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Carrier Estimation for QAM Receivers Using HOS Angelfire by alicejenny

VIEWS: 3 PAGES: 17

• pg 1
```									enemer@ieee.org

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

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

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
• 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

• 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

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

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) 
components of the noise
are small (zero) at the epok
 E  sin(L) 
  arg Rz ( L)  arctan s            
 Es  cos(L) 
[4]

• 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


• 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 


• 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 

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

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

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

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

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

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.

    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.