Peng et al. Journal of the European Optical Society-Rapid Publications
(2017) 13:21
Page 3 of 12
In classical PAM systems, the transmitted symbols are
expressed as:
¯
S
PAM
=
¯
S
T
,
¯
S
D
T
=
[
S
T
(0), ···S
T
(
N
T
− 1
)
, S
D
(0), ···
S
D
(N
D
− 1)
]
T
,
(1)
where
¯
S
T
is the length-N
T
vector of training symbols for
the equalizer;
¯
S
D
is the length-N
D
vector of PAM data
symbols.
When the PN sequence is inserted to the PAM symbols
to assist the channel estimation, the transmitted PN-PAM
symbols are written as:
¯
S
PN−PAM
=
¯ρ
PN
,
¯
S
D
T
=
[
ρ
PN
(0), ···ρ
PN
(
N
PN
− 1
)
,
S
D
(0), ···S
D
(
N
D
− 1
)
]
T
,
(2)
where ¯ρ
PN
is the vector of PN sequence for channel
estimation. The length of ¯ρ
PN
is N
PN
.
At the receiver side, the received PN sequence is used
to perform channel estimation. The PN sequence based
channel estimation for optical communications has been
initially introduced in [19]. The m-sequences are selected
as the PN sequences for channel estimations due to their
ease of generation and their associated low complexity.
The most significant benefit of using m-sequence for
channel estimation is its special circular autocorrelation
property. The circular autocorrelation of the m-sequence
is known as:
CR
j
=
1
N
PN
N
PN
−1
i=0
m
i
m
∗
[i+j]
N
PN
=
1 j = 0
−
1
N
PN
else
(3)
where m is the m-sequence, (·)
∗
is the complex conjugate,
[
·
]
N
PN
denotes modulo-N
PN
operation. With the help of
circular autocorrelation property shown in (3), the chan-
nel estimation can be simply obtained by performing time
domain correlation of known and received PN sequences:
¯
h =
1
N
PN
N
PN
−1
i=0
N
H
−1
l=0
h
l
ρ
i−l
+ w
i
· m
∗
[i+j]
N
PN
(4)
where w is the noise, N
H
is the channel length. In POF
channel model, the massive multi-path delay could be
modeled as discrete filter taps. Therefore, the maximal
channel multi-path delay in real POF channel could be
denoted by channel length N
H
with number of filter taps.
Finally, the accurate estimate of the channel impulse
response (CIR)
¯
h =[
h
0
,
h
1
, ···
h
N
H
−1
]
T
can be easily
obtained at a very low complexity cost [20]. According
to the analysis carried out in [19], the overall complexity
of the PN sequence-based channel estimation is
O(N
PN
·
log N
PN
), which is determined by the PN sequence length.
Minimum-phase pre-filtering
In communication systems, trellis-based equalizer can
effectively eliminate the inter-symbol interference (ISI)
after transmission over the channel. The maximum-
likelihood sequence estimation (MLSE) is recognized
as the optimal equalization algorithm in the sense of
sequence detection. As the decision is based on a
sequence of symbols, it can effectively avoid the error
propagation problem of DFE. However, it is worth noting
that for PAM with high orders modulations, the com-
putational complexity of MLSE equalizer dramatically
increases. The full MLSE equalizer becomes computa-
tionally prohibitive when the modulation order is high
and/or when the channel length is long. To avoid the
prohibitive complexity, a sub-optimal trellis-search based
equalizer, namely RSSE, is commonly used for its simplic-
ity in the hardware implementation.
Studies in [21, 22] show that, in order to obtain the
sub-optimal performance after trellis-based equalization,
discrete time minimum-phase overall impulse response
needs to be carried out previously. We employ an FFE
pre-filter to achieve the minimum-phase overall impulse
response. As an accurate CIR is obtained directly from
the PN sequence-based channel estimation, it is feasi-
ble to calculate the filter coefficients from the estimated
CIR. The coefficients of the pre-filter can be calculated in
closed-form from the estimated CIR
¯
h.
In [22], the coefficients of the minimum-phase pre-filter
are calculated by the linear prediction from the estimated
CIR. The linear prediction is realized by the well-known
Levinson-Durbin algorithms. Concretely, the pre-filter is
determined as follows:
F(z) = z
−N
H
H
∗
1/z
∗
(1 − P(z)),(5)
where H
∗
(1/z
∗
) is the time-reversed conjugated CIR,
(1 − P(z)) is the calculated linear prediction filter, z
−N
H
introduces a delay of N
H
samples.
The analysis of this minimum-phase pre-filter calcu-
lation shows that the overall computational complexity
of linear prediction method is significantly lower than
that of the minimum mean-squared error (MMSE)-DFE
method [22].
RSSE based equalization
In contrast to the MLSE equalizer where all possible
combinations of symbol sequences are compared with
received signal sequence, RSSE dramatically reduces the
number of candidates to be compared by applying con-
stellation partitioning and decision-feedback with early
decisions [23]. With a proper choice of the number of sur-
vivor states, the RSSE based equalizer can approach the
optimal performance offered by the MLSE equalizer.
More concretely, the entire symbol alphabet is divided
into subsets, and the search trellis is built based on these
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