Cascaded  mul+-­‐modulus    algorithm  for  8QAM  and  16QAM  

adap+ve  equaliza+on  

 Xiang  Zhou  

2

In

h xx

h xy

h yx

h yy

Out

x

y

Zx

Zy

Stochastic gradient algorithm

Each hij is a FIR filter

yxZ xyxxx ⊗+⊗= hhyxZ yyyxy ⊗+⊗= hh

yy

2

yy hhh

dd y

yy

εµ−→

xx

2

xx hhh

dd x

xxε

µ−→xy

2

xy hhh

dd x

xyε

µ−→

yx

2

yx hhh

dd y

yx

εµ−→

With stochastic gradient algorithm

Convergence parameter

εx

εy

Stochastic gradient algorithm

A closer look at the 2×2 adaptive equalizer

The key problem is how to calculate the feedback error For optimal performance, error should be zero for an ideal signal

3

Constant  modulus  algorithm  (CMA)    

22

,, RZ yxyx −=ε

Error signal calculation method

I

Q

8PSK

R

)(ˆ)()(h)(hyx knxnZkk yyyx −⋅+→ µε

)(ˆ)()(h)(hxy knynZkk xxxx −⋅+→ µε

)(ˆ)()(h)(hyy knynZkk yyyy −⋅+→ µε

)(ˆ)()(h)(hxx knxnZkk xxxx −⋅+→ µε

Convergence parameter

Reference circle

4

Cascaded multi-modulus algorithm for 8QAM

4 4

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Ideal 8-QAM signal

Q

R1

R2 I

Intermediate error ε1

I

Q

221

1RRA +

=

11 AZ −=ε

Modulus 1

Q

Final error ε2

I

Modulus 2

212 A−= εε

221

2RRA −

=

CMMA can achieve zero error for ideal 8QAM, resulting in a better SNR performance than the classic CMA

5

CMMA:  FIR  filter  tap  weight  update  equa+ons    

)(ˆ)(h)(hyx knxgkk yyyx −⋅+→ µε

)(ˆ)(h)(hxy knygkk xxxx −⋅+→ µε

)(sign)(sign ,1,, yxyxyx ZAZg ⋅−=

Input

h xx

h xy

h yx

h yy

CMMA Algorithm

output

21,, )( AAZabs yxyx −−=ε

x

y

Zx

Zy

CMMA Algorithm

)(ˆ)(h)(hyy knygkk yyyy −⋅+→ µε

)(ˆ)(h)(hxx knxgkk xxxx −⋅+→ µε

Where

Convergence parameter

Each hij is a FIR filter

Error signal

6 6

Performance comparison between the classic CMA and the new CMMA (an experimental result)

X Y 1.8×10-3 1.2×10-3

The classic CMA The new CMMA

After polarization recovery After polarization recovery

After phase recovery After phase recovery

After converge parameter optimization After converge parameter optimization

Refer to: X. Zhou et al, OFC2009, PDPB4

7  

Cascaded  mul+-­‐modulus  algorithm  (CMMA)  for  16QAM  

Intermediate    error  ε1  

Final  error  ε  

212 A−= εε 323 A−= εε

Intermediate      error  ε2  

I

Q  

I

Q  

I

Q  

11 AZ −=εI

Q  

( ) 321|| AAAZabs −−−=εEqua+on  for  error  calcula+on  

⎯  Error  signal  calcula+on  method  

CMMA  allow  zero  error  for  ideal  16QAM,  resul+ng  in  beMer  performance  

A1  A2   A3  

8  

Input

h  xx  

h  xy  

h  yx  

h  yy  

CMMA Algorithm

output

x

y  

Zx

Zy

CMMA Algorithm

Each hij is a FIR filter

CMMA  FIR  filter  tap  weight  update  equa+ons  for  16QAM    

)(ˆ)(h)(hyx knxgkk yyyx −⋅+→ µε

)(ˆ)(h)(hxy knygkk xxxx −⋅+→ µε

)(ˆ)(h)(hyy knygkk yyyy −⋅+→ µε

)(ˆ)(h)(hxx knxgkk xxxx −⋅+→ µε

Where

Convergence parameter

)(sign)(sign)(sign ,1,21,, yxyxyxyx ZAZAAZg ⋅−⋅−−=

( ) 321,, || AAAZabs yxyx −−−=ε