Accurate and spectral efficient channel estimation using inter-block precoding and superimposed pilots in optical OFDM systems

Changjian Guo; Yicheng Zheng; Han Zhang

doi:10.1364/OE.25.022228

1. Introduction

Optical orthogonal frequency-division multiplexing (OFDM) [1] is a promising modulation technique for next generation optical access networks that delivers high spectral efficiency and resilience against inter-symbol interference (ISI). Conventionally, in OFDM modulated systems, channel frequency response (CFR) is obtained via time and/or frequency training prior to signal equalization, which requires extra time/frequency resources allocated for training sequences, leading to the loss of bandwidth efficiency. To address this issue, superimposed pilot (SP) is utilized to enable an overhead-free OFDM system, where a periodic training sequence is arithmetically superimposed onto the data stream prior to transmission, either in frequency or time domain [2-3]. In SP-aided scheme, the CFR can be obtained by exploiting the first order statistics of the received signal. Therefore, no time slot or bandwidth is wasted for using pilots. However, some useful power of data sequence is allocated to SP, resulting in the signal-to-pilot interference (SPI). Generally, the data symbols can be assumed to be zero-mean with independent and identical distributions (i.i.d), whose statistical mean tends to zero when expectation is performed over a sufficient number of symbols. When the number of processed data blocks is limited, however, the residual data interference deteriorates the estimation accuracy. Various techniques have been proposed to eliminate the SPI, among which the data dependent superimposed training (DDST) and affine precoding have been considered in [4–10]. Specifically, DDST eliminates the SPI by nulling the frequency bins of the data symbols that are reserved for those of the SP [4-5]. This scheme is similar to frequency-division multiplexed (FDM) training based methods at the cost of distortion to the data symbols. In precoding based schemes, the data symbols are coded by a matrix that is orthogonal to the SP [6–10]. In this way, data and SP can be effectively separated after decoding of the received signals and SPI can be eliminated. Unfortunately, the design of precoding matrix must be orthogonal to the SP sequence, and the redundancy induced by precoding is directly proportional to the numbers of unknowns i.e., the channel taps and the number of antennas in wireless communication systems [6–10]. Besides, the design of precoding matrix is hard to meet the criteria of optimal pilot in practical relevance, where equi-powered and equi-spaced pilots are employed for channel estimation (CE) [11].

In this work, we propose a novel inter-block precoding based scheme, targeting on eliminating the SPI in quasi-static optical OFDM systems, where the CFR remain unchanged over multiple blocks. We show that by using a square m-sequence matrix of the same column-size as that of the data blocks, the SPI can be significantly reduced. While employing a Walsh-Hadamard (WH) matrix [12] of one more column-size than that of the data blocks, the SPI can be fully eliminated. The proposed scheme, in comparison with the conventional precoding schemes, has the following advantages: 1) The proposed scheme has no limit on the design of SP. That is, pilot employed for CE can be any periodic sequences, and is no longer necessary to be orthogonal to the precoding matrix. 2) No redundancy is induced when using an m-sequence matrix. Meanwhile, the redundancy induced by WH matrix is fixed to $1 / M + 1$ , where M is the number of uncoded OFDM blocks, and remains unchanged to the system unknowns. 3) No decoding process is required when performing CE, which entails a lower computation complexity in comparison with the conventional precoding schemes.

This paper is organized as follows: In section 2, we discuss the operating principle of the proposed precoding scheme. In section 3, we present the simulation and experimental results of a proof-of-concept IM/DD optical OFDM system. Finally, conclusion and discussion are provided in section 4.

2. Operating principle

Consider a multi-carrier block transmission system operating over an optical fiber channel. We assume the channel to be quasi static, i.e., time invariant over multiple OFDM blocks. This is the case in intensity modulated and directly detected (IM/DD) optical systems, e.g., short reach optical communication systems, passive optical networks (PON) and line of sight (LOS) based visible light communication (VLC) systems [13-14]. In coherent detected optical systems, it also holds reasonably provided that the frequency offset and phase noise is compensated before CE [15]. At the transmitter, we partition the data stream into M ( $M = 2^{n} - 1$ ) blocks of N symbols (subcarriers). The transmitted data in frequency-domain has the form:

S = [s_{1}, s_{2}, \dots, s_{N}] \in ℂ^{M \times N} .

In our proposed precoding and superimposed pilots (P-SP) scheme, the source matrix is firstly precoded by pre-multiplying with a matrix

P_{m} = [p_{1}, p_{2}, \dots, p_{M}] \in ℂ^{M \times M},

where

p_{k}

is a cyclic-shifted version of maximal length sequence (m-sequence)

p_{1}

. Alternatively, the precoding matrix

P_{m}

in (2) can be replaced by using a WH matrix, given by

P_{w} = [p_{1}, p_{2}, \dots, p_{M}] \in ℂ^{(M +1) \times M},

where

P_{w}

is a

(M + 1) \times (M + 1)

square WH matrix without the first column (with all entries + 1). The resulting data after precoding has the form

X = P S = [x_{1}, x_{2}, \dots, x_{N}] .

Prior to transmission, SP is arithmetically added onto to the precoded data, resulting in the combined signals in frequency-domain as

U = \sqrt{1 - ϕ} P S + C = [u_{1}, u_{2}, \dots, u_{N}],

where

C^{T} = [c_{1}, c_{2}, \dots, c_{M + 1}] \in ℂ^{N \times M + 1}

,

c_{1} = c_{2} = \dots = c_{M + 1}

for WH matrix coded case. Unlike previous work [6–10], we emphasize that the pilot matrix C in (5) can be any periodic pilots, without the limitation of being orthogonal to the precoding matrix P. In this paper, we employ the periodic sequence

c

with equal-spaced and equal-powered property, whose entries can be given by

c_{l} = \sqrt{ϕ} \exp (j 2 π l \times (l + 2) / P), l = 0, 1, ..., P - 1,

where

ϕ

and P(

P = N / Q

, P and Q are positive integers) are the power and period of the SP, respectively. Note that when P = N,

c

has a constant magnitude in both time- and frequency domains, otherwise, the corresponding SP is a sequence of pulse train of period Q in time-domain. At the transmitter, an inverse fast Fourier transform (IFFT) is used to generate the time-domain signal, which is padded with a cyclic prefix (CP) of length L to avoid ISI, and then sent for E/O conversion. At the receiver, assume perfect synchronization, the received frequency-domain signal after dropping CP and performing FFT, can be expressed as:

Y = H U + W = \sqrt{1 - ϕ} H (P S + C) + W = [y_{1}, y_{2}, \dots, y_{N}],

where

H

is the channel matrix, and

W

represents AWGN with a variance of

σ^{2}

. Typically, in an optical OFDM system using IM/DD,

H

can be considered as time-invariant over multiple OFDM blocks. In addition, the data symbols S are assumed to be zero mean with i.i.d. Denote

p_{0}

as a

(M + 1) \times 1

column vector with all entries of + 1, the expectation of Y is given by

E (Y) = \frac{1}{M + 1} p_{0}^{T} Y \in ℂ^{1 \times N} .

From Eq. (7) and Eq. (8), it can be seen that the SPI can be fully eliminated if

p_{0}^{T} P S = 0 \in ℂ^{1 \times N}

holds. Obviously, the expectation operator

p_{0}

is orthogonal to

P_{w}

, i.e.,

p_{0}^{T} p_{w_{k}} = 0,

k = 1, 2, \dots, M + 1

. When a WH precoding matrix is employed, the expectation of each subcarrier can be expressed as

\begin{matrix} E (y_{l}) = [1 / (1 / M + 1)] (\sqrt{1 - ϕ} p_{0}^{T} h_{l} (p_{l} S + c_{l}) + p_{0}^{T} w_{l}), \\ ≃ \sqrt{1 - ϕ} h_{l} c_{l} \end{matrix}

where

h_{l}

is the CFR of the l^th subcarrier, which is assumed to be constant during transmission of the data matrix S,

c_{l}

is a constant vector with all entries equal to

c_{l}

described in (6). Accordingly, the CFR

\overset{⌢}{H}

can be obtained by using either LS or MMSE criteria.

From Eq. (9), the data interference is eliminated by using WH precoding matrix. By analogy, when m-sequence matrix is used for precoding, $p_{0}^{T} p_{m_{k}} = 1, k = 1, 2, \dots, M$ , i.e., $p_{0}$ is not orthogonal to $P_{m}$ since there is one more ‘ + 1’ than ‘-1’ in each column of $P_{m}$ , To be specific, although m-sequence precoding is unable to fully eliminate the data interference, it can significantly reduce data interference due to $(1 / M) p_{0}^{T} p_{m_{k}} = 1 / M$ , as shown in Fig. 1.

Fig. 1 Statistical mean of the OFDM signals before adding SP (M = 255).

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After performing CE, the data vector $S$ can be recovered by multiplying the transpose of the precoding matrix after removing the SP

\hat{S} = P^{T} ({\overset{⌢}{H}}^{H} Y - C) .

The superscript ^H denotes the conjugate transposition. From Eq. (4) and Eq. (10), we have the following conclusions: 1) Unlike the existing precoding schemes [6–10], the SP matrix

C

can be any periodic sequences. There is no need for the design of SP to be orthogonal to the precoding matrix. 2) The redundancy of the proposed scheme is zero and a fixed

1 / M + 1

for m-sequence and WH matrix precoding, respectively. However, the redundancy induced by the existing precoding schemes is directly proportional to the numbers of unknowns (channel length, number of antennas, etc.). That is, the proposed scheme entails a much lower complexity in comparison with the existing schemes.

3. Experimental setup

The proposed P-SP scheme is experimentally demonstrated using an IM/DD optical OFDM system. The experimental setup is shown in Fig. 2. Four cases, i.e., preamble, WH matrix precoded P-SP, m-sequence precoded P-SP and conventional SP based CE schemes were considered for each run. The detailed parameters used in the simulations and experiments are summarized in Table. 1. The total number of subcarriers used for fast Fourier transform (FFT) is 512. The 2nd-241st subcarriers were modulated, while the 1st, 242nd – 272nd subcarriers were zero padded. In order to generate a real-valued OFDM signal, the complex conjugates of the 2nd – 241^nd subcarriers were placed to the 512th– 273rd subcarriers, respectively. After IFFT operation, the time- domain signal was padded with a cyclic prefix (CP) of size $N_{c p} = 8$ , parallel/serial converted, and then digital/analog (D/A) converted using an arbitrary waveform generator (AWG 7122C) operating at 20 Gsa/s. Adaptive bit and power loading were employed using Chow’s algorithm [16]. The output signals from the AWG were thenamplified and sent to an electro-absorption (EA) modulated DFB laser (EML) operating at1544.5 nm. For each transmission, the first OFDM block was used for timing synchronization [17], and the rest OFDM blocks were effective payloads. The optical signals were then amplified to 10-dBm and sent into 83.2-km of standard single mode fiber (SSMF) for transmission. At the receiver, a variable optical attenuator (VOA) was used to change the received optical power (ROP) prior to a pre-amplified receiver which consists of an Erbium doped fiber amplifier (EDFA) with a noise figure of about 4.3 dB and an optical band pass filter (OBPF) with an optical bandwidth of 0.8 nm. The pre-amplified optical signals were then detected via a photo detector (PD) with a bandwidth of 43 GHz. The electrical signals after PD were digitized using a 100Gsa/s real time oscilloscope (DPO73324D) and then processed off-line. The off-line DSP procedures include pre-filtering, timing synchronization, serial-to-parallel(S/P) conversion, FFT, channel estimation and equalization, decoding, demodulation, and error counting.

Fig. 2 Experimental setup.

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Table 1. Parameters Used for Experiments and Simulations

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4. Results and discussion

We firstly investigate the CE accuracy of the proposed scheme by simulation. The simulation parameters are shown in Table 1. The simulation results were averaged over 100 Monte Carlo simulations, each with $M$ OFDM blocks. Aside from amplified spontaneous emission (ASE) noises generated from the pre-amplified optical receiver, only chromatic dispersion (CD) was considered in the simulation, which was modeled using a truncated time-domain filter $h$ with 11 taps. The normalized mean square error (MSE) of the channel impulse response (CIR) is given by $M S E = {‖ \hat{h} - h ‖}^{2} / {‖ h ‖}^{2}$ . The signal to pilot power ratio (SPR) is defined as $S P R = - 10 \times \log_{10} (ϕ)$ where $ϕ$ is the pilot power ratio and the preamble overhead percentage for the SP- and preamble based CE approaches, respectively. Figure 3(a) – 3(c) illustrates the simulated MSE as a function of SPR, number of OFDM blocks (M), and SNR, respectively. From Fig. 3, we have the following observations: 1) The uncoded SP scheme has the worst MSE performance; 2) the WH matrix precoded P-SP scheme has almost the identical MSE performance to that of the preamble based scheme, since the SPI can be fully eliminated; 3) the MSE performance encounters an error floor when m-sequence based P-SP scheme is used, due to the non-orthogonal property between the expectation operator $p_{0}$ and the m-sequence matrix precoded data. Such error floor can be reduced by increasing the number of OFDM blocks.

Fig. 3 Simulated MSE performance as a function of (a) SPR; (b) number of OFDM blocks; (c) SNR.

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We then experimentally investigate the BER performance of the proposed scheme in an IM/DD optical communication system after transmitting 83.2-km of SSMF. Adaptive bit and power loading employing Chow’s algorithm were used, with a total transmission rate of 33-Gb/s. Firstly, the impact of M on the BER performance was examined, as shown in Fig. 4(a). One can see that: 1) The WH matrix coded P-SP scheme has the best BER performance; 2) both m-sequence and WH matrix coded P-SP scheme outperform the uncoded SP scheme; 3) the BER is inversely proportional to the number of OFDM blocks (M), owing to the fact that increasing M improves the performance of CE [see in Fig. 3(b)]; 4) for WH matrix coded P-SP scheme, the BER performance almost saturate when M ≥ 255; 5) the BER performance of the m-sequence coded P-SP scheme is worse than that that of the WH matrix coded P-SP one, although both the two schemes have almost identical MSE performance (see Fig. 3). The reasons are twofold: First, m-sequence precoding cannot fully eliminate the residual SPI, which exhibits as the dominate interference to data detection. Second, m-sequence precoding introduces self-interference since the m-sequences are not orthogonal to each other. Such induced self-interference leads to an error even in noise-free scenario.

Fig. 4 Experimental results. (a) BER as a function of number of OFDM blocks (M), with a data rate of 33-Gb/s and a received optical power (ROP) of −6 dBm; (b) BER as a function of the SPR, with a data rate of 33-Gb/s, M = 255 and an ROP of −6 dBm; (c) BER as a function of the ROP for 30-Gb/s adaptively bit and power loaded OFDM signal after 83.2-km SSMF transmission, with SPR = 15 dB.

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We also investigated the ratio of pilot power allocation for various SPRs in ranges of 5 to 30 dB. The following observations can be summarized from the results shown in Fig. 4(b): 1) The WH matrix coded P-SP scheme has the lowest BER when SPR is around 15 dB; 2) The BER performance of WH matrix coded P-SP scheme outperforms preamble based scheme when SPR > 15 dB. This can be explained as follows: for the preamble-based scheme, the CFR is obtained by averaging the estimated CFRs over k OFDM symbols (preambles), where k is the number of OFDM symbols, which depends on the SPR according to $S P R = - 10 \times \log_{10} (ϕ) = - 10 \times \log_{10} (k / (N + k))$ . That means, the performance of CFR estimation is directly proportional to k. Specifically, as SPR increases, k is reduced, i.e., when SPR>15 dB, the corresponding k < 8. In this case, the accuracy of the estimated CFR is reduced, leading to the degradation of BER performance. In m-sequence or WH matrix precoded cases, superimposed training is spread over the entire blocks. Increasing SPR can improve the performance of CFR estimation, but simultaneously reduce the data to SP power ratio. Therefore, the performance for using m-sequence and Walsh precoded case almost saturate when SPR lies in the range of 15~20dB. 3) Although decreasing SPR can offer a more accurate CE performance for m-sequences coded P-SP scheme [see Fig. 3(a)], the BER performance stays unchanged for a wide regime of SPR. This fact implies that the self-introduced noise by precoding is the dominate interference for high SNR case. 4) For the uncoded SP scheme, BER performance can be improved by decreasing the SPR value. It should be noted that, when a decision feedback approach is used for the conventional SP scheme, as plotted by the dashed curve shown in Fig. 4(b), the optimum SPR for uncoded SP scheme is around 8 dB.

Finally, we evaluated the BER performance of the IM/DD system for all aforementioned schemes as a function of the received optical power (ROP). A data rate of 30-Gb/s was used to ensure that a BER of around 2E-4 can be achieved. The result is shown in Fig. 4(c). It can be seen from Fig. 4(c) that, although the performance of the WH matrix coded P-SP scheme is comparable to that of the preamble based scheme when SPR = 15 dB, the former entails a much smaller preamble-size, and in turn, can offer a higher transmission efficiency. It should be noted that an error floor exists for the m-sequence matrix precoded P-SP scheme, due to the residual SPI as well as the self-interference introduced by non-orthogonal precoding.

5. Conclusions

We have proposed and experimentally demonstrated a spectral efficient CE approach for optical OFDM systems using inter-block precoding and SP. It has been confirmed through simulations and experiments that the WH matrix coded P-SP scheme can offer comparable performance as that of preamble based ones in terms of channel MSE and BER performance, while entailing a much smaller overhead-size.

Funding

Guangdong Natural Science Foundation (2014A030313430); Pearl River S&T Nova Program of Guangzhou (201710010028, 201610010199); National Natural Science Foundation of China (NSFC) (61471176); Science Foundation for Excellent Youth Scholars of Guangdong Province (YQ2015046); Science and Technology Planning Project of Guangdong Province (2017A010101015); and Special Project for Youth Top-notch Scholars of Guangdong Province (2016TQ03X100).

Acknowledgments

This work was performed in and supported in part by Joint International Research Laboratory of Optical Information.

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Items	Value
FFT size ( $N_{F F T}$ )	512
Modulated subcarriers ( $N_{c h}$ )	240
Samples used for CP ( $N_{c p}$ )	8
Modulation formats	Adaptive bit loading using BPSK and m-QAM (m = 4 - 256)
Signals used	OFDM
Blocks per frame (M)	255
AWG sample rate ( $f_{c l o c k}$ )	20 Gsa/s
ADC / DAC resolution for simulations	6 bits
Wavelength of optical carrier	1544.5 nm
Chromatic dispersion	16.7 ps/nm/km
Fiber length	83.2 km
Oscilloscope sample rate	100 Gsa/s

Accurate and spectral efficient channel estimation using inter-block precoding and superimposed pilots in optical OFDM systems

Abstract

1. Introduction

2. Operating principle

3. Experimental setup

4. Results and discussion

5. Conclusions

Funding

Acknowledgments

References and links

Cited By

Figures (4)

Tables (1)

Equations (10)

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