JOURNAL OF TELECOMMUNICATIONS, VOLUME 19, ISSUE 2, APRIL 2013 23

On solving the local minima problem of blind adaptive MIMO equalization

Y. BEN JEMAA and M. JAIDANE

Abstract— We consider in this paper the problem of blind MIMO-FIR channel equalization in a real context of digital transmission where the input data belongs to a finite alphabet set. A non linear equalizer with a Volterra structure is considered. A rigorous analysis using the classical approaches is difficult due to the non-linearity of the equation that governs the equalizer. Through a novel finite alphabet approach that we propose, we demonstrate that the problem of local minima is avoided when using, for adaptation, a particular family of algorithms including classical ones such as LMS. This was proven rigorously without using any assumptions in a digital transmission context when the input remains to a finite alphabet set.

Index Terms—Blind MIMO Equalization, Volterra structure, Adaptive algorithms, Local minima

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1 INTRODUCTIONlind  MIMO  (Multiple  Input  Multiple  Output)  equali-­‐‑zation   is   of   an   increasing   interest   research   topic   in  many   fields   such   as   new   digital   communications  

standards   and   multi-­‐‑sensors   applications   [1][2].   MIMO  channel   inversion   in   a   blind   way   -­‐‑without   any   training  input  sequences-­‐‑is  a  generic  and  difficult  problem.  Many  works  have  been  carried  in  blind  SISO  (Single  Input  Sin-­‐‑gle  Output)  and  recently  for  blind  MIMO  equalization  [3]  [4]  [5]  [6]  [7].  However,  to  our  knowledge,  no  exact  theo-­‐‑retical   analysis   in   a   real   digital   communication   system   -­‐‑when   the   inputs   remain   to   a   finite   alphabet   set   and   the  equalizer   is   a   finite   impulse   response   filter-­‐‑is   made.   In  particular,   the  convergence  towards   local  or  global  mini-­‐‑ma   is   not   well   investigated.   Many   blind   adaptive   algo-­‐‑rithms  have  been  proposed  for  SISO  and  MIMO  equaliz-­‐‑ers   [8]   [9]   [10]   but  without   being   confident   on   that   they  avoid   local   minima.   A   quadratic   minimization   criterion  associated  with  a  non  linear  equalizer  structure  seems  to  be   a   solution   for   non  minimum  phase   channel   equaliza-­‐‑tion   to   avoid   ill   convergence   but   no   exact   theoretical  analysis  allowed  to  verify  it.    While  a  DFE   (Decision  Feedback  Equalizer)   suffers   from  possible   error   propagation,   a   Volterra   structure   can   be  used  in  this  case  for  equalization.    

In   this   paper,   an   exact   performance   analysis   of   MIMO  blind   equalizer   is   presented  using   a   powerful   tool   dedi-­‐‑cated   for   the   digital   context.   Even   if   this   finite   alphabet  approach   is   not   dedicated   to   a   quantitative   analysis  [11][12],   we   can   deduce   powerful   qualitative   results.   In  particular,  using  this  exact  analysis,  we  prove  the  absence  of   local   minima   for   blind   MIMO   Volterra   equalizers  adapted  by  a   specific   family  of   algorithms.  This   is  made  for  minimum   or   non  minimum   phase  MIMO   FIR   chan-­‐‑nels.      In  order  to  assess  such  results,  this  paper  is  organized  as  follows:   in   section   2,   we   describe   the   MIMO   FIR   chan-­‐‑nel/adaptive  equalizer.   In  section  3,  we  present   the  finite  alphabet   approach   tailored   for   digital   transmission   con-­‐‑text;  we   use   this   approach   to   give   an   exact   convergence  analysis   of   this   blind  MIMO   equalizer.   In   section   4,   we  prove   that   the  MIMO   adaptive   Volterra   equalizers   con-­‐‑verge   to   unique   minima   and   then   the   problem   of   ill-­‐‑convergence   is   avoided   for   this   family   of   Blind   MIMO  Volterra  adaptive  equalizers.  

2 MIMO SYSTEM MODEL AND EQUALIZATION STRATEGY

2.1 MIMO FIR Channel / Adaptive non-linear Equalizer model

 Let  us  consider  an  M-­‐‑input  N-­‐‑output  Finite  Impulse  Filter  (MIMO   FIR)   communication   system   as   depicted   in  figure1.        

B

———————————————— Y.Ben  Jemâa  is  with  the  NationalEngineering  School  of  Sfax  ,  BP  1173  3038,  Tunisia  and  Signals  and  Systems  laboratory;        M.   Jaidane   is   with   the   National   Engineering   School   of   Tunis,   Le  belvedere1002,  Tunisia  and  Signals  and  Systems  laboratory;    

   

© 2013 JOT www.journaloftelecommunications.co.uk

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Fig. 1. Noiseless MIMO system and adaptive blind equalizer The   MIMO   channel   is   assumed   to   be   noiseless,   linear,  with  memory  and  block-­‐‑time-­‐‑invariant.  The  channel  out-­‐‑put   signal   sampled   at   the   symbol   rate   at   time   n   can   be  modeled  as:  

  Xn  =  F  An (1)

Where  Xn  =[x1[n],  x2[n],...,  xN  [n]]T   is  the  received  signal  at  time  n,  An  =[a1[n],a2[n],...,aM[n],a1[n−1],a2[n−2],...,aM  [n−L+1]]  is  the  transmitted  signal,  L  is  the  channel  memory  length  and  F  is  the  channel  matrix  with  dimension  N  ×  (ML).  All  inputs   {an}   remain   to   the  same  finite  alphabet  set   (for  ex-­‐‑ample  {±1},  {±1,  ±j}...)  such  as  QAM,  PSK  signals  depend-­‐‑ing  on  the  modulation.  For  demonstration  purpose,  equa-­‐‑tion   (1)  will   be   rewritten   as   follows   in   order   to   separate  the   memory   less   information   that   must   be   estimated   at  the   receiver   from   the   remaining   information   (with  memory):      Xn  =  JF  Bn  +  CF  Dn        (2) Where  Dn  =[d1[n],  d2[n],...,  dM  [n]]

T,  Bn=[a1[n],a2[n],...,aM[n]]T  ,  

di[n]=[ai[n−1],   ai[n−2],...,   ai[n−L+1]]T   ,   JF   and  CF   define   the  channel  matrices  with  JF  is  a  constant  matrix  with  dimen-­‐‑sion  N×M  and  CF  with  dimension  N×  M(L  −  1)  (F  =[JF,  CF]).  Matrix  CF   can  be  easily  determined   from  matrix  F  and   it  contains  all  channel  information.    

2.2 Blind MIMO Equalizer In  cases  of  non  minimum  phase  channel,  the  linear  equal-­‐‑izer  doesn’t  perform  well  (see  for  example  [13]).  To  over-­‐‑come  this  limitation,  non  linear  filters  are  used  in  order  to  identify   the   system   parameters.   Because   Volterra   series  can   completely   describe   the   input   and   output   transfer  characteristic   of   a   large   type   of   non   linear   system,   the  

Volterra  adaptive  filters  have  been  widely  applied  in  sev-­‐‑eral   important   applications.   In   particular,   adaptive  Volterra   filters   have   been   developed   for   channel   iden-­‐‑tification  and  equalization  [14][15].    If   we   consider   the   MIMO   Volterra   structure   of   order   v  and   memory   length   p,   the   relationship   between   the   kth  

output   yk[n]   and   the   input   is   given,   in   the   discrete   time  case,  by  the  following  equation  [16]:  

(3)

(3)

where  hkj1,   j2,..,ji  denotes  the  coefficients  of  the  order Volter-­‐‑ra   kernel   and   hk0   is   the   statistical   characteristic   corre-­‐‑sponding   to   yk[n].   This   equation   can   be   described   by   a  matrix  form  as  follows:  

Yn  =  JHXn  −  HnVn (4)

where    

Vn=[x1[n−1],  ·∙·∙·∙,  xN[n−1],  ·∙·∙·∙,  xN[n−p+1],  x1[n]2,  ·∙·∙·∙,  xN[n]2,    

x1[n]x1[n−1],  ·∙·∙·∙,xN[n−p+1]v]T  

is   a  vector  depending  on   the  non   linearity  degree  of   the  system,  Yn   =[y1[n],   y2[n],...,   yM   [n]]T   ,  Hn   is   the   estimated  parameters  matrix  of  the  equalizer  with  dimension  M×  Nβ  (β  =    ∑  Ci

p+i-­‐‑1  where  Ci

j  is  the  number  of  combinations  of  i  

objects   from   j   objects)   and   JH   is   a   constant   matrix   with  dimension  M  ×  N      

2.3 The adaptive algorithm family The   Bussgang   algorithms   and   the   higher   order   statistics  based   algorithms   are   the  mostly   used   in   blind   equaliza-­‐‑tion.   These   algorithms   have   local   minima   due   to   a   non  convex   cost   function   [4][6].   In   order   to   avoid   this   prob-­‐‑lem,   we   shall   propose   a   family   of   algorithms   to   update  the  Volterra  equalizer.  These  algorithms  are  described  by  the  following  recursion:    

Hn+1  =  Hn  +  µμYn  f  (VnH  )  

                                                               Yn          =  JHXn  –  HnVn                                                                  (5)    Where  Hn   is  the  matrix  equalizer  and  µμ   is  a  positive  step  size.  Function   f   allows   to   choose   the  adaptive  algorithm.  There  are  no  specific  conditions  on  this  function.    Note   that   the   particular   case,  when   the   function   f   is   the  identity   corresponds   to   the  minimization   of   a   quadratic  cost  function  based  on  the  energy  of  the  output  equalizer  vector  Yn  such  as  Jn  =  E(||Y  n  ||2),  where  E(.)  denotes  the  statistical  expectation  operator.  The   adaptive   algorithm   used   to   update   the   equalizer   in  

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this  case  is  the  LMS  algorithm.  According  to  equations  (2)  and   (5),   the   adaptation   is   governed   by   the   following   re-­‐‑cursion:                                Hn+1  =  Hn  +  µμ  (JH(JFBn  +  CFDn)  −  HnVn)  f  (Vn

H  )                (6)  

   We  will  study  the  statistical  convergence  of  this  recursion  in   terms  of   initial   conditions   in  order   to  analyze   the   ill   -­‐‑convergence  problem.    

3 LOCAL MINIMA ANALYSIS BY FINITE ALPHABET APPROACH

In  order   to   study   the   statistical   convergence  of   the   algo-­‐‑rithm,  we  describe  the  Volterra  equalizer  behavior,  in  the  mean   sense,   by   the   following   recursion   obtained   from  equation  (6):    E(Hn+1)  =  E(Hn)+  µμ  E((JH  (JF  Bn  +  CF  Dn)  −  HnVn)  f  (Vn

H  ))  

                                                     =  E(Hn(I  −µμVn  f  (Vn  H  )))+µμE(JH  JF  Bn  f(Vn  H)+JH  CF  Dn  f(Vn  H  ))                                                                                                          

(7)      Since  Vn  and  Hn  are  dependent,  it  is  difficult  to  solve  equa-­‐‑tion  (7)  [17].  To  overcome  this  difficulty,  we  here  propose  to  apply  an  original  approach  based  on  the  finite  alphabet  input  characteristics  [18].  This  allows  us  to  deduce  results  about   the   behavior   of   the   equalizer.   Especially,   we   will  answer  to  the  following  question:  can  we  solve  the  prob-­‐‑lem   of   convergence   towards   local  minima  with   the   pro-­‐‑posed  algorithms  family?     3.1 The Finite Alphabet formulation In   digital   transmission   context,   the   input   sequence   {an}  remains   in  a  finite  alphabet   set   {d1,d2,   ...,   dq},   for   example  {±1},  {±1,  ±3}  such  as  QAM  or  PSK  signals...  The  main  idea  of  the  proposed  convergence  analysis  is  to  express  vector  Vn  with  the  input  sequence  {an}  and  then  to  reformulate,  in  a  compact  form,  equation  (7).  Through  this  exact  reformu-­‐‑lation,  all  convergence  results  are  deduced,   in  particular,  the  unicity  of  the  blind  MIMO  equalizer  convergence.  The  vector  Vn  can  be  writhen  as:    

Where   F   =[JF,  CF]   is   the   channel   matrix   with   dimension  N×ML,  An  =  [Bn

T  ,DnT  ]T  ,  Ai,j  =  AiATj  and  Oi=[0,  ·∙·∙·∙  ,  1,  ·∙·∙·∙  ,  0,  ·∙·∙·∙]  

is  a  N  length  vector  of  zeros  except  the  ith  element  which  is  equal  to  1.  So,  we  have:                                                                          Vn  =  F An                                                                          (9)    The  vector  An  remains  to  a  finite  alphabet  set    A={ W1, W 2, ..., W N  }  of  cardinality  N  =  q

MLp.    

Consequently,  Vn   remains   in   a   finite   alphabet   set   which  depends   on   the   alphabet   of   the   input   and   the   channel  parameters.  Since  the  input  sequence  is  stationary,   it  can  be  modeled  by   a  discrete  Markov   chain   {θ(n)}  with  finite   state   space  {1,  2,  ...,  N  }  [19]:                                                                                  An  =  W  θ(n)                                                                (10)  

 This   Markov   chain   is   characterized   by   its   probability  transition   matrix  P   =[  Pij]   and   its   stationary   probability  vector  π(∞).  It  is  important  to  note  that  with  this  formula-­‐‑tion  we  can  consider  a  non-­‐‑equiprobable   input  sequence  such  as  a  particular  channel  encoder  output.  In  the  same  way,  Bn  and  Dn  can  be  expressed  with  the  vector  An:                            Bn  =  U  An                                                                                    Dn  =  V  An                                                                            (11)    where  U  =[IM  ,  O,...]  and  V  =[O  ...  IM(L−1),  O,...]  Where  O  is  a  matrix  of  zeros  and  I  is  the  identity  matrix.  Using  these  notations,  equation  (9)  and  equation  (11),  we  can  rewrite  equation  (7)  as:      E(Hn+1)  =  E(Hn  (I  −  µμ  F  An  f  (An

 H  F  H  )))    

                           +  E(µμJH  JF  U  An  f(An  H  F  H  

)+  µμJHF V  An  f(An  H  F  H  

))                                                        (12)  

 According  to  the  finite  alphabet  model  and  the  following  notations:                                                              Mn  =  I  −  µμ  F  An  f  (An

 H  F  H  )  

                       Zn    =  µμJH  JF  U  An  f(An  H F  H  

)+  µμJHF V  An  f(An  H  F  H  

)                            

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     the  recursion  (12)  describing  the  adaptive  Volterra  equal-­‐‑izer  mean  behavior  adapted  by  a  general   family  of  algo-­‐‑rithms  becomes:                                                            E(Hn+1)  =    E(Hn  Mn)+  E(Zn)                                                (13)  

  3.2 The proposed idea The   finite   alphabet   approach   consists,   since   there   is   N  possibilities   of   Wθ(n),   in   splitting   the   vector   E(Hn)   in   N  components  defined  by  :                                                                      qj(n)=  E(Hn  1θ(n)=j)                                                                  (14)    Where  1θ(n)=j  is  the  indicator  function.      So,  we  can  rewrite  E(Hn)  as  follows  :                                                        E(Hn)  =                         qj(n)                                                              (15)    Each  component  of  E(Hn)  is  governed  by  the  following  recursion:                                      qi  (n  +1)  =         E(Hn+11θ(n+1)=i  1θ(n)=j  )                            (16)        According  to  equation  (13),  we  have:    qi  (n  +1)  =   E(Hn  Mn  1θ(n+1)=i  1θ(n)=j)+E(Zn  1θ(n+1)=i  1θ(n)=j)                                                                                                      

(17)      Since  Mn  and  Zn  remain  also  in  a  finite  alphabet  sets  with  cardinality  N  and  they  are  respectively  {M1, M2, ..., MN

}  and  {  Z1, Z 2, ..., ZN

}  with  :    M  i  =  I  −  µμ F  W  i  f  (W  i

 H  F  H  )  

Z  i    =  µμJH  JF  U  W  i  f(W  i  H  F  H  

)+  µμJHF V  W  i    f(W  i  H  F  H  

)        (18)  

 We  have  then:    qi  (n+1)  =     E(Hn  1θ(n+1)=i  1θ(n)=j  )M  j  +Z  j  E(1θ(n+1)=i  1θ(n)=j  )                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                (19)    Since  the  input  is  characterized  by  a  Markov  chain,  we  have:      E(Hn  1θ(n+1)=i  1θ(n)=j  )  =  P  ji  E(Hn    1θ(n)=j  )  =  P  ji  qi  (n)  E(1θ(n+1)=i  1θ(n)=j  )  =  P  ji  πj(∞)                                                                                                      (20)    Hence,  

 qi  (n  +1)  =     qi  (n)  M  j  P  ji  +  Z  j P  ji  πj(∞)                                  (21)      In   order   to   rewrite   (21)   in   linear   form,  we   introduce   the  useful  notations:    

q~T  (n)  =[q1T(n),  q2T(n),  ...,  q  N

 T(n)]T  ,      

Z˜(n)  =[  Z1 ‘T

, Z2 ‘T

, ..., Z N

‘T  ]T  

 where        Z   i

  ‘T=         Z  j  P  ji  πj(∞)      Referring  to  [20]  in  order  to  use  Kronecker  product  prop-­‐‑erties,  we  can  write:      q˜(n  +1)  =  Γq˜(n)+  Z˜  

Γ            =  (P  T    ⊗  I)  diag  ((I  −  µμ  F *  Wi

 *  f(Wi  H  F  H  )*)  ⊗  I)        

(22)      This   elegant,   compact,   linear   and   deterministic   equation  will   replace   the   classical   intricate   convergence   equation  (7).  From  (22),  we  will  deduce  all  system  performances.  In  fact,   the  matrix  Γ  contains  all   relevant   information  about  the   Volterra   equalizer.   The   matrix   Γ,   which   completely  governs  the  quantity  q˜  then  E(Hn)  depends  explicitly  and  simply  on   the  finite  alphabet   input   through  the  alphabet  Wi   and   the   transition   matrix   P,   the   MIMO   channel  through  matrix  F  and  the  family  algorithms  through  func-­‐‑tion  f(.).    Since   the   recursion   (22)   is   linear   and   the   matrix   is   con-­‐‑stant   and   depends   only   on   the   step   size   and   the   input  statistics,  statistical  convergence  of  the  algorithm  and  the  possible   local   minima   problem   of   the   blind   adaptive  MIMO  equalizer  depend  only  on   the  eigenvalues  of  ma-­‐‑trix  Γ.    

4 EXACT ANALYSIS OF BLIND ADAPTIVE MIMO VOLTERRA EQUALIZER: CONVERGENCE WITHOUT LOCAL MINIMA

Referring to the equation (22), a critical step size µc exists and it verifies [2]:      If  µμ  ≤  µμc  then,  all  the  eigenvalues  of  matrix  Γ  are  less  than  1  and  the  algorithm  converges  in  the  mean  sense  [19].  The  critical  step  size  is  deduced  by  the  analysis  of  the  Γ  eigen-­‐‑values  as  a  function  of  µμ.   If  the  convergence  condition  is  satisfied,  we  determine  the  

(23)

27

steady  state  performances  (n  →∞)  by:             q˜∞  =  (I  −  Γ)-­‐‑1  Z˜                                                                            (24)  

 Since   quantities   Γ   and   Z˜are   constants,   the   value   q˜∞   is  unique.  We  can  then  deduce  that  the  matrix  equalizer  Hn  converges  in  the  mean  sens  to  a  unique  value  H∞  defined  as: H∞ = qj (∞) (25) where  qj  (∞)  are  the  components  of  vector  q˜∞.      This   important  result  confirms  the  absence  of   local  mini-­‐‑ma  of   the  proposed  algorithm  family.  The  result  already  demonstrated  is  presented  in  the  following  proposition.   4.1 Proposition The  Let  consider  a  MIMO  linear  with  memory  and  block-­‐‑time  invariant   channel   which   is   blindly   equalized   using   a  MIMO  adaptive  Volterra   equalizer  and  adapted  using  a  new  proposed  family   of   algorithms   described   by   the   equation   (5).   If   the   step  size  used   for  adaptation   is   less   than  a  critical   step  size  defined  by  equation  (23),  the  algorithm  converges  in  the  mean  sense  to  a  unique  value  and  the  problem  of  local  minima  is  avoided.      This  is  valid  for  any  finite  alphabet  although  asymmetric  or   non-­‐‑equiprobable   one,   any   MIMO   FIR   minimum   or  non  minimum  phase  channel.  No  condition  is  required  on  the  numbers  of  sensors  at  the  input  and  output.    

4.2 Numerical example In  figure  (2),  we  consider  a  linear  and  transversal  channel,  a  Volterra  structure  equalizer  with  order  2  and  memory  2,  the   alphabet   {±1}   and   the   quadratic   criterion  mentioned  before.   It   is   important   to   note   that   the   reached   criterion  value  is  the  same  for  different  initializations  of  the  equal-­‐‑izer,   this   can  prove   the  absence  of   local  minima.  For   the  plot,  we  have  considered  three  different  initializations.  

Fig.  2.  The  criterion  value  for  different  initializations:  ab-­‐‑

sence  of  local  minima  

5 CONCLUSION A  statistical  convergence  analysis  of  a  blind  MIMO  adap-­‐‑tive  Volterra  equalizer  adapted  by  a  new  family  of  algo-­‐‑rithms  is  developed  in  this  paper.  We  consider  a  general  MIMO   linear   noiseless   and  minimum   or   non   minimum  phase  channel.  Using  a  specific  approach  tailored  for  the  digital   transmission   context,   we   prove   theoretically   and  without   any  hypothesis   that   the  proposed   system  solves  the  problem  of  convergence  towards  local  minima.  

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Y. Ben Jemâa received a Ph.D. in Electrical Engineering from the National Engineering School of Tunis, Tunisia (ENIT) in 2003. In June 2012, she obtained the HDR in Telecommunications from the same school. She joined the National Engineering School of Sfax (ENIS) in 2000 where she is an associate Professor in Telecommu-nication Engineering. She is a member of the research Lab. U2S at ENIT. Her research interests incluse signal and image processing and digital communications. M. Jaidane received the M.S. degree in Electrical engineering from the National Engineering School of Tunis (ENIT), Tunisia, in 1980. From 1980 to 1987, she worked as a Research Engineer at the Sig-nal and System laboratory, CNRS/SUPELEC, France. She received the Doctorat d’Etat degree in 1987. Since 1987, she was with the ENIT, where she is currently a Full Professor. She is a Member of the signal and System Unit, ENIT. Her teaching and research inter-ests are in adaptive systems for digital communications and audio processing.