Abstract In this paper, I propose for the noiseless, real andtwo independent quadrature carrier case some approximatedconditions on the step-size parameter, on the equalizer’s taplength and on the channel power, related to the nature ofthe chosen equalizer and input signal statistics, for which ablind equalizer will not converge anymore. These conditionsare valid for type of blind equalizers where the error that isfed into the adaptive mechanism that updates the equalizer’staps can be expressed as a polynomial function of the equal-ized output of order three like in Godard’s algorithm. Sincethe channel power is measurable or can be calculated if thechannel coefficients are given, there is no need anymore tocarry out any simulation with various step-size parametersand equalizer’s tap length for a given equalization methodand input signal statistics in order to find the maximum step-size parameter for which the equalizer still converges.
Keywords Blind equalization · Blind deconvolution ·Convergence state
1 Introduction
It is well known that ISI (Intersymbol Interference) is a lim-iting factor in many communication environments where itcauses an irreducible degradation of the bit error rate (BER),thus imposing an upper limit on the data symbol rate. In orderto overcome the ISI problem, an equalizer is implemented inthose systems. Among the three types of equalizers, non-blind, semiblind and blind, the blind equalizer is chosen inmost cases due to the benefit of bandwidth saving and no need
M. Pinchas (B)Department of Electrical and Electronic Engineering,Ariel University Center of Samaria, Ariel, 40700 Israele-mail: [email protected]
of going through a training phase. Blind equalization algo-rithms are essentially adaptive filtering algorithms designedsuch that they do not require the external supply of a desiredresponse to generate the error signal in the output of theadaptive equalization filter [1]. The algorithm itself generatesan estimate of the desired response by applying a nonlineartransformation to sequences involved in the adaptation pro-cess [1]. The Bussgang algorithm is one of the three importantfamilies of blind equalization algorithms, where the nonlin-earity is in the output of the adaptive equalization filter [1].The nonlinearity is designed to minimize a cost function thatis implicitly based on higher-order statistics (HOS) accord-ing to one approach [2–4] or calculated directly according tothe Bayes rules [5–8]. Fast convergence speed and reachinga residual ISI where the eye diagram is considered to be openare the main requirements from a blind equalizer. Fast con-vergence speed may be obtained by increasing the step-sizeparameter. But increasing the step-size parameter too muchmight cause the equalizer not to converge at all.
Up to now, the range of the step-size parameter for whichthe equalizer still converges was obtained only for the meansquare error gradient (MSEG) and stochastic gradient (SG)algorithms [9]. But, the MSEG algorithm relies on knownchannel coefficients which in practical cases are unknown.The SG algorithm relies on known input signals. Thus, itcannot be considered as a blind equalizer.
In this paper, we derive the range of the step-size param-eter for which a blind equalizer belonging to the Bussgangalgorithm family will not converge. The obtained range isvalid for a type of blind equalizer where the error that is fedinto the adaptive mechanism that updates the equalizer’s tapscan be expressed as a polynomial function of the equalizedoutput of order three and where the gain between the equal-ized output and input sequence is equal to one. The derivationof the mentioned step-size parameter range is valid for the
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Fig. 1 Block diagram of abaseband communicationsystem
h[n]x[n]
c[n]
w[n]
y[n] z[n]
Equalizer
T ][
d[n]
Adaptive ControlAlgorithm
c[n+1]
++
+
-
noiseless, real and two independent quadrature carrier case.In addition, it depends on the input signal statistics, channelpower, equalizer’s tap length and on the nature of the chosenequalizer. From the obtained derivation, the maximum step-size parameter can be calculated for which the blind equalizerstill converges. Although we considered only the noiselesscase for carrying out our derivations, the noisy environmentis also addressed in this paper.
The paper is organized as follows: after having describedthe system under consideration in Sect. 2, the approximatedconditions for which a blind equalizer will not converge areintroduced in Sect. 3. In Sect. 4, simulation results are pre-sented, and the conclusion is given in Section 5.
2 System description
The system under consideration is illustrated in Fig. 1, wherewe make the following assumptions:
1. The input sequence x[n] belongs to a real or twoindependent quadrature carrier case constellation inputwhere x1[n] and x2[n] are the real and imaginary partsof x[n], respectively.
2. The unknown channel h[n] is a possibly non-minimum-phase linear time-invariant filter in which the transferfunction has no “deep zeros”, namely, the zeros lie suf-ficiently far from the unit circle.
3. The equalizer c[n] is a tap-delay line.4. The noise w[n] is an additive Gaussian white noise.5. The function T [·] is a memoryless nonlinear function
that satisfies the analyticity condition:T [z1[n] + j z2[n]] = T [z1[n]] + jT [z2[n]], wherez1[n] and z2[n] are the real and imaginary parts of theequalized output, respectively.
The sequence x[n] is transmitted through the channel h[n]and is corrupted with noise w[n]. Therefore, the equalizer’s
input sequence y[n] may be written as:
y[n] = x[n] ∗ h[n] + w[n] (1)
where “∗” denotes the convolution operation. The idealequalized output is expressed in [1] as:
z[n] = x[n − D]e jθ (2)
where D is a constant delay and θ is a constant phase shift.Therefore, in the ideal case we could write:
c[n] ∗ h[n] = δ[n − D]e jθ (3)
where δ is the Kronecker delta function. In this paper weassume that D = 0 and θ = 0, since D does not affect thereconstruction of the original input sequence x[n] and θ canbe removed by a decision device [1]. Since c[n] is unknown,we assume that some initial guess cg[n] has been selected forthe impulse response of the equalizer. We denote:
s[n] = cg [n] ∗ h [n] = δ [n] + ξ [n] (4)
where ξ [n] stands for the difference (error) between the idealvalue c[n] and the guess cg[n]. Convolving cg[n] with thereceived sequence y[n] and using (1), we obtain:
z [n] = y [n] ∗ cg [n]
= x [n] ∗ h [n] ∗ cg [n] + w [n] ∗ cg [n] (5)
substituting (4) into (5) yields:
z [n] = x [n] + p [n] + w [n] (6)
where p[n] is the convolutional noise, namely, the resid-ual intersymbol interference (ISI) arising from the differencebetween cg[n] and c[n] and w [n] = w [n]∗cg [n]. The ISI isoften used as a measure of performance in equalizers’ appli-cations, defined by:
I S I =∑
m |s[m]|2 − |s|2max
|s|2max(7)
where |s|max is the component of s, given in (4), having themaximal absolute value. Next, we consider the adaptation
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mechanism of the equalizer. There are two approachesto derive this adaptation scheme. According to the firstapproach, we define some estimator of x[n], d[n], which isproduced by the function T [z[n]]. Thus, the error signal is:
e [n] = T [z [n]] − z [n] . (8)
This error is fed into the adaptive mechanism, which updatesthe equalizer’s taps. In a compact filter vector notation, theupdated equation for the taps of the equalizer looks quitesimilar to the LMS algorithm for the non-blind case [10]:
ceq [n + 1] = ceq [n] + μ · e [n] y∗ [n] (9)
where ()∗ is the conjugate operation, μ is the step-size param-eter and ceq [n] is the equalizer vector where the input vector
is y[n] = [y[n] . . . y[n−N+1]]T and N is the equalizer’s tap
length. The operator ()T denotes for transpose of the function(). The main concern in this approach is to develop a properT [z[n]] that will produce a small error.
According to the second approach, a predefined cost func-tion F[n] that characterizes the intersymbol interference isdefined, see [2,11,12] and [13]. Minimizing this F[n] withrespect to the equalizer parameters will reduce the convo-lutional error. Minimization is performed with the gradientdescent algorithm that searches for an optimal filter tap set-ting by moving in the direction of the negative gradient -∇c F [n] over the surface of the cost function in the equalizerfilter tap space [10]. Thus, the updated equation is given by[10]:
ceq [n + 1] = ceq [n] + μ · (−∇ceq F [n])
= ceq [n] − μ∂ F [n]
∂z [n]y∗ [n] . (10)
It should be pointed out that the gradient of the function in(10) coincides with the partial derivative only if the param-eter space (which is a vector space, in this case) is endowedwith the standard Euclidean metric. If the space is a curvedRiemannian manifold endowed with a different metric, thestructure of the gradient may be very different. Note that by(8), (9) and (10):
T [z [n]] = z [n] − ∂ F [n]
∂z [n], (11)
thus choosing the cost function F[n] results in a correspond-ing choice of T [z[n]]. This approach is very popular andcost functions that were suggested by Godard [2] and others[13] proved themselves to overcome non-minimum phasechannels with low complexity. In this paper we adopt theadditional assumptions made in [14]:
1. The convolutional noise p[n], is a zero mean, whiteGaussian process with variance σ 2
p[n] = E[p[n]p∗[n]].2. The source signal x[n] is an independent non-Gaussian
signal with known variance and higher moments.
3. The convolutional noise p[n] and the source signal areindependent. Thus,
σ 2z [n]= E[z[n]z∗[n]]= E[x[n]x∗[n]]+E[p[n]p∗[n]]
4. No noise is added.5. ∂ F[n]
∂z[n] can be expressed as a polynomial function of theequalized output namely as P[z[n]] of order three.
6. The input sequence x[n] belongs to a real or two inde-pendent quadrature carrier case constellation input.
7. |s|2max = 1
Comments Assumptions 1 and 3 were also made in [1,5,15]and in [8]. It should be noted that the described model for theconvolutional noise p[n] is applicable during the latter stagesof the process where the process is close to optimality [8].According to [8], in the early stages of the iterative deconvo-lution process, the ISI is typically large with the result that thedata sequence and the convolutional noise are strongly corre-lated and the convolutional noise sequence is more uniformthan Gaussian [16]. However, satisfying equalization perfor-mance were obtained by [15] and others [7] in spite of thefact that the described model for the convolutional noise p[n]was used. These results [7,15] may indicate that the describedmodel for the convolutional noise p[n] can be used (maybenot in the optimum way) in the early stages where the “eyediagram” is still closed.
Since we consider the case where P[z[n]] = ∂ F[n]∂z[n] , (10)
may be written as:
ceq [n + 1] = ceq [n] + μ · e [n] y∗ [n]
= ceq [n] − μ∂ F [n]
∂z [n]y∗ [n]
= ceq [n] − μP [z [n]] y∗ [n] (12)
where P[z[n]] is the polynomial function of order three. Thereal part of P[z[n]] is according to [14] expressed as:
Pr [z[n]] =[a1 [xr + pr [n]] + a3 [xr + pr [n]]3
+a12 [xr + pr [n]] [xi + pi [n]]2]
(13)
where xr = x1[n], xi = x2[n], pr [n] and pi [n] are the realand imaginary parts of the convolutional noise p[n], respec-tively, and a1, a12, a3 are coefficients related to the chosentype of equalizer. In the latter stages where the blind equalizerhas converged, we may write that E[p2[n + 1]] ∼= E[p2[n]]where E[·] is the expectation operator. Since we deal withthe real or two independent quadrature carrier case, we mayassume as was done in [14] that for the two independentquadrature carrier case E[p2
r [n]] = E[p2i [n]]. Thus, in the
latter stages where the blind equalizer has converged we haveE[p2
r [n + 1]] ∼= E[p2r [n]]. In addition, for the case where
the blind equalizer does not converge we may write that
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Fig. 2 Cases A and B
E[p2r [n + 1]] �= E[p2
r [n]]. Recently [14], an approxima-tion for E[�p2
r ] = E[p2r [n + 1] − p2
r [n]] was obtained forthe above described assumptions:
E[�p2r ] ∼= B(D1 Bm3
p[n] + A1m2p[n] + B1m p[n] + C1 B)
where
A1 =(
B(
45σ 2xr
a23 + 18σ 2
xra3a12 + 6a1a3
+9σ 2xr
a212 + 2a1a12
)− 2 (3a3 + a12)
)
B1 =(
B
(
12(σ 2
xr
)2a3a12 + 6
(σ 2
xr
)2a2
12
+12σ 2xr
a1a3 + 4σ 2xr
a1a12 + a21 + 15E
[x4
r
]a2
3
+2E[x4
r
]a3a12 + E
[x4
r
]a2
12
)
−2(
a1 + 3σ 2xr
a3 + σ 2xr
a12
))
C1 =(
2(σ 2
xr
)2a1a12 + σ 2
xra2
1
+2E[x4
r
]σ 2
xra3a12 + E
[x4
r
]σ 2
xra2
12
+ 2E[x4
r
]a1a3 + E
[x6
r
]a2
3
)
B = μNσ 2x
k=R−1∑
k=0
|h [k]|2
D1 = 15a23 + 6a3a12 + 3a2
12 (14)
E[p2r [n]] = m p[n], E[(xr [n])2] = σ 2
xr, E[(xi [n])2] = σ 2
xi,
E[x[n]x∗[n]] = σ 2x and R is the channel length.
In the following, we will use (14) in order to derive someapproximated conditions on the step-size parameter, on the
equalizer’s tap length and on the channel power, related tothe nature of the chosen equalizer and input signal statistics,for which a blind equalizer will not converge.
3 The non-convergence case
In this section, we will find some conditions for which theequalizer will not converge. As we will show in the follow-ing, these conditions rely on the function B defined in (14),which is related to the step-size parameter, equalizer’s lengthand channel power and on the algorithm that is used via a1, a3
and a12.In the following, two major cases are considered denoted
as Case A and Case B. The main difference between Case Aand B lies on the parameter C1 given in (14). In normal opera-tion where the equalizer converges the equalizer may lead thesystem to perfect equalization performance from the residualISI point of view (I S I → 0) when C1 = 0 or leave the sys-tem with a residual ISI dependent on the input sequence sta-tistics when C1 �= 0. For further explanation on that subject,the reader may refer to [14]. Each case (A and B) may havesubcases as described in Fig. 2 where A11, B11 and �B aredefined in (21) and (22), respectively. The expression of �B
is defined in (30). The different cases and subcases dependon the input constellation statistics and on the algorithm thatis used via a1, a3 and a12. Therefore, for a given equalizationmethod, we may find ourselves in different cases and in dif-ferent subcases for different input sequences. In addition, fora given input constellation and different equalization meth-ods, we may have that each equalization method may leadto a different case or subcase. We start our derivations withCase A.
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SIViP (2012) 6:325–340 329
Case A First, we start to derive the relation between the con-volutional noise power m p[n] and ISI. By using (1), (4) and(5) we may write:
z[n] = cg[n] ∗ h[n] ∗ x[n] = s[n] ∗ x[n]⇓
E[z[n]2] = σ 2x
∑m |s(m)|2
(15)
Now we use the relation of σ 2p[n] = σ 2
z [n] − σ 2x together
with (15) to obtain:
σ 2p[n] = σ 2
z [n] − σ 2x = σ 2
x
[∑
m
|s(m)|2 − 1
]
(16)
which by the help of (7) may be expressed for |s|2max = 1 as:
σ 2p[n] = σ 2
x · I S I for |s|2max = 1. (17)
Please note that I S I is actually a function of n. For the real-valued case, we have m p[n] = σ 2
p[n] and σ 2x = σ 2
xr. For the
two independent quadrature carrier case, we have σ 2x = 2σ 2
xr
and σ 2p[n] = 2m p[n].
For the case of |D1 Bm3p[n]| � |A1m2
p[n]| (a case thatfits an easy channel, namely a channel where the ISI is rela-tively low (but the eye diagram is still very closed)), we mayneglect the product of D1 Bm3
p[n] in (14) and write that:
E[�p2r ] ∼= B(A1m2
p[n] + B1m p[n] + C1 B) (18)
Now, according to (18), the equalizer will not converge whenE
[�p2
r
] �= 0. Thus, for B �= 0 we may write:
A1m2p[n] + B1m p[n] + C1 B �= 0 (19)
Assuming C1 �= 0, the inequality of (19) can be obtained forthree cases:
Case A.1 Here we assume that A1 �= 0 and B1 �= 0. Thus,the left side of (19) can be considered as a second-order equa-tion according to m p[n] for which the inequality in (19) isobtained when � < 0 where � is defined as:
� = B21 − 4A1C1 B (20)
Note that B is a function of the step-size parameter, equal-izer’s length, channel power and source signal power. Sincewe are looking for some conditions on the step-size parame-ter, channel power, equalizer’s length and on the nature of thechosen algorithm (expressed via a1, a3 and a12), we rewrite
the expression for � (20) as a function of B :
� =(
B211 − 4C1 A11
)B2
+ (4C1 A12 − 2B11 B12) B + B212
where
A1 = B A11 − A12; B1 = B B11 − B12
A11 = 45σ 2xr
a23 + 18σ 2
xra3a12 + 6a1a3
+9σ 2xr
a212 + 2a1a12
A12 = 2 (3a3 + a12)
B11 = 12(σ 2
xr
)2a3a12 + 6
(σ 2
xr
)2a2
12
+12σ 2xr
a1a3 + 4σ 2xr
a1a12 + a21
+15E[x4
r
]a2
3 + 2E[x4
r
]a3a12 + E
[x4
r
]a2
12
B12 = 2(
a1 + 3σ 2xr
a3 + σ 2xr
a12
)(21)
Next we define:
�B = (4C1 A12 − 2B11 B12)2
−4(
B211 − 4C1 A11
)B2
12 (22)
Sol B2 = max
{− (4C1 A12 − 2B11 B12) − �B B
2(B2
11 − 4C1 A11) ,
− (4C1 A12 − 2B11 B12) + �B B
2(B2
11 − 4C1 A11)
}
Sol B1 = min
{− (4C1 A12 − 2B11 B12) − �B B
2(B2
11 − 4C1 A11) ,
− (4C1 A12 − 2B11 B12) + �B B
2(B2
11 − 4C1 A11)
}
(23)
where
�B B = √�B
and Sol B1 , Sol B
2 are the two solutions for B derived fromsetting � = 0 in (21).
Theorem The condition of � < 0 is obtained for two cases:
Case A.1.1 �B > 0 and(B2
11 − 4C1 A11)
< 0 and B is inthe range of: (B > Sol B
2 and B > 0)
Case A.1.2 �B > 0 and(B2
11 − 4C1 A11)
> 0 and B is inthe range of: ((Sol B
1 < B) and B > 0)
The proof is given in the Appendix.Since �B and
(B2
11 − 4C1 A11)
are independent with B,the range of B or equivalent the range of the step-size param-eter multiplied by the channel power, source signal power andequalizer’s tap length for which the equalizer will never enter
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to its convergence state can be calculated. Note that the rangeof B (given in cases A.1.1 and A.1.2) for which the equal-izer will not converge depends on the nature of the chosenequalizer via a1, a12 and a3 and on the input signal statistics.
Case A.2 Here we assume that A1 �= 0 and B1 = 0. There-fore, according to (20) we have that � < 0 for A1C1 > 0(note that B is a positive value). By using (21), the conditionon B can be derived:
B >A12
A11for A11C1 > 0 and A11C1 �= 0
or
B <A12
A11for A11C1 < 0 and A11C1 �= 0 (24)
Here again we have that the range of B for which the equal-izer will not converge depends on the nature of the chosenequalizer via a1, a12 and a3 and on the input signal statistics.
Case A.3 Here we assume that A1 = 0 and B1 = 0. Fromthe conditions of A1 = 0 and B1 = 0, we obtain by using(21):
A12
A11= B12
B11(25)
This expression (25) implies that the equalizer will not enterthe convergence state independent to B. In other words, itmeans that no convergence is obtained independent to thestep-size parameter, equalizer’s tap length and channel powerbut only due to the nature of the chosen equalizer (expressedvia a1, a3 and a12) combined with the statistics of the sourcesignal. This is obviously not a realistic case. In a realisticcase, the equalizer converges for values of B up to a pointwhere the value for B is too high.
Next we turn to Case B.
Case B Here we assume that C1 = 0. Setting C1 = 0 in(20) shields to � = B2
1 ≥ 0, which means that the equalizerwill always converge independent with B. This is obviouslynot true since we know from simulation results that startingfrom a specific step-size parameter that is considered as themaximum value for which we still have convergence , theequalizer will not converge. In order to solve the problem werecall (14) and set C1 = 0:
E[�p2r ] ∼= Bm p[n](D1 Bm2
p[n] + A1m p[n] + B1) (26)
As before, the equalizer will not converge if E[�p2r ] �= 0.
Therefore, non-convergence is obtained for Bm p[n] �= 0 and
D1 Bm2p[n] + A1m p[n] + B1 �= 0. (27)
Thus, the main idea here is as follows: obtaining the range forB for which (27) is made valid. Once that range is obtained,the maximum value of B for which we still have convergenceis also obtained. Note that B is a function of the step-sizeparameter. Now, choosing a higher value for the step-size
parameter causes having a higher level of ISI or equivalenthaving a higher value of m p[n]. Therefore, when the maxi-mum value or higher values for B are chosen, the assumptionof Bm p[n] �= 0 is made reasonable.
Assuming B1 �= 0, the inequality of (27) can be obtainedfor three cases:
Case B.1 Here we assume that D1 B �= 0 and A1 �= 0. Thus,the left side of (27) can be considered as a second-order equa-tion according to m p[n] for which the inequality in (27) isobtained when � < 0 where � is defined as:
� = A21 − 4B1 D1 B (28)
Since we are looking for some conditions on B dependingon the nature of the chosen algorithm (expressed via a1, a3
and a12) and on the input signal statistics, we rewrite theexpression for � (28) as a function of B with the help of(21):
� =(
A211 − 4D1 B11
)B2
+ (4D1 B12 − 2A11 A12) B + A212 (29)
Next we define:
�B = (4D1 B12 − 2A11 A12)2
−4(
A211 − 4D1 B11
)A2
12 (30)
SolB2 = max
{− (4D1 B12 − 2A11 A12) −
√�B
2(
A211 − 4D1 B11
) ,
− (4D1 B12 − 2A11 A12) +√
�B
2(
A211 − 4D1 B11
)
}
SolB1 = min
{− (4D1 B12 − 2A11 A12) −
√�B
2(
A211 − 4D1 B11
) , (31)
− (4D1 B12 − 2A11 A12) +√
�B
2(
A211 − 4D1 B11
)
}
where SolB2 , Sol
B1 are the two solutions for B derived from
setting � = 0 in (29). The condition of � < 0 is obtainedfor two cases that can be proved similar to what was done inthe Appendix for Case A.1:
Case B.1.1 �B > 0 and(
A211 − 4D1 B11
)< 0 and B is in
the range of: (B > SolB2 and B > 0)
Case B.1.2 �B > 0 and(
A211 − 4D1 B11
)> 0 and B is in
the range of: ((SolB1 < B) and B > 0)
Case B.2 Here we assume that D1 B �= 0 and A1 = 0.Therefore, according to (28) we have that � < 0 forD1 B1 > 0 (note that B is a positive value). By using (21),
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SIViP (2012) 6:325–340 331
the condition on B can be derived:
B >B12
B11for B11 D1 > 0 and B11 D1 �= 0
or (32)
B <B12
B11for B11 D1 < 0 and B11 D1 �= 0
Here again we have that the range of B for which the equal-izer will not converge depends on the nature of the chosenequalizer via a1, a12 and a3, and on the input signal statistics.
Case B.3 Here we assume that D1 B = 0 and A1 = 0.Therefore, in order to comply with (27) we demand thatB1 �= 0 which yields with the help of (21) to the follow-ing condition for B:
B �= B12
B11(33)
As in the previous case, the range of B for which the equal-izer will not converge depends on the nature of the chosenequalizer via a1, a12 and a3 and on the input signal statistics.But this case is actually not a realistic case. In a realistic casein contrast to what we have here, the equalizer converges forvalues of B up to a point where the value for B is too high.
4 Simulation
In this section, the main idea was to find via simulation themaximum value of B (14) for which a blind equalizer stillconverges and comparing it with the calculated maximumvalue obtained from the previous section. We used for thatpurpose Godard’s algorithm [2] which is widely used dueto its good equalization performance and low computationalburden. The equalizer taps for Godard’s algorithm [2] wereupdated according to:
cm [n + 1] = cm [n] − μG
(
|z [n]|2
− E[|x [n]|4]
E[|x [n]|2]
)
z [n] y∗ [n − m] (34)
where μG is the step-size. The values for a1, a12 and a3 corre-sponding to Godards’s [2] algorithm were defined as aG
1 , aG12
and aG3 , respectively, and are given by:
aG1 = − E
[|x (n)|4]
E[|x (n)|2] ; aG
12 = 1; aG3 = 1. (35)
In order to show that our derivations are valid not only forGodard’s algorithm [2], we needed another blind algorithmwith the following properties:
1. The algorithm leaves the system in the steady state with|s|2max = 1 (meaning that the gain between the equalizedoutput and input sequence is equal to one).
0 500 1000 1500 2000 2500 3000 3500 4000 4500−25
−20
−15
−10
−5
0
Iteration Number
ISI [
dB]
GodardNew
Fig. 3 Equalization performance comparison between Godard’s andNew algorithm for the 16QAM source input going through channel 1.The averaged results were obtained in 100 Monte Carlo trials for thenoiseless case. The equalizer’s tap length was set to 13 and μG =7e-5, μN = 1e-3
2. The error that is fed into the adaptive mechanism thatupdates the equalizer’s taps can be expressed as polyno-mial function of the equalized output of order three.
For that purpose, we developed a new blind algorithm:
cm [n + 1] = cm [n] − μN
(E
[(xr [n])4]
E[(xr [n])6] z3
r [n]
+ jE
[(xr [n])4]
E[(xr [n])6] z3
i [n] − z [n]
)
y∗ [n − m]
(36)
where μN is the step-size. The values for a1, a12 and a3 cor-responding to the new algorithm were defined as aN
1 , aN12 and
aN3 , respectively, and are given by:
aN1 = −1; aN
12 = 0; aN3 = E
[(xr [n])4]
E[(xr [n])6] (37)
In the following, we denote algorithm (36) as “New”. A per-formance comparison between Godard’s method with algo-rithm “New” for the 16QAM and 64QAM input constellationcan be found in Figs. 3 and 4, respectively. For the equaliza-tion performance comparison, the step-size parameters μG
and μN were chosen for fast convergence with low steady-state ISI.
It should be pointed out that in our simulation, the equal-izers were initialized by setting the center tap equal to oneand all others to zero. In addition, the reader may find ineach figure caption, the equalizer’s tap length, the chosenstep-size parameter, the chosen channel and input constella-tion making up the quantity B (14). Three input sources were
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0 1000 2000 3000 4000 5000 6000 7000−25
−20
−15
−10
−5
0
Iteration Number
ISI [
dB]
GodardNew
Fig. 4 Equalization performance comparison between Godard’s andNew algorithm for the 64QAM source input going through channel 1.The averaged results were obtained in 100 Monte Carlo trials for thenoiseless case. The equalizer’s tap length was set to 13 and μG =1.5e-6, μN = 1e-4
considered: A 16QAM source (a modulation using ± {1,3}levels for in-phase and quadrature components), a 64QAMsource (a modulation using ± {1,3,5,7} levels for in-phaseand quadrature components) and a QPSK source (a modu-lation using ± {1} levels for in-phase and quadrature com-ponents). Three different channels were considered.
Channel1 (initial ISI = 0.44): The channel parameterswere determined according to [12]:
h[n] ={
0 for n < 0; −0.4 for n = 0;0.84 · 0.4n−1 for n > 0
}.
Channel 1 is a non-minimum-phase linear time-invariantchannel.
Channel2 (initial ISI = 0.88): The channel parameterswere determined according to:h[n] = (0.4851,−0.72765,−0.4851). Channel 2 is a non-minimum-phase-linear time-invariant channel.
Channel3 (initial I S I = 0.933): the channel parametersare based on the digital microwave radio channel impulseresponse (channel 1) given in [17] where the complex-valuedbaseband channel response has been “fractionally sampled”at twice the baud interval (baud interval is 30 Mb). Thoseparameters were down decimated by 8 and normalized sothat hT h = 1. Thus, the total channel length was 38 (Fig. 5).
Figures 6 and 7 show the simulated performance of Godard’sequalization method for the 16QAM input case, namelythe ISI as a function of iteration number for channel 1.According to (22) and (23), �B > 0, Sol B
2∼= 0.03409 and
−6 −4 −2 0 2 4
−1.5−1
−0.50
0.51
1.5
Real Part
Imag
inar
y P
art
zero−plot of channel3 response
0 5 10 15 20 25 30 35 40−0.2
0
0.2
0.4
0.6
0.8real part of channel3image part of channel3
Fig. 5 Channel 3
0 2000 4000 6000 8000 10000 12000−6.5
−6
−5.5
−5
−4.5
−4
−3.5
Iteration Number
ISI [
dB]
Godard
Fig. 6 Equalization performance with Godard’s algorithm for the16QAM source input going through channel 1. The averaged resultswere obtained in 100 Monte Carlo trials for the noiseless case. Theequalizer’s tap length was set to 27 and μG = 1.2052e-4
< 0, we deal with Case A.1.1 wherenon-convergence is obtained for B > 0.03409. Figures 6and 7 are simulated for B ∼= 0.03 and B ∼= 0.033, respec-tively. According to Figs. 6 and 7, the equalizer converges forB ∼= 0.03 and B ∼= 0.033. Note that the equalizer convergesaccording to Fig. 6 in the right direction, while according toFig. 7 the equalizer started to converge in the right directionand went finally to the opposite one. If we would have sim-ulated for a little higher value for B (as is shown later on forthe 64QAM input case), the equalizer would have convergedalways to the opposite direction. But this is true until we reachB ∼= 0.034. It was found by simulation that for B ∼= 0.034,
Fig. 7 Equalization performance with Godard’s algorithm for the16QAM source input going through channel 1. The results wereobtained in 1 Monte Carlo trial for the noiseless case. The equalizer’stap length was set to 27 and μG = 1.3257e-4
the equalizer does not converge any more. Thus, we have ahigh correlation between the simulated result (B ∼= 0.034)and the calculated value (B > 0.03409). Figures 8, 9 and10 show the simulated performance of Godard’s equaliza-tion method for the 64QAM input case, namely the ISI as afunction of iteration number for channel 1. According to (22)and (23), �B > 0, Sol B
2∼= 0.00776 and Sol B
1∼= −0.0219.
Since A1 �= 0, C1 �= 0, B1 �= 0 and(B2
11 − 4C1 A11)
< 0,we deal with Case A.1.1 where non-convergence is obtainedfor B > 0.00776. Figures 8, 9 and 10 are simulated forB ∼= 0.0055, B ∼= 0.0065 and B ∼= 0.007, respectively.According to Figs. 8, 9 and 10, the equalizer converges forB ∼= 0.0055, B ∼= 0.0065 and B ∼= 0.007. Note that theequalizer converges according to Fig. 8 in the right directionwhile according to Figs. 9 and 10 the equalizer convergesto the opposite one. According to simulation results, theequalizer does not converge when B ∼= 0.0071. Thus, bycomparing the simulated result (B ∼= 0.0071) with the cal-culated value (B > 0.00776), we may conclude that theerror between them is small. Figures 11 and 12 show thesimulated performance of Godard’s equalization method forthe 16QAM input case, namely the ISI as a function of iter-ation number for channel 2. According to (22) and (23),�B > 0, Sol B
2∼= 0.03409 and Sol B
1∼= −0.3204. Since
A1 �= 0, C1 �= 0, B1 �= 0 and(B2
11 − 4C1 A11)
< 0, wedeal with Case A.1.1 where non-convergence is obtained forB > 0.03409. Figures 11 and 12 are simulated for B ∼= 0.027and B ∼= 0.032, respectively. According to Figs. 11 and 12,the equalizer converges for B ∼= 0.027 and B ∼= 0.032.Note that the equalizer converges according to Fig. 11 inthe right direction while according to Fig. 12 the equal-izer converges to the opposite one. According to simulationresults, the equalizer does not converge when B ∼= 0.033.
0 0.5 1 1.5 2 2.5x 10
4
−9
−8
−7
−6
−5
−4
−3
Iteration Number
ISI [
dB]
Godard
Fig. 8 Equalization performance with Godard’s algorithm for the64QAM source input going through channel 1. The averaged resultswere obtained in 100 Monte Carlo trials for the noiseless case. Theequalizer’s tap length was set to 27 and μG = 5.2609e-6
0 0.5 1 1.5 2 2.5
x 104
−8
−6
−4
−2
0
2
4
6
8
10
12
Iteration Number
ISI [
dB]
Godard
Fig. 9 Equalization performance with Godard’s algorithm for the64QAM source input going through channel 1. The results wereobtained in 1 Monte Carlo trial for the noiseless case. The equalizer’stap length was set to 27 and μG = 6.2174e-6
Thus, we have a high correlation between the simulated result(B ∼= 0.033) and the calculated value (B > 0.03409). Nextwe turn to the QPSK input case. That case fits to Case Bwhere A1 �= 0, B1 �= 0, D1 �= 0, B �= 0 and C1 = 0. Itturns out that according to Godard’s algorithm, �B<0 and(
A211 − 4D1 B11
)>0. This means that � > 0, which indi-
cates that there is a solution for m p for which the followingexpression D1 Bm2
p + A1m p + B1 is zero. In other words,there is a value for m p for which (26) is zero (the equalizer
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0 0.5 1 1.5 2 2.5
x 104
−4
−2
0
2
4
6
8
10
Iteration Number
ISI [
dB]
Godard
Fig. 10 Equalization performance with Godard’s algorithm for the64QAM source input going through channel 1. The averaged resultswere obtained in 100 Monte Carlo trials for the noiseless case. Theequalizer’s tap length was set to 27 and μG = 6.6957e-6
Fig. 11 Equalization performance with Godard’s algorithm for the16QAM source input going through channel 2. The averaged resultswere obtained in 100 Monte Carlo trials for the noiseless case. Theequalizer’s tap length was set to 27 and μG = 9.9988e-5
converges). Since �B < 0 and(
A211 − 4D1 B11
)> 0, we are
not able to define for Godard’s algorithm for the QPSK casea range for B for which the equalizer does not converge. Fig-ure 13 shows the simulated performance of Godard’s equal-ization method for the 16QAM input case, namely the ISI as afunction of iteration number for channel 3. According to (22)and (23), �B > 0, Sol B
2∼= 0.03409 and Sol B
1∼= −0.3204.
Since A1 �= 0, C1 �= 0, B1 �= 0 and(B2
11 − 4C1 A11)
< 0,we deal with Case A.1.1 where non-convergence is obtainedfor B > 0.03409. Figure 13 is simulated for B ∼= 0.034.According to Fig. 13, the equalizer converges for B ∼= 0.034.According to simulation results, the equalizer does not con-verge when B ∼= 0.0341. Thus, we have a high correlation
Fig. 12 Equalization performance with Godard’s algorithm for the16QAM source input going through channel 2. The results wereobtained in 100 Monte Carlo trials for the noiseless case. The equalizer’stap length was set to 27 and μG = 1.1850e-4
0 5000 10000 15000−2
0
2
4
6
8
10
12
Iteration Number
ISI [
dB]
Godard
Fig. 13 Equalization performance with Godard’s algorithm for the16QAM source input going through channel 3. The averaged resultswere obtained in 100 Monte Carlo trials for the noiseless case. Theequalizer’s tap length was set to 67 and μG = 5.0746e-5
between the simulated result (B ∼= 0.034) and the calculatedvalue (B > 0.03409).
Although our derivations were obtained for the noiselesscase only, we turn now testing them for the noisy situa-tion too. All the simulated cases described earlier in thispaper for the noiseless case, were simulated again for thenoisy one. In our simulated examples, we used signal-to-noise ratio (SNR) values suitable for the VDSL environ-ment. Note that for the 16QAM constellation, the minimumSNR is defined as 25 [dB] ([18,19]), but a good SNR allowsa 6 [dB] margin for robust operation [18]. Figures 14 and15 show the simulated performance of Godard’s equaliza-
Fig. 14 Equalization performance with Godard’s algorithm for the16QAM source input going through channel 1. The averaged resultswere obtained in 100 Monte Carlo trials for SN R = 25[d B]. Theequalizer’s tap length was set to 27 and μG = 1.2052e-4
Fig. 15 Equalization performance with Godard’s algorithm for the16QAM source input going through channel 1. The results wereobtained in 1 Monte Carlo trial for SN R = 25[d B]. The equalizer’stap length was set to 27 and μG = 1.3257e-4
tion method for the 16QAM input case, namely the ISI asa function of iteration number for channel 1 and SNR of25 [dB]. As before, we deal here with Case A.1.1 wherenon-convergence is obtained for B > 0.03409. Figures 14and 15 are simulated for B ∼= 0.03 and B ∼= 0.033, respec-tively. According to Figs. 14 and 15, the equalizer con-verges for B ∼= 0.03 and B ∼= 0.033. Note that the equal-izer converges according to Fig. 14 in the right directionwhile according to Fig. 15 the equalizer converges to theopposite one. According to Fig. 15, we have a high corre-lation between the simulated result (B ∼= 0.033) and thecalculated value (B ∼= 0.03409). Figure 16 shows the simu-lated performance of Godard’s equalization method for the
Fig. 16 Equalization performance with Godard’s algorithm for the16QAM source input going through channel 2. The results wereobtained in 100 Monte Carlo trials for SN R = 30[d B]. The equal-izer’s tap length was set to 27 and μG = 1.1850e-4
0 0.5 1 1.5 2 2.5
x 104
−9
−8
−7
−6
−5
−4
−3
Iteration Number
ISI [
dB]
Godard
Fig. 17 Equalization performance with Godard’s algorithm for the64QAM source input going through channel 1. The averaged resultswere obtained in 100 Monte Carlo trials for SN R = 25[d B]. Theequalizer’s tap length was set to 27 and μG = 5.2609e-6
16QAM input case, namely the ISI as a function of itera-tion number for channel 2 and SNR of 30 [dB]. Accord-ing to (22) and (23), �B > 0, Sol B
< 0, we deal with Case A.1.1 where non-convergence is obtained for B > 0.03409. Figure 16 issimulated for B ∼= 0.032. According to Fig. 16, the equal-izer converges for B ∼= 0.032. Here again, we have a highcorrelation between the simulated result (B ∼= 0.032) andthe calculated value (B ∼= 0.03409). Figures 17 and 18show the simulated performance of Godard’s equalizationmethod for the 64QAM input case, namely the ISI as a
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0 0.5 1 1.5 2 2.5
x 104
−4
−2
0
2
4
6
8
10
Iteration Number
ISI [
dB]
Godard
Fig. 18 Equalization performance with Godard’s algorithm for the64QAM source input going through channel 1. The averaged resultswere obtained in 100 Monte Carlo trials for SN R = 25[d B]. Theequalizer’s tap length was set to 27 and μG = 6.60009e-6
0 5000 10000 15000−2
0
2
4
6
8
10
12
Iteration Number
ISI [
dB]
Godard
Fig. 19 Equalization performance with Godard’s algorithm for the16QAM source input going through channel 3. The averaged resultswere obtained in 100 Monte Carlo trials for SN R = 25[d B]. Theequalizer’s tap length was set to 67 and μG = 4.62686e-5
function of iteration number for channel 1 and SNR of 25[dB]. According to (22) and (23), �B > 0, Sol B
< 0, we deal with Case A.1.1 where non-convergence is obtained for B > 0.00776. Figures 17 and18 are simulated for B ∼= 0.0055 and B ∼= 0.0069, respec-tively. According to Figs. 17 and 18, the equalizer convergesfor B ∼= 0.0055 and B ∼= 0.0069. Note that the equalizerconverges according to Fig. 17 in the right direction whileaccording to Fig. 18 the equalizer converges to the oppo-site one. Here again, we have a high correlation betweenthe simulated result (B ∼= 0.0069) and the calculated value
0 5000 10000 15000−2
0
2
4
6
8
10
12
Iteration Number
ISI [
dB]
Godard
Fig. 20 Equalization performance with Godard’s algorithm for the16QAM source input going through channel 3. The averaged resultswere obtained in 100 Monte Carlo trials for SN R = 30[d B]. Theequalizer’s tap length was set to 67 and μG = 4.77611e-5
(B ∼= 0.00776). Figures 19 and 20 show the simulated per-formance of Godard’s equalization method for the 16QAMinput case, namely the ISI as a function of iteration num-ber for channel 3 and SNR of 25 and 30 [dB], respectively.According to (22) and (23), �B > 0, Sol B
< 0, we deal with Case A.1.1 where non-convergence is obtained for B > 0.03409. Figures 19 and20 are simulated for B ∼= 0.031 and B ∼= 0.032, respec-tively. According to Figs. 19 and 20, the equalizer convergesfor B ∼= 0.031 and B ∼= 0.032, respectively. Accordingto simulation results, we have a high correlation betweenthe simulated result (B ∼= 0.032) and the calculated value(B ∼= 0.03409).
Up to now, we tested our derivations only for Godard’salgorithm [2]. In the following we test them for the “New”algorithm.
Figure 21 shows the simulated performance of the “New”algorithm for the 16QAM input case, namely the ISI asa function of iteration number for channel 1. Accordingto (22) and (23), �B > 0, Sol B
> 0, we deal with Case A.1.2 where non-convergence is obtained for B > 0.4497. Figure 21 issimulated for B ∼= 0.4. According to Fig. 21, the equal-izer converges for B ∼= 0.4. But, taking a higher value thanB ∼= 0.4, the equalizer does not converge. According to sim-ulation results, we have a high correlation between the simu-lated result (B ∼= 0.4) and the calculated value (B ∼= 0.4497).
Figures 22, 23 and 24 show the simulated performanceof the “New” algorithm for the 64QAM input case, namelythe ISI as a function of iteration number for channel 1. It
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0 500 1000 1500 2000 2500 3000 3500 4000 4500−13
−12
−11
−10
−9
−8
−7
−6
−5
−4
−3
Iteration Number
ISI [
dB]
New
Fig. 21 Equalization performance with the “New” algorithm for the16QAM source input going through channel 1. The averaged resultswere obtained in 100 Monte Carlo trials for the noiseless case. Theequalizer’s tap length was set to 27 and μN = 1.6069e-3
0 500 1000 1500 2000 2500 3000 3500 4000 4500−4
−2
0
2
4
6
8
Iteration Number
ISI [
dB]
New
Fig. 22 Equalization performance with the “New” algorithm for the64QAM source input going through channel 1. The averaged resultswere obtained in 100 Monte Carlo trials for the noiseless case. Theequalizer’s tap length was set to 53 and μN = 1.983642578125e-4
should be pointed out that the simulation was carried outfor the fixed point case (16 bit). According to (22) and (23),�B > 0, Sol B
2∼= 0.4285 and Sol B
1∼= −0.8381. Since A1 �=
0, C1 �= 0, B1 �= 0,�B > 0 and(B2
11 − 4C1 A11)
< 0,we deal with Case A.1.1 where non-convergence is obtainedfor B > 0.4285. Figures 22, 23 and 24 are simulated forB ∼= 0.4070. According to Figs. 22, 23 and 24, the equalizerconverges for B ∼= 0.4070. Note that according to Figs. 22,23 and 24 the equalizer started to converge in the right direc-tion and went finally to the opposite one. According to simu-lation results, the equalizer does not converge for B ∼= 0.42.Thus, we have a high correlation between the simulated result(B ∼= 0.4070) and the calculated value (B ∼= 0.4285).
0 500 1000 1500 2000 2500 3000 3500 4000 4500−4
−2
0
2
4
6
8
10
12
14
Iteration Number
ISI [
dB]
New
Fig. 23 Equalization performance with the “New” algorithm for the64QAM source input going through channel 1. The averaged resultswere obtained in 1 Monte Carlo trials for the noiseless case. The equal-izer’s tap length was set to 53 and μN = 1.983642578125e-4
0 500 1000 1500 2000 2500 3000 3500 4000 4500−4
−2
0
2
4
6
8
10
12
14
Iteration Number
ISI [
dB]
New
Fig. 24 Equalization performance with the “New” algorithm for the64QAM source input going through channel 1. The averaged resultswere obtained in 1 Monte Carlo trials for the noiseless case. The equal-izer’s tap length was set to 53 and μN = 1.983642578125e-4
It should be pointed out that the derived range of B forwhich the equalizer will not converge anymore is valid alsofor θ �= 0. The reason for it lies on the fact that the derivedrange of B was obtained through the expression E[�p2
r ] (14),which is a scaled version of �(I S I ) (�(I S I ) = I S I [n +1] − I S I [n]). According to (7), the I S I is a function of theabsolute value of cg[n]∗h[n]. Therefore, we obtain for θ �= 0the same value for the I S I as if θ = 0.
For a time-varying channel, the channel power∑k=R−1
k=0 |h (k)|2 may also be of time-varying nature. Thus,causing the parameter B to be time dependent. This meansthat also A1, B1 and the maximum value for the step-sizeparameter for which the equalizer still converges are time
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dependent. Obviously, if the changes in the channel poweroccur too often, we might be unable to measure the channelpower accurate and quickly enough. Thus, the calculation ofthe maximum value for the step-size parameter for which theequalizer still converges might be so inaccurate that therewas no point to derive it in the first place.
In blind equalization, typically convergence is understoodas the convergence of the equalizer to a point with ISI lowenough to have a good performance (usually, a dual modeis used, and after convergence a decision or radius directedalgorithm, starting from this low ISI level is able to com-pletely open the eye of the constellation). A situation as isshown in Figs. 7, 9 and 10 for instance is not useful at all.The final ISI is high, even worse than the initial ISI intro-duced by the channel alone. The equalizer is introducingadditional ISI. In this condition, a digital system does notwork at all. Although we can talk about convergence insuch cases, because the ISI finally achieves a stable level,from a blind equalization point of view, this situation isnot convergence to a desired solution. Thus, we may saythat by having the range of B for which a blind equal-izer does not converge, we have thus at the same timealso the upper bound for the step-size allowable for prac-tical use. But this is not a tight bound. It should be alsomentioned that the provided result is independent of thechannel (up to a gain factor). Therefore, working with nor-malized channels (having unity gain), the same bound isobtained for all channels, while typically the optimal step-size (for practical blind equalization) is specific for eachchannel.
It should be pointed out that if we deal with a case notgiven in the paper, then we are not able to say anything con-cerning the upper bound for the step-size parameter for whichthe equalizer still converges.
5 Conclusion
In this paper, we derived the range of the step-size param-eter multiplied by the equalizer’s tap length, channel powerand source signal variance (defined throughout the paperas parameter B, B = μNσ 2
x∑k=R−1
k=0 |h (k)|2) for which ablind equalizer will not converge. The derived range holdsfor a type of blind equalizer where the error that is fed into theadaptive mechanism which updates the equalizer’s taps canbe expressed as a polynomial function of the equalized out-put of order three and where the gain between the equalizedoutput and input sequence is equal to one. The obtained rangeis closely related to the nature of the chosen blind equalizerand is valid for the real and two independent quadrature car-rier case. According to simulation results carried out for twodifferent blind equalization methods for three different chan-nel characteristics and source inputs (16QAM and 64QAM),
high correlation was found between the calculated and sim-ulated maximum value for B for which the equalizer stillconverges. It should be pointed out that although our der-ivations were carried out for the noiseless case, satisfyingresults were also obtained for the noisy situation. For theQPSK case, we could not define the range for B for whichthe equalizer does not converge since the nature of the algo-rithm combined with the input signal statistics did not fallinto one of our derived conditions for which the equalizerdoes not converge. This case (QPSK case) will be handledin a future work.
Appendix
In this section, we proof the theorem from Section 3.The derived expression for E[�p2
r ] (14) is based on sev-eral assumptions valid for the convergence state. It is not validif the equalizer is far away from the convergence state. Thus,finding the range for the parameter B for which E[�p2
r ] �= 0means actually that a moment ago with a lower value for Bthe equalizer succeeded to reach the convergence state and
B1
BSol 2BSol
B1
BSol 2BSol
B1
BSol 2BSol
a
b
c
Fig. 25 Three options for case 2
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SIViP (2012) 6:325–340 339
1BSol 2
BSol
1BSol 2
BSol
1BSol 2
BSol
a
b
c
Fig. 26 Three options for case 3
now with the new value for B which is higher than the usedone from before, the equalizer does not converge anymore.
In the following, we go over all the possible cases thatmight bring � to be a negative value. There are four possiblecases that might lead to � < 0:
Case 1(B2
11 − 4C1 A11)
< 0 and �B < 0.This condition can be never fulfilled since �B (22) will
be a positive value if(B2
11 − 4C1 A11)
< 0.
Case 2 �B > 0 and(B2
11 − 4C1 A11)
< 0This case can be described in Fig. 25. Note that we are
dealing with the case of B > 0. Therefore, according toFig. 25, we may have � < 0 for positive values of B beingin the range of: ((B < Sol B
1 and B > 0) or (B > Sol B2 and
B > 0)). Please note that the condition of (B < Sol B1 and
B > 0) will never be fulfilled for realistic cases. In realis-tic cases, Sol B
1 < 0 where we have convergence before wechoose a value for B which is too high. Thus, for realisticcases we will see case c of Fig. 25. According to Fig. 25 casec, the equalizer converges for values of B which are less thanSol B
2 .
1 2B BSol Sol
1 2B BSol Sol
1 2B BSol Sol
Fig. 27 Three options for case 4
Case 3 �B > 0 and(B2
11 − 4C1 A11)
> 0This case can be described in Fig. 26. According to Fig. 26,
we may have � < 0 for positive values of B being inthe range of: ((Sol B
1 < B < Sol B2 ) and B > 0). Please
note that for realistic cases (Sol B1 > 0 (case a of Fig. 25)
where we have convergence before we choose a value for Bwhich is too high. According to Fig. 25 case a, the equal-izer converges again for values of B which are in the rangeof B > Sol B
2 . But for Sol B1 < B < Sol B
2 , the derivedexpression for E[�p2
r ] (14) is already not valid since theISI grows to infinity. Thus, we may conclude that for real-istic cases, non-convergence is obtained for (Sol B
1 > 0 andB > 0).
Case 4 �B = 0 and(B2
11 − 4C1 A11)
< 0This case can be described in Fig. 27. At this case
� < 0 for every positive B which is not equal toSol B
1 (Sol B1 = Sol B
2 ). This is obviously not a realisticcase. In a realistic case the equalizer converges for val-ues of B up to a point where the value for B is toohigh.
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Acknowledgments I would like to thank the anonymous reviewersfor their fruitful and detail comments that helped improve this paperconsiderably.
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