# DFT-Spread Spectrally Overlapped Hybrid OFDM-Digital Filter Multiple Access IMDD PONs
## 4 Upstream DFT-Spread Hybrid OFDM-DFMA PON Performance
### Impacts of Digital Filter Parameters on Maximum Aggregated Upstream Transmission Rates  
For the proposed PON simultaneously supporting two ONUs, the aggregated upstream signal transmission capacities versus the excess bandwidth parameter, \(\alpha\), are shown in Figure 6. In obtaining Figure 6, the digital filter length listed in Table 1 was adopted. In addition, adaptive bit-loading was applied to all subcarriers involved in each ONU for all of the cases of including and excluding DFT-spread. To implement adaptive bit-loading, the highest possible signal modulation formats within the range from DBPSK to 256-QAM were adaptively selected according to the channels' spectral characteristics to ensure that the BER across all subcarriers for each sub-band could reach the FEC limit of \(1\times 10^{-3}\). For each ONU sub-band, the achievable signal bit rate was \(R_{b}=f_{s}\sum_{k=1}^{N_{b}}n_{kb}/2(N_{s}+1)\big{(}1+C_{p}\big{)}M\), where \(n_{kb}\) is the number of binary bits conveyed by the \(k\)th subcarrier within one OFDM symbol period, \(N_{s}\) denotes the number of data-bearing subcarriers, and \(C_{p}\) indicates the overhead parameter associated with the cyclic prefix and training sequences. The excess bandwidth, \(\alpha\), was set to vary in the range \(0\leq\alpha\leq 1\). The ROPs were fixed at \(-6\) dBm for both cases. Figure 5: BER versus received optical power over a 25 km SSMF IMDD PON transmission system when a 7-bit resolution is considered. It can be seen in Figure 6 that, for both considered cases, the aggregate upstream signal transmission rate peaks at an \(\alpha\) factor value of 0.2, and then reduces steadily as \(\alpha\) increases above 0.2. Most importantly, for the case including DFT-spread, the maximum aggregate upstream signal transmission rate increases by up to 10% for \(\alpha\) = 0.2, compared to the case without DFT-spread. This performance enhancement is due to the overall improvement in SNRs across the subcarriers, thus allowing higher modulation formats to be used. To understand the physical mechanisms causing an optimum \(\alpha\) value of 0.2, the impact of \(\alpha\) on the digital filter responses should be observed. For low \(\alpha\) values! 0.2, the finite filter-length-induced filter magnitude response ripples can impact on performance. For \(\alpha\) = 0 and the up-sampling factor \(M\) = 2, the I-phase digital filter has a perfectly flat response [26], which is equivalent to a case where the digital filter length (channel-1) is reduced to one, while the Q-phase filter (channel-2) has significant frequency response ripples, as shown in Figure 7a,b, respectively [26]. The length of the filter mainly impacts the sharpness of the filter edge in the case of the Q-phase filter. On the other hand, as shown in Figure 7c,d, as \(\alpha\) starts to increase, there is a boost in the magnitude response of the I-phase filter and an attenuation in the magnitude response of the Q-phase filter. In addition the increasing distortion of the filter responses causes strong unwanted cross-channel-induced interference due to loss of orthogonality. It should be noted that the filter response distortions occur due to the aliasing effect as the employed subwavelength occupies the whole Nyquist band, and a narrower digital filter bandwidth can be observed when \(\alpha\) > 0 [26]. The increasing \(\alpha\) value also reduces the unwanted frequency response ripples in the Q-phase filter. These effects combine to give an overall increase in subcarrier SNR as \(\alpha\) increases from 0 to 0.2, beyond which there is an overall reduction in subcarrier SNR. To observe the effect of \(\alpha\) on subcarrier SNRs, with the simultaneous presence of both channels, the SNR performances for all subcarriers for the 25 km SSMF transmission are plotted in Figure 7e, utilizing the optimum \(\alpha\) parameter value of 0.2 and zero excess bandwidth (\(\alpha\) = 0) for performance comparisons. In obtaining this figure, the ROP was fixed at \(-\)6 dBm.