In optical fiber communication systems, PAM transmission with adaptive DFE has been widely used. However, as the optical fiber channels are quite stable, it is possible to obtain the optimal DFE coefficients with a reduced transmission overhead. Furthermore, as DFE has the drawback of the error propagation, it is necessary to introduce a novel decision mechanism to replace the direct decision in DFE. In this section we propose a novel PAM transmission scheme for optical fiber communications. To achieve satisfactory performance, the reception of this novel scheme includes three main components: the PN sequence based channel estimator, a minimum-phase receiver filter, and a reduced-state sequence estimation(RSSE) based equalizer. More details of the proposed scheme is presented in following part of the section.

### PN sequence based channel estimation

The SCM based transmissions enjoy the advantage of low PAPR over MCM ones and are therefore attractive for optical fiber transmissions. In the meantime they normally require high channel estimation accuracy in order to prevent the error propagation phenomenon in the reception. The PN sequence-based channel estimation was known for its merits of low complexity and high accuracy in wireless communication scenarios [18]. It was recently proved to be very efficient for in DMT transmission over optical fiber [19]. This motivates us to investigate the application of PN sequence based channel estimation for SCM-based optical fiber transmissions.

In classical PAM systems, the transmitted symbols are expressed as:

$$ \begin{aligned} \bar{S}_{\text{PAM}}&=\left[\bar{S}_{\mathrm{T}},\bar{S}_{\mathrm{D}}\right]^{T}=\left[S_{\mathrm{T}}(0), \cdots S_{\mathrm{T}}\left(N_{\mathrm{T}}-1\right),S_{\mathrm{D}}(0), \cdots\right.\\&\qquad\qquad\qquad\quad\left.S_{\mathrm{D}}(N_{\mathrm{D}}-1)\right]^{T}, \end{aligned} $$

(1)

where \(\bar {S}_{\mathrm {T}}\) is the length- *N*
_{T} vector of training symbols for the equalizer; \(\bar {S}_{\mathrm {D}}\) 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:

$$ \begin{aligned} \bar{S}_{\mathrm{PN-PAM}}=\left[\bar{\rho}_{\text{PN}},\bar{S}_{\mathrm{D}}\right]^{T}&=\left[\rho_{\text{PN}}(0),\cdots \rho_{\text{PN}}\left(N_{\text{PN}}-1\right),\right.\\ &\left.\quad S_{\mathrm{D}}(0),\cdots S_{\mathrm{D}}\left(N_{\mathrm{D}}-1\right)\right]^{T}, \end{aligned} $$

(2)

where \(\bar {\rho }_{\text {PN}}\) is the vector of PN sequence for channel estimation. The length of \(\bar {\rho }_{\text {PN}}\) 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:

$$ \text{CR}_{j}=\frac{1}{N_{\text{PN}}}\sum_{i=0}^{N_{\text{PN}}-1}m_{i}m^{*}_{[i+j]_{N_{\text{PN}}}}= \left\{ \begin{array}{cl} 1 &j=0\\ -\frac{1}{N_{\text{PN}}} &else\\ \end{array}\right. $$

(3)

where *m* is the m-sequence, (·)^{∗} is the complex conjugate, \(\left [\cdot \right ]_{N_{\text {PN}}}\) denotes modulo- *N*
_{PN} operation. With the help of circular autocorrelation property shown in (3), the channel estimation can be simply obtained by performing time domain correlation of known and received PN sequences:

$$ \widetilde{\bar{h}}=\frac{1}{N_{\text{PN}}}\sum_{i=0}^{N_{\text{PN}}-1}\left(\sum_{l=0}^{N_{\mathrm{H}}-1}h_{l}\rho_{i-l}+w_{i}\right)\cdot m^{*}_{[i+j]_{N_{\text{PN}}}} $$

(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) \(\widetilde {\bar {h}}=[\widetilde {h}_{0},\widetilde {h}_{1},\cdots \widetilde {h}_{N_{\mathrm {H}}-1}]^{T}\) 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 \(\mathcal {O}(N_{\text {PN}}\cdot \log N_{\text {PN}})\), 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 computational complexity of MLSE equalizer dramatically increases. The full MLSE equalizer becomes computationally 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 simplicity 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 feasible 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 \(\widetilde {\bar {h}}\).

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:

$$ \widetilde{F}(z)=z^{-N_{\mathrm{H}}}H^{\ast}\left(1/z^{\ast}\right)(1-P(z)), $$

(5)

where *H*
^{∗}(1/*z*
^{∗}) is the time-reversed conjugated CIR, (1−*P*(*z*)) is the calculated linear prediction filter, \(\phantom {\dot {i}\!}z^{-N_{\mathrm {H}}}\) introduces a delay of *N*
_{H} samples.

The analysis of this minimum-phase pre-filter calculation 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 constellation partitioning and decision-feedback with early decisions [23]. With a proper choice of the number of survivor 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 subsets. The subsets need to be chosen such that the Euclidean distance of symbols within each subset is maximized. Once the survival path is determined, the hard decision within the subset is made directly so that only one survival candidate is reserved in each set, while others are simply discarded for the following search. The selection among subsets is not taken in the current step. Therefore, the number of overall trellis states involved in the search is reduced to \(Z=\prod ^{N_{\mathrm {C}}}_{k=1}J_{k}\), where *J*
_{
k
} is the number of subsets preserved for the symbol *k* steps before the currently detecting symbol, *N*
_{C} is the constraint length which is chosen according to the number of significant channel paths, and can be less than the overall channel length *N*
_{H}. It is worth noting that through the choice of *J*
_{
k
}, the performance and complexity of RSSE equalizer can achieve arbitrary trade-off between the optimal MLSE equalizer and the simple DFE equalizer [22]. For example, when *J*
_{
k
}=*M*, 1≤*k*≤*N*
_{C}, the RSSE equalizer becomes the MLSE equalizer for PAM-*M* modulation. Similarly, when *J*
_{
k
}=1, 1≤*k*≤*N*
_{C}, since there is only one subset preserved for decision, the RSSE equalizer is turned into a DFE equalizer.