relation of ISI[n] =
m
p
[n]
σ
2
x
r
to:
ISI[n] << min
1
σ
2
x
r
r
B
1
BD
,
1
σ
2
x
r
B
1
A
1
(6)
Now, we may say that if the above condition holds,
the obtained expression in (4) for the ISI as a func
tion of time is approximately valid. Please note that
for the very low ISI case, the convergence time of an
equalizer is much faster compared to the case where
the initial ISI is considered high. In addition, accord
ing to (Lee and Messerschmitt, 1997), the best rate of
convergenceis dependent on the number of ﬁlter coef
ﬁcients. The more coefﬁcients (in the equalizer), the
longer it takes for the coefﬁcients to converge. The
more coefﬁcients there are, the more ”noise” is intro
duced into the adaptation of each coefﬁcient by the
simultaneous adaptation of the other coefﬁcients (Lee
and Messerschmitt, 1997). Now, let us go back to the
function of γ in (5). It can be seen that γ functions
as an compensation factor between the very low ISI
condition (6) and any other given initial ISI. There
fore, when the initial ISI is much higher than given in
(6), γ will slow down the convergence rate.
Next we turn to calculate the total iteration number
that takes to enter the convergence state. The ex
ponent from (5) can be written as follows: e
γ
BtB
1
∆t
=
e
t
∆t
γBB
1
= e
−
t
τ
where τ = 
∆t
γBB
1
. Next we assume that
for t = 8τ the equalizer has approximately reached
its steady state position. Note that for a simple RC
circuit (one capacitor and one resistor), it is often as
sumed that the capacitor is approximatelyfull charged
or discharged after 5τ = 5RC seconds. Since we are
looking for a more accurate solution we choose in
stead of 5τ, 8τ which was found by simulation tri
als leading to satisfying results. Thus we may write:
t = 8τ = 8
∆t
γBB
1
 Now let the sampling time be ∆t.
Thus we may write that t = n∆t = 8
∆t
γBB
1
 from which
we obtain (for ∆t 6= 0) the total number of iteration
required for convergence:
n =
f
ISI(0)L
min
1
σ
2
x
r
r
B
1
BD
,
1
σ
2
x
r
B
1
A
1
8
B

1
B
1
 (7)
4 SIMULATION
In the following we use Godard’s equalizer (Go
dard, 1980) and the 16QAM constellation (a mod
ulation using ± {1,3} levels for inphase and
quadrature components) as the source. The equal
izer taps for Godard’s equalizer (Godard, 1980)
were updated according to: c
l
[n+ 1] = c
l
[n] −
µ
G
z[n]
2
−
E
[
x[n]
4
]
E
[
x[n]
2
]
z[n] y
∗
[n− l] where µ
G
is the
stepsize and l is the equalizer’s tap length. The val
ues for a
1
, a
12
and a
3
corresponding to Godards’s
(Godard, 1980) algorithm were deﬁned as a
G
1
, a
G
12
and
a
G
3
respectively and were given by: a
G
1
= −
E
[
x[n]
4
]
E
[
x[n]
2
]
,
a
G
12
= 1 and a
G
3
= 1. Two different channels were con
sidered.
Channel1 (initial ISI = 0.44): The channel param
eters were determined according to (Shalvi and Wein
stein, 1990): h
n
= 0 for n < 0; −0.4 for n =
0 0.84· 0.4
n−1
for n > 0.
Channel2 (initial ISI = 0.5): The channel parame
ters were determined according to (Fiori, 2001):
h
n
= (−0.0144, 0.0006, 0.0427, 0.0090,
−0.4842, −0.0376, 0.8163, 0.0247, 0.2976, 0.0122,
0.0764, 0.0111, 0.0162, 0.0063)
For Channel1 and Channel2 an equalizer with 13 and
21 taps was used respectively. In the simulation,
the equalizer was initialized by setting the center tap
equal to one and all others to zero. Fig. 2 and Fig. 3
show the simulated performance of Godard’s equal
ization method for the 16QAM input case, namely
the ISI as a function of iteration number for vari
ous stepsize parameters, channel characteristics and
equalizer’s tap length, compared with the calculated
ISI as a function of iteration number (5) proposed in
this paper. According to Fig. 2 and Fig. 3, the ap
proximated closedform expression for the ISI as a
function of time (or iteration number) (5), ﬁts very
well the simulated results. Next, the expression for
the total number of iteration required for convergence
(7) was calculated for each simulation:
Case I – Described in Figure 2. The calculated
number of iteration required for convergence accord
ing to (7) is 3135.
Case II – Described in Figure 3. The calculated
number of iteration required for convergence accord
ing to (7) is 1788.
According to Fig. 2 and Fig. 3, there is a high
correlation between the simulated and calculated (7)
results for the number of iteration required for con
vergence.
Next we turn to the noisy case situation. Fig.4
shows the simulated performance of Godard’s equal
ization method for the 16QAM input case, namely the
ISI as a function of iteration number for various SNR
values, compared with the calculated ISI as a func
tion of iteration number (5) proposed in this paper.
AN APPROXIMATED EXPRESSION FOR THE CONVERGENCE TIME OF ADAPTIVE BLIND EQUALIZERS
413