Dispersion tolerance enhancement using an improved offset-QAM OFDM scheme

Jian Zhao; Paul D. Townsend

doi:10.1364/OE.23.017638

1. Introduction

Offset quadrature amplitude modulation (offset-QAM) orthogonal frequency division multiplexing (OFDM) has attracted much attention in optical transmission systems [1–7]. The main principle difference of offset-QAM OFDM from conventional OFDM (C-OFDM) is that the quadrature tributary is delayed by half a symbol period with respect to the in-phase tributary, and for each tributary, instead of inter-carrier interference (ICI) free operation, crosstalk can exist but is controlled to have π/2 phase difference from the desirable signal so that the signal can still be recovered. In practice, this principle can be implemented in either the optical or electrical domains. In the electronic-domain implementation, it has been shown that similar to C-OFDM, the discrete Fourier transform (DFT) can be applied for efficient multiplexing and demultiplexing [6]. Compared to C-OFDM [8,9], DFT-based offset-QAM OFDM may greatly relax the length of cyclic prefix (CP) for dispersion compensation and achieve ~20% increase in the net data rate for the same transmission reach. Furthermore, this technology significantly reduces the memory length of pulse-shaping filter and exhibits lower complexity than Nyquist FDM [10].

Although the dispersion tolerance of the conventional offset-QAM OFDM scheme was numerically and experimentally investigated previously, the fundamental mechanism limiting this dispersion tolerance was not elucidated. In this paper, we analytically study the influence of dispersion on the system performance, and show that in conventional offset-QAM OFDM, the phase difference between the desirable signal and the crosstalk from adjacent subcarriers is a function of dispersion so that the orthogonality cannot be maintained when the dispersion increases to a certain value. This effect limits the dispersion tolerance in the scale of T²∙N/π², where T and N are the time interval between samples and the number of subcarriers, respectively. We propose a novel scheme to overcome this limitation and significantly improve the dispersion tolerance by N times. It is shown that the proposed scheme exhibits advantages of greatly improved spectral efficiency, larger dispersion tolerance, and/or reduced complexity compared to CP-based C-OFDM [8,9] and reduced-guard-interval (RGI) OFDM [11,12]. These advantages make the proposed scheme a promising solution for long-haul and submarine optical transmission systems with distances up to ~10,000 km.

2. Principle

2.1. Dispersion limitation of the conventional offset-QAM OFDM scheme

In this subsection, we will show how dispersion limits the performance of the conventional offset-QAM OFDM scheme when CP is not used. Different from the previous works [6,7], the dispersion considered in this paper is large and one-tap equalization is no longer sufficient for channel compensation. Figure 1 shows the multiplexing and demultiplexing diagram of the conventional scheme. Offset-4QAM format and 128 subcarriers are adopted in the illustration. Two bi-polar binary phase shift keying (BPSK) data streams are encoded and mapped into subcarriers. For the in-phase tributary, the phases of even subcarriers are set to be 0 (or π) while those of odd subcarriers are set to be π/2 (or 3π/2). Conversely, for the quadrature tributary, the phases of odd subcarriers are set to be 0 (or π) while those of even subcarriers are set to be π/2 (or 3π/2). The quadrature tributary is then delayed by half a symbol period, N/2, with respect to the in-phase tributary, where N is the number of samples per OFDM symbol (or the number of subcarriers). An inverse fast Fourier transform (IFFT) is applied to generate time-domain samples from the in-phase tributary at times i⋅N, and from the quadrature tributary at times (i + 1/2)⋅N, where i is an integer. The generated outputs pass through finite impulse response (FIR) filters for pulse shaping before parallel-to-serial (P/S) conversion. Assuming that a_i,n and s(i⋅N + k) are the n^th subcarrier data in the frequency domain and the k^th sample in the time domain in the i^th OFDM symbol, s(i⋅N + k) can be derived as:

\begin{array}{l} s (i \cdot N + k) = s^{r e a l} (i \cdot N + k) + j \cdot s^{i m a g} (i \cdot N + k) \\ = \sum_{p = - \infty}^{+ \infty} \sum_{n = - N / 2 + 1}^{N / 2} a_{p, n}^{r e a l} \exp (j π n / 2) \cdot \exp (2 π j (p \cdot N + k) \cdot n / N) \cdot h_{f i l t e r} (i \cdot N + k - p \cdot N) \\ + \sum_{p = - \infty}^{+ \infty} \sum_{n = - N / 2 + 1}^{N / 2} a_{p, n}^{i m a g} \exp (j π (n + 1) / 2) \cdot \exp (2 π j (p \cdot N + k) \cdot n / N) \cdot h_{f i l t e r} (i \cdot N + k - N / 2 - p \cdot N) \\ k = - N / 2 + 1, - N / 2 + 2... N / 2 - 1, N / 2 \end{array}

In Eq. (1), we employ the periodic property and set the ranges of n and k to be [-N/2 + 1 N/2], rather than [0 N-1], to facilitate mathematical derivations. h_filter(i⋅N + k), -∞ < i < + ∞, represents the impulse response of the k^th FIR filter in Fig. 1. Note that for each k, h_filter(i⋅N + k) requires only one sample per OFDM symbol, with the memory length as short as two [6].

Fig. 1 Principle of DFT-based implementation for offset-QAM OFDM.

Download Full Size | PDF

At the receiver, after coherent detection and analogue-to-digital conversion, the received signal is serial-to-parallel (S/P) converted in the digital domain with the access time of N/2, that is, the in-phase tributary accesses the sampled points from times i⋅N while the quadrature tributary accesses the sampled points from times (i + 1/2)⋅N. The outputs pass through FIR filters, and an FFT is applied to transform the signals to the frequency domain. Without the loss of generality, we only derive the output of the FFT for the in-phase tributary of the m^th subcarrier in the i^th OFDM symbol, $b_{i, m}^{r e a l}$ :

b_{i, m}^{r e a l} = \sum_{k = - N / 2 + 1}^{N / 2} \sum_{q = - \infty}^{+ \infty} \exp (- 2 π j k \cdot m / N) \cdot r (q \cdot N + k) \cdot h_{r e c e i v e r_f i l t e r} ((i - q) \cdot N - k)

where r(q⋅N + k) is the received time-domain signal:

r (q \cdot N + k) = \sum_{d = - \infty}^{+ \infty} \sum_{e = - N / 2 + 1}^{+ N / 2} s (d \cdot N + e) h_{c} (q \cdot N + k - d \cdot N - e)

h_{receiver_filter}((i-q)⋅N-k) is the impulse response of the k^th receiver matched filter and is equal to h_filter(q⋅N + k-i∙N). h_c(q∙N + k-d∙N-e) represents the impulse response of the channel, including the contributions from the back-to-back response, dispersion, phase noise etc. In the previous works [6,7], the channel response on a subcarrier is assumed to be a constant and so can be compensated using one-tap equalization. This assumption corresponds to the case that the memory length of h_c(q∙N + k-d∙N-e) is a small percentage of the period of one OFDM symbol, and is no longer valid for large dispersion values. Figure 2(a) shows the real and imaginary parts of the dispersion-induced frequency response, as well as the spectral profile of subcarriers. The accumulated dispersion, the sampling rate, and the number of subcarriers in the figure are 42,500 ps/nm, 80 GS/s, and 128, respectively. It can be clearly seen that the frequency response oscillates rapidly. The dispersion imposed on a subcarrier is no longer a constant and so one-tap equalization is not sufficient to compensate the frequency response.

Fig. 2 (a) Real (dashed) and imaginary (dotted) parts of the dispersion-induced frequency response, as well as the spectral profile of subcarriers (solid) in offset-QAM OFDM. (b) Relationship between the signal and the ICI in the constellation diagram. In (b), the in-phase tributary of a subcarrier in offset-4QAM OFDM is illustrated.

Download Full Size | PDF

In order to find out the influence of this effect, we write the channel frequency response at a frequency with the distance of d/(TN), d < 0.5, to the center of the m^th subcarrier, as:

\begin{array}{l} H_{c} ((m + d) / (T N)) = H_{b} ((m + d) / (T N)) \cdot \exp (j \cdot β_{2} L / 2 \cdot (^{2 π / (T N)) 2} \cdot (^{m + d) 2}) \\ \approx H_{b} (m / (T N)) \cdot \exp (j \cdot (α \cdot m^{2} + α \cdot 2 m d + α \cdot d^{2})) \end{array}

where H_c(∙) is the Fourier transform of h_c(∙). β₂L is the accumulated dispersion and T is the time interval between samples. α = β₂∙L∙(2π/(TN))²/2. In Eq. (4), we separate H_c(∙) into two effects. H_b(∙) represents the contribution of the back-to-back response. In practice, this response does not change rapidly over frequency so that H_b((m + d)/(TN)) ≈H_b(m/(TN)). The second effect arises from dispersion and consists of three terms in exp(∙): 1) the first term represents the well-known multiplication constant imposed on each subcarrier in conventional OFDM systems and can be simply compensated using one-tap equalization; 2) the second term, as shown more clearly later, introduces different time delays for different subcarriers; 3) the last term represents the dispersion-induced pulse broadening effect on each subcarrier. Generally, if the last term is considered, there is no closed-form expression for the received signal r(q⋅N + k). However, we will show in Appendix I that the dispersion tolerance, β₂∙L, using the conventional decoding scheme scales with T²∙N/π², while that limited by the third term is in the scale of T²∙N²/π² which is much larger than T²∙N/π². Therefore, we can neglect this term at this stage and keep in mind this scale. In the next subsection, we will propose a novel decoding scheme which can remove the limitation scale of T²∙N/π² in the conventional decoding algorithm, so that this term becomes dominant and the scale of the dispersion tolerance is therefore enhanced to T²∙N²/π². By neglecting the third term in exp(∙) on the right-hand side of Eq. (4), we can derive the received signal as:

\begin{array}{l} b_{i, m}^{r e a l} = H_{b} (m / (T N)) \cdot \exp (j π m / 2 + j \cdot α \cdot m^{2}) \cdot (A \cdot a_{i, m}^{r e a l} + I_{i, m}^{r e a l} + j \cdot \exp (j \cdot α \cdot m) \cdot c_{m + 1}^{r e a l} \\ + j \cdot \exp (- j \cdot α \cdot m) \cdot c_{m - 1}^{r e a l} + j \cdot c_{m}^{i m a g} + j \cdot \exp (j \cdot α \cdot m) \cdot c_{m + 1}^{i m a g} + j \cdot \exp (- j \cdot α \cdot m) \cdot c_{m - 1}^{i m a g}) \end{array}

The detailed derivation is provided in Appendix I. In Eq. (5), A,

I_{i, m}^{r e a l}

,

c_{m + 1}^{r e a l}

,

c_{m - 1}^{r e a l}

,

c_{m}^{i m a g}

,

c_{m + 1}^{i m a g}

,

c_{m - 1}^{i m a g}

are all real values, and represent the signal attenuation coefficient, inter-symbol interference (ISI), the crosstalk from the in-phase tributary of the (m + 1)^th and (m-1)^th subcarriers and the quadrature tributary of the m^th, (m + 1)^th, and (m-1)^th subcarriers, respectively. In practice, the term H_b(m/(TN))∙exp(jπm/2 + j∙α∙m²) can be compensated via channel equalization [7]. The key information provided by Eq. (5) is that in additional to ISI, all crosstalk terms have a coefficient of either exp(j∙α∙m) or exp(-j∙α∙m). Figure 2(b) depicts the relationship between the signal and the ICI in the constellation diagram. When |α∙m| << 1, exp(j∙α∙m) ≈1 and Eq. (5) simplifies to that derived in our previous works. The desirable signal is real while all crosstalk terms are imaginary, so that the signal can be correctly decoded by extracting the real part after channel equalization. However, for a large value of dispersion, i.e. a large α∙m, the crosstalk is no longer orthogonal to the signal. For example, all crosstalk terms would become real and the signal and the crosstalk are in-phase when α∙m = π/2. The dispersion tolerance of the conventional scheme is therefore limited by the condition of |α∙m| < 1. From the definition of α ( = β₂∙L∙(2π/(T∙N))²/2) and the range of m (-N/2 + 1 < m < N/2), we can conclude that it is the high-frequency subcarriers that limit the system performance, and the dispersion tolerance, β₂∙L, scales with T²∙N/π². As an example, for 80-GS/s sampling rate and 128 subcarriers, the transmission reach over standard single-mode fiber (SSMF) is around 90 km, which matches the simulation results in section 3.

The above analysis is based on single polarization but it can be readily extended to polarization-division multiplexed (PDM) systems. Polarization mode dispersion (PMD) should be considered in these systems. However, similar to the back-to-back frequency response H_b(∙), the memory length of the PMD is much smaller than the time period of one OFDM symbol. Consequently, this effect only introduces a constant to each subcarrier in Eq. (5) and can be compensated using one-tap equalization.

2.2. Proposed offset-QAM OFDM scheme

Subsection 2.1 shows how dispersion limits the performance of the conventional offset-QAM OFDM scheme when CP is not applied. This limitation occurs when the memory length of the channel impulse response is comparable to the time period of one OFDM symbol so that the frequency response on a subcarrier is not a constant. From another point of view, this limitation is due to the second term in exp(∙) in Eq. (4), which is not included in the mathematical model of our previous works. Physically, this term represents that different subcarriers have different delays under dispersion and one DFT window is not sufficient to decode all subcarriers. In this subsection, we will propose a new decoding scheme which greatly enhances the scale of the dispersion tolerance from T²∙N/π² to T²∙N²/π². Figure 3 shows the block diagram of the scheme. After coherent detection and analogue-to-digital conversion with a sampling rate of 1/T, the received signal is split into x paths, with the k^th path experiencing a time delay of kNT/x or a sample delay of kN/x in the digital domain. These delayed versions are then processed using FIR filters and the FFT individually, with each processing unit similar to that of the conventional offset-QAM OFDM scheme. Figure 4 gives an example of 2 paths and 7 subcarriers to illustrate the principle of the proposed scheme. Without loss of generality, we still focus on the decoding of the in-phase tributary of the signal. When there is no dispersion, all subcarrier data in the i^th OFDM symbol, $a_{i, n}^{r e a l}$ , -N/2 + 1 ≤ n ≤ N/2, are decoded by the first processing unit (with zero delay) at the time i∙N. In the presence of a large value of dispersion, the pulse of a subcarrier (denoted as m) may delay into the time window of other processing units or even other OFDM symbol periods. $a_{i, m}^{r e a l}$ in the i^th OFDM symbol should be decoded by the k^th processing unit at the time (D + i)∙N if (DNT + kNT/x) is the closest value to the relative delay of the m^th subcarrier. For example, for 4 paths (path 0–3) with sample delays of 0, N/4, N/2, and 3N/4, when a subcarrier exhibits a delay of 1.1NT, this subcarrier is decoded by path 0 and the decoded data at the time i∙N in the digital domain corresponds to that in the (i-1)^th OFDM symbol at the transmitter. When (DNT + kNT/x) is ideally equal to the delay of the m^th subcarrier, we can analytically prove that the desirable signal and the crosstalk are orthogonal regardless of the dispersion:

\begin{array}{l} b_{i, m}^{r e a l} = H_{b} (m / (T N)) \cdot \exp (j π m / 2 - j \cdot α \cdot m^{2}) \cdot \\ (a_{i, m}^{r e a l} + j \cdot c_{m + 1}^{r e a l} + j \cdot c_{m - 1}^{r e a l} + j \cdot c_{m}^{i m a g} + j \cdot c_{m + 1}^{i m a g} + j \cdot c_{m - 1}^{i m a g}) \end{array}

The detailed derivation is provided in Appendix II. Comparison between Eq. (5) and Eq. (6) shows that the coefficient of either exp(j∙α∙m) or exp(-j∙α∙m) on the crosstalk terms is removed. In this case, the system performance is limited by the broadening effect of the pulse of each subcarrier, i.e. the third term in exp(∙) in Eq. (4) and the dispersion tolerance scale is, therefore, enhanced from T²∙N/π² to T²∙N²/π².

Fig. 3 Block diagram of the proposed offset-QAM OFDM decoding scheme.

Download Full Size | PDF

Fig. 4 An example to illustrate the principle of the proposed scheme (x = 2 and 7 subcarriers).

Download Full Size | PDF

Equation (6) assumes that for an arbitrary m^th subcarrier, there is a DFT window which can ideally align with the pulse of that subcarrier after dispersion. In practice, this condition cannot be satisfied because the number of paths x (e.g. 4 or 8) is significantly smaller than that of the subcarriers N (e.g. 128), and the delays of these paths are fixed (e.g. 4 paths with delays of 0, N/4, N/2, and 3N/4 samples). When there is a timing misalignment between the pulse of a subcarrier and the optimal DFT window for this subcarrier, the performance of the decoded signal on this subcarrier is degraded similarly to that in the conventional offset-QAM OFDM system (analytically studied in Eq. (5) and numerically studied in [6]). The extent of the degradation depends on the number of paths x, and the maximum misalignment is half the delay interval between paths. For example, the maximum misalignment is N/8 and N/16 for 4 and 8 paths, respectively. On the other hand, the complexity of subcarrier demultiplexing (the FIR filters and FFTs in the boxes in Fig. 3) increases linearly with x so its value should be chosen to balance the performance and complexity. In the previous works [6,7], it was shown the conventional offset-QAM OFDM could achieve the same performance as CP-OFDM with ~20% CP. It implies that a resolution of 4 or 8 paths may be sufficient to avoid performance degradation. It can also be deduced that the proposed scheme does not require symbol synchronization because the timing misalignment is always below half the delay interval between paths. This is in contrast to the conventional offset-QAM OFDM and CP-OFDM, in which symbol synchronization is essential to ensure correct decoding.

After parallel processing, the outputs of all processing units are sent into a decision unit, which re-maps the outputs of the processing units to the appropriate index of the decoded subcarriers and symbols. For example, if the relative delay of a subcarrier is 2.5TN, this subcarrier should be decoded by path 2 in a 4-path system, and the decoded data at the time i∙N in the digital domain corresponds to that in the (i-2)^th OFDM symbol at the transmitter. The mapping rule can be obtained at the initial system setup stage. After the re-mapping unit, phase and channel are compensated and the signal is sent for data decoding. These DSP algorithms are the same as those in the conventional offset-QAM OFDM schemes [7], and so the complexity increment of the proposed scheme only occurs at the subcarrier demultiplexing stage, i.e. the FIR filters and FFTs in Fig. 3.

It is noted that the proposed scheme does not offer performance benefits in C-OFDM. This is because offset-QAM OFDM recovers the in-phase and quadrature tributaries separately, under the principle that when the signal is real (or imaginary), the crosstalk terms can be controlled to be imaginary (or real) regardless of dispersion, as shown in Eq. (6). However, C-OFDM recovers the in-phase and quadrature tributaries at the same time point. If there are crosstalk terms due to dispersion-induced ISI and/or ICI, the system performance would be degraded. This can be seen more clearly in the simulation results in the next section.

3. Simulation setup and results

We simulated 224-Gb/s PDM offset-QPSK OFDM and CP-based C-OFDM to illustrate the advantages of the proposed scheme. The sampling rate of digital-to-analogue converters (DACs) was 80 GS/s. The number of subcarriers varied from 16 to 1024. For each case, the number of zero-padded subcarriers in the high-frequency region was controlled such that the signal line rate (including the CP in C-OFDM) was fixed to be 224 Gb/s. In offset-QAM OFDM, the FIR filters created a set of square-root-raised-cosine (SRRC) functions with different roll-off factors. After DACs, low-pass filters with 40-GHz bandwidth were used to remove the aliasing. The peak-to-average-power ratio of the electrical signal was controlled to be 10 dB. The signals of two polarizations were modulated, combined, and launched into fiber links. Nonlinearity in the fiber was not considered so as to isolate the effect of dispersion. At the receiver, the signals were filtered by an 80-GHz optical filter and coherently detected. The bandwidth of the receiver front end was 60 GHz and the sampling rate of analogue-to-digital converters was 80 GS/s. The number of simulated QPSK symbols was 200,000.

Figure 5(a) shows the time index (normalized by N) to decode different subcarriers. This time index is obtained using training sequences at the system setup stage. In theory, this index is equal to D + k/x as discussed above. It is seen that for a dispersion of 42,500 ps/nm, the subcarriers in one OFDM symbol may span over 12 OFDM symbols due to the delays of different subcarriers. In conventional CP-OFDM, a CP length of ~12 symbols should be used to enable the compensation of this dispersion value, resulting in significantly reduced net spectral efficiency. However, the proposed algorithm can support this dispersion without any CP. In practice, the integer part of the delay of a subcarrier with respect to the time period of one OFDM symbol does not influence the performance, provided that the index of the transmitted and decoded symbols can be re-mapped correctly. In contrast, the number of paths determines the maximum timing misalignment between the pulse of a subcarrier and the optimal DFT window for this subcarrier. Figure 5(b) shows the index of paths to decode different subcarriers (circles) and the remainder of normalized delays after taking the integer part (solid). The reminder of the normalized delay, in theory, is (D_a – DNT)/(NT), where D_a is the actual delay of a subcarrier. It can be seen from Fig. 5(b) that the proposed algorithm indeed selects the path k with (DNT + kNT/x) being the closest value to the actual delay of a subcarrier D_a, and the maximum misalignment is half the delay interval between paths.

Fig. 5 (a) Normalized time index to decode different subcarriers. (b) The index of paths to decode different subcarriers (circles) and the remainder of normalized delays of subcarriers (solid). The number of subcarriers is 128 (that for data modulation is 90). The dispersion is 42,500 ps/nm. The roll-off factor of the SRRC function and x are 0.5 and 4, respectively.

Download Full Size | PDF

Figure 6(a) depicts the required OSNR at a BER of 10⁻³ versus dispersion for the conventional CP-OFDM and offset-QAM OFDM with conventional and the proposed decoding algorithms. CP is not used for any of the cases shown in the figure. It can be seen that conventional offset-QAM OFDM can improve the dispersion tolerance by a factor of ~3 compared to CP-OFDM with zero CP. The proposed algorithm can further improve the tolerance by ~N times, from 1,300 ps/nm to 160,000 ps/nm at 20-dB OSNR. This agrees with the theoretical prediction obtained in the previous section. The performance of the proposed scheme depends on the number of paths. Figure 6(b) shows the required OSNR versus the number of paths for 42,500 and 85,000 ps/nm. Increasing the number of paths reduces the maximum timing misalignment between the pulse of a subcarrier and the corresponding DFT window, and thus mitigates the performance degradation. It can be seen that x of 8 can eliminate the penalty, while x of 4 reduces the complexity by half at the expense of ~1-dB OSNR penalty. Additional results show that the proposed scheme can also enhance the dispersion tolerance by ~N times for higher-level formats such as offset-16QAM. However, these formats are not investigated in this paper because they may not be suitable to long-haul transmission due to their higher OSNR requirement and lower tolerance to nonlinear effects.

Fig. 6 (a) Required OSNR at a BER of 10⁻³ versus dispersion for conventional CP-OFDM (C) and offset-QAM OFDM (O) when CP is not used. (b) Required OSNR at a BER of 10⁻³ versus the number of paths for the proposed scheme. In (a) and (b), the number of subcarriers is 128. The roll-off factor of the pulse in offset-QAM OFDM is 0.5.

Download Full Size | PDF

Figure 7(a) depicts the required OSNR versus dispersion for different pulse roll-off factors and memory lengths of the FIR filters in the proposed scheme. We also simulated the cases using the roll-off factor of 0.5 and 1, and the memory length of 80. The results are similar to those with the memory length of 2, and so are not plot in the figure. The result implicitly verifies that the dispersion tolerance of the proposed scheme is limited by the dispersion-induced pulse broadening effect on each subcarrier. When the roll-off factor is 1, the spectral width of each subcarrier is large and so the dispersion tolerance is low. Reducing the roll-off factor to 0.5 improves the performance due to a narrower spectral width of each subcarrier. In these two cases, a memory length of 2 is sufficient to create the time-domain pulse [6]. A roll-off factor of 0 results in the best dispersion tolerance, but in this case, the pulse is a sinc function with a long oscillating tail, and a large memory length is required for the FIR filters to create this function, resulting in higher complexity.

Fig. 7 (a) Required OSNR at a BER of 10⁻³ versus dispersion for different pulse roll-off factors and memory lengths of the FIR filters. The cases using the roll-off factor of 0.5 and 1, and the memory length of 80 are similar to those with the memory length of 2, and so are not plot in the figure. (b) Required OSNR at a BER of 10⁻³ versus the normalized synchronization error. In (a) and (b), the number of subcarriers is 128 and x = 4.

Download Full Size | PDF

Figure 7(b) shows the required OSNR versus the synchronization error (normalized by the time period of one OFDM symbol). The system has four paths, with delays of 0 + e_s, N/4 + e_s, N/2 + e_s, and 3N/4 + e_s samples, where e_s is the synchronization error and is in the range of [-N/8 N/8]. It is clearly seen that the system performance is insensitive to the synchronization error because the maximum misalignment is always less than half the delay interval between paths. This implies that the proposed scheme does not require symbol synchronization and therefore offers additional benefits in practical implementations.

Figure 8(a) depicts the required OSNR versus dispersion for different number of subcarriers. It is interesting to see that the dispersion tolerance indeed scales with N², with tolerance values increased by a factor of ~4 each time the number of subcarriers is doubled. For 256 subcarriers, the dispersion tolerance can be increased to ~640,000 ps/nm at an OSNR of 20 dB. This is equivalent to around 37,600-km SSMF, which covers all applications in optical transmission systems. Figure 8(b) shows the dispersion tolerance at 20-dB OSNR versus the number of subcarriers for the conventional CP-OFDM with different lengths of CP and offset-QAM OFDM with and without the proposed algorithm. Conventional offset-QAM OFDM exhibits a similar tolerance as that of CP-OFDM with 18.75% CP and these tolerances scale linearly with the number of subcarriers. In practice, a large number of subcarriers can be used to improve the dispersion tolerance but at the expense of reduced tolerance to frequency offset and higher complexity. In addition, even when 1024 subcarriers are used, the tolerance of the conventional systems is still limited below ~1,000 km. In contrast, the proposed scheme exhibits greatly enhanced tolerance even compared to CP-OFDM with 100% CP, and this advantage increases as the number of subcarriers grows because the dispersion tolerance scales with the square of the number of subcarriers. The elimination of additional CP would result in significant enhancement in the net spectral efficiency. Figure 9(a) shows the normalized spectral efficiency of CP-OFDM and the proposed scheme when the roll-off factor of the signal pulse is 0.5. The normalized spectral efficiency is calculated by:

S E_{n e t} = N / (N_{C P} + N)

where the required CP length in the digital domain, N_CP, is:

N_{C P} = | β_{2} L \cdot Δ ω | / T

For example, for 5000-km of SSMF with 17 ps/km/nm (β₂ = 21.7 ps²/km) and the 224-Gb/s offset-QPSK OFDM signal, we calculate that the required CP length is 3054. When the number of subcarriers is 128, the CP overhead is 2400% and the normalized spectral efficiency is 0.04. Therefore, as shown in Fig. 9(a), the spectral efficiency is degraded significantly for CP-OFDM. In contrast, the proposed system can maintain zero CP (or 100% normalized spectral efficiency) within its dispersion tolerance.

Fig. 8 (a) Required OSNR at a BER of 10⁻³ versus dispersion for the proposed scheme with different number of subcarriers. (b) Dispersion tolerance at an OSNR of 20 dB versus the number of subcarriers for CP-OFDM (C) using different lengths of CP and offset-QPSK OFDM (O) without CP. In (a) and (b), the pulse roll-off factor is 0.5 and x = 4.

Download Full Size | PDF

Fig. 9 (a) Normalized spectral efficiency versus dispersion for 224-Gb/s PDM QPSK CP-OFDM and the proposed PDM offset-QPSK OFDM with 128 and 256 subcarriers. (b) Required complexity of RGI-OFDM and the proposed scheme. The number of subcarriers and the point size of the FDE are 128 and 2048, respectively.

Download Full Size | PDF

We have shown that the proposed scheme can significantly improve the dispersion tolerance of the conventional offset-QAM OFDM, and exhibit greatly enhanced spectral efficiency compared to CP-OFDM. In the literature, methods are also proposed to reduce the length of CP in conventional CP-OFDM. A typical scheme is reduced-guard interval (RGI) OFDM based on frequency domain equalization (FDE). In this scheme, an additional FDE is applied before subcarrier demultiplexing with the point size of the FDE significantly larger than the number of subcarriers. Figure 9(b) shows the required complex multiplications of RGI-OFDM and the proposed scheme. The number of subcarriers is 128 and the FDE point size is 2048. In the figure, we only consider the complexity of the demultiplexing stage because the complexities of phase and channel estimation etc. are similar. For the proposed scheme, there are x paths with each consisting of N FIR filters and an FFT. Assuming that the complexity of an N-point FFT is N/2∙log₂(N), the total complexity is Nx/2∙log₂(N) + 3Nx/2 if the memory length of the FIR filters is 2. In this calculation, we use the symmetric property of the pulse shape. It can be seen that this complexity is insensitive to dispersion. On the other hand, the complexity of the RGI-OFDM is N/2∙log₂(N) + (N_FDE∙log₂(N_FDE) + N_FDE)∙N/(N_FDE – N_CP), where N_FDE is the point size of the FDE. This calculation is based on the overlap-and-add principle in the RGI-OFDM [11]. The complexity of one FDE operation is fixed as (N_FDE∙log₂(N_FDE) + N_FDE). However, the FDE blocks have to overlap, and the net number of equalized samples in each FDE operation is N_FDE – N_CP. Consequently, the FDE-induced complexity per OFDM symbol is (N_FDE∙log₂(N_FDE) + N_FDE)∙N/(N_FDE – N_CP). Figure 9(b) shows that the complexity of RGI-OFDM increases as distance increases, and is higher than that of the proposed scheme with x = 4 and 8 when the dispersion value is larger than 20,000 and 40,000 ps/nm, respectively. In addition, RGI-OFDM cannot support dispersion values larger than 60,000 ps/nm even with 2048 FDE point size, which also introduces a large latency to the system. This verifies the advantage of the proposed scheme over RGI-OFDM.

4. Conclusions

We have found analytically the fundamental mechanism limiting the dispersion tolerance of the conventional offset-QAM OFDM scheme and shown that its tolerance scales with T²∙N/π². We have proposed a novel scheme which can overcome this limitation and significantly enhance the dispersion tolerance by a factor of N so that the tolerance of the proposed scheme can approach that of one subcarrier. Numerical simulations show that the proposed scheme can support a 224-Gb/s PDM offset-4QAM OFDM signal over ~160,000 ps/nm dispersion (equivalent to ~9,400 km) without any CP under 128 subcarriers, and this tolerance can be further enhanced by a factor of 4 using 256 subcarriers. We also show that the net spectral efficiency and complexity of this scheme do not change within its dispersion tolerance, and hence it exhibits advantages of improved spectral efficiency, larger dispersion tolerance, and/or reduced complexity compared to CP-based C-OFDM and RGI-OFDM. This makes the proposed scheme a very promising solution for long-distance optical transmission.

Appendix I

For a large dispersion value, the impulse response of the channel, h_c(q∙N+k-d∙N-e), may span over multiple OFDM symbols. Assuming the convolution of the OFDM symbol pulse h_filter(∙) and h_c(∙) only has non-zero values within (-MN/2 MN/2], where M is a sufficiently large integer, we can write h_c(q∙N+k-d∙N-e) as:

\begin{array}{l} h_{c} (q \cdot N + k - d \cdot N - e) \\ = \sum_{g = - M N / 2 + 1}^{M N / 2} H_{b} (g / (M N T)) \cdot \exp (j \cdot α \cdot {(g / M)}^{2}) \cdot \exp (2 π j (q \cdot N + k - d \cdot N - e) \cdot g / (M N)) \end{array}

where α = β₂∙L∙(2π/(T∙N))²/2 depends on the accumulated dispersion β₂∙L and the time interval between samples T. H_b(∙) and exp(j∙α∙(g/M)²) are discussed in Eq. (4) and represent the frequency response of the back-to-back and dispersion, respectively. We combine Eqs. (1), (3), and (9), and obtain the received signal as:

\begin{array}{l} r (q \cdot N + k) = \sum_{p = - \infty}^{+ \infty} \sum_{n = - N / 2 + 1}^{N / 2} (a_{p, n}^{r e a l} \cdot \exp (j π n / 2) \cdot \sum_{g = - M N / 2 + 1}^{M N / 2} H_{f i l t e r} (g - M n) \cdot H_{b} (g / (M N T)) \\ \cdot \exp (j \cdot α \cdot (^{g / M) 2}) \cdot \exp (2 π j ((q \cdot N + k) g + (M n - g) \cdot p N) / (M N)) \\ + a_{p, n}^{i m a g} \cdot \exp (j π (n + 1) / 2) \cdot \sum_{g = - M N / 2 + 1}^{M N / 2} H_{f i l t e r} (g - M n) \cdot H_{b} (g / (M N T)) \\ \cdot \exp (j \cdot α \cdot (^{g / M) 2}) \cdot \exp (2 π j ((q \cdot N + k) g + (M n - g) (p N + N / 2)) / (M N))) \end{array}

where H_filter(∙) is the Fourier transform of h_filter(∙). We further define g-Mn = g₁. Note that in offset-QAM OFDM, there is no spectral overlap between subcarriers with more than one subcarrier distance (e.g. the m^th and (m ± 2)^th subcarriers) [1]. Therefore, H_filter(g₁) is non-zero only in the range of [-M M]. Therefore, we can re-write Eq. (10) as:

\begin{array}{l} r (q \cdot N + k) = \sum_{p = - \infty}^{+ \infty} \sum_{n = - N / 2 + 1}^{N / 2} (a_{p, n}^{r e a l} \cdot \exp (j π n / 2) \cdot \exp (2 π j k n / N) \cdot H_{b} (n / (N T)) \\ \cdot \sum_{g_{1} = - M}^{M} H_{f i l t e r} (g_{1}) \cdot \exp (j \cdot α \cdot (^{n + g_{1} / M) 2}) \cdot \exp (2 π j ((q \cdot N + k - p \cdot N) \cdot g_{1}) / (M N)) \\ + a_{p, n}^{i m a g} \cdot \exp (j π (n + 1) / 2)) \cdot \exp (2 π j k n / N) \cdot H_{b} (n / (N T)) \\ \cdot \sum_{g_{1} = - M}^{M} H_{f i l t e r} (g_{1}) \cdot \exp (j \cdot α \cdot (^{n + g_{1} / M) 2}) \exp (2 π j ((q \cdot N + k - p \cdot N - N / 2) \cdot g_{1}) / (M N))) \end{array}

In Eq. (11), we assume that H_b(∙) does not change rapidly over frequency and so is a constant within a subcarrier, i.e. H_b((Mn + g₁)/(TMN)) ≈H_b(n/(TN)). This is valid in practice because the memory length of the back-to-back response is much smaller than the period of one OFDM symbol. In contrast, the dispersion-induced frequency response is not a constant within a subcarrier. We expand the dispersion for each subcarrier n:

\exp (j \cdot α \cdot {(n + g_{1} / M)}^{2}) = \exp (j \cdot α \cdot n^{2} + 2 j \cdot α \cdot n \cdot g_{1} / M + j \cdot α \cdot {(g_{1} / M)}^{2})

As explained in subsection 2.1, the third term on the right-hand side of Eq. (12) is neglected and we combine Eqs. (11) and (12) to obtain the received signal as:

\begin{array}{l} r (q \cdot N + k) = \sum_{p = - \infty}^{+ \infty} \sum_{n = - N / 2 + 1}^{N / 2} \exp (2 π j k n / N) \cdot H_{b} (n / (N T)) \\ (a_{p, n}^{r e a l} \cdot \exp (j π n / 2) \cdot \exp (j \cdot α \cdot n^{2}) \cdot h_{f i l t e r} (q \cdot N + k - p \cdot N + α \cdot N n / π) \\ + a_{p, n}^{i m a g} \cdot \exp (j π (n + 1) / 2) \cdot \exp (j \cdot α \cdot n^{2}) \cdot h_{f i l t e r} (q \cdot N + k - p \cdot N + α \cdot N n / π - N / 2)) \end{array}

Equation (13) provides a closed-form expression for the received signal affected by dispersion. We then replace r(q⋅N + k) in Eq. (2) and obtain:

\begin{array}{l} b_{i, m}^{r e a l} = \exp (j π m / 2 + j \cdot α \cdot m^{2}) \cdot H_{b} (m / (N T)) \\ \cdot \sum_{k_{1} = - \infty}^{+ \infty} \sum_{p = - \infty}^{+ \infty} (a_{p, m}^{r e a l} \cdot h_{f i l t e r} (k_{1} + (i - p) \cdot N + α \cdot N m / π) \cdot h_{f i l t e r} (k_{1}) \\ + j \cdot a_{p, m}^{i m a g} \cdot h_{f i l t e r} (k_{1} + (i - p - 0.5) \cdot N + α \cdot N m / π) \cdot h_{f i l t e r} (k_{1})) \\ + \exp (j π (m + 1) / 2 + j \cdot α \cdot (^{m + 1) 2}) \cdot H_{b} ((m + 1) / (N T)) \\ \cdot \sum_{k_{1} = - \infty}^{+ \infty} \sum_{p = - \infty}^{+ \infty} (a_{p, m + 1}^{r e a l} \cdot h_{f i l t e r} (k_{1} + (i - p) \cdot N + α \cdot N (m + 1) / π) \cdot h_{f i l t e r} (k_{1}) \cdot \exp (2 π j k_{1} / N) \\ + j \cdot a_{p, m + 1}^{i m a g} \cdot h_{f i l t e r} (k_{1} + (i - p - 0.5) \cdot N + α \cdot N (m + 1) / π) \cdot h_{f i l t e r} (k_{1}) \cdot \exp (2 π j k_{1} / N)) \\ + \exp (j π (m - 1) / 2 + j \cdot α \cdot (^{m - 1) 2}) \cdot H_{b} ((m - 1) / (N T)) \\ \cdot \sum_{k_{1} = - \infty}^{+ \infty} \sum_{p = - \infty}^{+ \infty} (a_{p, m - 1}^{r e a l} \cdot h_{f i l t e r} (k_{1} + (i - p) \cdot N + α \cdot N (m - 1) / π) \cdot h_{f i l t e r} (k_{1}) \cdot \exp (- 2 π j k_{1} / N) \\ + j \cdot a_{p, m - 1}^{i m a g} \cdot h_{f i l t e r} (k_{1} + (i - p - 0.5) \cdot N + α \cdot N (m - 1) / π) \cdot h_{f i l t e r} (k_{1}) \cdot \exp (- 2 π j k_{1} / N)) \end{array}

where k₁ = k + (q-i)⋅N. The six terms on the right-hand side of Eq. (14) represent the influence of the in-phase and quadrature tributaries of the m^th, (m + 1)^th, and (m-1)^th subcarriers, respectively. For the signal term (the in-phase tributary of the m^th subcarrier), we can obtain:

\begin{array}{l} \exp (j π m / 2 + j \cdot α \cdot m^{2}) \cdot H_{b} (m / (N T)) \cdot \sum_{k_{1} = - \infty}^{+ \infty} \sum_{p = - \infty}^{+ \infty} a_{p, m}^{r e a l} \cdot h_{f i l t e r} (k_{1} + (i - p) \cdot N + α \cdot N m / π) \cdot h_{f i l t e r} (k_{1}) \\ = \exp (j π m / 2 + j \cdot α \cdot m^{2}) \cdot H_{b} (m / (N T)) \cdot (a_{i, m}^{r e a l} \cdot \sum_{k_{1} = - \infty}^{+ \infty} h_{f i l t e r} (k_{1} + α \cdot N m / π) \cdot h_{f i l t e r} (k_{1}) \\ + \sum_{k_{1} = - \infty}^{+ \infty} \sum_{p \neq i}^{} a_{p, m}^{r e a l} \cdot h_{f i l t e r} (k_{1} + (i - p) \cdot N + α \cdot N m / π) \cdot h_{f i l t e r} (k_{1})) \\ = \exp (j π m / 2 + j \cdot α \cdot m^{2}) \cdot H_{b} (m / (N T)) \cdot (A \cdot a_{i, m}^{r e a l} + I_{i, m}^{r e a l}) \end{array}

where A and

I_{i, m}^{r e a l}

are real values. It is shown in Eq. (15) that even in the absence of crosstalk from adjacent subcarriers, the received signal

b_{i, m}^{r e a l}

would experience an amplitude reduction on the desirable signal

a_{i, m}^{r e a l}

by a factor of A, and inter-symbol interference (ISI) from adjacent symbols,

I_{i, m}^{r e a l}

, unless m is equal to zero. This is because different subcarriers have different time delays under dispersion and a fixed FFT window cannot align with the pulses of all subcarriers. For subcarriers with m ≠ 0, the sampling point is no longer at the center of the pulse, resulting in amplitude reduction of the desirable signal and ISI. On the other hand, the crosstalk from the in-phase tributary of the (m + 1)^th subcarrier can be re-written as:

\begin{array}{l} \exp (j π (m + 1) / 2 + j \cdot α \cdot (^{m + 1) 2}) \cdot H_{b} ((m + 1) / (N T)) \cdot \\ \sum_{k_{1} = - \infty}^{+ \infty} \sum_{p = - \infty}^{+ \infty} a_{p, m + 1}^{r e a l} \cdot h_{f i l t e r} (k_{1} + (i - p) \cdot N + α \cdot N (m + 1) / π) \cdot h_{f i l t e r} (k_{1}) \cdot \exp (2 π j k_{1} / N) \\ \approx j \cdot \exp (j π m / 2) \cdot (^{- 1) i - p} \cdot \exp (j α \cdot (^{m + 1) 2}) \cdot H_{b} (m / (N T)) \cdot \exp (- j \cdot α \cdot (m + 1)) \cdot \sum_{k_{2} = - \infty}^{+ \infty} \sum_{p = - \infty}^{+ \infty} a_{p, m + 1}^{r e a l} \\ \cdot h_{f i l t e r} (k_{2} + \frac{(i - p) N + α \cdot N (m + 1) / π}{2}) h_{f i l t e r} (k_{2} - \frac{(i - p) N + α \cdot N (m + 1) / π}{2}) \exp (2 π j k_{2} / N) \\ = j \cdot \exp (j π m / 2 + j \cdot α \cdot m^{2}) \cdot H_{b} (m / (N T)) \cdot \exp (j \cdot α \cdot m) \cdot c_{m + 1}^{r e a l} \end{array}

Here we assume that H_b(m/(TN)) ≈H_b((m + 1)/(TN)) for the back-to-back response, which is valid in practice because this response does not change rapidly over frequency and the spectral width of one OFDM subcarrier is narrow.

c_{m + 1}^{r e a l}

represents the amount of crosstalk from the in-phase tributary of the (m + 1)^th subcarrier. Provided that the signal pulse in offset-QAM OFDM, h_filter(∙), is an even function [1], h_filter(k₂ + (i-p)N/2 + αN(m + 1)/2/π)∙h_filter(k₂-(i-p)N/2-αN(m + 1)/2/π) can also be proved to be an even function such that

c_{m + 1}^{r e a l}

is real. Similarly, we can obtain other terms in Eq. (14) and re-write

b_{i, m}^{r e a l}

as:

\begin{array}{l} b_{i, m}^{r e a l} = H_{b} (m / (T N)) \cdot \exp (j π m / 2 + j \cdot α \cdot m^{2}) \cdot (A \cdot a_{i, m}^{r e a l} + I_{i, m}^{r e a l} + j \cdot \exp (j \cdot α \cdot m) \cdot c_{m + 1}^{r e a l} \\ + j \cdot \exp (- j \cdot α \cdot m) \cdot c_{m - 1}^{r e a l} + j \cdot c_{m}^{i m a g} + j \cdot \exp (j \cdot α \cdot m) \cdot c_{m + 1}^{i m a g} + j \cdot \exp (- j \cdot α \cdot m) \cdot c_{m - 1}^{i m a g}) \end{array}

where

c_{m + 1}^{r e a l}

,

c_{m - 1}^{r e a l}

,

c_{m}^{i m a g}

,

c_{m + 1}^{i m a g}

,

c_{m - 1}^{i m a g}

are all real values.

Appendix II

From Eq. (14), we know that the delay of the m^th subcarrier is α∙Nm/π. Therefore, we decode $a_{i, m}^{r e a l}$ using the delayed version of the received time-domain signal r(q⋅N+k-α∙Nm/π):

b_{i, m}^{r e a l} = \sum_{k = - N / 2 + 1}^{N / 2} \sum_{q = - \infty}^{+ \infty} \exp (- 2 π j k \cdot m / N) \cdot r (q \cdot N + k - α \cdot N m / π) \cdot h_{f i l t e r} (q \cdot N + k - i \cdot N)

Similar to Eq. (13), r(q⋅N + k-α∙Nm/π) is derived as:

\begin{array}{l} r (q \cdot N + k - α \cdot N m / π) = \sum_{p = - \infty}^{+ \infty} \sum_{n = - N / 2 + 1}^{N / 2} \exp (2 π j k n / N - 2 j α \cdot m n) \cdot H_{b} (n / (N T)) \cdot \\ (a_{p, n}^{r e a l} \cdot \exp (j π n / 2) \cdot \exp (j \cdot α \cdot n^{2}) \cdot h_{f i l t e r} (q \cdot N + k - p \cdot N + α \cdot N (n - m) / π) \\ + a_{p, n}^{i m a g} \cdot \exp (j π (n + 1) / 2) \cdot \exp (j \cdot α \cdot n^{2}) \cdot h_{f i l t e r} (q \cdot N + k - p \cdot N + α \cdot N (n - m) / π - N / 2)) \end{array}

We then combine Eqs. (18) and (19) to obtain:

\begin{array}{l} b_{i, m}^{r e a l} = \sum_{n = m - 1, m, m + 1} \exp (j π n / 2 + j \cdot α \cdot n^{2} - 2 j \cdot α \cdot m n) \cdot H_{b} (n / (N T)) \\ \sum_{k_{1} = - \infty}^{+ \infty} \sum_{p = - \infty}^{+ \infty} (a_{p, n}^{r e a l} \cdot h_{f i l t e r} (k_{1} + (i - p) \cdot N + α \cdot N (n - m) / π) \cdot h_{f i l t e r} (k_{1}) \cdot \exp (2 π j k_{1} (n - m) / N) \\ + j \cdot a_{p, m}^{i m a g} \cdot h_{f i l t e r} (k_{1} + (i - p - 0.5) \cdot N + α \cdot N (n - m) / π) \cdot h_{f i l t e r} (k_{1}) \cdot \exp (2 π j k_{1} (n - m) / N)) \end{array}

The terms in Eq. (20) can be analyzed using the manipulations similarly to Eqs. (15)-(16), and we can therefore obtain Eq. (21):

\begin{array}{l} b_{i, m}^{r e a l} = H_{b} (m / (T N)) \cdot \exp (j π m / 2 - j \cdot α \cdot m^{2}) \cdot \\ (a_{i, m}^{r e a l} + j \cdot c_{m + 1}^{r e a l} + j \cdot c_{m - 1}^{r e a l} + j \cdot c_{m}^{i m a g} + j \cdot c_{m + 1}^{i m a g} + j \cdot c_{m - 1}^{i m a g}) \end{array}

Acknowledgments

This work was supported by the Science Foundation Ireland under grant number 11/SIRG/I2124, 12/IA/1270 and 13/TIDA/I2718, and EU 7th Framework Program under grant agreement 318415 (FOX-C).

References and links

1. J. Zhao and A. D. Ellis, “Offset-QAM based coherent WDM for spectral efficiency enhancement,” Opt. Express 19(15), 14617–14631 (2011). [CrossRef] [PubMed]

2. S. Randel, A. Sierra, X. Liu, S. Chandrasekhar, and P. J. Winzer, “Study of multicarrier offset-QAM for spectrally efficient coherent optical communications,” in Proc. European Conference on Optical Communication (2011), paper Th.11.A.1. [CrossRef]

3. F. Horlin, J. Fickers, P. Emplit, A. Bourdoux, and J. Louveaux, “Dual-polarization OFDM-OQAM for communications over optical fibers with coherent detection,” Opt. Express 21(5), 6409–6421 (2013). [CrossRef] [PubMed]

4. Z. Li, T. Jiang, H. Li, X. Zhang, C. Li, C. Li, R. Hu, M. Luo, X. Zhang, X. Xiao, Q. Yang, and S. Yu, “Experimental demonstration of 110-Gb/s unsynchronized band-multiplexed superchannel coherent optical OFDM/OQAM system,” Opt. Express 21(19), 21924–21931 (2013). [CrossRef] [PubMed]

5. M. Xiang, S. Fu, M. Tang, H. Tang, P. Shum, and D. Liu, “Nyquist WDM superchannel using offset-16QAM and receiver-side digital spectral shaping,” Opt. Express 22(14), 17448–17457 (2014). [PubMed]

6. J. Zhao, “DFT-based offset-QAM OFDM for optical communications,” Opt. Express 22(1), 1114–1126 (2014). [CrossRef] [PubMed]

7. J. Zhao, “Channel estimation in DFT-based offset-QAM OFDM systems,” Opt. Express 22(21), 25651–25662 (2014). [CrossRef] [PubMed]

8. B. Liu, L. Zhang, X. Xin, and J. Yu, “None pilot-tones and training sequence assisted OFDM technology based on multiple-differential amplitude phase shift keying,” Opt. Express 20(20), 22878–22885 (2012). [CrossRef] [PubMed]

9. B. Inan, S. Adhikari, O. Karakaya, P. Kainzmaier, M. Mocker, H. von Kirchbauer, N. Hanik, and S. L. Jansen, “Real-time 93.8-Gb/s polarization-multiplexed OFDM transmitter with 1024-point IFFT,” Opt. Express 19(26), B64–B68 (2011). [CrossRef] [PubMed]

10. R. Schmogrow, S. Ben-Ezra, P. C. Schindler, B. Nebendahl, C. Koos, W. Freude, and J. Leuthold, “Pulse-shaping with digital, electrical and optical filters–a comparsion,” IEEE J. Lightwave Technol. 31(15), 2570–2577 (2013). [CrossRef]

11. X. Liu, S. Chandrasekhar, B. Zhu, P. J. Winzer, A. H. Gnauck, and D. W. Peckham, “448-Gb/s reduced-guard-interval CO-OFDM transmission over 2000 km of ultra-large-area fiber and five 80-GHz-Grid ORADMs,” IEEE J. Lightwave Technol. 29(4), 483–490 (2011). [CrossRef]

12. Q. Zhuge, C. Chen, and D. V. Plant, “Dispersion-enhanced phase noise effects on reduced-guard-interval CO-OFDM transmission,” Opt. Express 19(5), 4472–4484 (2011). [CrossRef] [PubMed]

Dispersion tolerance enhancement using an improved offset-QAM OFDM scheme

Abstract

1. Introduction

2. Principle

2.1. Dispersion limitation of the conventional offset-QAM OFDM scheme

2.2. Proposed offset-QAM OFDM scheme

3. Simulation setup and results

4. Conclusions

Appendix I

Appendix II

Acknowledgments

References and links

Cited By

Figures (9)

Equations (21)

Optics Express