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

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

Shafayat Abrar
Electrical Engineering Department,
sabrar@comsats.edu.pk, shafayat1972@yahoo.com

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 .
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
v(n)
x(n)
a(n)    Channel                      Equalizer       y(n)              a(n)     cording to
C(z)        +                 W(z)
Decision
Device

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
∂wn
signal:
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
n
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
ˆ
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-
ˆ
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 ]]
n
wn+1 =                                                           (5)     The resulting error function (say local) ψ L [yn ] can be given
wn ,               if sgn[ψ1 [yn ]] = sgn[ψ2 [yn ]]
as
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)
ˆ
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
ˆ
the symbols) of the constellation can be considered as global          be understood that due to multiple contours exhibited by
ˆ
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
n
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
2
follows:
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)
n
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
I
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
n
ψG [yn ] = yn − R1 sgn[yn ], where R1 =              .   (14)                 IV. DIAMOND-CONTOUR STOP-AND-GO
E[|an |]
ALGORITHM
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
n
computed as:
where RR and RI are computed as E[a 2 ]/E[|an,R |] and
n,R
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)
n,I
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)
Weerackody−Kassam’s
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
−5               0              5
−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
n
|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
√                               √                                                                =        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).
ALGORITHM
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
1
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
−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)
4(a).
−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-
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
Thaiupathump−Kassam’s
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

3
where RR = E[|aR |                          ]/E[a2 ]
and RI = E[|aI |
R
3
]/E[a2 ].
I
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
(a)
2      4
−4
−4       −2     0
(b)
2        4

a
replaced with |ˆ R,n | and |ˆ I,n |, respectively, to get a new
a
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
QAM.
−4
−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).
−5
[6] G. Picchi and G. Prati. “Blind Equalization and Car-
−10
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,
−25
Circular contour SAG                                       July 1992.
−30
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–
10
Circular contour SAG                1875, Nov. 1980.
5
[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
−5
Diamond contour SAG
[11] M. J. Ready and R. P. Gooch. “Blind Equalization
Line contour SAG
Square contour SAG
−10
Int. Conf. Acoust. Speech Signal Process., pages 1699–
−15
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.
Fig. 8.     Convergence traces for MSE.                                    [13] K. Wesolowsky. “Self-Recovering Adaptive Equaliza-
tion Algorithms for Digital Radio and Voiceband Data
Modems”. Proc. European Conf. Circuit Theory and
IX. CONCLUSION
Design, pages 19–24, 1987.
In this work, a framework for stop-and-go based blind                                                          [14] K.N. Oh and Y.O. Chin. “Modiﬁed Constant Modu-
equalization algorithms is presented. Based on the proposed                                                            lus Algorithm: Blind Equalization and Carrier Phase
framework, three new stop-and-go algorithms are proposed.                                                              Recovery Algorithm”. Proc. 1995 IEEE Int. Conf.
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(point-contour) “Stop-and-Go Decision-Directed Algorithm”                                                         [15] J. Yang, J.-J. Werner, and G.A. Dumont. “The Mul-
[6] is the best among other stop-and-go algorithms. A                                                                  timodulus Blind Equalization Algorithm”. IEEE Intl.
detailed convergence analysis of these algorithms is under                                                             Conf. on DSP, 1:127–130, 1997.
study.                                                                                                            [16] S. Abrar, A. Zerguine, and M. Deriche. “Soft Con-
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