Stop-and-Go Algorithms for Blind Channel Equalization in - PDF by rfb16446


									              Stop-and-Go Algorithms for Blind Channel Equalization in QAM Data
                                   Communication Systems

                                                                              Shafayat Abrar
                                                            Electrical Engineering Department,
                                                        COMSATS-IIT (CIIT), Islamabad, Pakistan.

   Abstract: This paper presents a framework of stop-and-go                            input data {an }, each element of which belongs to a complex
algorithms for the blind equalization of QAM data commu-                               alphabet A of QAM symbols. The data sequence {a n } is
nication systems. Based on the proposed framework, three                               sent through a complex LTI channel whose output x n is
new stop-and-go algorithms are presented. It is discussed                              observed by the receiver. The function of the blind equalizer
that each of the existing and proposed stop-and-go algorithm                           at the receiver is to estimate the original data {a n } from
forces the equalizer output to match some statistical contour                          the received signal x n . The input/output relationship of the
on QAM constellation space. Using computer simulations, it                             QAM system can be written as:
is shown that the Picchi and Prati’s “stop-and-go decision-                                                    K−1
directed algorithm”, which forces the equalizer output to                                               xn =         an−iT ci + νn ,                      (1)
lie on point contours, is the best among other stop-and-go                                                     i=0
algorithms.                                                                            where T is the symbol (or baud) period and K is the
   Keywords: Blind equalization, adaptive equalizers, stop-                            length of channel impulse-response. The channel noise ν n is
and-go, decision-directed, QAM.                                                        assumed to be stationary, Gaussian, and independent of the
                                                                                       channel input a n . Denote the equalizer parameter vector with
                         I. INTRODUCTION                                               N + 1 elements as wn = [w0,n , w1,n , · · · , wN,n ]T , where the
   In most digital communication systems, inter-symbol in-                             superscript T represents transpose. In addition, define the
terference (ISI) occurs due to bandwidth limited channel or                            received signal vector as x n = [xn , xn−1 , · · · , xn−N +1 ]T .
multipath propagation. Channel equalization is one of the                              The output signal of the equalizer is thus y n = wn xn =
techniques to mitigate the effect of ISI. Adaptive algorithms                          yR,n + yI,n , where R and I represent the real and imaginary
are used to initialize and adjust equalizer coefficients when a                         parts of yn , respectively. If {h} = {c} ∗ {w} represents
channel is unknown and possibly time-varying. Convention-                              the overall channel-equalizer impulse response. The channel
ally, initial setting of the equalizer tap weights is achieved                         output xn can be expressed as:
by a training sequence before data transmission.
                                                                                        xn     =        an−iT hi + νn = h0 an +              ai hn−iT + νn .
   However, when sending a training sequence is impractical                                         i                                  i=0
or impossible, it is desirable to equalize a channel without
the aid of a training sequence. Equalizing a channel without                                                                     signal+ISI+noise

training mode is known as blind equalization. A typical                                In blind equalization, the channel input a n−D is unavailable,
                                                                                       and thus different minimization criteria are explored. The
                                                                                       crudest blind equalization algorithm is the decision-directed
                                                                                       scheme that updates the adaptive equalizer coefficients ac-
       a(n)    Channel                      Equalizer       y(n)              a(n)     cording to
                C(z)        +                 W(z)

                                                                                                   wn+1 = wn − µ(yn − D[yn ])x∗
                                                                                                                              n                           (2)
                                                                                       where D[yn ] is the closest symbol to yn . Under high ISI,
                                          Blind Algorithm
                                                                                       the convergence behavior of decision-directed equalizer is
                                                                                       very poor. Better blind adaptive equalization algorithms are
         Fig. 1.   Baseband model of communication system.                             designed to minimizing special non-MSE cost functions that
                                                                                       do not directly involve the input a n while still reflect the
blind equalization setup is depicted in Fig. 1 using a simple                          current level of ISI in the equalizer output. Define the mean
system diagram. The complex baseband model for a typical                               cost function as:
QAM (quadrature amplitude modulated) data communication
                                                                                                           J(w) = E[Ψ[yn ]]                               (3)
system consists of an unknown linear time-invariant (LTI)
channel cn which represents the physical inter-connection                              where Ψ[·] is a scalar function of the equalizer output. J(w)
between the transmitter and the receiver at baseband. The                              should be specified such that at its minimum, the correspond-
transmitter generates a sequence of complex-valued random                              ing wn results in a minimum ISI or MSE equalizer. Using
  (3), the stochastic gradient descent minimization algorithm            symbols. Let us consider the following error function (say
  is easily derived as                                                   global), which is used in adaptation process to minimize
                         ∂Ψ[yn ]                                         the difference between the statistics of y n and some pre-
     wn+1 = wn − µ ·             = wn − µ · Ψ [yn ]x∗ .
                                                    n                    calculated statistics (RR and RI ) of the transmitted QAM
  Let ψ be the first derivative of Ψ. ψ is often called the error
  function since it replaces the prediction error in the LMS                    ψG [yn ] = (g1 (yR,n , yI,n ) − RR ) · g2 (yR,n , yI,n )
  adaptation. The resulting blind equalization algorithm can                         + (g3 (yR,n , yI,n ) − RI ) · g4 (yR,n , yI,n ).        (6)
  be written as:
                                                                         where g1 (·), g2 (·), g3 (·) and g4 (·) are zero-memory (and
                wn+1 = wn − µ · ψ[yn ]x∗ .
                                       n                        (4)
                                                                         preferably continuous) functions. The resulting weight adap-
  Thus the design of the blind equalizer thus translates into the        tation process is wn+1 = wn − µ · ψG [yn ]x∗ . RR and RI
  selection of a suitable function Ψ (or ψ) such that the local          are positive (dispersion constants) and are computed so that:
  minima of J(w) correspond to a significant removal of ISI
  from the equalizer output y n .                                                               E[ψG [an ]a∗ ] = 0.
                                                                                                           n                                  (7)

         II. STOP-AND-GO BLIND EQUALIZATION                              The error functions of almost all existing stochastic gradient
                     ALGORITHMS                                          based blind equalization algorithms can be mapped onto (6).
     Given the standard form of the blind equalization algo-                Now we want to design an error function ψ L [yn ] that
  rithm in (4), it is apparent that the convergence characteristics                      ˆ
                                                                         incorporates a n (the local information). There are many ways
  of blind algorithms are largely determined by the sign of              to do it. One way is to allow dual mode; that is based on
  the error signal ψ[y n ]. In order, for the coefficients of a           the error level between y n and an , the equalizer will switch
  blind equalizer, to converge to the vicinity of the optimum            between certain blind and decision-directed adaptations, as
  minimum-MSE (MMSE) solution as achieved by LMS adap-                   reported in [2]. So the algorithm switches between a suitable
  tation (under supervision), the sign of its error signal should                                          ˆ
                                                                         ψG [yn ] and ψL [yn ] = yn − an . Another method exploits
  agree with the sign of the LMS prediction error y n − an−D             ˆ
                                                                         an directly in the weight adaptation process without going
  most of the time. Slow or ill convergence can occur if the                                                           ˆ
                                                                         into decision-directed mode [3]. It embeds a n and dispersion
  sign of the two errors differ sufficiently often.                       constants (R) together, such that ψ G [yn ] becomes function
     The idea behind the stop-and-go algorithms is to allow the                                                ˆ
                                                                         of both dispersion constant(s) and a n .
  adaptation “to go” only when the error function is more likely            The third approach, which is the subject of this paper, is
  to have the correct sign for the gradient descent direction.           “stop-and-go” scheme. In this approach, the adaptation pro-
  Given several criteria for blind equalization, one can expect          cess apparently uses only a n and yn from startup to final con-
  a more accurate descent direction when more than one of                vergence. It is achieved by replacing the dispersion constants
  the existing algorithms agree on the sign (direction) of the                                                                   ˆ
                                                                         RR and RI with some suitable nonlinear functions of a n . For
  error functions. When the error signs differ for a particular          example, replacing R R and RI with g1 (|ˆR,n |, |ˆI,n |) and
                                                                                                                       a      a
  output sample, parameter adaptation is “stopped”. Consider             g3 (|ˆR,n |, |ˆI,n |) in (6) will incorporate local information
                                                                              a        a
  two algorithms with error functions ψ 1 and ψ2 . The following         in weight adaptation process. Note that, it will result in an
  stop-and-go algorithm can be devised [1]:                              increase in the number of contours and as a result, the QAM
                                                                         symbols in alphabet A lie on (at least) one or more contours.
          wn − µψ1 [yn ]x∗ , if sgn[ψ1 [yn ]] = sgn[ψ2 [yn ]]
wn+1 =                                                           (5)     The resulting error function (say local) ψ L [yn ] can be given
          wn ,               if sgn[ψ1 [yn ]] = sgn[ψ2 [yn ]]
  Error functions ψ 1 and ψ2 should be selected such that they
  maximize reliable regions and make most of the local and the                                              a        a
                                                                       ψL [yn ] = (g1 (yR,n , yI,n ) − g1 (|ˆR,n |, |ˆI,n |)) · g2 (yR,n , yI,n )
  global knowledge of the constellation. Given the equalizer            +  (g3 (yR,n , yI,n ) − g3 (|ˆR,n |, |ˆI,n |)) · g4 (yR,n , yI,n )
                                                                                                      a        a                              (8)
  output yn , the closest symbol an = D[yn ] can be considered
  as a local information; while the size (number of alphabets),          Note that ψL [yn ] forces yn to lie on the contour which also
  shape (square or cross) and energy (mean distance between              contains the closest symbol a n = D(yn ) on it. It can easily
  the symbols) of the constellation can be considered as global          be understood that due to multiple contours exhibited by
  information. a n is termed local as it may change from one             ψL [yn ], the steady-state misadjustment offered by ψ L [yn ]
  output to another; while the size, shape and energy are fixed           is much lower compared to that of ψ G [yn ]. If ψG [yn ] is
  and don’t depend on any specific value of y n . Most of the             capable of removing ISI, then the use of ψ L [yn ] will be
  stochastic gradient descent algorithms employ error functions          beneficial, only when the sign of the two error functions,
  which exhibit global information. They compute an estimate             ψG [yn ] and ψL [yn ], match. If ψL [yn ] is incorporated in the
  of an , by doing some nonlinear operation on the current               weight adaptation process then the real and the imaginary
  equalizer output y n such that the certain statistics of y n are       parts of ψL [yn ] should be weighted with binary flags f R
  forced to match with global statistics of the transmitted data         and fI , respectively, to indicate the sign match. Flags, f R
and fI , are obtained as follows:                                    wn+1 = wn − µ [ψL [yn ]R · fR +  ψL [yn ]I · fI ] x∗ , where
                                                                     ψL [yn ] is computed as decision-directed error:
                  1 + sgn[ψL [yn ]R ] · sgn[ψG [yn ]R ]
             fR =                                           (9)
                                    2                                                             ˆ                ˆ
                                                                                ψL [yn ] = yR,n − aR,n +  (yI,n − aI,n )                                                        (16)
                  1 + sgn[ψL [yn ]I ] · sgn[ψG [yn ]I ]
             fI =                                          (10)      Note that (16) can be expressed in an equivalent form as
The resulting weight adaptation rule is,
                                                                  ψL [yn ] = yR,n − |ˆR,n | sgn[yR,n ] + (yI,n − |ˆI,n |sgn[yI,n ]) (17)
                                                                                     a                             a
wn+1 = wn − µ · (ψL [yn ]R · fR +  ψL [yn ]I · fI )x∗ . (11)
                                                                      Comparing (15) and (17); it can be observed that Picchi and
where subscripts R and I denote the real and imaginary
components, respectively. Since ψ L [yn ] is being used in                       Benveniste−Rudget’s
                                                                         Reduced Constellation Algorithm (1983)
                                                                                                                    Picchi−Prati’s Point−Contour Stop−and−Go Algorithm (1987)
                                                                                                                   Reliable regions for yR
                                                                                                                                                      Reliable regions for y
weight adaptation process, the values of R R and RI need                   4                                  4                                  4

not to satisfy (7); instead, they can be selected such that the            2                                  2                                  2

sign of the two error functions agree most of the time. It is              0                                  0                                  0

observed that if R R and RI are selected as the outermost                 −2                                 −2                                 −2
contour, then the reliable regions can be maximized.
                                                                          −4                                 −4                                 −4
                                                                           −4     −2      0       2      4    −4       −2     0      2      4    −4    −2      0     2       4

              RR = max[g1 ({|aR,n |}, {|aI,n |})]          (12)                           (a)                                 (b)                             (c)

              RI = max[g3 ({|aR,n |}, {|aI,n|})]           (13)                        Fig. 2.        Contours in (a) RCA (b)-(c) PC-SAGA

The above mentioned scheme (Equations (6), (8), (9), (10),           Prati algorithm is a stop-and-go version of RCA obtained by
(11), (12) and (13)) can be applied to any stochastic gradient       replacing RR and RI with |ˆR,n | and |ˆ I,n |, respectively.
                                                                                                    a            a
descent based blind equalization scheme to develop its stop-         Similar to RCA, which forces y n to belong to one of the
and-go version. In subsequent sections, we will review two           four possible point contours, the Picchi and Prati algorithm
existing stop-and-go blind equalization algorithms based on          forces yn to belong to the nearest constellation symbol. It can
this framework and three new stop-and-go algorithms will             be said that it modifies the RCA’s 4-point contours into an
be presented.                                                        M-point contours. Picchi and Prati algorithm is thus named
                                                                     Point-Contour Stop-and-Go Algorithm (PC-SAGA). Based on
III. POINT-CONTOUR STOP-AND-GO ALGORITHM                             the proposed framework in (12) or (13), we compute R R =
   The first blind equalizer for multilevel PAM signals was           RI = β for QAM as
introduced by Sato [4]. In essence, it is identical to the
                                                                                β      max[{|a |}] = max[{|aI |}]
decision-directed algorithm when the PAM input is binary                                √ R
(±1). For M -level PAM signals, it is defined by the error                               M − 1,       for square QAM
                                                                                        3 M − 1, for cross QAM.
function                                                                                                             (18)

                                              E[a2 ]                                     2    2
  ψG [yn ] = yn − R1 sgn[yn ], where R1 =              .   (14)                 IV. DIAMOND-CONTOUR STOP-AND-GO
                                              E[|an |]
The Sato algorithm was extended to complex signals (QAM)
                                                                        The first complexity-efficient (signed-error) blind equal-
by Benveniste et al. [5] by separating signals into their real
                                                                     izer was proposed by Weerackody et al. [7]. The
and imaginary parts as
                                                                     Weerackody-Kassam hard-limited algorithm (WK-HLA) is
ψG [yn ] = yn,R − RR sgn[yn,R ] + (yn,I − RI sgn[yn,I ]) (15)       specified by wn+1 = wn − µψG [yn ]x∗ , where ψG [yn ] is
                                                                     computed as:
where RR and RI are computed as E[a 2 ]/E[|an,R |] and
E[a2 ]/E[|an,I |], respectively. The resulting weight adap-        ψG [yn ] = sgn[|yR,n | + |yI,n | − R] · (sgn[yR,n ] +  sgn[yI,n ]) (19)
tation algorithm is called reduced constellation algorithm           This algorithm attempts to drive the equalizer output to reside
(RCA), as it attempts to resolve the output of the channel           on a 45o rotated-square (diamond) contour as depicted in Fig.
to belong to one of the four statistical symbols of a reduced        3(a). The final equalizer output is obtained by the removal of
constellation. Those four points are (R R , RI ), (RR , −RI ),       this rotation. It has been reported in [7] that it performs better
(−RR , −RI ) and (−RR , RI ).                                        than CMA (if properly initialized). Kim et al. [8] observed
   Picchi and Prati [6] developed the first ever stop-and-go          that Weerackody-Kassam algorithm can be transformed into
algorithm for blind equalization. They observed that a simple        a stop-and-go algorithm. The ψ L [yn ] proposed in Kim’s
decision-directed adaptation can open a closed eye, provided         diamond-contour stop-and-go algorithm is as follows:
the adaptation is stopped for the small proportion of the
instances when the decision-directed and the RCA errors                    ψL [yn ] = sgn[|yR,n | + |yI,n | − |ˆR,n | − |ˆI,n |]
                                                                                                               a         a
have different sign. The weight adaptation rule is given as                             · (sgn[yR,n ] +  sgn[yI,n ])                                                            (20)
                                                                                            where Rk is the square of the radii of the nearest constel-
                                                       Kim et al. Diamond−Contour
                  Hard−Limited Algorithm (1992)       Stop−and−Go Algorithm (1996)
              5                                   5
                                                                                            lation symbol for each equalizer output. To improve the
                                                                                            RDE’s convergence, different techniques have been sug-
              0                                   0
                                                                                            gested. In [12], the RDE is modified by incorporating stop-
                                                                                            and-go flags such that adaptation takes place only when the
                                                                                            sign of ψG [yn ] = (|yn |2 − R) and ψL [yn ] = (|yn |2 −
              −5               0              5
                                                    −5             0            5
                                                                                            {D[yn ]}2 ) match. It was named Stop-and-Go Decision-
                               (a)                                (b)
                                                                                            Directed Multiple-modulus Algorithm (SAG-DDMMA). The
       Fig. 3.      Contours in (a) WK-HLA (b) Kim’s DC-SAGA.                               resulting reliable regions are depicted in Fig. 4(c).
                                                                                               In this paper, a straightforward “stop-and-go” version of
                                                                                            CMA, similar to one given in [12], is proposed. The pro-
Due to 45o rotation, the sliced symbol a n = aR,n +  aI,n is
                                          ˆ      ˆ        ˆ                                 posed Circular-Contour Stop-and-Go Algorithm (CC-SAGA)
obtained differently as a n = D[yn ·e−π/4 ]·eπ/4 . The single
                         ˆ                                                                  is based on the framework described in Section II, and is
flag f is obtained by comparing the sign of [|y R,n | + |yI,n | −                            given as wn+1 = wn − µ · ψL [yn ] · f · x∗ , where f is 1
|ˆR,n |−|ˆI,n |] and [|yR,n |+|yI,n |−R]. It should be noted that
 a       a                                                                                  when the sign of ψ G [yn ] (22) and ψ L [yn ] (23) match, and 0
no closed form expression for the computation of R has been                                 otherwise. The value of R in ψ G [yn ], is computed as
reported in [8]. However, based on the proposed framework,
the value of R (that maximizes the reliable regions) can be                                           R = max[{Rk }] = max[{|a2 | + |a2 |}]
                                                                                                                               R      I
obtained as:                                                                                                        2 · β2,              square QAM
              √                               √                                                                =        2    1                      (24)
        R = 2 · max[{|aR |}, {|aI |}] = 2 · β               (21)                                                         β−      + β 2 , cross QAM
                                                                                                                        3    3
where β is obtained from (18). Fig. 3(b) depicts multiple                                   where β is computed form (18). The proposed algorithm
diamond contours generated by Kim’s stop-and-go algorithm.                                  differ from [12] in the formulation of reliable regions; oth-
Kim’s algorithm is thus named Diamond-Contour Stop-and-                                     erwise, the weight adaptation process is similar. The reliable
Go Algorithm (DC-SAGA).                                                                     regions formed by the proposed algorithm are depicted in
      V. CIRCULAR-CONTOUR STOP-AND-GO                                                       Fig. 4(d).
                                                                                                         Godard’s Algorithm [p=2] (1980)     Ready−Gooch (1990)
                                                                                                        Constant Modulus Algorithm (1983) Radius−Directed Equalization
   The Godard algorithm [9] is one of the best known blind                                                5                                    5

equalization algorithms and is a stochastic gradient algorithm
for the cost function, J p = 2p E(|y(n)|p − R)2 , where p ∈
{1, 2, · · ·} and E denotes statistical expectation. The corre-                                           0                                    0

sponding algorithm is w n+1 = wn − µyn |yn |p−2 (|yn |2 −
R)x∗ . The constant modulus algorithm (CMA) proposed in
     n                                                                                                                                   (a)                                   (b)
[10] is a special case of Godard algorithm for p = 2. The                                                −5
                                                                                                          −5            0            5
                                                                                                                                            −5                0            5

corresponding algorithm is w n+1 = wn −µψG [yn ]x∗ , where
                                                       n                                                 Cheolwoo−Hong Stop−and−Go                 Proposed Circular−Contour
ψG [yn ] is given as                                                                                     Multimodulus Algorithm (1998)
                                                                                                          5                                    5
                                                                                                                                                    Stop−and−Go Algorithm

                      ψG [yn ] = yn (|yn |2 − R)                                     (22)
and R = E[|a(n)|4 ]/E[|a(n)|2 ]. CMA forces the equalizer                                                 0                                    0

output yn to reside on a circular contour as depicted in Fig.
                                                                                                                                         (c)                                   (d)
                                                                                                         −5                                −5
   A decision-directed type CMA was proposed in [11],                                                     −5            0            5      −5                0            5

called radius-directed equalization (RDE), for QAM signals
                                                                                            Fig. 4.    (a) CMA, (b) RDE, (c) SAG-DDMMA, (d) Proposed CC-SAGA.
based on the known modulus (circular contour) of the con-
                                                   √ radii
stellation symbol radii. For example, 16 QAM has three √
 √ √            √
( 2, √10 and √
√                 18); and 32 QAM has five radii ( 2, 10,                                    VI. LINE-CONTOUR STOP-AND-GO ALGORITHM
   18, 26 and 34). The algorithm uses the error between
the equalizer output modulus and the nearest symbol radius                                     The constant modulus algorithm (CMA) is blind to car-
to update the equalizer weights, as depicted in Fig. 4(b).                                  rier phase offset error. The CMA has been modified to
It provides faster convergence than the CMA. However, the                                   incorporate phase information, it resulted in multimodulus
convergence of RDE is not guaranteed, as it operates totally                                algorithm (MMA). The multimodulus algorithm was pro-
in decision-directed mode. RDE uses error function ψ L [yn ]                                posed independently by many authors [13], [14], [15]. This
which is given as,                                                                          algorithm minimizes the dispersion of real and imaginary
                                                                                            parts, yR,n and yI,n , and forces them to lie on straight-line
                     ψL [yn ] = yn (|yn |2 − Rk )                                    (23)   contours. MMA error function ψ G [yn ] is given as ψG [yn ] =
       2     2            2     2
yR,n (yR,n −RR )+ yI,n (yI,n −RI ). Recently, a complexity                                                           QAM constellations. Figure 6(a) depicts the square-contour
efficient MMA is proposed, named soft-constraint satisfac-                                                             generated by TW-GSCA for 16-QAM constellation.
tion multimodulus algorithm (SCS-MMA) [16], [17]. The
SCS-MMA error function ψ G [yn ] is expressed as
                                                                                                                                                                             Proposed Square−Contour
                                                                                                                                     Square−Contour Algorithm (2003)          Stop−and−Go Algorithm
                                                                                                                                     4                                 4

  ψG [yn ] = yR,n (|yR,n | − RR ) +  yI,n (|yI,n | − RI )                                                     (25)                  2                                 2

where RR = E[|aR |                          ]/E[a2 ]
                                   and RI = E[|aI |
                                                                                                        ]/E[a2 ].
                                                                                                                                     0                                 0

The (straight) line-contours exhibited by SCS-MMA are                                                                                −2                                −2

shown in Fig. 5(a). In order to obtain a stop-and-go version
of SCS-MMA, the dispersion constants R R and RI are                                                                                  −4
                                                                                                                                      −4     −2     0
                                                                                                                                                           2      4
                                                                                                                                                                        −4       −2     0
                                                                                                                                                                                              2        4

replaced with |ˆ R,n | and |ˆ I,n |, respectively, to get a new
error function ψ L [yn ] as follows                                                                                        Fig. 6.        Contours in (a) TW-GSCA (b) Proposed SC-SAGA.

                            a                            a
ψL [yn ] = yR,n (|yR,n | − |ˆR,n |) +  yI,n (|yI,n | − |ˆI,n |) (26)                                                    In order to obtain a stop-and-go version of TW-GSCA,
The proposed Line-Contour Stop-and-Go Algorithm (LC-                                                                  the dispersion constant R p in TW-GSCA are replaced with
SAGA) can be expressed as w n+1 = wn −µ [ψL [yn ]R · fR +                                                             (|ˆR,n + aI,n | + |ˆR,n − aI,n |)p to get ψL [yn ] as follows
                                                                                                                        a      ˆ         a      ˆ
 ψL [yn ]I ·fI ] x∗ , where fR and fI are binary-flags which
                   n                                                                                                      ψL [yn ] = {(|yR,n + yI,n | + |yR,n − yI,n |)p
indicate the match of signs of the real and the imaginary parts                                                                    −(|ˆR,n + aI,n | + |ˆR,n − aI,n |)p }
                                                                                                                                      a      ˆ         a      ˆ
of ψG [yn ] and ψL [yn ], respectively. ψG [yn ] and ψL [yn ] are
                                                                                                                                · (|yR,n + yI,n | + |yR,n − yI,n |)p−1
computed from (25) and (26), respectively. For the proposed
algorithm for QAM signals, R R and RI are computed as the                                                                       · {sgn[yR,n + yI,n ] + sgn[yR,n − yI,n ]
outermost line contour as follows:                                                                                              + (sgn[yR,n + yI,n ] − sgn[yR,n − yI,n ])}                                (29)
      RR = RI = max[{|aR |}] = max[{|aI |}] = β                                                                (27)   The proposed Square-Contour Stop-and-Go Algorithm (SC-
                                                                                                                      SAGA) can be expressed as w n+1 = wn − µ · f · ψL [yn ]x∗ ,  n
where β is computed form (18). The reliable regions obtained                                                          where f is the binary-flag which indicates the match of
by the proposed algorithm are depicted in Figure 5(b) for 16-                                                         signs of ψG [yn ] and ψL [yn ]. For the proposed algorithm, the
QAM.                                                                                                                  dispersion constant R is computed as the outermost square
                                                           Proposed Line−Contour Stop−and−Go Algorithm
                                                                                                                      contour as follows:
   Abrar et al. Soft Constraint Satisfaction
       Multimodulus Algorithm (2004)           Reliable regions for yR
                                                                                                                                R = 2 · max[{|aR |}, {|aI |}] = 2 · β
                                                                                     Reliable regions for yI
     4                                       4                                4                                                                                                                            (30)
     2                                    2                                    2
                                                                                                                      where β is computed form (18). The reliable regions obtained
     0                                    0                                    0                                      by the proposed algorithm are depicted in Figure 6(b) for 16-
    −2                                  −2                                    −2
     −4     −2       0      2       4
                                        −4                                    −4
                                                                                                                                    VIII. SIMULATION RESULTS
                                         −4      −2       0      2       4     −4     −2       0        2      4
                    (a)                                  (b)                                  (c)
                                                                                                                         In this section, the performance of existing and pro-
  Fig. 5.        Contours in (a) SCS-MMA and (b)-(c) Proposed LC-SAGA.                                                posed stop-and-go algorithms are compared. In simulations,
                                                                                                                      a complex-valued seven taps transversal equalizer was used
                                                                                                                      and it was initialized so that the center tap was set to one
          VII. SQUARE-CONTOUR STOP-AND-GO                                                                             and other taps were set to zero. The channel used in the
                     ALGORITHM                                                                                        simulation was taken from [6]. The signal to noise ratio
   Recently, Thaiupathump and Kassam presented an inter-                                                              (SNR) was taken as 30dB at the input of the equalizer.
esting family of generalized square-contour algorithms (TW-                                                           The residual ISI [19] and MSE are measured for 16-QAM
GSCA) [18]. Unlike WK-HLA (as described in Section IV),                                                               signaling and compared as performance parameters. Each
these algorithms are free from unnecessary 45 o rotation. The                                                         trace is the ensemble average of 200 independent runs with
weight adaptation process is w n+1 = wn − µψG [yn ]x∗ ,    n
                                                                                                                      random initialization of noise and data source.
where ψG [yn ] is computed as,                                                                                           Fig. 7 depicts traces of residual ISI convergence. Examin-
                                                                                                                      ing this result, we observe that PC-SAGA and DC-SAGA are
     ψG [yn ] = ((|yR,n + yI,n | + |yR,n − yI,n |)p − Rp )                                                            relatively the fastest and the slowest converging algorithms,
                 · (|yR,n + yI,n | + |yR,n − yI,n |)p−1                                                               respectively. The SC-SAGA is next to the PC-SAGA in
                 · {sgn[yR,n + yI,n ] + sgn[yR,n − yI,n ]                                                             performance. Fig. 8 depicts traces of MSE convergence,
                                                                                                                      where we observe that PC-SAGA is consistently performing
                 + (sgn[yR,n + yI,n ] − sgn[yR,n − yI,n ])}                                                   (28)
                                                                                                                      the best and SC-SAGA is next to it. While the CC-SAGA
(Refer to [18] for the computation of R p .) The beauty of                                                            is performing worst by giving nonconverging MSE floor
this algorithm is that it forces the equalizer outputs to reside                                                      because of its incapability to remove (almost 45 o ) phase-
on a square-contour; which is a desirable contour for square                                                          offset error (introduced by the channel).
                                                                                                                   [6] G. Picchi and G. Prati. “Blind Equalization and Car-
                                                                                                                       rier Recovery using a ‘Stop-and-Go’ Decision-Directed
                                                                                                                       Algorithm”. IEEE Trans. Commun., COM-35:877–887,
     Residual ISI (dB)

                         −15                                                                                           1987.
                                         Diamond contour SAG                                                       [7] V. Weerackody and S.A. Kassam. A Simple Hard-
                         −20                     Point contour SAG                                                     Limited Adaptive Algorithm for Blind Equalization.
                                                        Line contour SAG                                               IEEE Trans. Circuit and Systems-II, 39(7):482–487,
                                                            Circular contour SAG                                       July 1992.
                                                                 Square contour SAG                                [8] Youngkyun Kim, Sungjo Kim, and Mintaig Kim. “The
                                                                                                                       Derivation of a New Blind Equalization Algorithm”.
                         −35                                                                                           ETRI Journal, 18(2):53–60, July 1996.
                            0   1000   2000      3000     4000     5000    6000    7000   8000    9000    10000
                                                              Iterations                                           [9] D.N. Godard. “Self-Recovering Equalization and Car-
                                                                                                                       rier Tracking in Two-Dimensional Data Communica-
                                       Fig. 7.     Convergence traces for SER.                                         tions Systems”. IEEE Trans. Commun., COM-28:1867–
                                                                                   Circular contour SAG                1875, Nov. 1980.
                                                                                                                  [10] J.R. Treichler and B.G. Agee. “A New Approach to
                                                                                                                       Multipath Correction of Constant Modulus Signals”.
                           0                                                                                           IEEE Trans. acoust. Speech Signal Process., ASSP-
     MSE (dB)

                                                                                                                       31:459–471, 1983.
                                              Point contour SAG
                                                                                   Diamond contour SAG
                                                                                                                  [11] M. J. Ready and R. P. Gooch. “Blind Equalization
                                                  Line contour SAG
                                                        Square contour SAG
                                                                                                                       Based on Radius Directed Adaptation”. Proc. IEEE
                                                                                                                       Int. Conf. Acoust. Speech Signal Process., pages 1699–
                                                                                                                       1702, April 1990.
                                                                                                                  [12] C.W. You and D.S. Hong. “Nonlinear Blind Equaliza-
                         −20                                                                                           tion Schemes Using Complex-Valued Multilayer Feed-
                                1000   2000      3000     4000     5000    6000    7000   8000    9000    10000
                                                               Iterations                                              forward Neural Networks”. IEEE Trans. Neural Net-
                                                                                                                       work, 9(6):1442–1455, Nov. 1998.
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                                                                                                                       tion Algorithms for Digital Radio and Voiceband Data
                                                                                                                       Modems”. Proc. European Conf. Circuit Theory and
                                         IX. CONCLUSION
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