters has been completely gone through. If it has, one exits in 390. If not, the parameters of θ are modified in 385 and one returns to the calculation step 360. The values of the estimated parameters, {circumflex over (θ)}, are retrieved from the memory.
The estimation method according to the second embodiment as described above has, as will be seen later, a lower error rate than the estimation method through auto-correlation known from the prior art.
An estimation method according to a third embodiment is, however, preferred for low signal-to-noise ratios.
This method assumes that the modulation symbols ak,n can be likened to independent complex random variables, identically distributed, circular, symmetrical and Gaussian. It will be recalled that a complex random variable is circular, symmetrical and Gaussian if its real and imaginary components are independent Gaussian variables with the same variance. In that case, the vector y defined by (5) is itself a vectorial, circular, symmetrical, Gaussian random variable, with covariance matrix:

E(yyH)=2σa2HθHθH+2N0IM  (18)

The conditional probability density of y relative to θ in the Gaussian hypothesis, also called Gaussian conditional probability and denoted pg(y|θ), can then be expressed as follows:




  
    
      
        
          p
          ⁡
          
            (
            
              y
              |
              θ
            
            )
          
        
        =
        
          
            
              1
              
                
                  
                    (
                    
                      2
                      ⁢
                      π
                    
                    )
                  
                  M
                
                ⁢
                
                  det
                  ⁡
                  
                    (
                    
                      
                        2
                        ⁢
                        
                          σ
                          a
                          2
                        
                        ⁢
                        
                          H
                          θ
                        
                        ⁢
                        
                          H
                          θ
                          H
                        
                      
                      +
                      
                        2
                        ⁢
                        
                          N
                          0
                        
                        ⁢
                        
                          I
                          M
                        
                      
                    
                    )
                  
                
              
            
            ⁢
            exp
          
          -
          
            
              1
              2
            
            ⁢
            
              
                
                  y
                  H
                
                ⁡
                
                  (
                  
                    
                      
                        σ
                        a
                        2
                      
                      ⁢
                      
                        H
                        θ
                      
                      ⁢
                      
                        H
                        θ
                        H
                      
                    
                    +
                    
                      
                        N
                        0
                      
                      ⁢
                      
                        I
                        M
                      
                    
                  
                  )
                
              
              
                -
                1
              
            
            ⁢
            y
          
        
      
    
    
      
        (
        19
        )
      
    
  




The given probability distribution (19) differs from the actual distribution inasmuch as the modulation symbols generally belong to a constellation of points. In practice, however, that modeling is satisfactory subject to a correct choice of the variance σu2.
Moreover, if I is the unit matrix with a given size and A is any matrix with the same size, we have the following properties:

det(I+AAH)=det(I+AHA) and  (20)

(I+AAH)−1=I−A(I+AHA)−1AH  (21)

Given properties (20) and (21), the Gaussian conditional probability density (19) can be reformulated, to within a multiplicative coefficient:




  
    
      
        
          
            p
            g
          
          ⁡
          
            (
            
              y
              |
              θ
            
            )
          
        
        ∝
        
          
            1
            
              det
              ⁡
              
                (
                
                  
                    2
                    ⁢
                    
                      σ
                      a
                      2
                    
                    ⁢
                    
                      H
                      θ
                      H
                    
                    ⁢
                    
                      H
                      θ
                    
                  
                  +
                  
                    2
                    ⁢
                    
                      N
                      0
                    
                    ⁢
                    
                      I
                      KN
                    
                  
                
                )
              
            
          
          ⁢
          
            exp
            (
            
              
                
                  σ
                  a
                  2
                
                
                  2
                  ⁢
                  
                    N
                    0
                  
                
              
              ⁢
              
                y
                H
              
              ⁢
              
                
                  
                    H
                    θ
                  
                  ⁡
                  
                    (
                    
                      
                        
                          σ
                          a
                          2
                        
                        ⁢
                        
                          H
                          θ
                          H
                        
                        ⁢
                        
                          H
                          θ
                        
                      
                      +
                      
                        
                          N
                          0
                        
                        ⁢
                        
                          I
                          KN
                        
                      
                    
                    )
                  
                
                
                  -
                  1
                
              
              ⁢
              
                H
                θ
                H
              
              ⁢
              y
            
            )
          
        
      
    
    
      
        (
        22
        )
      
    
  




Looking for the maximum of the Gaussian conditional probability pg(y|θ) amounts to looking for the minimum of −log pg(y|θ). The vector of the parameters of the OFDM signal that corresponds to the likelihood maximum is then given by:




  
    
      
        
          θ
          ^
        
        =
        
          
            
              arg
              ⁢
              
                
              
              ⁢
              min
            
            θ
          
          ⁢
          
            (
            
              Ω
              ⁡
              
                (
                
                  
                    -
                    log
                  
                  ⁢
                  
                    
                  
                  ⁢
                  
                    
                      p
                      g
                    
                    ⁡
                    
                      (
                      
                        y
                        |
                        θ
                      
                      )
                    
                  
                
                )
              
            
            )
          
        
      
    
    
      
        (
        23
        )
      
    
  




where Ω is a cost function that increases with −log pg (y|θ).

According to (22), an example of a cost function leading to the likelihood maximum, in the Gaussian hypothesis, can be written:




  
    
      
        
          Ω
          ⁡
          
            (
            
              
                -
                log
              
              ⁢
              
                
              
              ⁢
              
                
                  p
                  g
                
                ⁡
                
                  (
                  
                    y
                    |
                    θ
                  
                  )
                
              
            
            )
          
        
        =
        
          
            det
            ⁡
            
              (
              
                
                  2
                  ⁢
                  
                    σ
                    a
                    2
                  
                  ⁢
                  
                    H
                    θ
                    H
                  
                  ⁢
                  
                    H
                    θ
                  
                
                +
                
                  2
                  ⁢
                  
                    N
                    0
                  
                  ⁢
                  
                    I
                    KN
                  
                
              
              )
            
          
          -
          
            
              
                σ
                a
                2
              
              
                2
                ⁢
                
                  N
                  0
                
              
            
            ⁢
            
              y
              H
            
            ⁢
            
              
                
                  H
                  θ
                
                ⁡
                
                  (
                  
                    
                      
                        σ
                        a
                        2
                      
                      ⁢
                      
                        H
                        θ
                        H
                      
                      ⁢
                      
                        H
                        θ
                      
                    
                    +
                    
                      
                        N
                        0
                      
                      ⁢
                      
                        I
                        KN
                      
                    
                  
                  )
                
              
              
                -
                1
              
            
            ⁢
            
              H
              θ
              H
            
            ⁢
            y
          
        
      
    
    
      
        (
        24
        )
      
    
  




As in the second embodiment, when the length of the observation window is a whole multiple of the duration of an OFDM symbol, and more generally when the approximation (14) is valid, the cost function (24) can take the simplified form:




  
    
      
        
          
            Ω
            ⁡
            
              (
              
                
                  -
                  log
                
                ⁢
                
                  
                
                ⁢
                
                  
                    p
                    g
                  
                  ⁡
                  
                    (
                    
                      y
                      |
                      θ
                    
                    )
                  
                
              
              )
            
          
          =
          
            
              KN
              ⁢
              
                
              
              ⁢
              
                log
                ⁡
                
                  (
                  
                    ω
                    a
                    2
                  
                  )
                
              
            
            -
            
              
                
                  σ
                  a
                  2
                
                
                  
                    N
                    0
                  
                  ⁢
                  
                    ω
                    a
                    2
                  
                
              
              ⁢
              
                y
                H
              
              ⁢
              
                H
                θ
              
              ⁢
              
                H
                θ
                H
              
              ⁢
              y
            
          
        
        ⁢
        
          

        
        ⁢
        with
      
    
    
      
        (
        25
        )
      
    
  
  
    
      
        
          ω
          a
          2
        
        =
        
          
            2
            ⁢
            
              σ
              a
              2
            
            ⁢
            
              q
              ⁡
              
                (
                
                  1
                  +
                  
                    D
                    N
                  
                
                )
              
            
          
          +
          
            2
            ⁢
            
              
                N
                0
              
              .
            
          
        
      
    
    
      
        (
        26
        )
      
    
  




The calculation of the cost function according to (24) or (25) requires that the values of N0 and σa be known. The noise level, N0, may be estimated either through knowledge of the noise temperature in the band, or by measurements done in the absence of a signal. The signal level σa may be estimated by measuring the power level in the band of interest, equal to N0+σa2.
FIG. 4 diagrammatically illustrates a method for the blind estimation of an OFDM signal parameter according to a second embodiment of the invention.
Steps 410 to 450 are identical to steps 310 to 350, respectively, and therefore will not be described again.
Upon each iteration of the loop, one calculates, in 460, the matrix Hθ, then the matrix HθHθH. The value of η is then deduced therefrom taken from the function by the cost function Ω, for the current vector θ, according to expression (24) or (25). In the illustrated example, expression (25) has been used.
In 470, one tests whether this value is less than Ωmin. If it is, Ωmin is up dated by η in 475 and the corresponding values of the parameters θ are stored in memory. One then tests, in 480, whether the range of parameters has been completely gone through. If it has, one exits in 490. If not, one modifies the parameters of θ in 485 and returns to the calculation step 460. The values of the estimated parameters, {circumflex over (θ)}, are the values present in the memory.
FIG. 5 shows the estimation error rate of an OFDM signal parameter, here the OFDM symbol duration NTc as a function of the signal-to-noise ratio.
The curve designated by COR is that obtained using the estimation method based on the auto-correlation method of the OFDM signal, as known from the prior art.
The curve designated by DML is that based on the estimation method according to the alternative of the second embodiment of the invention (expression (16)), that designated by ADML pertaining to the simplified version (expression (17)).
The curve designated by GML is that based on the estimation method according to the third embodiment of the invention (expression (24)), that designated by AGML pertaining to the simplified version (expression (25)).
It can be seen that regardless of the signal-to-noise ratio level, the estimation methods DML, GML as well as the simplified versions thereof ADML and AGML, are higher performing than that of the prior art.
Moreover, the GML and AGML methods yield a more reliable estimation than the DML and ADML methods, in particular for low signal-to-noise ratios. One may therefore consider opting for one or the other method according to the signal-to-noise ratio or the processing time.

## Claims

1. A method of blind estimation of at least one modulation parameter (θ) of an OFDM signal comprising a plurality of OFDM symbols, the method comprising:
baseband sampling the OFDM signal for a given observation period (T0) to obtain a plurality of samples;
calculating, for a plurality of values of the parameter and a block of modulation symbols (ak,n) of a modulation constellation, a cost function (Ω) depending on a distance between a vector y made up of the plurality of samples and a vector Hθa, wherein a comprises the symbols of the block and Hθ comprises a matrix representing the modulation of the sub-carriers of the OFDM multiplex by the modulation symbols, the matrix depending on the parameter to be estimated; and
averaging the cost function for each value of the parameter on a set of blocks of possible symbols to obtain an averaged value (Ea(Ω(∥y−Hθa∥))) of the cost function, the estimation of the parameter being obtained as the value of the parameter ({circumflex over (θ)}) corresponding to the lowest averaged value





  
    (
    
      
        θ
        ^
      
      =
      
        
          
            arg
            ⁢
            
              
            
            ⁢
            min
          
          θ
        
        ⁢
        
          
            E
            a
          
          ⁡
          
            (
            
              Ω
              ⁡
              
                (
                
                  
                  
                    y
                    -
                    
                      
                        H
                        θ
                      
                      ⁢
                      a
                    
                  
                  
                
                )
              
            
            )
          
        
      
    
    )
  
  .

2. The method according to claim 1, wherein the modulation parameter(s) of the OFDM signal is (are) the number (N) of sub-carriers of that signal and/or the duration of the prefix (DTc) of an OFDM symbol and/or the useful duration (NTc) of an OFDM symbol.

3. A method of blind estimation of at least one modulation parameter (θ) of an OFDM signal comprising a plurality of OFDM symbols, the method comprising:
baseband sampling the OFDM signal for a given observation period (T0) to obtain a plurality M of samples;
calculating, for a plurality of values of the parameter, a cost function (Ω) depending on a distance between a vector y made up of the plurality of samples and a vector HθHθ†y wherein Hθ comprises a matrix representing the modulation of the sub-carriers of the OFDM multiplex by a block of modulation symbols, the matrix depending on the parameter to be estimated, and wherein Hθ† comprises the pseudo-inverse matrix of Hθ; and
estimating the parameter being obtained as the value of the parameter ({circumflex over (θ)}) corresponding to the lowest value of the cost function of the plurality of values of the parameter





  
    (
    
      
        θ
        ^
      
      =
      
        
          
            arg
            ⁢
            
              
            
            ⁢
            min
          
          θ
        
        ⁢
        
          
            E
            a
          
          ⁡
          
            (
            
              Ω
              ⁡
              
                (
                
                  
                  
                    y
                    -
                    
                      
                        H
                        θ
                      
                      ⁢
                      a
                    
                  
                  
                
                )
              
            
            )
          
        
      
    
    )
  
  .

4. The estimation method according to claim 3, wherein the pseudo-inverse matrix Hθ† is calculated as




  
    H
    θ
    H
  
  
    q
    ⁡
    
      (
      
        1
        +
        
          D
          /
          N
        
      
      )
    
  




wherein HθH comprises the conjugated transpose of the matrix Hθ, q comprises a test value of the sampling rate of the OFDM signal, D and N are respectively a prefix length test value and a useful length test value expressed in number of chips (Tc).

5. A method for of blind estimation of at least one modulation parameter θ of an OFDM signal comprising a plurality of OFDM symbols, the method comprising:
baseband sampling the OFDM signal for a given observation period (T0) to obtain a plurality of samples;
calculating, for a plurality of values of the parameter, a cost function Ω, a decreasing function of the Gaussian conditional probability pg(y|θ), where y comprises a vector made up of the plurality of samples and pg(y|θ) is obtained from a Gaussian distribution of the modulation symbols modulating the various sub-carriers of the OFDM multiplex for the plurality of OFDM symbols; and
estimating the parameter being obtained as the value of the parameter ({circumflex over (θ)}) corresponding to the lowest value of the cost function on the plurality of values of the parameter.

6. The method according to claim 5, wherein the cost function is a function of −log pg(y|θ).

7. The estimation method according to claim 6, wherein the cost function can be calculated using the expression:




  
    Ω
    ⁡
    
      (
      
        
          -
          log
        
        ⁢
        
          
        
        ⁢
        
          
            p
            g
          
          ⁡
          
            (
            
              y
              |
              θ
            
            )
          
        
      
      )
    
  
  =
  
    
      det
      ⁡
      
        (
        
          
            2
            ⁢
            
              σ
              a
              2
            
            ⁢
            
              H
              θ
              H
            
            ⁢
            
              H
              θ
            
          
          +
          
            2
            ⁢
            
              N
              0
            
            ⁢
            
              I
              KN
            
          
        
        )
      
    
    -
    
      
        
          σ
          a
          2
        
        
          2
          ⁢
          
            N
            0
          
        
      
      ⁢
      
        y
        H
      
      ⁢
      
        
          
            H
            θ
          
          ⁡
          
            (
            
              
                
                  σ
                  a
                  2
                
                ⁢
                
                  H
                  θ
                  H
                
                ⁢
                
                  H
                  θ
                
              
              +
              
                
                  N
                  0
                
                ⁢
                
                  I
                  KN
                
              
            
            )
          
        
        
          -
          1
        
      
      ⁢
      
        H
        θ
        H
      
      ⁢
      y
    
  



where Hθ comprises a matrix representing the modulation of the sub-carriers of the OFDM multiplex by the modulation symbols, the matrix depending on the parameter to be estimated, HθH comprises the conjugated transpose of the matrix Hθ, IKN comprises the unit matrix, σa2 comprises an estimation of the variance of the modulation symbols, N0 comprises an estimation of the noise density, and det(.) comprises a determinant.

8. The method according to claim 6, wherein the cost function is obtained from the expression:



  
    
      Ω
      ⁡
      
        (
        
          
            -
            log
          
          ⁢
          
            
          
          ⁢
          
            
              p
              g
            
            ⁡
            
              (
              
                y
                |
                θ
              
              )
            
          
        
        )
      
    
    =
    
      
        KN
        ⁢
        
          
        
        ⁢
        
          log
          ⁡
          
            (
            
              ω
              a
              2
            
            )
          
        
      
      -
      
        
          
            σ
            a
            2
          
          
            
              N
              0
            
            ⁢
            
              ω
              a
              2
            
          
        
        ⁢
        
          y
          H
        
        ⁢
        
          H
          θ
        
        ⁢
        
          H
          θ
          H
        
        ⁢
        y
      
    
  




  where




  
    
      ω
      a
      2
    
    =
    
      
        2
        ⁢
        
          σ
          a
          2
        
        ⁢
        
          q
          ⁡
          
            (
            
              1
              +
              
                D
                N
              
            
            )
          
        
      
      +
      
        2
        ⁢
        
          N
          0
        
      
    
  


where Hθ comprises a matrix representing the modulation of the sub-carriers of an OFDM multiplex by the modulation symbols, the matrix depending on the parameter to be estimated, HθH comprises the conjugated transpose of the matrix Hθ, IKN is comprises the unit matrix, σa2 comprises an estimate of the variance of the modulation symbols, N0 comprises the noise density, q comprises a test value of the sampling rate of the OFDM signal, D and N are respectively a prefix length test value and a useful length test value expressed in number of chips (Tc).