# Spectrally Efficient Multiplexing - OFDM

This is a continuation from the previous tutorial - ** linear pulse propagation**.

Single-carrier modulation (SCM) has been the de facto modulation choice and has long been implemented in commercialized products for both long- and short-reach optical fiber communications. The SCM gains its popularity due to low hardware complexity for relatively low-speed (10 Gb/s or less) communication systems.

In the recent decade, multicarrier modulation (MCM), which has higher spectral efficiency (SE) than the conventional SCM, has been argued as a promising alternative to satisfy the exponential growth of the Internet traffic.

Several schemes that utilize MCM have been demonstrated since 2006. For instance, direct-detection optical orthogonal frequency-division multiplexing (DDO-OFDM) is proposed for short-reach applications, and coherent optical OFDM (CO-OFDM) for long-haul transmission.

While the implementation (hardware and software) of DDO-OFDM is simpler, CO-OFDM promises higher SE, better receiver sensitivity, and polarization-dispersion resilience. Lab demonstrations confirm CO-OFDM with a data rate higher than 1-Terabit per second (Tb/s) can be delivered over 600-km standard single-mode fiber (SSMF).

OFDM is one of the variations of MCM in which the data information is carried over multiple low-rate subcarriers. It has been widely understood that OFDM has the robustness against channel dispersion and has its ease of phase and channel estimation in a time-varying environment.

However, due to the nature of MCM, OFDM has the disadvantages of high peak-to-average power ratio (PAPR) and high sensitivity to frequency offset and phase noise.

In order to fully discuss the applications of OFDM in optical communication, we present an introduction to the fundamentals of OFDM, including its basic mathematical formulation, discrete Fourier transform (DFT) implementation, cyclic prefix (CP), spectral efficiency (SE), and PAPR characteristics.

We review two major schemes of OFDM: CO-OFDM and DDO-OFDM that are popular and have been widely adopted by the optical communications. Furthermore, we show a novel variant of OFDM, which is called DFT-spread OFDM system (DFT-S OFDM), with many interesting features that can significantly improve the system performance and SE. Finally, we show a few OFDM-based superchannel transmission technologies to achieve high SE for high-speed optical transports.

## 1. OFDM Basics

OFDM is a special form of a broader class of MCM. The principle of OFDM is to transmit the information through a large number of orthogonal subcarriers. The OFDM signal in time domain consists of a continuous stream of OFDM symbols with a regular period \(T_s\). The OFDM baseband signal \(s(t)\) is written as

\[\tag{5.1}s(t)=\boldsymbol{\sum}_{i=-\infty}^{+\infty}\boldsymbol{\sum}_{k=-N_{sc}/2+1}^{k=N_{sc}/2}c_{ki}\exp(j2\pi{f_k}(t-iT_s))f(t-iT_s)\]

\[\tag{5.2}f_k=\frac{k-1}{t_s},\qquad\Delta{f}=\frac{1}{t_s}\]

\[\tag{5.3}f(t)=\begin{cases}1,\quad(-\Delta_G\lt{t}\le{t_s})\\0,\quad(t\le-\Delta_G,t\gt{t_s})\end{cases}\]

where \(c_{ki}\) is the \(i\)th information symbol at the \(k\)th subcarrier, \(f(t)\) is the pulse waveform of the symbol, \(f_k\) is the frequency of the subcarrier, and \(\Delta{f}\) is the subcarrier spacing, and \(N_{sc}\), \(\Delta_G\), and \(t_s\) are the number of OFDM subcarriers, guard interval (GI) length, and observation period, respectively.

The optimum detector for each subcarrier could use a filter that matches the subcarrier waveform, or a correlator matched to the subcarrier as shown in Figure 5.1. Therefore, the detected information symbol \(\vec{c}_{ki}\) at the output of the correlator is given by

\[\tag{5.4}\vec{c}_{ki}=\int_0^{T_s}r(t-iT_s)s_k^*dt=\int_0^{T_s}r(t-iT_s)\exp(-j2\pi{f_k}t)dt\]

where \(r(t)\) is the received time-domain signal, \(s_k\) is the \(k\)th subcarrier waveform, and the * stands for complex conjugate. The classical MCM uses nonoverlapped band-limited signals, and can be implemented with a bank of large number of oscillators and filters at both transmit and receive ends.

The major disadvantage of nonoverlapped MCM is that it requires excessive bandwidth. This is because in order to design the filters and oscillators cost-efficiently, the channel spacing has to be multiples of the symbol rate, greatly reducing the spectral efficiency.

On the contrary, OFDM employs overlapped yet orthogonal signal set. This orthogonality originates from the straightforward correlation between any two subcarriers, given by

\[\tag{5.5}\begin{align}\delta_{kl}&=\frac{1}{T_s}\int_0^{T_s}s_ks_l^*dt=\frac{1}{T_s}\int_0^{T_s}\exp(j2\pi(f_k-f_l)t)dt\\&=\exp(j\pi(f_k-f_l)T_s)\frac{\sin(\pi(f_k-f_l)T_s)}{\pi(f_k-f_l)T_s}\end{align}\]

If the following condition

\[\tag{5.6}f_k-f_l=m\frac{1}{T_s}\]

is satisfied, the two subcarriers are orthogonal to each other. This signifies that these orthogonal subcarrier sets, with their frequency spaced at multiple of inverse of the symbol rate can be recovered with the matched filters (Eq. 5.4) without intercarrier interference (ICI), in spite of strong signal spectral overlapping.

One of the enabling techniques for OFDM is the insertion of cyclic prefix (CP), which is also known as guard interval (GI). Cyclic prefix was proposed to resolve the channel dispersion-induced intersymbol interference (ISI) and ICI. Figure 5.2 shows insertion of a cyclic prefix by cyclic extension of the OFDM waveform into the guard interval, \(\Delta_G\). As shown in Figure 5.2, the waveform in the guard interval is essentially an identical copy of that in the DFT window, with time-shifted by “\(t_s\)” behind.

It can be seen that, if the maximum delay spread of multipath fading is smaller than the guard time \(\Delta_G\), the CP can perfectly accommodate the ISI. In the context of optical transmission, the delay spread due to the chromatic dispersion (CD) among the subcarriers should not exceed the guard time, and the fundamental condition for the complete elimination of ISI in optical medium is thus given by

\[\tag{5.7}\frac{c}{f^2}|D_t|\cdot{N_{sc}}\cdot\Delta{f}\le\Delta_G\]

where \(f\) is the frequency of the optical carrier, \(c\) the speed of light, \(D_t\) the total accumulated chromatic dispersion in units of ps/km, and \(N_{sc}\) is the number of subcarriers.

## 2. Coherent Optical OFDM (CO-OFDM)

CO-OFDM offers good performance in the sense of receiver sensitivity, spectral efficiency, and robustness against polarization dispersion, but requires high complexity in transceiver design.

In the open literature, CO-OFDM was first proposed by Shieh and Athaudage, and the concept of the coherent optical multiple-input-multiple-output (MIMO)-OFDM was formalized by Shieh et al..

The early CO-OFDM experiments were carried out by Shieh et al. for a 1000 km SSMF transmission at 8 Gb/s, and by Jansen et al. for 4160 km SSMF transmission at 20 Gb/s. The principle and transmitter/receiver design for CO-OFDM are given in the following sections.

### 2.1. Principle of CO-OFDM

#### 2.1.1. Coherent-Detection and Optical OFDM

The synergies between coherent optical communications and OFDM are twofold. OFDM enables channel and phase estimation for coherent detection in a computationally efficient way.

Coherent detection provides linearity in radio frequency (RF)-to-optical (RTO) up-conversion and optical-to-RF (OTR) down-conversion, much needed for OFDM. Consequently, CO-OFDM is a natural choice for optical transmission in the linear regime. A generic CO-OFDM system is depicted in Figure 5.3.

In general, a CO-OFDM system can be divided into five functional blocks including

- RF OFDM transmitter
- RTO up-converter
- The optical channel
- The OTR down-converter
- The RF OFDM receiver.

The detailed architecture for RF OFDM transmitter/receiver has already been shown in Figure 5.3, which generates/recovers OFDM signals either in baseband or an RF band. Let us assume for now a linear channel where optical fiber nonlinearity is not considered.

It is apparent that the challenges for CO-OFDM implementation are to obtain a linear RTO up-converter and linear OTR down-converter. It has been proposed and analyzed that by biasing the Mach–Zehnder modulators (MZMs) at null point, a linear conversion between the RF signal and optical field signal can be achieved.

Meanwhile, by using coherent detection, a linear transformation from "optical signal" to RF (or baseband electrical) signal can be achieved. Therefore, combining such a composite system cross RF and optical domain, a linear channel can be constructed where OFDM can perform its best role of mitigating channel dispersion impairment in both RF and optical domains. In this section, we use the term “RF domain” and “electrical domain” interchangeably.

As shown in Figure 5.4, coherent detection uses a 2 ×4 90\(^\circ\) optical "hybrid mixer" and a pair of balanced photodetectors. The main purposes of coherent detection are (i) to linearly recover the In-Phase (I) and Quadrature (Q) components of the incoming signal, and (ii) to suppress or cancel the common mode noise.

Using a six-port 90\(^\circ\) hybrid, signal detection and analysis have been realized in RF domain for decades, and its application to single-carrier coherent optical systems can be found. In what follows, in order to illustrate its working principle, we perform an analysis of down-conversion via coherent detection assuming ideal condition for each component shown in Figure 5.4.

The purpose of the four output ports of the 90\(^\circ\) optical hybrid is to generate a 90\(^\circ\) phase shift between I and Q components, and 180\(^\circ\) phase shift between balanced detectors. Ignoring imbalance and loss of the optical hybrid, the output signals \(E_{1-4}\) can be expressed as

\[\tag{5.8}\begin{align}E_1&=\frac{1}{\sqrt{2}}[E_s+E_\text{LO}],\quad{E_2}=\frac{1}{\sqrt{2}}[E_s-E_\text{LO}]\\E_3&=\frac{1}{\sqrt{2}}[E_s-jE_\text{LO}],\quad{E_4}=\frac{1}{\sqrt{2}}[E_s+jE_\text{LO}]\end{align}\]

where \(E_s\) and \(E_\text{LO}\) are, respectively, the electric field of the incoming signal and local oscillator (LO) signal. We further decompose the incoming signal into two components: (i) the received signal free from the amplified spontaneous noise (ASE), \(E_r(t)\), and (ii) the ASE noise, \(n_o(t)\), namely

\[\tag{5.9}E_s=E_r+n_o\]

We first study how the I component of the photodetected current is generated, and the Q component can be derived accordingly. The I component is obtained by using a pair of the photodetectors, PD1 and PD2 in Figure 5.4, whose photocurrent \(I_{1-2}\) can be described as

\[\tag{5.10}I_1=|E_1|^2=\frac{1}{2}\{|E_s|^2+|E_\text{LO}|^2+2\text{Re}\{E_sE_\text{LO}^*\}\}\]

\[\tag{5.11}I_2=|E_2|^2=\frac{1}{2}\{|E_s|^2+|E_\text{LO}|^2-2\text{Re}\{E_sE_\text{LO}^*\}\}\]

\[\tag{5.12}|E_s|^2=|E_r|^2+|n_o|^2+2\text{Re}\{E_rn_o^*\}\]

\[\tag{5.13}|E_\text{LO}|^2=I_\text{LO}(1+I_\text{RIN}(t))\]

where \(I_\text{LO}\) and \(I_\text{RIN}(t)\) are the average power and relative intensity noise (RIN) of the LO laser, and “Re” or “Im” denotes the real or imaginary part of a complex signal. For simplicity, the photodetection responsivity is set to unity.

The three terms at the right-hand side of (5.12) represent the signal-to-signal beat noise (SSBN), signal-to-ASE beat noise, and ASE-to-ASE beat noise. Because of the balanced detection, using (5.10) and (5.11), the I component of the photocurrent becomes

\[\tag{5.14}I_I(t)=I_1-I_2=2\text{Re}\{E_sE_\text{LO}^*\}\]

Now, the noise suppression mechanism is completely revealed: the three noise terms in (5.12) and the RIN noise in (5.13) from a single detector are completely cancelled via balanced detection. Meanwhile, it has been shown that coherent detection can be performed by using a single photodetector, but at the cost of reduced dynamic range.

In a similar fashion, the Q component from the other pair of balanced detectors can be derived as

\[\tag{5.15}I_Q(t)=I_3-I_4=2\text{Im}\{E_sE_\text{LO}^*\}\]

Using the results of (5.14) and (5.15), the complex detected signal \(\tilde{I}(t)\) consisting of both I and Q components becomes

\[\tag{5.16}\tilde{I}(t)=I_I(t)+jI_Q(t)=2E_sE_\text{LO}^*\]

From (5.16), the linear down-conversion process via coherent detection becomes quite clear; the complex photocurrent \(\tilde{I}(t)\) is in essence a linear replica of the incoming complex signal that is frequency down-converted by a local oscillator frequency. Thus, with linear coherent detection at receiver and linear generation at transmitter, complex OFDM signals can be readily transmitted over the optical fiber channel.

#### 2.1.2. Digital Signal Processing (DSP) of CO-OFDM

In Section 2.1.1, we have introduced the concept of CO-OFDM with focus on the principle of coherent detection. Here, we revisit the principle of CO-OFDM from experimental and signal processing point of view.

Figure 5.5 shows a conceptual diagram of a complete CO-OFDM system.

The function of the OFDM transmitter is to map the data bits into each OFDM symbol, and generate the time series by inverse discrete Fourier transform (IDFT) expressed in (Eq. 5.1), including insertion of the guard interval. The digital signal is then converted to analog signal through the digital-to-analog converter (DAC), and filtered with a low-pass filter (LPF) to remove the alias signal.

In Figure 5.5, direct up-conversion architecture is used where RF OFDM transmitter outputs a baseband OFDM signal. The subsequent RTO up-converter transforms the baseband signal to the optical domain using an optical IQ modulator comprising a pair of MZMs with a 90\(^\circ\) phase offset. The baseband OFDM signal is directly up-converted to the optical domain given by

\[\tag{5.17}E(t)=\exp(j\omega_{\text{LD1}}t+\phi_{\text{LD1}})\cdot{s_B(t)}\]

where \(\omega_\text{LD1}\) and \(\phi_\text{LD1}\), respectively, are the angular frequency and the phase of the transmitter laser. The up-converted signal \(E(t)\) traverses the optical medium with the impulse response \(h(t)\), and the received optical signal becomes

\[\tag{5.18}E(t)=\exp(j\omega_\text{LD1}t+\phi_\text{LD1})s_B(t)\otimes{h(t)}\]

where “\(\otimes\)” stands for convolution. The optical OFDM signal is then fed into the OTR down-converter where the optical OFDM signal is converted to RF OFDM signal. There are two ways to do the down-conversion. One is direct down-conversion architecture where the intermediate frequency (IF) is near DC. The other is to first down-convert the signal to RF domain with an IF and then down-convert to baseband. The IF signal can be expressed as

\[\tag{5.19}r(t)=\exp(j\omega_\text{off}t+\Delta\phi)r_0(t),\quad{r_0}(t)=s_B(t)\otimes{h(t)}\]

\[\tag{5.20}\omega_\text{off}=\omega_\text{LD1}-\omega_\text{LD2},\quad\Delta\phi=\phi_\text{LD1}-\phi_\text{LD2}\]

where \(\Delta\omega_\text{off}\) and \(\Delta\phi\) are, respectively, the angular frequency offset and phase offset between transmit and receive lasers.

In the RF OFDM receiver, the down-converted OFDM signal is first sampled with an analog-to-digital converter (ADC). Then, the signal needs to go through three levels of synchronizations before the symbol decision can be made.

The three levels of synchronizations are (i) FFT window synchronization where OFDM symbol is properly delineated to avoid ISI, (ii) frequency synchronization, namely, frequency offset \(\omega_\text{off}\) needs to be estimated and compensated, and (iii) the subcarrier recovery, where each subcarrier channel is estimated and compensated.

Assuming successful completion of DFT window synchronization and frequency synchronization, the RF OFDM signal through DFT of the sampled value of Eq. (5.19) becomes

\[\tag{5.21}r_{ki}=e^{\phi_i}h_{ki}c_{ki}+n_{ki}\]

where \(r_{ki}\) is the received information symbol, \(\phi_i\) is the OFDM symbol phase (OSP) or common phase error (CPE), \(h_{ki}\) is the frequency domain channel transfer function, and \(n_{ki}\) is the noise. The third synchronization of subcarrier recovery involves estimation of OSP \(\phi_i\) and the channel transfer function \(h_{ki}\).

Once they are known, an estimated value of \(c_{ki}\), \(\hat{c}_{ki}\) is given by zero-forcing method as

\[\tag{5.22}\hat{c}_{ki}=\frac{h_{ki}^*}{|h_{ki}|^2}e^{-i\phi_i}r_{ki}\]

\(\hat{c}_{ki}\) is used for symbol decision or to recover the transmitter value \(c_{ki}\), which can subsequently be mapped back to the original transmitted digital bits.

This description of CO-OFDM processing has so far left out the pilot-subcarrier or training-symbol insertion where a proportion of the subcarriers or all the subcarriers in one OFDM symbol are known values to the receiver. The purpose of these pilot subcarrier or training symbol is to assist the above-mentioned three-level synchronization. Another important aspect of the CO-OFDM signal processing not involved is the error-correction coding involving the error-correction encoder/decoder and the interleaver/de-interleaver.

#### 2.1.3. Polarization-Mode Dispersion (PMD) Supported CO-OFDM

It is well-known that optical fiber can support two polarization modes. The propagation of an optical signal is influenced by the polarization effects including polarization coupling, polarization-dependent loss (PDL), and polarization-mode dispersion (PMD).

By utilizing the MIMO algorithm in digital signal processing (DSP), the capacity of CO-OFDM system can be doubled by using polarization-division multiplexing (PDM), and the impact of PMD can be digitally removed from the signal.

As shown in Figure 5.6, a two-input-two-output (TITO) scheme of CO-OFDM is usually applied to support polarization-multiplexed transmission in the presence of PMD. It consists of two sets of CO-OFDM transmitter and receiver, each pair for one polarization.

In such a scheme, because the transmitted OFDM information symbol \(\vec{c}_{ki}\) can be considered as polarization modulation or polarization multiplexing, the capacity is thus doubled compared with single-input-single-output (SISO) scheme.

As the impact of the PMD is to simply rotate the subcarrier polarization, it can be treated by channel estimation and constellation reconstruction. Therefore, the doubling of the channel capacity will not be affected by PMD. Due to the polarization-diversity receiver employed at the receive end, TITO scheme does not need polarization tracking at the receiver.

Similar to the single-polarization OFDM signal model described in Section 5.2, the OFDM time-domain signal, s(t) is described using Jones vector given by Shieh et al.

\[\tag{5.23}\pmb{s}(t)=\boldsymbol{\sum}_{i=-\infty}^{+\infty}\boldsymbol{\sum}_{k=-N_{sc}/2+1}^{N_{sc}/2}\pmb{c}_{ik}\boldsymbol{\prod}(t-iT_s)\exp(j2pf_k(t-iT_s))\]

\[\tag{5.24}\pmb{s}(t)=\begin{bmatrix}s_x\\s_y\end{bmatrix},\quad\pmb{c}_{ik}=\begin{bmatrix}c_{ik}^x\\c_{ik}^y\end{bmatrix}\]

\[\tag{5.25}f_k=\frac{k-1}{t_s}\]

\[\tag{5.26}\boldsymbol{\prod}(t)=\begin{cases}1,\quad(-\Delta_G\lt{t}\lt{t_s})\\0,\quad(t\le-\Delta_G,t\gt{t_s})\end{cases}\]

where \(s_x\) and \(s_y\) are the two polarization components in the time domain, \(\pmb{c}_{ik}\) is the transmitted OFDM symbol in the form of Jones vector for the \(k\)th subcarrier in the \(i\)th OFDM symbol, \(c_{ik}^x\) and \(c_{ik}^y\) are the two polarization elements for \(\pmb{c}_{ik}\), \(f_k\) is the frequency for the \(k\)th subcarrier, \(N_{sc}\) is the number of OFDM subcarriers, and \(T_s\), \(\Delta_G\), and \(t_s\) are the OFDM symbol period, guard interval length, and observation period, respectively.

The Jones vector \(\pmb{c}_{ik}\) is employed to describe generic OFDM information symbol regardless of the methods of the OFDM transmitter polarization configuration. In particular, the \(\pmb{c}_{ik}\) encompasses various modes of the polarization generation including single polarization, polarization multiplexing, and polarization modulation, as they all can be represented by the two-element Jones vector \(\pmb{c}_{ik}\). The transmitted Jones vectors can be recovered from the received Jones vectors by using training symbols.

We select a guard interval long enough to handle the fiber dispersion including PMD and CD. This time margin condition is given by

\[\tag{5.27}\frac{c}{f^2}|D_t|\cdot{N_{sc}}\cdot\Delta{f}+\text{DGD}_\text{max}\le\Delta_G\]

where \(f\) is the frequency of the optical carrier, \(c\) is the speed of light, \(D_t\) is the total accumulated chromatic dispersion in units of ps/pm, \(N_{sc}\) is the number of the subcarriers, \(\Delta{f}\) is the subcarrier channel spacing, and \(\text{DGD}_\text{max}\) is the maximum budgeted differential-group-delay (DGD), which is about 3.5 times of mean PMD to have sufficient margin.

Assuming long-enough symbol period, we arrive at the received symbol given by

\[\tag{5.28}\vec{c}_{ik}^{\;'}=e^{j\phi_i}\cdot{e^{j\Phi_D(f_k)}}\cdot{T_k}\cdot\vec{c}_{ik}+\vec{n}_{ik}\]

\[\tag{5.29}T_k=\boldsymbol{\prod}_{i=1}^{N}\exp\left\{\left(-\frac{1}{2}j\vec{\beta}_l{f_k}-\frac{1}{2}\vec{\alpha}_l\right)\vec{\sigma}\right\}\]

\[\Phi_D(f_k)=\pi\cdot{c}\cdot{D_t}\cdot{f_k^2}/f_\text{LD1}^2\]

where \(\pmb{c}_{ik}'=\left[c_{ik}'^x\;c_{ik}'^y\right]^T\) is the received information symbol in the form of the Jones vector for the kth subcarrier in the ith OFDM symbol, \(\pmb{n}_{ik}=\left[n_{ik}^x\;n_{ik}^y\right]^T\) is the noise including two polarization components, \(\pmb{T}_k\) is the Jones matrix for the fiber link, \(\Phi_D(f_k)\) is the phase dispersion owing to the fiber chromatic dispersion, and \(\phi_i\) is the OSP noise owing to the phase noises from the lasers and RF local oscillators (LOs) at both the transmitter and receiver. \(\phi_i\) is usually dominated by the laser phase noise.

## 3. Direct-Detection Optical OFDM (DDO-OFDM)

Direct-detection optical OFDM (DDO-OFDM) usually has simpler transmitter/ receiver than CO-OFDM, thus lower system cost. It has many variants that trade-off between the spectral efficiency and the system cost from a broad range of applications.

For instance, the first report of the DDO-OFDM takes advantage of the fact that the OFDM signal is more immune to the impulse clipping noise seen in CATV networks. Another example is single-sideband (SSB)-OFDM, which has been recently proposed by Djordjevic and Vasic and Lowery and Armstrong for long-haul transmission.

Tang et al. have proposed an adaptively modulated optical OFDM (AMO-OFDM) that uses bit and power loading showing promising results for both multimode fiber and short-reach SMF fiber links.

The common feature for DDO-OFDM is the use of a simple square-law photodiode at the receiver. DDO-OFDM can be divided into two categories according to how the optical OFDM signal is generated:

- Linearly mapped DDO-OFDM (LM-DDO-OFDM) where the optical OFDM spectrum is a replica of baseband OFDM
- Nonlinearly mapped DDO-OFDM (NLM-DDO-OFDM) where the optical OFDM spectrum does not display a replica of baseband OFDM.

In what follows, we discuss the principles and design choices for these two categories of direct-detection OFDM systems.

### 3.1. Linearly Mapped DDO-OFDM

As shown in Figure 5.7, the optical spectrum of an LM-DDO-OFDM signal at the output of the optical OFDM (O-OFDM) transmitter is a linear copy of the RF OFDM spectrum plus an optical carrier, which usually occupies 50% of the overall power.

The position of the main optical carrier can be one OFDM spectrum bandwidth away or right at the end of the OFDM spectrum. Formally, such type of DDO-OFDM can be described as

\[\tag{5.30}s(t)=e^{j2\pi{f_0}t}+\alpha{e}^{j2\pi(f_0+\Delta{f})t}\cdot{s_B}(t)\]

where \(s(t)\) is the optical OFDM signal, \(f_0\) is the main optical carrier frequency, \(\Delta{f}\) is guard band between the main optical carrier and the OFDM band (Figure 5.7), and \(\alpha\) is the scaling coefficient that describes the OFDM band strength related to the main carrier. \(s_B(t)\) is the baseband OFDM signal given by

\[\tag{5.31}s_B=\boldsymbol{\sum}_{k=-\frac{1}{2}N_{sc}+1}^{\frac{1}{2}N_{sc}}c_ke^{j2\pi{f_kt}}\]

where \(c_k\) and \(f_k\) are, respectively, the OFDM information symbol and the frequency for the \(k\)th subcarrier. For explanatory simplicity, only one OFDM symbol is shown in (5.31). After the signal passing through fiber link with chromatic dispersion, the OFDM signal can be approximated as

\[\tag{5.32}\begin{align}r(t)&=e^{j(2\pi{f_0}t+\Phi_D(-\Delta{f})+\phi(t))}+\alpha{e}^{j(2\pi(f_0+\Delta{f})t+\phi(t))}\\&\quad\cdot\boldsymbol{\sum}_{k=-\frac{1}{2}N_{sc}+1}^{\frac{1}{2}N_{sc}}c_{ik}e^{(j2\pi{f_k}t+\Phi_D(f_k))}\end{align}\]

\[\tag{5.33}\Phi_D(f_k)=\pi\cdot{c}\cdot{D_t}\cdot{f_k^2}/f_0^2\]

where \(\Phi_D(f_k)\) is the phase delay due to chromatic dispersion for the \(k\)th subcarrier. \(D_t\) is the accumulated chromatic dispersion in the unit of ps/pm, \(f_0\) is the center frequency of optical OFDM spectrum, and \(c\) is the speed of light.

At the receiver, the photodetector can be modeled as a square-law detector and the resultant photocurrent signal is

\[\tag{5.34}\begin{align}I(t)&\propto|r(t)|^2=1+2\alpha\text{Re}\left\{e^{j2\pi\Delta{f}t}\boldsymbol{\sum}_{k=-\frac{1}{2}N_{sc}+1}^{\frac{1}{2}N_{sc}}c_{ik}e^{(j2\pi{f_kt}+\Phi_D(f_k)-\Phi_D(-\Delta{f}))}\right\}\\&\quad+|\alpha^2|\boldsymbol{\sum}_{k_1=-\frac{1}{2}N_{sc}+1}^{\frac{1}{2}N_{sc}}\boldsymbol{\sum}_{k_2=-\frac{1}{2}N_{sc}+1}^{\frac{1}{2}N_{sc}}c_{k_2}^*c_{k_1}e^{(j2\pi(f_{k_1}-f_{k_2})t+\Phi_D(f_{k_1})-\Phi_D(f_{k_2}))}\end{align}\]

The first term is a DC component that can be easily filtered out. The second term is the fundamental term consisting of linear OFDM subcarriers that are to be retrieved. The third term is the second-order nonlinearity term that needs to be removed.

There are several approaches to minimize the penalty due to the second-order nonlinearity term:

- (A) Offset SSB-OFDM. Sufficient guard band is allocated such that the linear term and the second-order nonlinearity of the RF spectra are nonoverlapping. As such, the third term in Equation 5.34 can be easily removed using an RF or DSP filter.
- (B) Baseband Optical SSB-OFDM. \(\alpha\) coefficient is reduced as much as possible such that the distortion as result of the third-term is reduced to an acceptable level.
- (C) Subcarrier interleaving. From Equation 5.34, it follows that if only odd subcarriers are filled, that is, \(c_k\) is nonzero only for the odd subcarriers, the second-order intermodulation will be at even subcarriers, which are orthogonal to the original signal at the odd subcarrier frequencies. Subsequently, the third-term does not produce any interference.
- (D) Iterative distortion reduction. The basic idea is to go through a number of iterations of estimation of the linear term, and compute the second-order term using the estimated linear term, and removing the second-order term from the right side of Equation 5.34.

There are advantages and disadvantages among all these four approaches. For instance, Approach B has the advantage of better spectral efficiency, while sacrificing receiver sensitivity. Approach D has both good spectral efficiency and receiver sensitivity, but has a burden of computational complexity.

Figure 5.8 shows one offset SSB-OFDM proposed by Lowery et al.. They show that such DDO-OFDM can mitigate enormous amount of chromatic dispersion up to 5000 km standard SMF (SSMF) fiber.

The proof-of-concept experiment was demonstrated by Schmidt et al. from the same group for 400 km DDO-OFDM transmission at 20 Gb/s.The simulated system is 10 Gb/s with 4QAM modulation with the baud rate around 5 GHz.

In the electrical OFDM transmitter, the OFDM signal is up-converted to an RF carrier at 7.5 GHz generating an OFDM band spanning from 5 to 10 GHz. The RF OFDM signal is fed into an optical modulator.

The output optical spectrum has the two OFDM sidebands that are symmetric across the main optical subcarrier. An optical filter is then used to filter out one OFDM sideband.

This SSB is crucial to ensure that there is one-to-one mapping between the RF OFDM signal and the optical OFDM signal. The power of the main optical carrier is optimized to maximize the sensitivity. At the receiver, only one photodetector is used.

The RF spectrum of the photocurrent is depicted as an inset in Figure 5.8. It can be seen that the second-order intermodulation, the third-term in Equation 5.34 is from DC to 5 GHz, whereas the OFDM spectrum, the second term in Equation 5.34, spans from 5 to 10 GHz.

As such, the RF spectrum of the intermodulation does not overlap with the OFDM signal, signifying that the intermodulation does not cause detrimental effects after proper electrical filtering.

### 3.2. Nonlinearly Mapped DDO-OFDM (NLM-DDO-OFDM)

The second class of DDO-OFDM is nonlinearly mapped OFDM. Instead of linearly mapping the electric field (baseband OFDM) to the optical field, NLM-DD-OFDM aims to obtain a linear mapping between baseband OFDM and optical intensity.

For simplicity, we assume the generation of NLM-DDO-OFDM using the direct modulation, the waveform after the direct modulation can be expressed as

\[\tag{5.35}E(t)=e^{j2\pi{f_0t}}A(t)^{1+jC}\]

\[\tag{5.36}A(t)\equiv\sqrt{P(t)}=A_0\sqrt{1+\alpha\text{Re}(e^{(j(2\pi{f_{\text{IF}}}t)}\cdot{s_B}(t))}\]

\[\tag{5.37}s_B(t)=\boldsymbol{\sum}_{k=-\frac{1}{2}N_{sc}+1}^{\frac{1}{2}N_{sc}}c_ke^{j2\pi{f_kt}}\]

\[\tag{5.38}m\equiv\alpha\sqrt{\boldsymbol{\sum}_{k=-\frac{1}{2}N_{sc}+1}^{\frac{1}{2}N_{sc}}|c_k|^2}\]

where \(E(t)\) is the optical OFDM signal, \(A(t)\) and \(P(t)\) are the instantaneous amplitude and power of the optical OFDM signal, \(c_k\) is the transmitted information symbol for the \(k\)th subcarrier, \(C\) is the chirp constant for the direct-modulated DFB laser, \(f_\text{IF}\) is the IF for the electrical OFDM signal for modulation, \(m\) is the optical modulation index, \(\alpha\) is a scaling constant to set an appropriate modulation index \(m\) to minimize the clipping noise, and \(s_B(t)\) is the baseband OFDM signal.

Assuming the chromatic dispersion is negligible, the detected current is

\[\tag{5.39}I(t)=|E(t)|^2=|A|^2=A_0(1+\alpha\text{Re}(e^{(j2\pi{f_\text{IF}t})}\cdot{s_B}(t)))\]

Equation 5.39 shows that the photocurrent contains a perfect replica of the OFDM signal \(s_B(t)\) with a DC current. We also assume that the modulation index \(m\) is small enough so that the clipping effect is not significant.

Equation 5.39 shows that by using NLM-DDO-OFDM without chromatic dispersion, the OFDM signal can be perfectly recovered. The fundamental difference between the NLM- and LM-DDO-OFDM can be gained by studying their respective optical spectra.

Figure 5.9 shows the optical spectra of NLM-DDO-OFDM using (i) direct modulation with the chirp coefficient C of 1 in (5.35) and modulation index \(m\) of 0.3 in (5.38) and (ii) offset SSB-OFDM.

It can be seen that, in sharp contrast to SSB-OFDM, NLM-DDO-OFDM has a multiple of OFDM bands with significant spectral distortion, indicating the nonlinear mapping from the baseband OFDM to the optical OFDM. The consequence of this nonlinear mapping is fundamental.

Any type of dispersion (such as chromatic dispersion, polarization dispersion, and modal dispersion) occurs in the link result in the fact that the linear baseband OFDM signal can no longer be recovered; namely, any dispersion will cause the nonlinearity for NLM-DD-OFDM systems.

In particular, unlike SSB-OFDM, the channel model for direct-modulatedOFDMis no longer linear under any form of optical dispersion. Subsequently, NLM-DD-OFDM only fits short-haul application such as multimode fiber for local area networks (LAN), or short-reach single-mode fiber (SMF) transmission.

This class of optical OFDM has attracted attention recently due to its low cost. Some notable works of NLM-DD-OFDM are experimental demonstrations and analysis of optical OFDM over multimode fibers and compatible SSB-OFDM (CompSSB) proposed by Schuster et al. to achieve higher spectral efficiency than offset SSB-OFDM.

## 4. Self-Coherent Optical OFDM

While long-haul networks have witnessed a capacity evolution to multi-Terabit during the last decade with the revival of coherent communications, short-reach networks within distance of hundreds of kilometers need to increase their capacity per wavelength beyond 40 or even 100 Gb/s to meet the ever-increasing traffic demand.

Different from long-haul communications, these short-reach networks require massive number of transceivers across diverse geographic areas. Direct detection (DD) can significantly lower the expense compared with the coherent counterpart, making it suitable for short-reach applications.

However, the conventional single-ended PD-based direct-detection systems cannot undertake the task due to two fundamental bottlenecks: (i) chromatic dispersion (CD)-induced signal fading due to the lack of receiver phase diversity, which limits the transmission distance and (ii) second-order nonlinearity due to photodetection, which limits the system capacity.

A multitude of solutions have been proposed to overcome the problems, among which a host of self-coherent (SCOH) systems have attracted most attention. The SCOH systems send both the modulated signal (S) and the carrier (C) at the transmitter; at the receiver, the carrier serves as a reference and beats with the signal during photodetection, namely the SCOH detection.

SCOH provides the following advantages compared with conventional DD:

- The receiver sensitivity increases dramatically with the help of carrier; and it is even not surprising that several SCOH receivers achieve the phase diversity.
- The signal and the carrier are generated with the same laser source, which guarantees that the phase between S and C are coherent with each other at the transmitter; therefore, SCOH naturally mitigates the systems vulnerability (especially the coherent optical OFDM) to the laser frequency offset and phase noise.
- The SCOH system is capable of being compatible with the nowadays powerful DSP technology, which provides the flexibility when deployed to the short-reach applications.

In this tutorial, we review the current status of the SCOH systems. Depending on the receiver structure, we divide them into two categories:

- DD based on single-ended photodetector (PD)
- DD based on balanced receiver.

### 4.1. Single-Ended Photodetector-Based SCOH

Traditional direct-detection (DD) scheme uses intensity modulation at the transmitter. The modulator is driven by a real-valued RF signal, leading to a Hermitian symmetric optical spectrum.

This method wastes half of the optical spectral efficiency; more importantly, it gives rise to the problem of CD-induced power fading, which limits the transmission distance.

The most straightforward approach to avoid the problem is to use SSB field modulation, which is adopted by a set of OFDM-based SCOH systems as illustrated in Figure 5.10.

The offset-SSB scheme shown in Figure 5.10(a) first up-converts the RF signal with an electrical IQ mixer; then uses the output to drive an MZM, which is biased off the null point to provide an optical carrier.

The frequency gap between the signal (S) and the carrier (C) is reserved for the second-order SSBN, whose bandwidth is equal to that of the signal. The electrical spectral efficiency (E-SE) is sacrificed by half in this case.

To increase the E-SE, the virtual-SSB OFDM [42] is proposed shown by Figure 5.10(b). Virtual-SSB arranges one inserted RF tone at the leftmost OFDM subcarrier; then transfers the signal along with the RF tone to a complex signal by the IFFT; the real and imaginary parts of the signal drive an optical I/Q modulator biased at the null point.

The original optical carrier is thus suppressed, and the new optical carrier is provided by the inserted RF tone located at the left edge of the signal spectrum.

The RF transmitter structure is simplified compared with offset-SSB; moreover, virtual-SSB does not reserve the frequency gap between the S and C, which effectively doubles the E-SE.

To eliminate the SSBN, an iterative SSBN estimation and cancellation technique is introduced at the receiver using DSP, which sacrifices the system computational complexity mainly induced by the iterative FFT operation inside the cancellation algorithm.

As the electrical bandwidth is the dictating factor for the transponder cost, it is necessary to apply the double sideband (DSB) modulation to further increase the E-SE.

The block-wise phase switching (BPS) DD follows the idea. In Figure 5.10(c), for two identical consecutive signal blocks, the phase of the main carrier is switched by 90\(^\circ\) or 180\(^\circ\) at the transmitter; while the receiver recovers the DSB signal with phase diversity. It is noted that BPS is suitable for both OFDM and single-carrier systems.

### 4.2. Balanced Receiver-Based SCOH

After field modulation is introduced into SCOH schemes, it becomes meaningful for DD receivers to achieve the phase diversity. Balanced receiver-based DD provides the following advantages at the sacrifice of system expense:

- Phase-diverse signals can be recovered;
- The SSBN can be eliminated spontaneously by the balanced photodetector instead of the iterative SSBN cancellation, which significantly increases the computational complexity.

A classical SCOH enabling coherent-like detection at the receiver without LO is as follows: (i) transmitter sends the carrier (C) along with the signal (S); (ii) at reception, C and S are first separated into two paths, which are then served as the two inputs of the standard coherent receiver.

The polarization multiplexing (POL-MUX) DD first realizes this by separating the S and C in frequency domain by a narrow bandwidth LPF, as shown in Figure 5.11(a).

A frequency gap between the S and C needs to be reserved for (i) laser wavelength drift and (ii) filter bandwidth. In fact, the laser may have a frequency drift of 10 GHz; moreover, it is quite expensive for current commercial filter to achieve a low-pass bandwidth within 10 GHz.

Therefore, the POL-MUX DD has a huge system cost while still wastes the SE conspicuously. The signal carrier interleaved (SCI) DD avoids this problem by separating the S and C in time domain.

Two signal blocks are followed by one carrier block at the transmitter. At reception, the stream is divided into two paths as shown in Figure 5.11(b), while the lower path is delayed by one block length. Therefore, the S and C from different paths can be mixed with each other at the coherent receiver. SCI-DD sacrifices the SE by one-third.

### 4.3. Stokes Vector Direct Detection

Recently, the Stokes vector (SV) DD is proposed, which at the first time achieves 100% SE with reference to the single polarization-modulated coherent detection.

Compared with POL-MUX coherent systems, SV-DD has the following advantages: the transmitter only requires one-polarization modulation; the receiver does not need an LO; the DSP is much simpler due to using less number of FFTs, and the laser frequency offset and phase noise need not be tracked.

SV-DD achieves 100% spectral efficiency with reference to the single-polarization coherent detection. SV-DD receiver is polarization independent. Although several prior DD schemes have demonstrated dual-polarization modulation and reception, the transmitter complexity approaches that of the POL-MUX coherent systems.

Using the SV-DD scheme, it has experimentally demonstrated the first successful direct detection of 160 Gb/s single-wavelength single-polarization-modulated signal after transmission over 160-km SSMF.

In the SV-DD scheme, the signal (S) and carrier (C) are, respectively, placed onto two orthogonal polarizations at the transmitter. The signal can be expressed by a Jones vector \(\pmb{J}=[S\;C]^T\), where a vector or matrix is represented in bold font and superscript “\(T\)” stands for transpose.

Converting the Jones vector to the Stokes space, we arrive at the Stokes vector \(\pmb{S}=[s_1,s_2,s_3]^T=[|s|^2-|c|^2,\text{Re}(s\cdot{c}^*),\text{Im}(s\cdot{c^*})]^T\), where \(\text{Re}( )\) and \(\text{Im}( )\) represent real and imaginary parts of a complex number.

This Stokes vector \(\pmb{S}\) can be detected as follows: the signal is split with a polarization beam splitter (PBS) into two outputs of \(X\) and \(Y\), respectively. Since the signal polarization has been randomly rotated in the fiber, the received signals \(X\) and \(Y\) are the mixture of the transmitted signals of \(S\) and \(C\).

One of the critical tasks of SV-DD is to acquire this polarization rotation (PR) between the input and output signals. Compared with the coherent detection for which the PR is accomplished in the Jones space, SV-DD is done in the Stokes space by the subsystem shown in Figure 5.11(c).

Both \(X\) and \(Y\) are split with 3 dB couplers, which are identified as ports 1 and 4, respectively. Ports 1 and 4 are fed into a balanced PD directly, resulting in the output of \(|X|^2-|Y|^2\), which is the first component of SV.

Ports 2 and 3 are fed into a standard balanced receiver consisting of an optical hybrid and two balanced PDs. The output of the balanced receiver is \(\text{Re}(X\cdot{}Y^*)\) and \(\text{Im}(X\cdot{}Y^*)\), respectively, which are the second and third components of SV. For simplicity, the above-mentioned analysis omits some simple scaling constants for coupler and photodiode outputs.

To recover the signal, the remaining task is to acquire the 3×3 SV rotation matrix (RM) of the channel and rotate the Stokes vectors at the receiver back to those at the transmitter.

The effectiveness of the SV-DD algorithm is fully revealed: (i) combining the second and third components of SV, we have the final output of \(S\cdot{}C^*\) that has full phase diversity of signal \(S\), from which the input signal \(S\) is fully recovered without being affected by the CD induced fading and (ii) the nonlinearity term is completely lumped into the first SV component without affecting the recovered signals, which are derived from second and third SV components.

SV-DD signal is complex modulated and no frequency gap is required. Therefore, SV-DD has an electrical SE of four times of offset-OFDM and two times of virtual single-sideband (VSSB) or block-wise phase switching (BPS). This makes SV-DD an ideal DD format for achieving high data rate with reasonable electrical bandwidth.

## 5. Discrete Fourier Transform Spread OFDM System (DFT-S OFDM)

Optical communication has rapidly advanced toward 1-Tb/s and beyond transport. As the available bandwidth of SSMF is limited, high spectral efficiency (SE) becomes an important issue.

CO-OFDM has become one of the promising candidates due to its high SE and resilience to linear channel impairments such as CD. Experimental demonstration at data rate of 1-Tb/s and beyond has been achieved using either Nyquist wavelength-division multiplexing (WDM) or CO-OFDM.

Despite many promising features, CO-OFDM system suffers from high PAPR, which leads to inferior tolerance to fiber nonlinearity compared with SC system, and has become an obstacle to its practical implementation in long-haul transmission systems.

Although specialty fibers such as ultra-large area fiber (ULAF) or low-loss low-nonlinearity pure silica core fiber (PSCF) with Raman amplification has been suggested to further extend the reach of transmission systems, it could be either more expensive or not compatible with the deployed links.

To solve the nonlinear tolerance problem, DFT-S OFDM has recently been proposed with an attractive feature of much reduced PAPR. DFT-S OFDM is called single-carrier frequency-division multiplexing (SC-FDM) that has been incorporated into the 3GPP-LTE standard in uplink for the next generation mobile system with many interesting features.

Furthermore, benefited from the sub-band or subwavelength accessibility of CO-OFDM, properly designed multiband DFT-S OFDM (MB-DFT-S OFDM) potentially have better nonlinearity tolerance over either conventional CO-OFDM or SC system for ultra-high-speed transmission.

The nonlinearity advantage of MB-DFT-S OFDM has been verified through simulation. In addition, optical transmission experiments utilizing the DFT-S OFDM or SC-FDM have been demonstrated very recently by several groups, which shows a potential advantage of better nonlinear tolerance and high SE. The reconfigurable optical add/drop multiplexer (ROADM) functionally has also been demonstrated on SC-FDM superchannel.

### 5.1. Principle of DFT-S OFDM

The DSP at the transmitter and receiver of a DFT-S OFDM system is shown in Figure 5.12. For comparison, the signal processing of conventional OFDM is also illustrated.

Similar to the conventional OFDM, signal processing in DFT-S OFDM is repetitive in a few different time intervals called blocks. At the input to the transmitter, a baseband modulator transforms the binary serial input data to a multilevel modulation formats such as M-ary phase-shift keying (M-PSK) or M-ary quadrature amplitude modulation (M-QAM).

The most commonly used modulation formats in OFDM system include binary phase-shift keying (BPSK), QPSK, 16QAM, and 64QAM. The modulation format can be made adaptive by the system to match the current channel conditions, and thereby the transmission data rate.

The transmitter next groups the modulation symbols \(X_n\) into many OFDM blocks (serial to parallel), each containing \(N\) symbols. After that, the first unique step in DFT-S OFDM is an \(N\)-point DFT before the subcarrier mapping operation to produce a frequency-domain representation \(X_k\) of the input symbols.

Then, each of the \(N\)-point DFT outputs is mapped to one of the \(M\) (>\(N\)) subcarriers in a conventional OFDM that can be transmitted. As in conventional OFDM, the typical value of \(M\) must be a power of 2 (e.g., 64, 128, or 256).

The choice of \(N\) in DFT-S OFDM must follow the relationship \(N=M/Q\), which means \(N\) must be an integer submultiple of \(M\). \(Q\) is defined as the bandwidth expansion factor of the symbol sequence.

DFT-S OFDM can handle \(Q\) simultaneous transmissions without co-channel interference (CCI) when each terminal is allocated \(N\) symbol per block. The result of the subcarrier mapping is the set \(\tilde{X}_l\) (\(l =0,1,2, … ,M-1\)) of complex subcarrier amplitudes, where the \(N\) of amplitudes are nonzero.

As in conventional OFDM, an \(M\)-point IDFT transforms the subcarrier amplitudes to a complex time-domain signal \(\tilde{X}_m\). Each \(\tilde{X}_m\) then modulates a single frequency carrier and all the modulated symbols are transmitted sequentially.

The transmitter then inserts CP in order to provide a guard time to prevent ISI. The modulated DFT-S OFDM signal is then launched into a wireless or fiber-optic channel for transmission.

After transmission, the receiver first transforms the time-domain received signal into the frequency domain via DFT, de-maps the subcarriers, and then performs frequency-domain equalization to remove the channel distortion.

Minimum mean square error (MMSE) frequency-domain equalization method is generally preferred over zero forcing (ZF) due to the robustness against noise. Subsequently, the equalized symbols are transformed back from frequency- to time-domain via IDFT, and finally the detection and decoding is performed.

In DFT-S OFDM, since the DFT size \(M > N\), several approaches have been proposed to the mapping of transmission symbols \(X_k\) to DFT-SOFDM subcarriers. These approaches can be divided into two categories: distributed and localized.

Distributed subcarrier mapping means the DFT outputs of the input data are allocated over the entire bandwidth with the unused subcarriers filled with zeros, resulting in a noncontinuous comb-shaped spectrum. The well-known interleaved DFT-S OFDM (IDFT-S OFDM), or so-called interleaved SC-FDMA (IFDMA) is at special case of distributed DFT-S OFDM.

On the contrary, localized subcarrier mapping means consecutive subcarriers are occupied by the DFT outputs of the input data, resulting in a continuous spectrum that occupies a fraction of the total available bandwidth.

For IDFT-S OFDM, time symbols are simply a repetition of the original input symbols with a systematic phase rotation applied to each symbol in the time domain. Therefore, the PAPR of IDFT-S OFDM signal is the same as in the case of a conventional single carrier signal.

In the case of localized DFT-S OFDM (LDFT-S OFDM), or so-called localized SC-FDMA (LFDMA), the time signal has exact copies of input time symbols in \(N\) sample positions. The other \(M-N\) time samples are weighted sums of all the symbols in the input block.

As we can see from Figure 5.12, the first obvious difference between conventional OFDM and DFT-S OFDM is the additional pair of \(N\)-point DFT/IDFT in the DFT-S OFDM, with DFT in the transmitter and IDFT in the receiver.

The second fundamental difference between DFT-S OFDM and conventional OFDM is in the receiver equalization and detection processes. In conventional OFDM, since the data symbol is carried by individual subcarriers, channel equalization, channel inversion, and data detection is performed individually on each subcarrier.

Channel coding or power/rate adaptation is required for OFDM to protect individual subcarriers if there are nulls in the channel spectrum, which would severely degrade the system performance since there is essentially no way to recover the data affected by the null.

In the case of DFT-S OFDM, channel equalization and inversion is done similarly in the frequency domain but data detection is performed after the frequency-domain-equalized data are reverted back to time domain by IDFT.

Hence, it is more robust to spectral nulls compared with conventional OFDM since the noise is averaged out over the entire bandwidth. Additional advantages of DFT-S OFDM include less sensitivity to the carrier frequency offset (CFO), less sensitivity to the laser phase noise, and less nonlinear distortion due to the much reduced PAPR, while conventional OFDM suffers due to the multicarrier nature of OFDM.

### 5.2. Unique-Word-Assisted DFT-S OFDM (UW-DFT-S OFDM)

Unique-word (UW) was first proposed for single-carrier frequency-domain equalization (SC-FDE) systems and has been extensively studied in wireless communications. The data pattern structure of UW-DFT-S OFDM for two polarizations is illustrated in Figure 5.13.

The unique-words (UWs), normally comprising a Zadoff–Chu (ZC) sequence and an optional GI is inserted periodically at both ends of the payload. The Zadoff–Chu sequence is generated with the following equation:

\[\tag{5.40}x_u(n)=\begin{cases}e^{-j\frac{\pi{un}(n+1)}{N_\text{ZC}}}\quad\text{if }N_\text{ZC}\text{ is odd}\\e^{-j\frac{\pi{un^2}}{N_\text{ZC}}}\quad\quad\text{if }N_\text{ZC}\text{ is even}\end{cases}\quad(0\le{n}\le{N_\text{ZC}-1})\]

where \(x_u(n)\) is the Zadoff–Chu sequence, \(N_\text{ZC}\) is sequence length, and \(u\) is an integer relatively prime of \(N_\text{zc}\).

A similar OFDM symbol structure can be drawn for the second polarization by using different UWs. The reason to use two UWs within one OFDM symbol is the compatibility with polarization diversity.

The first and second UWs are orthogonal to each other when combining the two OFDM symbols for two polarizations in a Jones vector form. The two UWs for the two polarizations (see Figure 5.13), \(\begin{pmatrix}\text{UW}_{x1}\\\text{UW}_{y1}\end{pmatrix}\) and \(\begin{pmatrix}\text{UW}_{x2}\\\text{UW}_{y2}\end{pmatrix}\) are given by

\[\tag{5.41}\begin{pmatrix}\text{UW}_{x1}&\text{UW}_{x2}\\\text{UW}_{y1}&\text{UW}_{y2}\end{pmatrix}=\begin{pmatrix}\text{UW}&-\text{cshift}(\text{UW})^*\\\text{cshift}(\text{UW})&\text{UW}^*\end{pmatrix}\]

where cshift(\(\cdot\)) denotes a circular shift of the sequence by half of the sequence length and “\(\ast\)” denotes complex conjugate. The circular shift ensures that UWs for two polarizations, for example, UW\(_{x1}\) and UW\(_{y1}\) are uncorrelated so long as the channel length is shorter than half of the unique-word length. The short UWs in each OFDM symbol can be used for multiple purposes: timing synchronization, channel estimation, phase estimation, and so on.

## 6. OFDM-Based Superchannel Transmissions

In the past few decades, optical communication systems have evolved rapidly thanks to the advance of electronics such as high-speed DSP and fiber-optic technology.

For high-speed optical transmission systems, multiplexing technique for the optical signals has been extensively studied and implemented. One of the most significant schemes is the WDM, in which the signals are multiplexed in wavelength (or frequency) domain as wavelength channels.

The WDM transmission technique can be classified by the frequency spacing between the adjacent channels as the coarseWDM (CWDM), dense WDM (DWDM), or even high-density WDM (e.g., superchannel transmission). These patterns are characterized by the ratio between the channel spacing \(\Delta{f}\) and the modulation symbol rate of the channel \(B\).

Coherent detection and DSP enable signals with extremely narrow guard band (or no guard band) can be recovered with minimum sacrifice of receiver sensitivity. The associated technologies in single-carrier cases are termed as “quasi-Nyquist” WDM representing the scenario that \(1\le\Delta{f}/B\le1.2\), “Nyquist” WDM for \(\Delta{f}/B=1\), and “super-Nyquist” WDM for \(\Delta{f}/B\lt1\).

Given that signals are densely packed, spectral overlapping may possibly affect the system performance. Thus, usually prefiltering, either done optically or electronically, is required to mitigate crosstalk. For instance, optical prefiltering has been utilized in “quasi-Nyquist” WDM and “super-Nyquist” WDM, and electronic prefiltering has been used.

One of the very elegant ways to fundamentally eliminate the crosstalk among different wavelength channels and achieve the “Nyquist” condition is to perform OFDM modulation. The OFDM modulation guarantees crosstalk-free reception of symbol-rate-spaced channels without any prefiltering.

The “OFDM conditions” essentially mean the following: First, the carrier spacing should be equal to the symbol rate, which requires the carriers on which the modulation is imprinted to be frequency locked. Second, to prevent ISI, the time window for modulation and demultiplexing should be aligned. Failing to do so results in large crosstalk and destroys the orthogonality condition.

It is worth noting that after long distance transmission while the fiber dispersion becomes significant, the symbols of neighboring carriers can be severely displaced. In this scenario, dispersion compensation can be adopted to rewind the orthogonality condition. Third, sufficient bandwidth is needed at the transmitter and the receiver to modulate each subcarrier, due to the fact that the spectral representative of the modulated symbols is usually a sinc function. In other words, sufficient oversampling is required to capture most of the sinc function for each of the modulated subcarriers.

### 6.1. No-Guard-Interval CO-OFDM (NGI-CO-OFDM) Superchannel

In the conventional CO-OFDM, the freedom to select the OFDM symbol length (namely the size of FFT, NFFT) is constrained by the fiber dispersion, particularly the CD and PMD.

The CP inserted between OFDM symbols offers an effective approach to remove the ISI induced by the CD and PMD. We use the percentage of GI (GI%) to characterize the SE penalty caused by the GI, defined as the ratio between the GI length \(N_\text{GI}\) and \(N_\text{FFT}\).

A given GI% limits the minimum value of \(N_\text{FFT}\) for a fixed dispersion value. Although the GI% can be simply reduced by increasing the OFDM symbol length, this will inevitably increase the CO-OFDM vulnerability to the fiber nonlinearity, laser phase noise, and CFO. In short, there is a trade-off between choosing a small \(N_\text{FFT}\) to achieve better receiver sensitivity and a small GI% to achieve larger SE.

For the conventional CO-OFDM demonstrated in previous works, the GI% was typically kept in the range of 10–25%. The length of GI becomes larger for systems with higher baud rate and longer transmission distance, making GI one of the dominant factors to limit the system SE.

To increase the SE as large as possible, the no-guard-interval (NGI) CO-OFDM is proposed in 2008. Instead of using the GI to remove the ISI induced by the dispersion, NGI-CO-OFDM compensates the CD and PMD at the receiver by enabling the DSP, namely the electronic dispersion compensation (EDC).

The transmitter configuration maintains the same as the conventional CO-OFDM, in which the PDM is applied to double the spectral efficiency. The receiver front-end consists of a polarization-diversity optical hybrid, four balanced photodetectors (B-PDs), and four ADCs.

The outputs of the ADCs are the in-phase and quadrature part of the dual polarizations. The receiver back-end contains several unique DSP procedures for NGI-CO-OFDM, as shown in Figure 5.14.

The back-end first applies the EDC to the signals of the two polarizations, respectively, by using the overlap frequency domain equalization, which includes several FFT and IFFT pairs.

Then, carrier separation is conducted by shifting each subcarrier to the baseband. The data sequence of each subcarrier is obtained by the FFT. The polarization demultiplexing as well as the channel equalization is fulfilled by the adaptive equalizer, such as the least mean square (LMS) algorithm and the constant modulus algorithm (CMA), which is widely applied in the single-carrier systems.

The remaining DSP is the same as the conventional CO-OFDM. The carrier recovery includes the frequency offset compensation and the phase estimation. The final symbol decision can be made after all the DSP procedure and the BER can be calculated.

The DSP of NGI-CO-OFDM has two major differences compared with the conventional CO-OFDM. First, the CD compensation is added before any further DSP; second, the adaptive equalization is adopted for the polarization demultiplexing and channel estimation, instead of the classic training symbol (TS)-aided equalization in CO-OFDM.

Alternatively, a modified NGI-CO-OFDM, called zero GI (ZGI) CO-OFDM, uses the training symbols with CP, followed by the data symbols without CP. Therefore, the channel estimation can maintain its accuracy without being affected by the dispersion.

The channel estimation is conducted (in the form of a 2 × 2 matrix for each subcarrier) from the OFDM demodulator once the TSs have been processed, and then the PMD compensation can be achieved by applying the inverse of the channel matrix.

The effectiveness of NGI-CO-OFDM has been verified by several remarkable experiments. Sano et al. demonstrate the 13.4-Tb/s (134 × 111-Gb/s/ch) NGI-CO-OFDM transmission over 3600 km of SMF in 2008; while Liu et al. demonstrate the transmission of a 1.2-Tb/s 24-Carrier NGI-CO-OFDM super-channel over 7200-km of ultra-large-area fiber (ULAF) in 2009.

NGI-COOFDM provides an effective approach to maximize the system SE, without any need to consider the large dispersion induced by the ultra-long transmission distance and wide optical bandwidth.

However, NGI-CO-OFDM significantly sacrifices the receiver computational complexity. Therefore, we would rather conclude that there is always a trade-off between the SE and the computational complexity, and NGI-CO-OFDM offers one extreme scheme in terms of SE.

### 6.2. Reduced-Guard-Interval CO-OFDM (RGI-CO-OFDM) Superchannel

Section 5.1 described the implementation of NGI-CO-OFDM scheme for the improvement of the SE. NGI-CO-OFDM removes the GI completely and compensates the fiber dispersion at the receiver using blind channel equalization. However, since no GI is added to the OFMD symbol, NGI-CO-OFDM suffers from the ISI caused by the transmitter bandwidth limitation, and complex equalization algorithm is needed at the receiver to compensate the effect of PMD.

For a more cost-effective DSP at the receiver, RGI-CO-OFDM scheme is proposed. In this scheme, the ISI with short memory, such as the transmitter bandwidth limitations or PMD-induced ISI, is accommodated by a short length of GI between the adjacent OFDM symbols, while the ISI with long memory, such as CD-induced ISI, is compensated at the receiver using EDC-based on DFT, IDFT and overlap-add. By using RGI-CO-OFDM, the DSP complexity can get greatly reduced compared with NGI-CO-OFDM, and the GI length and OFDM size can be much reduced compared with the conventional OFDM.

Consider a 112-Gb/s PMD-OFDM system over 1500-km SSMF (D = 17 ps/ nm/km) with baud rate of 56-GHz. For conventional OFDM, the DFT size is chosen to be 2048, and the GI length is chosen to be 512 samples to accommodate the CD, leading to the GI% of 25% (512/2048).

For RGI-CO-OFDM, four samples of GI (71.4 ps) is used to accommodate the instantaneous DGD, and the DFT size can be shortened to 128. The GI% is dramatically reduced to 3.13% (4/128) and the subcarrier spacing is increased by a factor of 16, relaxing the requirements on frequency locking between transmitter laser and receiver OLO.

A data rate of 448-Gb/s with SE of 7 b/s/Hz has been demonstrated using RGI-CO-OFDM with 16QAM modulation. The experimental setup in is shown in Figure 5.15.

The DFT of size is 128 with 75 filled with data. Four samples of GI are added into each symbol, resulting in a symbol length of 132. Training symbols (TSs) were inserted at the beginning of each OFDM frame. Inset (a) of Figure 5.15 shows the TSs after the PDM.

The first two TSs are used for frame synchronization, and the last two are for channel estimation. The sampling rate 10 GS/s is used, resulting in 22.4 Gb/s OFDM signal with a spectral bandwidth of 6.016 GHz.

The generated OFDM signal is then shifted by 6.016 GHz and combined with a time-delayed OFDM signal in the original frequency, forming a 44.8-Gb/s signal consisting of two decorrelated bands, as illustrated in inset (b) of Figure 5.15.

The 2-band signal is then expanded by a 5-comb generator to form a 10-band 224-Gb/s signal. A PDM emulator is used to enable the PDM, achieving the bit rate of 448-Gb/s signal within a bandwidth of 60.16 GHz.

The signal is launched into a transmission loop, consisting of four Raman-amplified 100-km ULAF spans. At the receiver, two optical local oscillators (OLOs) have to be used to recover the entire 448-Gb/s signal due to the ADC bandwidth limitation as shown in the inset (c) of Figure 5.15; 50-GS/s ADCs are used to collect the data.

New DSP module is added to compensate self-phase modulation (SPM) and CD through a multistep FFT-based algorithm, similar to that for single-carrier transmission.

Figure 5.16(a) shows the measured bit error ratio (BER) as a function of the optical signal-to-noise ratio. At BER=3.8 × 10\(^{-3}\), which is the threshold for 7% forward error correction (FEC), the required OSNR for the 448-Gb/s signal is 25 dB. Figure 5.16(b) shows the \(Q^2\) factor as a function of transmission distance. The reach distance is improved by 25% using the fiber nonlinearity compensation (NLC). With NLC, the mean BER of the 448-Gb/s signal is below 3 × 10\(^{-3}\) after 2000-km transmission.

### 6.3. DFT-S OFDM Superchannel

DFT-S OFDM superchannel is another potential candidate for high SE optical transports. The nonlinear advantage of multiband DFT-S OFDM for fiber transmission has been theoretically predicted and analyzed, enabling longer reach or higher SE (by using higher-order modulation) compared with conventional optical OFDM system at the same FEC threshold.

In multiband DFT-S OFDM systems, each sub-band is essentially filled with a digitally generated single-carrier signal. It has been numerically studied that DFT-S OFDM signal processes lower PAPR compared with conventional OFDM signal. For instance, the PAPR value of 7.5 dB occupies probability higher than 99.9%, and this PAPR is 3.2 dB lower than the value in conventional OFDM with the same probability.

Furthermore, one of the important findings of DFT-S OFDM for optical transmission is that there exists an optimal bandwidth within which the sub-bands should be partitioned. The insertion of UWs and the partitioned subcarrier mapping can slightly alter the performance of the DFT-S OFDM, but nevertheless the advantage of reduced PAPR remains significant. In the following section, we show experimental demonstration of 1-Tb/s PDM-QPSK UW-DFT-S OFDM transmission over 80-km span engineering SSMF and EDFA-only amplification that is compatible with most of the deployed links.

The experimental setup of 1-Tb/s UW-DFT-S OFDM system is shown in Figure 5.17. Our laser sources are 16 external-cavity lasers (ECLs) with low laser linewidth (<100 kHz) combined together and fed into an optical intensity modulator to impress three tones on each wavelength. The tone spacing is set at 6.5625 GHz driven by a synthesizer. The wavelength spacing of all the ECLs is carefully controlled and stabilized at ∼20.1875 GHz (1.615 nm).

Inset (i) of Figure 5.17 shows the generated densely spaced 48 tones monitored at point (i) using a high-resolution (0.01 nm) optical spectrum analyzer (OSA). After tone generation, the optical carrier is split into two equal branches by a 3-dB polarization-maintaining (PM) coupler and two arbitrary waveform generators (AWGs) are used to drive two IQ modulators to modulate different data patterns on the two polarizations. The baseband spectra for the data pattern are shown in the inset (ii) of Figure 5.17.

After IQ modulation, the optical outputs on the two polarizations are multiplexed with a polarization beam combiner. The optical spectrum of generated 16-channel PDM-OFDM signal occupying a bandwidth of 323 GHz is monitored at point (iii), shown as inset (iii) of Figure 5.17.

The 16-channel OFDM signal is then launched into a recirculating loop, which consists of two spans of 80-km SSMF with loss compensated by EDFAs. The received OFDM signal after transmission is shown in inset (iv) of Figure 5.17.

At the receiver, a 10-GHz optical filter is used to filter out one band each time, and the optical signal is converted to the electrical domain by an optical coherent receiver. The baseband signal is then received by a four-channel Tektronix oscilloscope at 50-GSa/s sampling rate. The DSP at the transmitter and receiver is shown in Figure 5.18.

The transmitted data pattern is generated following the procedure as shown in the “transmitter” block of Figure 5.18. In order to facilitate a stable comparison, time-domain signals of DFT-S OFDM and conventional OFDM are cascaded digitally in MATLAB before loading onto the AWGs. The parameters of DFT-S OFDM and conventional OFDM are for DFT-S, middle 2625/4096 subcarriers are filled with data while the center 65 subcarriers around DC are nullified to avoid performance degradation due to DC leakage, occupying a bandwidth of 6.409 GHz/band. For conventional OFDM the middle 83/128 subcarriers are filled with data while 3 subcarriers around DC are nullified, occupying a bandwidth of 6.484 GHz/band.

The difference between UW-DFT-S OFDM and conventional OFDM is that the 2560 data subcarriers in UW-DFT-S OFDM are mapped from DFT precoded UW-assisted data pattern, which will be described later, while in conventional OFDM the 80 data subcarriers are directly mapped with QPSK data pattern. After IFFT to convert data from frequency to time domain, a 128-point cyclic prefix (CP) is appended before each symbol.

The dissimilar number of subcarriers used in conventional OFDM and UW-DFT-S OFDM is because that conventional OFDM can only compensate CPE within one OFDM symbol and thus imposes a constraint to the use of long symbol unless other complicated phase noise compensation methods such as RF-pilot tone are used.

However for ultra-long-haul transmission, a large number of subcarriers are preferred or else the CP overhead is too much (in conventional OFDM the overhead of CP is more than 50%, while in DFT-S OFDM the overhead of CP is reduced to only 3%).

At receiver, the received four data streams \(I_x\), \(Q_x\), \(I_y\), and \(Q_y\), are first converted from analog to digital by ADCs, and then combined to complex signal and timing synchronized using pilot symbols. The frequency offset is then estimated and compensated, also using the pilot symbols.

The time domain signal is then converted from serial to parallel followed by the removal of CP. Subsequently, the data are transformed from time to frequency domain by a 4096-point DFT.

Channel estimation and equalization are first performed with the assistance of short- and long-UW pattern training sequence. Then, phase noise compensation is realized with the assistance of short UWs in each OFDM symbol using a novel channel estimation and phase estimation method, which is discussed later. The UWs are then removed followed by the payload data decision, QPSK demodulation and finally BER calculation for the performance evaluation.

The raw data rate of our UW-DFT-S OFDM signal is 1.2-Tb/s (6.25 GHz × 48 band × 2 bit/s × 2 pol), and the net data rate is 1.0-Tb/s after excluding all the overheads. The net spectral efficiency is 3.1 bit/s/Hz.

Figure 5.19 shows the BER sensitivity for optical back-to-back of 1-band (with 1 laser), 3-band (with 1 laser), and 48-band (with 16 lasers) PDM-QPSK UW-DFT-S OFDM system corresponding to a raw data rate of 25-Gb/s, 75-Gb/s, and 1.2 Tb/s.

The required OSNR for DFT-S OFDM system is similar to conventional OFDM system measured at 22.9 dB for a BER of 4.6 × 10\(^{-3}\) (7% FEC). This is 17 dB more than that of a single-band system and only 0.9 dB away from the theoretical value.

Figure 5.20 shows the \(Q\) factor performance against launch power. The optimum launch power is 8 dBm for conventional OFDM and 9 dBm for DFT-S OFDM, which agrees well with our simulated value. A noticeable 0.6 dB improvement in \(Q\) factor is observed for DFT-S OFDM compared with conventional OFDM.

Figure 5.21 shows the measured BER against the transmission distance for DFT-S OFDM and conventional OFDM at launch powers of 8 and 9 dBm. It can be seen that the maximum possible transmission distance at a BER of 4.6 × 10\(^{-3}\) (7% FEC) is 8300 and 7300 km for DFT-S OFDM and conventional OFDM, respectively, which shows a 20% increase in reach for DFT-S OFDM. If BER of 2 × 10\(^{-2}\) 20% FEC) is used, the reach of DFT-S OFDM can be extended to more than 10,000 km.

Finally, the BER performance of all 48 bands is measured at the launch power of 9 dBm and transmission distance of 8000 km (80 km × 100) as shown in Figure 5.22. The BERs of all bands in DFT-S OFDM are below the 7% FEC threshold, whereas for conventional OFDM all bands have crossed the BER limit.

## 7. Summary

This tutorial describes the basic concept of OFDM theory and its application in optical transmission systems.

In Sections 2 and 3, principles of coherent optical OFDM and direct-detection optical OFDM are provided. The system configuration, BER and \(Q\) performance, and tolerance to channel impairments including the fiber attenuation, chromatic dispersion, PMD, and fiber nonlinearities are provided.

In Section 4, a novel variant of OFDM called DFT-S OFDM is introduced. The basic figures of merit of DFT-S OFDM including the PAPR advantage and unique-word-assisted DSP for multiple purposes (timing synchronization, channel estimation, phase estimation, etc.) are discussed.

In Section 5, OFDM-based superchannel transmission technologies are reviewed. The concept of superchannel transmission is introduced and how it can increase the system spectral efficiency is explained.

The most recent process in OFDM superchannel transmission, including the reduced-guard-interval OFDM superchannel, no-guard-interval OFDM superchannel, and DFT-S OFDM superchannel is provided. Experimental results are given and their performances are compared with the conventional OFDM system.

The next tutorial gives a ** detailed introduction to binary optics**.