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Carrier Estimation for QAM Receivers Using HOS Angelfire by alicejenny

VIEWS: 3 PAGES: 17

									enemer@ieee.org




      New algorithmic approach for estimating the frequency
      and phase offset of a QAM carrier in AWGN conditions
      Using HOC




                                                              Jump to first page
                   • Context and Motivation
                   • Problem statement
                   • Carrier Estimation with HOC
                         • Frequency offset
                         • Phase offset
                   • Performance analysis
                         • Simulation Data
                         • Analysis
                   • Conclusion




E. Nemer   -   2                                   Jump to first page
         cos2 ( f c  f )t   
                                                    K samples
                                                      / symbol
                                                                 I
                                                 Rx
                                   LPF
                                                 Nyq
                                                                 n
                                                                                    PLL
r (t )   (t )
                                                                             FFE                FBE
                                                                 Q
                                                 Rx              n                 Slicer
                                      LPF
                                                 Nyq                                         Equalizer


           sin2 ( f c  f )t   
                                                    Carrier
                                                    Estimation

      Often done in 2 stages :
               First :
                           blind (no reliance on apriori symbol knowledge)
                           does not require a proper timing estimation
                           uses the output of the RxNyq (oversampled)
               Second :
                           decision-directed
                           can only correct for small offset
                           uses typically 1 sample / symbol
  E. Nemer        -   3                                                            Jump to first page
    • Non-decision directed recovery


                              ML-based schemes

                              Feed-forward , feedback




 • Criteria in designing algorithms

                     • Performance degradation in low SNR conditions


                     •Complexity issues : simplified ML approaches are often used


                     • Issues related to acquisition time, hang up are key in burst receivers




E. Nemer   -   4                                                            Jump to first page
   • Inherent suppression of Gaussian and symmetric processes
                      - Symmetric process                           --> third cumulant = zero

                      - Gaussian process                             --> All cumulants > order 2 are = zero

                            y(n) = x(n) + g(n)                --> HOS (y) = HOS (x)




                                                                                                           Linear ? HOS = 0
   • Detection of Non-linearity and phase coupling                            Gaussian          H(f)
                                                                                                           HOS != 0 ? Non-linear !
                                                 C4
   • Peculiar boundaries              -2<     -------------------
                                                            2         < inf
                                                C2


                        -2          -0.75        0                        3                              inf
                                                                                                                       Normalized
                                                                                                                       kurt
                   rand                     Gaussian                                                   Impulse
                                                                        Laplacian                      process
                   binary     2 sinusoids
                              Unif. phase

E. Nemer   -   5                                                                                               Jump to first page
         cos2 ( f c  f )t   
                                                             K samples
                                                               / symbol
                                                                                                I
                                      LPF                  Rx                                   n
                                                           Nyq
r (t )   (t )

                                                                                               Q
                                                           Rx                                  n
                                       LPF
                                                           Nyq


           sin2 ( f c  f )t   
                                                                              z n  sn e j ( 2 f Ts n  )   n
                                                             Carrier
                                                             Estimation


                          Assuming:
                                 Symbol rate (but not timing) is known
                                 Channel is equalized
                                 Statistics of the sent symbols are known (but not exact values)


                           Estimate the frequency offset


  E. Nemer        -   6                                                                        Jump to first page
A demodulated M-PSK signal can be represented as :                                                   Freq offset

                                                                     n  0,1,...,N - 1
               z n  sn e j ( n  )   n
                                                        Freq offset:         2  f  Ts


The freq offset may be estimated from the autocorrelation of Zn
at lag L, assumed to be a multiple of K, the symbol time



                          
Rz ( L)  E z n z n - L  Es  e jL  rI ( L)  rQ ( L)
                  *
                                                                                        
                                                                              E sn sn  L  E s
                                                                                    *




                                      Es  sin(L)              
      arg Rz ( L)  arctan                                    
                             Es  cos(L)  rI ( L)  rQ ( L) 
                                                                            In-phase and quadrature
                                                                            components of the noise
                                                                            are small (zero) at the epok
                                    E  sin(L) 
             arg Rz ( L)  arctan s            
                                    Es  cos(L) 
                                                                                                                       [4]


E. Nemer   -     7                                                                                Jump to first page
 • Generalized M-QAM signal at the receiver output :
                                                                                                  Freq offset

               z n  sn e j ( n  )   n    n  0,1,...,N - 1
                                                            Freq offset:      2  f  Ts

 • Consider the 1D slice of the 4th order cumulant


                C ( L)  Ez                  - Ez  - 2Ez z 
                                     2 2*               2    2               *      2
                     4                 z
                                     n n L             n                  n n L


 • Where the sub-terms are :


                 Z n  L  S n  L  e j  2 n  2 L  2 
                   2         2




                                             
               E Z n Z n  L  E S n S n  L  e - jL
                       *               *




                                        
               E Z n Z n *  E S n S n *  e -2 jL
                   2   2         2   2
                                                    

E. Nemer   -     8                                                                             Jump to first page
 • Assume the diagonal symbols are used (S1, S3)
                S1  a  ja            S 2  a - ja
                S 3  -a - ja          S 4  -a  ja                               Freq offset

 • Then higher order moments can be expressed in terms of the signal energy

             
           E S n S n *  1 .5 E s
               2   2
                               2
                                                                   
                                                E S n S n  L  1 .5 E s
                                                    2   2*             2



           E S            
                      S n  Es
                        *
                    n
                                                               
                                                 E S n S n  L  - Es
                                                         *




 • The (normalized) 4th-order cumulant becomes :


           K 
                    C4 ( L)
                            
                    C4 (0) 1.5
                              1
                                   
                                1  0.5e - j 2 L       
           K
                 1
                1.5
                                 
                    0.5  cos2 (L) - j sin(L) cos(L)     
 • From which , the frequency offset may be deduced

                                       ˆ    1            imag( K )         
                                         - arctan
                                            L       real ( K ) - 0.5 / 1.5 
                                                                            

E. Nemer   -    9                                                               Jump to first page
 • Consider the 1D slice of the 3rd order cumulant, defined as :
                                                                                                    Phase offset
                              *
                                  
               C 3 ( L)  E Z n Z n  L
                                  2
                                                  
 • and can be shown to be (for diagonal symbols) :

                                         
               C 3 ( L)  E S n S n  L  e j 2 L  n  
                              * 2



               C 3 ( L)  2a 3 cos(x) - sin( x)  jsin( x)  cos(x)                     x  2L  n  
 • Consider the sum of the real and imaginary parts :

               T  real C3 ( L)  imag C3 ( L)
               T  4a 3  cos(2L  n   )

 • and the normalized sum (by the energy) :                       T
                                                                 E s3 / 2
                                                                                   
                                                                                  2 2  cos(2L  n   )


 • If the frequency offset is zero (or known), then the phase may be estimated as:


                                                    ˆ            2 T 
                                                     arccos        3/ 2 
                                                                  2 Es 

E. Nemer   -   10                                                                                 Jump to first page
                                                     Effect of Nyquist Filter and Channel


    The Cumulants at the output of a channel / filter
    may be written in terms of that at the input and the coefficients :

                                                          CY ( p )  CW ( p ) (hk ) p
                           W                    Y                                              k

                                      h(k)




     Relations between Normalized Cumulants
                                                                  (h )     k
                                                                                    p


                                               K Y ( p, q )        k
                                                                                    p/q
                                                                                          KW ( p, q )
                                                                 (h )
                                                                k
                                                                        k
                                                                                q




E. Nemer   -   11                                                                                  Jump to first page
    Fc = 5 MHz
    Fsym = 160 ksym/s
    Block size : 100 symbols
                                                                                      Freq offset
                Method        SNR         Df=200 Hz   2 kHz            10 kHz

                2nd order     40          199 Hz      2000 Hz          10000
                              dB                                       Hz
                4th order                 200         2007             10026

                2nd order     20          190         2002             9999

                4th order                 198         2009             10024

                2nd order     15          205         1982             9992

                4th order                 202         1990             10016

                2nd order     8           221         1954             9992

                4th order                 198         1972             10016




                    4th order yields better                  … but not at high freq offsets
                    accuracy at small freq offsets

E. Nemer   -   12                                                                  Jump to first page
               Complex
               sinusoid                   Gaussian noise



                    z n  sn e j ( n  )   n

                                                           
                                                        E zn zn L 
                                                           2  2*
                                                                              1
                                                                               N
                                                                                     z       2
                                                                                              n
                                                                                                   2*
                                                                                                  zn L
                                                                                    n 1: N



  When computing the HOC of Zn, using time averages, added terms occur due to :

       • Bias                                  - Occurs when frame length not an integer number of periods
         (sine wave)


       • Bias                                  - Time estimator is only asymptotically unbiased.
         (Gaussian noise)                      - Need to define a new unbiased estimator of 4th stat


       • Variance                              - Function of underlying process (noise) energy
         Gaussian noise                        - May be reduced by increasing segment length




E. Nemer   -   13                                                                       Jump to first page
                                                                1
   Bias : Case
   of a Sine
                                             M L  2 , 3, 4   
                                                                N
                                                                      L
                                                                      zn
                                                                    n 1:N
   wave

                       - Occur when the frame length not an integer number of periods

                       - Can be reduced by computing HOC over several segments and averaging them.




                a 2  2 sin TwN  cosTw[ N - 1]                        2nd and 4th order
           M2                                         1                  moments of a sine wave
                2              N sin Tw                


                a 4  2 sin 2TwN cos2Tw[ N - 1]  4  8 sin TwN cosTw[ N - 1]  2  
           M4                                                                            6
                16              N sin 2Tw                         N sin Tw               

                                                                                        T: sample time
                                                                                        w : frequency
                                                                                        Q: Phase

    - More bias terms are present in the 4th order moment than the 2nd order




E. Nemer     -   14                                                                    Jump to first page
     Bias : Case
     of Gaussian
     Noise
                             - Time-average based estimator is only asymptotically unbiased.

                             - Need to define a new unbiased estimator of 4th stat :


                      
 C4 (0)  E zn zn * - 3 E zn
             2 2           2
                                         2
                                                        C4 (0)  M 4 - 3M 2 
                                                                                            2



                                                                      1
                                                               M4 
                                                                      N
                                                                            z
                                                                           n 1: N
                                                                                     2 2*
                                                                                      z
                                                                                     n n




                                         
                                                                       1
                                E C 4 ( 0)  C 4 ( 0)
                                                               M2 
                                                                       N
                                                                            z z
                                                                           n 1: N
                                                                                       *
                                                                                     n n



                                       Biased !



     Define a (new) unbiased estimator :
                                                                 2
                                                   UC 4 (0)  1   M 4 - 3M 2 
                                                                                  2

                                                               N

           Valid only for white Gaussian noise

E. Nemer     -   15                                                                             Jump to first page
           Variance :
           Case
           of Gaussian
           Noise
                                   - Function of underlying process (noise) energy

                                   - May be reduced by increasing segment length


                                                                  1
                                               M L  2 , 3, 4   
                                                                  N
                                                                        L
                                                                        zn
                                                                      n 1:N



                                   2 2                    15 3                            96 4
                      Var M 2              Var M 3                       Var M 4  
                                    N                        N                                N




                                 The segment size N has to be significantly increased (by 48  ) in order
                                                                                                     2

                                 to bring the variance of the 4th order moment at par with the 2nd order.




E. Nemer     -   16                                                                            Jump to first page
               New estimators for the carrier phase and frequency offset were
                developed based on newly established expressions of the 3rd and 4th-
                order cumulants of the demodulated QAM signal.

               The higher order estimator is more robust to noise for small values of
                frequency offsets, though it is not the case for larger ones.

               Clearly the improvement depends on the ability to find better ways to
                compute the HOS in a way to reduce the large bias and variance,
                when using a finite data set.

               more noise-robust methods for (blind) carrier estimation may be
                developed based on higher order statistics.




E. Nemer   -   17                                                           Jump to first page

								
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