Channel estimation in DFT-based offset-QAM OFDM systems

Jian Zhao

doi:10.1364/OE.22.025651

1. Introduction

Multicarrier techniques have attracted much interest for high-speed optical communication systems [1–9], due to their higher spectral efficiency and enhanced tolerance to dispersion. Two common multicarrier systems are conventional orthogonal frequency division multiplexing (C-OFDM) [1–5] and Nyquist frequency (or wavelength) division multiplexing (N-FDM/N-WDM) [6–9]. The former utilizes a sinc-function spectrum to achieve sub-channel orthogonality while the latter employs filters at transmitters/receivers to create a rectangular spectral profile. However, with either technique, there is a long oscillating tail in the frequency or time domain, resulting in disadvantages such as vulnerability to intercarrier interference (ICI) in C-OFDM and a long memory length for pulse shaping in N-FDM.

Recently, offset quadrature amplitude modulation (offset-QAM) OFDM, whose basic principle was introduced in wireless communications [10], has attracted much attention in optical transmission systems [11–16]. This technology can be implemented in either the optical or electrical domains. In the electronic-domain implementation, it is shown that similar to C-OFDM, discrete Fourier transform (DFT) can be applied for efficient offset-QAM OFDM multiplexing/demultiplexing [16]. One-tap equalizers can be used for channel compensation, and phase estimation can be realized via pilot tones. Compared to C-OFDM, DFT-based offset-QAM OFDM may greatly relax the length of guard interval (GI) for dispersion compensation and achieve 23% increase in net data rate under the same transmission reach [16]. On the other hand, this technology significantly reduces the memory length of the pulse-shaping filter and results in lower complexity than N-FDM.

However, channel estimation in offset-QAM OFDM is different from that in C-OFDM [17,18], and, as identified in this paper, can be the key issue to limit the system performance if the conventional least square (LS) method is employed. This is attributed to the residual crosstalk terms in the decoded offset-QAM signal although they are orthogonal to the desirable data. Therefore, particular investigation is required on this issue to ensure the advantages of offset-QAM OFDM over C-OFDM and N-FDM in the applications of optical communications. In this paper, we derive a close-form expression for the decoded offset-QAM OFDM signal, from which the influence of residual crosstalk on channel estimation can be identified. We propose and investigate four channel estimation methods, which vary in terms of performance, complexity, and tolerance to system impairments, in order to find out the optimal design. It is theoretically and experimentally verified that simple and effective channel estimation can be realized in offset-QAM OFDM, and can operate seamlessly with other algorithms (synchronization, phase estimation etc). The study confirms that offset-QAM OFDM can be a very promising solution for optical communication systems and networks.

2. Principle

2.1. Closed-form expression of decoded offset-QAM OFDM signal

Figure 1 shows the multiplexing/demultiplexing principle of DFT-based offset-QAM OFDM. Offset-16QAM format is adopted in the illustration. Two bi-polar four-amplitude-shift-keying (4-ASK) data are encoded with Gray coding. 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 symbol period, T/2, with respect to the in-phase tributary. 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 and N is the number of samples per OFDM symbol. 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) is 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 (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 (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(i⋅N + k), -∞ < i < + ∞, represents the impulse response of the k^th FIR filter, with one sample per OFDM symbol for each k. The required sampling rate of digital-to-analogue converters for s(i⋅N + k) is N/T, the same as that in C-OFDM without additional requirement on the hardware speed.

Fig. 1 Principle of DFT-based implementation for offset-16QAM OFDM.

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At the receiver, the received signal is serial-to-parallel (S/P) converted with the access time of T/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 ((i - q) \cdot N + k) \cdot h_{r e c e i v e r, k} (q \cdot N)

where r(i⋅N + k) is the received time-domain signal. We firstly assume an ideal channel so that r(i⋅N + k) = s(i⋅N + k). h_receiver,k(q⋅N) is the impulse response of the k^th receiver FIR filter. For proper decoding, h_receiver,k(q⋅N) is set to be h(q⋅N-k). Therefore, the remaining problem is to design h(i⋅N-k) to ensure the subcarrier orthogonality. In C-OFDM and N-FDM, the pulse shape should be a rectangular and a sinc function, respectively. In offset-QAM OFDM, the requirement for the design of pulse shape or signal spectral profile to achieve subcarrier orthogonality can be greatly relaxed: a) the signal pulse is an even function; b) the signal pulse satisfies the inter-symbol interference (ISI) free criterion; c) there is no spectral overlap between subcarriers with more than one subcarrier distance (e.g. the m^th and (m ± 2)^th subcarriers). We firstly apply condition c) to Eq. (2) and derive

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

in Eq. (2) as:

\begin{array}{l} b_{i, m}^{r e a l} = \sum_{k_{1} = - \infty}^{+ \infty} \sum_{p = - \infty}^{+ \infty} a_{p, m}^{r e a l} \exp (j π m / 2) \cdot h ((i - p) \cdot N - k_{1}) \cdot h (k_{1}) \\ + \sum_{k_{1} = - \infty}^{+ \infty} \sum_{p = - \infty}^{+ \infty} a_{p, m + 1}^{r e a l} \exp (j π (m + 1) / 2) \cdot h ((i - p) \cdot N - k_{1}) \cdot h (k_{1}) \cdot \exp (2 π j k_{1} / N) \\ + \sum_{k_{1} = - \infty}^{+ \infty} \sum_{p = - \infty}^{+ \infty} a_{p, m - 1}^{r e a l} \exp (j π (m - 1) / 2) \cdot h ((i - p) \cdot N - k_{1}) \cdot h (k_{1}) \cdot \exp (- 2 π j k_{1} / N) \\ + \sum_{k_{1} = - \infty}^{+ \infty} \sum_{p = - \infty}^{+ \infty} a_{p, m}^{i m a g} \exp (j π (m + 1) / 2) \cdot h ((i - p) \cdot N - N / 2 - k_{1}) \cdot h (k_{1}) \\ + \sum_{k_{1} = - \infty}^{+ \infty} \sum_{p = - \infty}^{+ \infty} a_{p, m + 1}^{i m a g} \exp (j π (m + 2) / 2) \cdot h ((i - p) \cdot N - N / 2 - k_{1}) \cdot h (k_{1}) \cdot \exp (2 π j k_{1} / N) \\ + \sum_{k_{1} = - \infty}^{+ \infty} \sum_{p = - \infty}^{+ \infty} a_{p, m - 1}^{i m a g} \exp (j π (m) / 2) \cdot h ((i - p) \cdot N - N / 2 - k_{1}) \cdot h (k_{1}) \cdot \exp (- 2 π j k_{1} / N) \end{array}

where k₁ = q⋅N-k. The six terms on the right-hand side of Eq. (3) represent the influence of the in-phase tributary of the m^th, (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. Condition b) is then applied to further simplify Eq. (3). Because the pulse shape satisfies ISI free criteria, the first term on the right-hand side of Eq. (3) can be re-written as:

\sum_{k_{1} = - \infty}^{+ \infty} \sum_{p = - \infty}^{+ \infty} a_{p, m}^{r e a l} \cdot \exp (j π m / 2) \cdot h ((i - p) \cdot N - k_{1}) \cdot h (k_{1}) = a_{i, m}^{r e a l} \cdot \exp (j π m / 2)

On the other hand, the second term on the right-hand side of Eq. (3) can be obtained as:

\begin{array}{l} \sum_{k_{1} = - \infty}^{+ \infty} \sum_{p = - \infty}^{+ \infty} a_{p, m + 1}^{r e a l} \exp (j π (m + 1) / 2) \cdot h ((i - p) \cdot N - k_{1}) \cdot h (k_{1}) \cdot \exp (2 π j k_{1} / N) \\ = j \cdot \exp (j π m / 2) \cdot (^{- 1) i - p} \cdot \sum_{k_{2} = - \infty}^{+ \infty} \sum_{p = - \infty}^{+ \infty} a_{p, m + 1}^{r e a l} \cdot h ((i - p) \cdot N / 2 - k_{2}) \cdot h ((i - p) \cdot N / 2 + k_{2}) \cdot \exp (2 π j k_{2} / N) \\ = j \cdot c_{m + 1}^{r e a l} \cdot \exp (j π m / 2) \end{array}

where

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

is real. In Eq. (5), we use condition a): if h(⋅) is an even function, it is readily derived that h((i-p)⋅N/2-k₂)⋅h((i-p)⋅N/2 + k₂) is also an even function. From Eqs. (4) and (5), we find that the decoded signal is real while the crosstalk value from the in-phase tributary of the (m + 1)^th subcarrier is imaginary. It is noted that the crosstalk can be from ... (i-2)^th, (i-1)^th, i^th, (i + 1)^th, (i + 2)^th ... symbols of adjacent subcarriers (m + 1). Similar manipulation can be carried out to derive other terms on the right-hand sides of Eq. (3) and

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

can be written as:

b_{i, m}^{r e a l} = (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}) \cdot \exp (j π m / 2)

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}

represent 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.

2.2. Channel estimation in offset-QAM OFDM systems

From Eq. (6), it can be seen that in an ideal case, the desirable signal is real while crosstalk value is imaginary. Therefore, data can be correctly decoded if the phase shifts applied to different subcarriers at the transmitters, exp(jπm/2), are reversed and the real part is extracted:

a_{i, m, e s t}^{r e a l} = r e a l {b_{i, m}^{r e a l} \cdot \exp (- j π m / 2)} = a_{i, m}^{r e a l}

However, Eq. (6) does not consider the channel response. In practice, the responses of electrical drivers, modulators, optical filters, and photodiodes, as well as dispersion and timing errors in the synchronization process, would influence the performance. When the number of subcarriers is large such that the system response on the signal spectrum of individual subcarriers is approximately a constant, we can generalize Eq. (6) as:

\begin{array}{l} b_{i, m}^{r e a l} = \exp (j π m / 2) \cdot \exp (j φ_{i}) \cdot (a_{i, m}^{r e a l} \cdot H (ω_{m}) + j \cdot c_{m + 1}^{r e a l} \cdot H (ω_{m + 1}) + j \cdot c_{m - 1}^{r e a l} \cdot H (ω_{m - 1}) \\ + j \cdot c_{m}^{i m a g} \cdot H (ω_{m}) + j \cdot c_{m + 1}^{i m a g} \cdot H (ω_{m + 1}) + j \cdot c_{m - 1}^{i m a g} \cdot H (ω_{m - 1})) \end{array}

where H(ω_m) = H_b-t-b(ω_m)⋅exp(jβ₂ω_m²L/2)⋅exp(-jτω_m). φ_i is the common phase error (CPE) in the i^th symbol. H(ω_m) represents the overall system response at the frequency of the m^th subcarrier ω_m, and includes the contributions from the back-to-back response, H_b-t-b(ω_m), the dispersion, and the time delay. L and β₂ are the fiber length and the second-order dispersion value, respectively. τ represents the value of time delay. In practice, H(ω_m) ≈H(ω_{m + 1}) ≈H(ω_m-1) under commonly used number of subcarriers in OFDM systems, e.g. 128/256/512 subcarriers. Equation (8) also assumes that there is no ISI between OFDM symbols. From Eq. (8),

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

is only influenced by the adjacent subcarriers (m + 1) and (m-1), whose time delay in the presence of dispersion,| β₂L(ω_{m + 1}-ω_m-1)|, is much smaller than that in C-OFDM, | β₂L(ω_{-N/2 + 1}-ω_N/2)|, especially for a large number of N. Consequently, the ISI in each offset-QAM OFDM subcarrier is significantly reduced and the technology may support transmission without guard interval. In [16], it was shown that offset-QAM OFDM could have 23% increment in net data rate over C-OFDM to achieve the same transmission reach. Note that offset-QAM OFDM is different from reduced-guard-interval (RGI) C-OFDM [19], which is still based on the combination of C-OFDM and N-FDM.

In C-OFDM, because there is no crosstalk between subcarriers, channel response can be estimated using the LS method based on time-domain averaging, pilot tones [17], or intra-symbol frequency-domain averaging [18]. In offset-QAM OFDM, despite the orthogonality to the desirable signal, the crosstalk terms in Eq. (8) would influence the channel estimation performance when the conventional LS method is applied. From Eq. (8), it is deduced that if the phase of the channel response at ω_m is not correctly recovered, the nonlinear operation of real{⋅} would filter out part of signal power and pass through part of crosstalk power, resulting in performance degradation. Therefore, the phase terms in $b_{i, m}^{r e a l}$ have to be firstly compensated, and then the amplitude of the channel response is estimated by:

H_{e s t} (ω_{m}) = \frac{1}{M} \sum_{p = 1}^{M} r e a l {b_{p, m}^{r e a l} \cdot \exp (- j π m / 2) \cdot \exp (- j β_{2} L ω_{m}^{2} / 2) \cdot \exp (j τ ω_{m}) \cdot \exp (- j φ_{p})} / a_{p, m}^{r e a l}

where M is the number of averaging symbols and should be sufficiently large to mitigate the noise impact. In practice, the CPE φ_i can be estimated using pilot tones before channel estimation. However, Eq. (9) requires accurate estimation of the dispersion β₂L and places stringent requirement on the symbol synchronization algorithm in the offset-QAM OFDM system. More importantly, it is found in the experiment that even when the parameters β₂L and τ are precisely determined (e.g. via manual optimization), there is still a penalty. This penalty may be from imperfect channel estimation using Eq. (9), which does not consider the residual phase shifts in the responses of optoelectronic devices, i.e. H_b-t-b(ω_m). These residual phases are unknown in practice and have different values for different ω_m.

To solve this problem, we firstly propose to improve the channel estimation performance by appropriately designing the training symbols (TSs). From Eq. (8), it can be deduced that the crosstalk terms on the m^th subcarrier are only from the (m-1)^th and the (m + 1)^th subcarriers. Therefore, if the training data are inserted every two subcarriers in the TSs, the crosstalk terms may be avoided. However, further design consideration is required. To clearly see this, we study two kinds of TSs, named modifed LS-1 (M-LS-1) and M-LS-2, respectively: 1) Odd subcarriers are set to be real-valued while even subcarriers are set to be zero in the 1st, 3rd, 5th… TSs. Conversely, in the 2nd, 4th, 6th… TSs, even subcarriers are set to be real-valued while odd subcarriers are set to be zero. 2) Odd subcarriers are set to be real-valued while even subcarriers are set to be zero in the 1st, 2nd, …, M/2th TSs. Conversely, even subcarriers are set to be real-valued while odd subcarriers are set to be zero in the (M/2 + 1)^th, (M/2 + 2)^th, …, M^th TSs, where M is the number of TSs. Channel estimation is realized by using:

H_{e s t} (ω_{2 m - 1}) = \frac{2}{M} \sum_{p \in P_{2 m - 1}}^{} b_{p, 2 m - 1}^{r e a l} \cdot \exp (- j π (2 m - 1) / 2) \cdot \exp (- j φ_{p}) / a_{p, 2 m - 1}^{r e a l}

H_{e s t} (ω_{2 m}) = \frac{2}{M} \sum_{p \in P_{2 m}}^{} b_{p, 2 m}^{r e a l} \cdot \exp (- j π (2 m) / 2) \cdot \exp (- j φ_{p}) / a_{p, 2 m}^{r e a l}

where the channel response for odd and even subcarriers are estimated individually using corresponding TSs, P_2m-1 and P_2m. M-LS-1 and M-LS-2 can have significantly different performance although only the position of TSs for estimating the response of odd and even subcarriers varies. It is attributed to the fact that the signal pulses between OFDM symbols overlap, and by using M-LS-1, channel estimation performance is still degraded by the crosstalk from adjacent subcarriers in adjacent offset-QAM OFDM symbols. In contrast, M-LS-2 may mitigate the crosstalk and approach the optimal performance. It is noted that the validity of Eq. (10) is based on the assumption that the orthogonality maintains in the presence of dispersion, timing delay etc, which is true in practice as shown in the experiment.

In addition to the design of TSs, we also propose a method to track the unknown phase shift at each frequency, ω_m, before the nonlinear operator real{⋅}. Assuming the estimated channel response is H_est(ω_m), we have:

a_{i, m, e s t}^{r e a l} = r e a l {b_{i, m}^{r e a l} \cdot \exp (- j π m / 2) \cdot H_{e s t, i} (ω_{m})}

H_{e s t, i} (ω_{m}) = H_{e s t, i - 1} (ω_{m}) + e \cdot δ_{i} (ω_{m})

δ_{i} (ω_{m}) = 0.5 \times δ_{i - 1} (ω_{m}) + 0.5 \times (a_{i - 1, m}^{r e a l} - a_{i - 1, m, e s t}^{r e a l}) \cdot (\frac{\partial a_{i, m, e s t}^{r e a l}}{\partial r e a l {H_{e s t, i} (ω_{m})}} + j \frac{\partial a_{i, m, e s t}^{r e a l}}{\partial i m a g {H_{e s t, i} (ω_{m})}})

where e is a parameter to control the estimation speed. The channels in practical optical communication systems are commonly static, and so the estimation process is only required at the initial stage. However, this method is also applicable to a slow-varying optic-fiber channel to track the channel. We will show that when applied in offset-QAM OFDM systems, this method can greatly relax the requirements for symbol synchronization and prior estimation of dispersion parameters compared to the conventional LS method. In addition, it precisely estimates unknown residual phases in the responses of optoelectronic devices H_b-t-b(ω_m) and overcomes the performance limit of Eq. (9). Because Eq. (11) uses modified least-mean-square (LMS) algorithm after the nonlinear operation real{⋅} to track the correct phase shifts at each ω_m, we call this method M-LMS-1 in this paper.

Equation (11) may mitigate the impact of crosstalk in Eq. (8) even when these crosstalk terms exit without special design of the TSs. In all proposed methods above (M-LS-1, M-LS-2, and M-LMS-1), the receiver FIR filter is designed to be h_receiver,k(q⋅N) = h(q⋅N-k) and channel estimation does not produce update to the coefficients of the pulse-shaping FIR filters. However, it is unknown if the presence of channel response in the hardware as well as dispersion, delay etc may possibly break this orthogonality. In order to clarify this issue, we further study a more completed algorithm (named M-LMS-2) which includes the response of the FIR filters in the channel estimation:

{\vec{h}}_{r e c e i v e r, k, i} = {\vec{h}}_{r e c e i v e r, k, i - 1} + ε \cdot {\vec{κ}}_{k, i}

{\vec{κ}}_{k, i} = 0.5 \times {\vec{κ}}_{k, i - 1} + 0.5 \times (a_{i - 1, m}^{r e a l} - a_{i - 1, m, e s t}^{r e a l}) \cdot c o n j (H_{e s t, i - 1} (ω_{m}) \cdot {\vec{r}}_{k, i - 1} \cdot F_{m, k})

where

{\vec{h}}_{r e c e i v e r, k, i}

= [h_receiver,k(-l⋅N/2) ... h_receiver,k(l⋅N/2)] with l being the memory length of the FIR filter.

{\vec{r}}_{k, i}

= [r((i-l/2)⋅N + k) … r((i + l/2)⋅N + k)], representing the received time-domain signal vector. F_m,k is the element in the m^th row and the k^th column of the Fourier matrix. conj(⋅) presents the conjugate. Equation (12) ensures that the FIR filters at the receiver are also optimized in the presence of channel response, and so avoids possible performance degradation due to the loss of the orthogonality. Note that for N subcarriers and M TSs, the complexities of M-LS-1, M-LS-2, and M-LMS-1 scale with N⋅M, while that of M-LMS-2, when N subcarriers and M TSs are fully used, scales with (l + 1)⋅N²⋅M, resulting in higher implementation complexity.

3. Experimental setup

Experiments were carried out to investigate the performance of different channel estimation methods. Figure 2 shows the experimental setup. The IFFT and FFT used 128 points, of which 102 subcarriers were used for 16QAM data modulation. The six subcarriers in the zero-frequency region were not modulated, allowing for AC-coupled amplifiers and insertion of pilot tones for phase estimation. The positions of pilot tones were close to DC in order to reduce the influence of dispersion. The twenty subcarriers in the high-frequency region were zero-padded to avoid aliasing. The FIR filter created a set of squared raised-cosine function (SRRC) functions with different roll-off values, or a 3rd-order super-Gaussian function. In all cases, the 3-dB bandwidth of the shaped spectrum was the same as the subcarrier spacing. The offset-QAM OFDM symbol sequence began with a start-of-frame (SOF) symbol designed based on the principle similar to [5] to enable symbol synchronization. In M-LS-1 and M-LS-2, the TSs for channel estimation were designed as discussed above. In conventional LS, M-LMS-1, and M-LMS-2 methods, special design was not required. The number of TSs was fixed to be 80 in all methods to ensure that optimal performance could be obtained for fair comparison. Inset of Fig. 2 shows the spectrum of an offset-QAM OFDM signal when the SRRC function with a roll-off coefficient of 0.5 was used. It can be seen that offset-QAM OFDM avoided the long spectral tails, and so exhibited greatly suppressed side lobes. The generated signal was clipped with a peak-to-average power ratio of 11 dB before downloaded to an arbitrary waveform generator with 12-GS/s digital-to-analogue converters (DACs). The signal line rate including forward error correction was 38 Gb/s.

Fig. 2 Experimental setup of coherent optical offset-16QAM OFDM. Inset shows its spectrum.

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A laser with 6-kHz linewidth was used to generate the optical carrier. The electrical OFDM signal was fed into an optical I/Q modulator with a peak-to-peak driving swing of 0.5V_π to avoid nonlinear distortion. Single polarization was adopted in the experiment but the scheme could be readily extended to dual-polarization systems using the same polarization multiplexing/demultiplexing principle as C-OFDM. The generated optical signal was amplified by an erbium doped fiber amplifier (EDFA), filtered by a 4-nm optical band-pass filter (OBPF), and transmitted over a recirculating loop comprising 60-km single-mode fiber (SMF) with 14-dB fiber loss. The noise figure of the EDFA was 6 dB and another 0.8-nm OBPF was used in the loop to suppress the amplified spontaneous emission noise. The launch power per span was around −9 dBm to avoid the nonlinear effects. At the receiver, the optical signal was detected with a pre-amplified coherent receiver. A variable optical attenuator (VOA) was used to vary the optical signal-to-noise ratio (OSNR) for the bit error rate (BER) measurements. The pre-amplifier was followed by an OBPF with a 3-dB bandwidth of 0.64 nm, a second EDFA, and another OBPF with a 3-dB bandwidth of 1 nm. The second EDFA at the receiver ensured fixed input powers into photodiodes, and eliminated potential influence of thermal noise. A polarization controller (PC) was used to align the polarization of the filtered OFDM signal before entering the signal path of a 90° optical hybrid. The optical outputs of the hybrid were connected to two balanced photodiodes with 40-GHz 3-dB bandwidths, amplified by 40-GHz electrical amplifiers, and captured using a 50-GS/s real-time oscilloscope. The total number of measured 16QAM symbols was 240,000. The received signal was up-sampled, synchronized, and then serial-to-parallel (S/P) converted, before passed through the receiver FIR filters. An FFT was applied to transform the signals to the frequency domain. Pilot tones were extracted for phase estimation. Different channel estimation methods were employed and the signals were then equalized using one-tap equalizers based on the estimated channel response.

4. Results and discussions

Figure 3 shows the BER versus the received OSNR for offset-16QAM OFDM with different channel estimation methods, when the signal spectral profile is (a) a SRRC function with the roll-off coefficient of 0.5 and (b) a 3rd-order Gaussian function. The figure shows that the conventional LS method exhibits significant performance degradation even when the phase noise, synchronization, and dispersion (0 km in this case) in Eq. (9) are compensated or manually optimized. This penalty might be attributed to the unknown phase shifts in the hardware, including optical filters, amplifiers etc. Figure 4 shows the system amplitude and phase responses at 0 km under three different time delays. The responses were obtained using the M-LMS-1 method as the phase could not be precisely estimated using the conventional LS method. The curves for 0-ps time delay represents the response at the time point obtained using the synchronization algorithm [5], and those for 10- or −10-ps time delays are obtained by shifting this time point by 10 or −10 ps. It can be clearly seen that the amplitude response is nearly the same for all three cases. However, the phase shifts are not zero even at 0 km and vary significantly with the time delays. This non-zero phase shifts break the assumption of Eq. (9). Consequently, the conventional LS method exhibits a large penalty and is also expected to be sensitive to the timing errors (shown later). The use of the M-LS-1 method improves the performance somewhat. However, although the even subcarriers in the 1st, 3rd, 5th… TSs are set to be zero, the estimated responses at odd subcarriers using the 1st, 3rd, 5th… TSs are still degraded by crosstalk from even subcarriers in the 2nd, 4th, 6th… TSs, due to overlapped signal pulses. M-LS-2 can mitigate this effect, and so shows significantly improved performance. The required OSNR to achieve a BER of 10⁻³ is around 12.6 dB, which is close to the theoretical limit of 12.3 dB for a 38-Gbit/s 16QAM signal. On the other hand, M-LMS-1 and M-LMS-2 exhibit slightly better performance than the M-LS-2 method. This slight improvement might be due to their adaptive capability to track the optimal operation condition. It is also shown that M-LMS-2 does not result in significant performance improvement compared to M-LS-2 and M-LMS-1 while introducing additional implementation complexity. It implies that in the experiment, once the transmitter and receiver FIR filters are decided and implemented in the digital signal processing, the orthogonality can be maintained regardless of the response of the hardware and the channel. Consequently, M-LS-2 and M-LMS-1 are sufficient to approach the optimal performance at lower implementation complexity. Note that the validity is based on a large subcarrier number such that the channel response on individual subcarriers is approximately a constant. When the number of subcarriers is small or the channel response varies significantly over the frequency, M-LMS-2 might be more effective. The last conclusion we can draw from Fig. 3 is that the use of either the SRRC or the Gaussian function as the signal spectrum results in similar performance. This is in contrast to C-OFDM and N-FDM, where a sinc function and a rectangular function should be used as the signal spectral profile to ensure the orthogonality.

Fig. 3 BER versus the received OSNR using different channel estimation methods when (a) a SRRC function and (b) a 3rd-order Gaussian function are used as the signal spectrum.

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Fig. 4 (a) Amplitude and (b) phase responses of the system at 0 km obtained using the M-LMS-1 method under −10-ps (dashed), 0-ps (solid), and 10-ps (dotted) time delays.

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Figure 5(a) shows BER versus the roll-off coefficient of the receiver filter when that of the transmitter filter is 0.5. In order to ensure the orthogonality, the FIR filters at the transmitter and the receiver should satisfy h_receiver,k(q⋅N) = h(q⋅N-k). M-LS-2 and M-LMS-1 do not have the capability to recover the orthogonality if this assumption is broken. On the other hand, M-LMS-2 is more tolerant to the loss of channel orthogonality. It is noted that in a relatively static metro or long-distance optical communication system, the FIR filters can be known beforehand, and so the benefit of M-LMS-2 over M-LS-2 and M-LMS-1 is limited while additional complexity is induced. However, M-MLS-2 may have advantages in secure access or metro optical networks. In the following, we will focus on M-LS-2 and M-LMS-1 and their comparison with the conventional LS method. Figure 5(b) shows the BER versus the memory length of the pulse-shaping filter at 0 km when a SRRC function with a roll-off coefficient of 0.5 is used as the signal spectrum. As expected, the time-domain pulse of the offset-QAM OFDM signal has significantly suppressed tails, and so a memory length as short as two is sufficient to achieve the near-optimal performance. This is in contrast to the N-FDM where a memory length longer than 40 is required to create the sinc-function pulse shape with long tails [16]. It is also seen in the figure that the performance improvement of M-LS-2 and M-LMS-1 over the LS method is transparent to the memory length of the pulse-shaping filters.

Fig. 5 (a) BER versus roll-off coefficient of the receiver filter for different channel estimation method when the roll-off coefficient of the transmitter filter is 0.5. (b) BER versus the memory length of the pulse-shaping filter when a SRRC function with a roll-off coefficient of 0.5 is used as the signal spectrum. In (a) and (b), the OSNR is 14 dB.

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Figure 6(a) shows BER versus the received OSNR for offset-QAM OFDM after 0-, 300-, and 600-km transmission. In the figures, the GI length is zero. It can be clearly seen that offset-QAM OFDM without any GI exhibits negligible transmission penalty after 300 and 600 km. The required OSNR to achieve a BER of 10⁻³ at 600 km is around 13.2 dB, which has < 1-dB penalty compared to that for 0 km. This is in contrast to C-OFDM where the requirement for the GI length is much more restricted. Figure 6(b) shows BER versus the fiber length using different channel estimation methods. It is confirmed that performance improvements of the M-LS-2 and M-LMS-1 methods are not sensitive to the fiber length, and more than one order of magnitude in BER reduction is observed for all fiber lengths.

Fig. 6 (a) BER versus the received OSNR (dB) for M-LS-2 and M-LMS-1 at different fiber lengths. (b) BER versus the fiber length for LS, M-LS-2, and M-LMS-1 methods. The OSNR values for 0, 120, 240, 360, 480, 600, 720 km are 15.9, 15.2, 16.1, 15.4, 15.7, 16.1, and 15.5 dB, respectively.

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The benefits of M-LS-2 and M-LMS-1 are not limited to pure performance improvement. In previous results, the performance of the conventional LS method is obtained by manually optimizing the dispersion parameters and time delays. These, however, significantly reduce the flexibility and increase the operation complexity in practical implementation. Even in a relatively static optical transmission channel, dispersion parameters may not be precisely obtained and the time delays depend on the performance of synchronization algorithms. To investigate these issues, Fig. 7(a) shows the BER versus residual dispersion for different channel estimation methods. It is clearly shown that the conventional LS method is very sensitive to the residual dispersion. It implies that precise dispersion information is required beforehand or timing-consuming searching is needed at the receiver to obtain the minimized BER. The use of M-LS-2 and M-LMS-1 can greatly relax this restriction. The figure shows that BER variation using these two methods is negligible for [-800ps/nm 800ps/nm]. Figure 7(b) shows BER versus timing error for LS, M-LS-2, and M-LMS-1. From Fig. 4(b), it is seen that different time delays may influence the phase response significantly. The conventional LS method cannot obtain this phase response, and so shows large performance degradation when the synchronization is not perfectly realized. On the other hand, M-LS-2 and M-LMS-1 are not sensitive to the timing error, and the performance is almost the same when timing errors are in the range of [-10ps 10ps]. For the synchronization algorithm used in this paper, the standard deviation of synchronization timing errors can be less than 2.2 ps for OSNR values as low as 3 dB [5]. This is well within the tolerance range of the M-LS-2 and M-LMS-1 methods, and the offset-QAM OFDM system can operate automatically with the optimal BER performance. This confirms that effective and simple channel estimation can be realized in offset-QAM OFDM systems, and work seamlessly with other algorithms (synchronization, phase estimation etc).

Fig. 7 (a) BER versus residual dispersion at 600 km. (b) BER versus timing error at 600 km. In (a) and (b), the OSNR is 15.1 dB.

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5. Conclusions

DFT-based offset-QAM OFDM has attracted much attention for its advantages over C-OFDM and N-FDM in optical communications. However, conventional LS channel estimation degrades the system performance and hinders its applications in optical systems and networks. In this paper, we derive a closed-form expression for the decoded offset-QAM OFDM signal and identify the performance degradation in offset-QAM OFDM systems when the conventional LS method is applied. We have proposed and investigated several methods, namely M-LS-1, M-LS-2, M-LMS-1, and M-LMS-2, to find out the optimal design in terms of performance and complexity. It is shown that M-LS-2 and M-LMS-1 methods can realize a performance close to the theoretical limit at low complexity. The study ensures simple and effective channel estimation in offset-QAM OFDM systems and enables this technology very promising for optical communications.

Acknowledgments

This work was supported by the Science Foundation Ireland under grant number 11/SIRG/I2124 and 13/TIDA/I2718, Irish Research Council New Foundations 2013 CDAMT, and EU 7th Framework Program under grant agreement 318415 (FOX-C).

References and links

1. X. Q. Jin, R. P. Giddings, and J. M. Tang, “Real-time transmission of 3 Gb/s 16-QAM encoded optical OFDM signals over 75 km SMFs with negative power penalties,” Opt. Express 17(17), 14574–14585 (2009). [CrossRef] [PubMed]

2. 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]

3. Q. Yang, Z. He, Z. Yang, S. Yu, X. Yi, and W. Shieh, “Coherent optical DFT-spread OFDM transmission using orthogonal band multiplexing,” Opt. Express 20(3), 2379–2385 (2012). [CrossRef] [PubMed]

4. E. Giacoumidis, A. Tsokanos, C. Mouchos, G. Zardas, C. Alves, J. L. Wei, J. M. Tang, C. Gosset, Y. Jaouen, and I. Tomkos, “Extensive comparison of optical fast OFDM and conventional OFDM for local and access networks,” J. Opt. Commun. Netw. 4(10), 724–733 (2012). [CrossRef]

5. J. Zhao, S. K. Ibrahim, D. Rafique, P. Gunning, and A. D. Ellis, “Symbol synchronization exploiting the symmetric property in optical fast OFDM,” IEEE Photon. Technol. Lett. 23(9), 594–596 (2011). [CrossRef]

6. X. Zhou, L. Nelson, P. Magill, B. Zhu, and D. Peckham, “8x450-Gb/s, 50-GHz-spaced, PDM-32QAM transmission over 400km and one 50GHz-grid ROADM,” in Proc. Optical Fiber Communications Conference (2012), post-deadline paper PDPB3.

7. G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist WDM and CO-OFDM in high-speed PM-QPSK systems,” IEEE Photon. Technol. Lett. 22(15), 1129–1131 (2010). [CrossRef]

8. J. Zhao and A. D. Ellis, “Electronic impairment mitigation in optically multiplexed multicarrier systems,” J. Lightwave Technol. 29(3), 278–290 (2011). [CrossRef]

9. Z. Dong, X. Li, J. Yu, and N. Chi, “6×144 Gb/s Nyquist WDM PDM-64QAM generation and transmission on a 12-GHz WDM grid equipped with Nyquist band pre-equalization,” J. Lightwave Technol. 30(23), 3687–3692 (2012). [CrossRef]

10. B. R. Saltzberg, “Performance of an efficient parallel data transmission system,” IEEE Trans. Commun. Technol. 15(6), 805–811 (1967). [CrossRef]

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

12. 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]

13. 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]

14. 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]

15. 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). [CrossRef] [PubMed]

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

17. L. Liu, X. Yang, and W. Hu, “Chromatic dispersion compensation using two pilot tones in optical OFDM systems,” in Proc. Asia Communications and Photonics Conference (2011), PDP 830937.1–6. [CrossRef]

18. X. Liu and F. Buchali, “Intra-symbol frequency-domain averaging based channel estimation for coherent optical OFDM,” Opt. Express 16(26), 21944–21957 (2008). [CrossRef] [PubMed]

19. A. Tolmachev and M. Nazarathy, “Filter-bank based efficient transmission of reduced-guard-interval OFDM,” Opt. Express 19(26), B370–B384 (2011). [CrossRef] [PubMed]

Channel estimation in DFT-based offset-QAM OFDM systems

Abstract

1. Introduction

2. Principle

2.1. Closed-form expression of decoded offset-QAM OFDM signal

2.2. Channel estimation in offset-QAM OFDM systems

3. Experimental setup

4. Results and discussions

5. Conclusions

Acknowledgments

References and links

Cited By

Figures (7)

Equations (16)

Optics Express