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Impact of resampling interpolation FIR filter in the practical Kramers-Kronig receiver

Yuyang Liu, Anxu Zhang, Lipeng Feng, Kai Lv, Hao Liu, Guangnan Su, Xia Sheng, Yadong Gong, Xiaoli Huo, Junjie Li, Chengliang Zhang, Yan Li, and Jian Wu

Open Access Open Access

Abstract

The practical Kramers-Kronig (KK) receiver has been a competitive receiving technique in the data-center, medium reach, and even long-haul metropolitan networks. Nevertheless, an extra digital resampling operation is required at both ends of the KK field reconstruction algorithm due to the spectrum broadening caused by adopting the nonlinear function. Generally, the digital resampling function can be implemented by using linear interpolation (LI-ITP), the Lagrange cubic interpolation (LC-ITP), the spline cubic interpolation (SC-ITP), time-domain anti-aliasing finite impulse response (FIR) filter method (TD-FRM) scheme, and fast Fourier transform (FFT)-based scheme. However, the performance and the computational complexity analysis of different resampling interpolation schemes in the KK receiver have not been thoroughly investigated yet. Different from the interpolation schemes of conventional coherent detection, the interpolation function of the KK system is followed by the nonlinear operation, which will broaden the spectrum significantly. Due to the frequency-domain transfer function of different interpolation schemes, the broadened spectrum will have a potential spectrum aliasing, which will cause serious inter-symbol interference (ISI) and further impair the KK phase retrieval performance. We experimentally investigate the performance of different interpolation schemes under different digital up-sampling rates (i.e. the computational complexity) as well as the cut-off frequency, the tap number of the anti-aliasing filter, and the shape factor of the TD-FRM scheme in a 112-Gbit/s SSB DD 16-QAM system over 1920-km Raman amplification (RFA)-based standard single-mode fiber (SSMF). The experimental results involve that the TD-FRM scheme outperforms other interpolation schemes and the complexity is reduced by at least 49.6%. In fiber transmission results, take 20% soft decision-forward error correction (SD-FEC) of 2×10−2 as the threshold, the LI-ITP and LC-ITP schemes only reach 720-km while others can reach up to 1440-km.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

With the explosive growth of data traffic in recent years, the demand for low-cost, low-complexity, and high-spectrum efficiency in optical communication system is rapidly increasing. Direct-detection (DD) has drawn great attention owing to its low energy consumption, low-cost, and easy implementation in short-reach, intermediate-haul, and even long-haul metropolitan networks [1–4]. However, due to the phase information being discarded during the square-law detection using a single photodetector (PD), the performance gap between DD and conventional coherent detection is quite increased, which further leads to the limited spectral efficiency (SE) and transmission distance. The optical single-sideband (SSB) DD technology has been a competitive transmission scheme, which not only doubles the spectral efficiency, but also eliminates the power fading in comparison with the optical double-sideband (DSB) modulation [5–7]. However, a nonlinear impairment called signal-signal beat interference (SSBI) is inevitable in the SSB DD system without increasing the carrier-to-signal power ratio (CSPR) or increasing the frequency gap between the carrier and the SSB signals, etc. [8–10]. Recently, Kramers-Kronig (KK) receiver has been proposed as a DD reception technology, which combines the advantages of the high SE of the coherent receiver and the low-cost of the DD receiver. As long as the minimum phase condition is satisfied, the KK receiver can reconstruct the full optical complex field digitally from the intensity detected by the single PD through the Hilbert transform or the Hilbert FIR (HT-FIR) filter [11–14].

Till now, KK reception-based experiments have achieved an information data rate up to 909-Tbit/s and a fiber transmission distance up to 1440-km [5,14,15]. Besides, the practical implementation of the KK receiver has also been widely elaborated in the aspects of the KK commercial transmission validation, parallelized block-wise KK implementation, and analog low-latency KK optical SSB receiver, which shows great practical performance in data-center interconnects, short-medium distance, and even long-haul metropolitan networks [16–19]. In general, due to the nonlinear function that exists in the KK receiver, an extra digital up-sampling operation is necessary before phase retrieval, after performing look-up table (LUT)-based logarithm, root, trigonometric functions, and the HT-FIR filter-based phase recovery, the digital down-sampling operation is adopted [20,21]. The digital resampling has been studied in many fields of digital signal processing (DSP) techniques, such as clock recovery (CR), channel equalization, etc. [22–26]. Generally, there are three interpolation methods: linear and nonlinear function-based interpolation, anti-aliasing FIR filter-based interpolation, and FFT-based interpolation, their performance, computational complexity, and implementation are also not the same [27–29]. However, the performance of the resampling interpolation in the KK receiver has not been investigated yet. Different from the interpolation schemes of the conventional coherent detection, the interpolation function of the KK system is followed by the nonlinear logarithm operation, which will change the spectrum of the up-sampled signal significantly [6]. Due to the frequency-domain transfer function of different interpolation schemes, the spectrum of the up-sampled signal will have a potential spectrum aliasing, which will cause a serious aliasing noise and the inter-symbol interference (ISI) and further impair the KK phase retrieval performance.

In this paper, the noise mechanism, computational complexity, and performance of different interpolation functions in the practical Kramers-Kronig receiver are investigated and analyzed in a 112-Gbit/s SSB DD 16-QAM system over 1920-km Raman amplification (RFA)-based standard single-mode fiber (SSMF). The experimental results show that the time-domain anti-aliasing FIR filter method (TD-FRM) has the optimal resampling performance and outperforms other interpolation schemes, and the complexity is reduced by at least 49.6% when a similar BER performance is obtained. Moreover, the impact of the tap number, normalized cut-off frequency, and shape factor of the Kaiser-window-shaped anti-aliasing FIR low-pass filter (LPF) in the TD-FRM scheme is also performed, the results show that when the tap number and the normalized cut-off frequency are larger than 7, and the ratio of the bandwidth to the digital sampling rate, superior performance of the TD-FRM scheme can be obtained. Finally, fiber transmission results are shown and elaborated.

2. Principle of interpolation filter in KK receiver

2.1 Analog and digital up-sampling rate versus log-induced spectrum broadening

The principle of the KK receiver has been studied in detail in the previous work [12]. Considering the spectrum broadening caused by the nonlinear functions, an extra digital up-sampling operation is generally required before the KK phase retrieval process. The digital up-sampling can be easily implemented by the linear and nonlinear interpolation function, the time-domain (TD), and frequency-domain (FD) schemes. Figure 1(a) shows the schematic diagram of the signals after the digital up-sampling operation, wherein, fa is half of the analog sampling rate, fN is half of the Nyquist sampling rate, and fd is half of the digital up-sampling rate. Then, the logarithm operation is performed and expanded according to the KK principle and Taylor expansion [30]:

$$\begin{aligned} &\ln \sqrt {I(t)} = \frac{1}{2}\ln [{I(t)} ]= \frac{1}{2}\ln {|{A + s(t)} |^\textrm{2}}\textrm{ = }\frac{1}{2}\ln \{{[{A + s(t)} ][{A + {s^ \ast }(t)} ]} \}\\ &= \ln A + \underbrace{{\frac{1}{2}\left[ {\frac{{s(t)}}{A} + \frac{{{s^ \ast }(t)}}{A}} \right]}}_{{first - order}} + \underbrace{{\frac{1}{2}\left[ {\frac{{{s^2}(t)}}{{2{A^2}}} + \frac{{{s^ \ast }^2(t)}}{{2{A^2}}}} \right]}}_{{second - order}} + \underbrace{{\frac{1}{2}\left[ {\frac{{{s^3}(t)}}{{3{A^3}}} + \frac{{{s^ \ast }^3(t)}}{{3{A^3}}}} \right]}}_{{third - order}} + \cdots \end{aligned}$$
$$\begin{aligned} F\left[ {\ln \sqrt {I(t)} } \right] &= \ln A\cdot \delta (f) + \underbrace{{\frac{1}{2}\left\{ {\frac{{S(f)}}{A} + \frac{{{{[{S( - f)} ]}^ \ast }}}{A}} \right\}}}_{{first - order}} + \underbrace{{\frac{1}{2}\left\{ {\frac{{S(f) \ast S(f)}}{{2{A^2}}} + \frac{{{{[{S( - f)} ]}^ \ast } \ast {{[{S( - f)} ]}^ \ast }}}{{2{A^2}}}} \right\}}}_{{second - order}}\\ &\quad +\underbrace{{\frac{1}{2}\left\{ {\frac{{S(f) \ast S(f) \ast S(f)}}{{3{A^3}}} + \frac{{{{[{S( - f)} ]}^ \ast } \ast {{[{S( - f)} ]}^ \ast } \ast {{[{S( - f)} ]}^ \ast }}}{{3{A^3}}}} \right\}}}_{{third - order}} + \cdots \end{aligned}$$
where A is the continuous wave (CW), s(t) is the up-sampled data-carrying signal and s*(t) is the conjugation of s(t), Eq. (2) is the Fourier transform of Eq. (1). In Eq. (1), s(t) is composed of the original data-carrying signal m(t) and the additional noise n(t). Thus, the 2-order auto-convolution can be calculated as follows:
$$\begin{aligned} S(f) \ast S(f) &= [{M(f) + N(f)} ]\ast [{M(f) + N(f)} ]\\ &= \underbrace{{M(f) \ast M(f)}}_{{\textrm{useful part}}} + \underbrace{{2M(f) \ast N(f) + N(f) \ast N(f)}}_{{\textrm{aliasing part}}} \end{aligned}$$
where * is the convolution operation.

 figure: Fig. 1.

Fig. 1. Schematic diagram of (a) up-sampled signals, (b) each component of the 2-order auto convolution and (c) the 2-order auto convolution.

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Figure 1(b) shows the schematic diagram of each component of Eq. (2) after the logarithm operation. Similarly, the third-order and higher-order auto-convolution can also be calculated and performed. Figure 1(c) shows the schematic diagram of 2-order auto-convolution, it can be observed that when the unsatisfactory interpolation schemes are adopted, the signal-to-noise convolution component and noise-to-noise auto-convolution component would be aliased into the useful part. However, we notice that when the analog sampling rate is higher than twice the Nyquist sampling rate, the aliasing part of 2-order auto-convolution can be ignored, but such a high analog sampling rate is not suitable for the cost-effective system like the KK receiver. Thus, we will elaborate the impact of different interpolation schemes in the following sections.

2.2 Different interpolation schemes-based resampling

The linear interpolation (LI-ITP), the Lagrange cubic interpolation (LC-ITP), and the spline cubic interpolation (SC-ITP) schemes have been discussed in detail in [25–27,31–34]. The LI-ITP is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points. The LC-ITP refers to calculating the interpolation points based on a given set of input signals via the Lagrange polynomial. The SC-ITP using not-a-knot end conditions and the interpolated value at a query point is based on a cubic interpolation of the values at neighboring grid points in each respective dimension. We then introduce the time-domain FIR filter-based resample method (TD-FRM) as Fig. 2 shows [35]. The TD-FRM scheme first inserts zeros (i.e. zero-insertion) by a factor of R2 followed by an anti-aliasing FIR filter and then discard samples (i.e. decimation operation) by a factor of R1 described in [28,29]. R1 and R2 are the denominator and numerator of the fractional sampling rate ratio Ru, respectively.

 figure: Fig. 2.

Fig. 2. Basic structure of the time-domain FIR filter-based resample method (TD-FRM). R1 and R2 are the denominator and numerator of the fractional sampling rate ratio, respectively.

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Taking the sampling rate from 80-GHz to 120-GHz as an example, R1 and R2 are firstly calculated:

$${R_u} = \frac{{{R_2}}}{{{R_1}}} = \frac{{120\textrm{ - }GHz}}{{80\textrm{ - }GHz}} = \frac{3}{2}$$

Then, the received signal x[n] with a length of Np is up-sampled by inserting zeros with a factor of R2 = 3 and the output signal s[n] with a length of R2·Np is expressed as follows:

$$s[n] = \left\{ {\begin{array}{cc} {x[(n - 1)/{R_2} + 1],}&{n \in [1,{R_2} + 1, \cdots ,{N_p} \cdot {R_2} + 1]}\\ {0,}&{otherwise} \end{array}} \right.$$

Thus, the spectrum of the received signals will be copied and broadened as Fig. 3(a) shows. It should be noted that the spectrum of the signal is captured from the experiments with CSPR = 12 dB. Since every 10 points of the spectrum are smoothed, the spectrum is the envelope of the actual signal spectrum.

 figure: Fig. 3.

Fig. 3. (a) The electrical spectrums of the received signals x[n] (blue dotted line) and the up-sampled signals s[n] (orange solid line), respectively. (b) The electrical spectrums of the filtered signal u[n] (blue solid line), the Kaiser-window-shaped anti-aliasing LPF hk[n] (orange dotted line) and the down-sampled signal y[n] (red solid line).

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Then, the anti-aliasing FIR filter with the impulse response of hk[n] is adopted to eliminate the effect of the copied spectrum, which employs an Nk-tap Kaiser-window-shaped low-pass FIR filter (LPF) with a normalized cut-off frequency of fco and fco is generally calculated as follows [35]:

$${f_{co}} = \frac{\pi }{{\max ({R_1},{R_2})}}$$
where max(·) refers to the maximum value of R1 and R2. Here, fco equals to π/3. The performance of the Kaiser-window-shaped anti-aliasing LPF will be introduced in detail in the experimental results part.

After performing anti-aliasing LPF to the up-sampled signals, the filtered signals are finally discarded samples by R1. The spectrums of the Kaiser-window-shaped LPF, the signals after LPF, and the final resampled signals y[n] are shown in Fig. 3(b).

Besides, the frequency-domain FFT-based resample method (FD-FRM) can also be implemented to realize digital sampling rate conversion. Figure 4 shows the basic structure of the FD-FRM scheme. The FD-FRM scheme first calculates the resample ratio Ru to determine the up-sample or down-sample operation. Up-sampling is adopted as long as Ru is greater than 1 and vice versa. After performing the Np-point (Np is length of the received signal x[n]) fast Fourier transform (FFT), the zero-padding (ZP) or the samples cutting (SC) is employed in the signals when the up-sample or the down-sample is determined [29]. Finally, the inverse fast Fourier transform (IFFT) operation is performed.

 figure: Fig. 4.

Fig. 4. Basic structure of the frequency-domain FFT-based resample method (FD-FRM). ZP: zero-padding. SC: samples cutting. FFT: fast Fourier transform. IFFT: inverse fast Fourier transform.

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Taking the sampling rate from 80-GHz to 120-GHz as an example, R1 and R2 are first calculated as 2 and 3, respectively. Since Ru is greater than 1, an FD zero-padding up-sampling is performed on the received signal x[n]. Zeros are added to the high-frequency part of the positive frequency and the low-frequency part of the negative frequency of the signal [29].

On the other hand, FD samples cutting (SC) is employed when Ru is less than 1. Considering the sample rate from 120-GHz to 80-GHz with R1 and R2 are 3 and 2, FFT is performed first followed by a sample cutting unit. Similar to the FFT-based up-sampling, the high-frequency part of the positive frequency and the low-frequency part of the negative frequency of the signal are cut [29].

3. Experimental setup and offline DSP

3.1 Experimental setup

We investigate the performance of different interpolation schemes in a 112-Gbit/s single sideband (SSB) direct detection (DD) KK reception-based system as Fig. 5 shows. At the transmitted-side, a 28-GBaud 16-QAM signal is generated in the 65-GSa/s arbitrary waveform generator (AWG, Keysight M8195A, 3-dB bandwidth of 25-GHz), after amplifying by two linear electric amplifiers (EAs), the signals are then sent to the I/Q modulator (Fujitsu FTM 7961EX/301, 3-dB bandwidth of 22-GHz). A tunable four-channel narrow linewidth laser is used as the laser source (Keysight N7714A, linewidth < 100-kHz), and the output laser is the same polarization. Laser 1 with a center frequency of 193.4-THz is used as the signal optical carrier and Laser 2 with a center frequency of 193.383-THz is used as a direct current (DC) component. The DC component after polarization-maintaining erbium-doped fiber amplifier (PM-EDFA) is coupled with the optical signal using a polarization-maintaining coupler (PMC) to generate the minimum phase signal (MPS). It should be noted that two laser sources are used to adjust the CSPR value more conveniently. Before sending the MPS signals to the optical fiber transmission link, an EDFA and a variable optical attenuator (VOA) are used to adjust the optical launch power. The fiber link is composed of a multi-span cascaded Raman amplifier (RFA) and 80-km standard single-mode fiber (SSMF). After fiber transmission, the signal is amplified by another EDFA to compensate for the power loss, and an optical band-pass filter (OBPF) with a 3-dB bandwidth of 0.4-nm to remove the out-of-band noise and other undesired frequency components. Finally, after detected by an AC-coupled photodetector (PD) with a 3-dB bandwidth of 70-GHz, the signal is captured by a digital sampling oscilloscope (DSO, Lecroy LabMaster 10-36Zi-A, 3-dB bandwidth of 36-GHz) with an analog sampling rate of 80-GSa/s.

 figure: Fig. 5.

Fig. 5. (a) Experimental setup. (b) Transmitter-side DSP. (c) Different interpolation schemes-based KK reception. (d) Receiver-side DSP. PRBS: pseudo-random bit sequence, AWG: arbitrary waveform generator, IQ-MZM: I/Q Mach-Zehnder modulator, EDFA: erbium-doped fiber amplifier, OBPF: optical band-pass filter, PD: photodetector, DSO: digital sampling oscilloscope, DSP: digital signal processing.

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3.2 Transmitted-side and received-side offline DSP

At the transmitted-side, a 14-order pseudo-random bit sequence (PRBS) with a length of 214-1 is used for QAM signal generation and then Gray-mapped. After up-sampling with a factor of 2 samples per symbol (2-SPS) and root-raised cosine (RRC) shaping with a roll-off factor of 0.2, the signal is sent to the AWG.

At the received-side, the signal is firstly processed by different interpolation schemes-based KK receiver and RRC-based matched filter with a roll-off factor of 0.2, and then in-phase and quadrature (I/Q) imbalance compensation based on the Gram–Schmidt orthogonalization procedure (GSOP) algorithm is used for possible I/Q imbalance caused by the IQ modulator or other optoelectronics (O/E) devices. The Hilbert FIR filter used in the KK receiver is realized by the frequency-domain sampling approach mentioned in [14,36]. It should be noted that to isolate the impact of different interpolation schemes from the Gibbs phenomenon, the tap number of the HT-FIR filter is 33 [30]. Next, the frequency-domain electrical dispersion compensation (FD-EDC) and Gardner algorithm-based T/2-clock recovery (CR) are used. Besides, channel equalization based on the multi-module algorithm (MMA) with a 15-tap number is employed to compensate for the residual inter-symbol interference (ISI). 4-order frequency-domain frequency offset estimation (FOE) and Viterbi-Viterbi phase estimation (VVPE) are used for linewidth compensation and possible phase offset due to the DSP algorithm. Finally, BER is calculated using 10 million points after decision-directed extended Kalman filter (DD-EKF), QAM de-mapping and decision.

4. Experimental results and discussions

4.1 Back-to-back case

At first, we investigate the BER performance versus CSPR when different interpolation schemes are considered in the optical back-to-back (B2B) case as Fig. 6 shows. It should be noted that the initial tap number of the Kaiser anti-aliasing filter of TD-FRM and the HT-FIR filter are 9 and 33, respectively. The digital up-sampling rate is 112-GHz (4 samples per symbol, 4-SPS) and the analog sampling rate of DSO is 80-GHz, thus the up-sampling (UP) ratio Ru is 7/5. For all interpolation schemes, the BER performance is improved as CSPR increases, and tends to be unchanged when CSPR is larger than 12 dB. Due to the limited bandwidth of the radio frequency (RF) cable, the CSPR demand will be higher [30]. Besides, the BER performance of CSPR = 14 dB is little deteriorated, which is caused by the modulator bias jitter, or the polarization of the DC component and the signal is not exactly the same due to the fiber disturbance, and further resulting in some power loss when the signal and the DC beat in the square-law detection [30,37]. On the one hand, the performance gap between the LI-ITP and the LC-ITP schemes with other interpolation schemes is due to the joint large inter-symbol-interference (ISI) as mentioned above and the aliasing noise-induced poor KK phase retrieval and we will discuss in detail in the following part.

 figure: Fig. 6.

Fig. 6. The BER performance versus CSPR with different interpolation schemes when digital sampling rate is 112-GHz (4-SPS) in B2B case.

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Figure 7 shows the electrical spectrum of the up-sampled signals, after logarithm operation, aliasing noise, and after root operation when different interpolation schemes with a digital up-sampling rate of 168-GHz (6-SPS) are used, original received intensity signal with an analog sampling rate of 80-GHz (black line) and TD-FRM with a digital up-sampling rate of 168-GHz (blue line and green line) are used as references. It should be noted that a 6-SPS up-sampling rate is used to enrich the observed spectrum component. The aliasing noise mentioned in the principle part can be observed, and the aliasing noise power of the LI-ITP, LC-ITP, SC-ITP, and FD-FRM schemes is decreased in turn, resulting in a gradually better BER performance. The aliasing noise will be aliased into the original data-carrying signals according to Eq. (1) to Eq. (3) and hard to remove by the subsequent DSP algorithm like MMA, this is similar to the Nyquist sampling theorem, if signal aliasing occurs, it is unable to recover the original signal through filtering [38]. Besides, the high-frequency part of the log-signal after LI-ITP will be suppressed due to the unsatisfactory transfer function of the LI-ITP scheme, which also impairs the BER performance. However, this suppression will be degraded with the LC-ITP, SC-ITP, and FD-FRM schemes as the transfer function of which are better than that of the LT-ITP scheme. After performing the HT-FIR filter, the phase can be extracted from the up-sampled intensity signal.

 figure: Fig. 7.

Fig. 7. Electrical spectrum of the up-sampled signals, after logarithm operation, aliasing noise, and after root operation with a digital up-sampling rate of 168-GHz (6-SPS) are used, original received intensity signal with an analog sampling rate of 80-GHz (black line) and TD-FRM with a digital up-sampling rate of 168-GHz (blue line and green line) are used as references: (a) LI-ITP and TD-FRM, (b) LC-ITP and TD-FRM, (c) SC-ITP, (d) FD-FRM.

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In terms of the intensity of the signal, the performance is also impaired by the spectral broadening due to the root operation:

$$\begin{array}{l} \sqrt {I(t)} = \exp \left[ {\ln \sqrt {I(t)} } \right] = \exp \left\{ {\ln \sqrt {[{A + s(t)} ][{A + {s^ \ast }(t)} ]} } \right\}\\ = \exp \left\{ {\ln A + \underbrace{{\frac{1}{2}\left[ {\frac{{s(t)}}{A} + \frac{{{s^ \ast }(t)}}{A}} \right]}}_{{first - order}} + \underbrace{{\frac{1}{2}\left[ {\frac{{{s^2}(t)}}{{2{A^2}}} + \frac{{{s^ \ast }^2(t)}}{{2{A^2}}}} \right]}}_{{second - order}} + \cdots } \right\} \end{array}$$
where A is the continuous wave (CW), s(t) is the up-sampled data-carrying signal and s*(t) is the conjugation of s(t). Similarly, the intensity of the signal will be impacted by the same reason: aliasing noise and the high-frequency part suppression-induced BER degradation [38,39].

Nevertheless, the LI-ITP and LC-ITP schemes can still be used in the KK receiver because there are algorithms for bandwidth-limited compensation and inter-symbol-interference (ISI) cancellation in the practical post-compensation DSP algorithms, such as the multi-module algorithm (MMA) [40]. Besides, the power of the noise-to-noise part and the noise-to-signal aliasing part is not that large, the BER performance will not be that unacceptable. However, when the fiber transmission distance increases, which also implies that the optical-to-signal noise ratio (OSNR) decreases, the LI-ITP and the LC-ITP schemes may not work well normally, which will be discussed in detail in the optical fiber transmission experiment part.

For the FD-FRM and the TD-FRM schemes, the performance gap between them is mainly caused by two reasons: one is the edge effect of the FD-FRM scheme, in which some error occurs at the beginning and end of the interpolated time samples when the FFT operation is performed. The other is that the FD-FRM scheme can be equivalent to using a rectangular window function in the frequency domain, that is convoluting a sinc(x) function in the time domain, which further leads to time-domain signal sequence distortion and reduces the signal-to-noise ratio of the interpolated signal [14,38,41]. For the TD-FRM scheme, although the spectrum will be broadened and copied after the time-domain zero-padding operation, due to the excellent performance of Kaiser-window-shaped anti-aliasing FIR LPF, the useless components of the high-frequency will be eliminated, and further resulting in a better BER performance. The performance analysis of the Kaiser-window-shaped FIR low-pass filter (LPF) will be performed and introduced in detail as follows.

4.2 Computational complexity analysis

We also investigate the BER performance versus digital up-sampling rate considering different interpolation schemes as Fig. 8 shows. It should be noted that the CSPR is kept at 13 dB. The digital up-sampling rate is closely related to the computational complexity as Fig. 9 shows. Due to the Gibbs phenomenon of the HT-FIR filter, there is an optimal digital up-sampling rate, that is 4 samples per symbol (4-SPS, 112-GHz), and the optimal digital up-sampling rate is the same for all interpolation schemes [30]. It can be observed that the TD-FRM scheme outperforms the LI-ITP and the LC-ITP schemes and has a significant BER reduction by more than an order of magnitude at all sampling rates. Besides, compared with the SC-ITP and the FD-FRM scheme, the TD-FRM scheme also has significant BER advantages at the optimal sampling rate of 4-SPS.

 figure: Fig. 8.

Fig. 8. The BER performance versus samples per symbol (SPS) with different interpolation schemes when tap number of the HT-FIR filter is 33.

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 figure: Fig. 9.

Fig. 9. Computational complexity (Nm) versus parallel block length (Np) with different sampling rate and interpolation schemes.

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Next, Table 1 shows the complexity comparison of different interpolation schemes and we also perform the computational complexity comparison of different interpolation schemes-based KK receiver considering the digital up-sampling rate, and length of the received signal as shown in Fig. 9. It should be noted that the computational complexity can be represented as the number of multipliers in different interpolation-based KK reception [6,14]. Due to the block-wise operation, the edge effect mitigation using the overlap approach (OLA) is adopted with a commonly used OLA-saved length of (Nf-1)/4, Nf is the length of the HT-FIR filter, and Nf is 33. Besides, the length of the received signal is equal to the parallel block length after serial-to-parallel (S/P) converter, and the tap number of the Kaiser-window-shaped anti-aliasing FIR filter is 9. It can be observed that although the complexity of the LI-ITP scheme is the lowest, its performance is also unsatisfactory. The LC-ITP and the SC-ITP schemes have the highest complexity, but the SC-ITP also has an optimal BER performance. Moreover, the TD-FRM and the FD-FRM schemes have excellent BER performance, but the FD-FRM scheme has a higher computational complexity. Finally, the computational complexity of the TD-FRM is reduced by 49.6% and 54.2% compared with the SC-ITP and FD-FRM, respectively, when a similar BER performance is obtained as Fig. 8 and Fig. 9 show. Therefore, the comprehensive performance of the TD-FRM scheme is more acceptable.

Tables Icon

Table 1. Complexity comparison of different interpolation schemes [6,14]a

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4.3 Impact of Kaiser-window-shaped FIR LPF in TD-FRM scheme

Considering the BER performance and the computational complexity, the TD-FRM is the optimal interpolation scheme in the KK receiver, but the performance of the TD-FRM is also impacted by the tap number, cut-off frequency, and the shaping factor of the Kaiser-window-shaped anti-aliasing FIR LPF. The Kaiser-window-shaped anti-aliasing FIR LPF is obtained by convolution of the Kaiser-window function and a low-pass filter.

Kaiser-window function is an optimal window function due to the adjustment of the window-shaping, and the Kaiser-window-shaped FIR LPF has been implemented in the TD-FRM scheme as an anti-aliasing filter. The tap coefficient of the Kaiser-window function is:

$${w_k}[n] = \frac{{{I_0}(\beta )}}{{{I_0}(\alpha )}} = \frac{{{I_0}\left( {\alpha\sqrt {1 - {{\left( {\frac{{n - {N_k}/2}}{{{N_k}/2}}} \right)}^2}} } \right)}}{{{I_0}(\alpha )}}$$
where Nk is the tap number of the Kaiser-window function, α is the shaping factor and I0(β) is the zeroth-order modified Bessel function of the first kind [28,38]. To obtain a Kaiser-window that represents a FIR filter with side-lobe attenuation of αs dB, use the following α:
$$\alpha = \left\{ {\begin{array}{cc} {0.112({\alpha_s} - 8.7)}&{{\alpha_s} > 50dB}\\ \begin{array}{l} 0.5842{({\alpha_s} - 8.7)^{0.4}} + \\ 0.07886({\alpha_s} - 21) \end{array}&{21dB < {\alpha_s} < 50dB}\\ 0&{{\alpha_s} < 21dB} \end{array}} \right.$$

It should be noted that increasing α widens the main lobe and decreases the amplitude of the sidelobes (i.e., increases the attenuation). Thus, an Nk-tap Kaiser-window-shaped FIR low-pass filter (LPF) with a normalized cut-off frequency fco can be obtained:

$${h_k}[n] = {w_k}[n]{h_{LPF}}[n]$$
where hLPF[n] is a low-pass filter with a cut-off frequency of fco, fco can be calculated as Eq. (6) shows. The digital down-sampling converts the sampling rate to the 2 samples per symbol, i.e. 56-GHz in our system.

Thus, we investigate the BER performance versus the tap number of the Kaiser-window-shaped low-pass FIR LPF under different digital up-sampling rate as Fig. 10 shows, shaping factor is kept at 5, the cut-off frequency is obtained as Eq. (6) and the tap number of the HT-FIR filter is 33. It can be observed that the BER performance is improved as the tap number of the Kaiser LPF (Nk) increases and then tends to be stable when Nk is larger than 7. Figure 11 shows the electric spectrums of the signals after Kaiser LPF u[n], the frequency transfer function of the Kaiser LPF hk[n], and the signals after down-sampling y[n], the signals before Kaiser LPF s[n] is used as a reference. The meaning of u[n], y[n], s[n] have been performed in Fig. 2. It should be noted that 7/17/31-tap Kaiser FIR LPF with a digital up-sampling rate of 112-GHz (4-SPS) is used (yellow circle in Fig. 10). Although the poor in-band flatness caused by the lower tap number of Kaiser LPF will impair the spectrum of the signals, the ripple of the 7-tap Kaiser LPF is about 1.2 dB in the signal’s frequency interval, and which can also be compensated by the subsequent MMA algorithm mentioned in the experiment setup section.

 figure: Fig. 10.

Fig. 10. BER performance versus the tap number of the Kaiser-window-shaped low-pass FIR LPF with different sampling rate.

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 figure: Fig. 11.

Fig. 11. Electric spectrum of the signals after Kaiser LPF u[n], the frequency transfer function of the Kaiser LPF hk[n] and the signals after down-sampling y[n], the signals before Kaiser LPF s[n] is used as a reference.

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We then investigate the BER performance versus cut-off frequency fco and shape factor α of the Kaiser-window-shaped anti-aliasing LPF when the 33-tap HT-FIR filter is used as Fig. 12 shows. The bandwidth of the signal is 33.8-GHz = (1 + 0.2) × 28-GHz + 0.2-GHz, and the ratio of the bandwidth to the digital sampling rate is about 0.3, 0.2, and 0.15 for the 4-SPS, 6-SPS, and 8-SPS. The BER tends to be stable when the normalized cut-off frequency is larger than 0.28, 0.2, and 0.14 for the digital sampling rate of 4-SPS, 6-SPS, and 8-SPS, almost the same as the ratio of the bandwidth to the digital sampling rate.

 figure: Fig. 12.

Fig. 12. The BER performance versus shape factor α and cut-off frequency fco.

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Figure 13(a) shows the electrical spectrums of the signals before, with, and after different cut-off frequency fco-based TD-FRM schemes when the digital up-sampling rate is 112-GHz (4-SPS) and the shape factor is 3. It can be observed that the recovered signal after digital up-sampling with fco = 0.23 suffers more low-frequency and high-frequency suppression than that of fco = 0.28, which seriously affects the BER performance. Thus, the optimal normalized cut-off frequency fco is approximately equal to the ratio of the bandwidth to the digital sampling rate, other sampling rates can also be analyzed similarly. Besides, although increasing the shape factor will improve the attenuation of the stopped-band, it will also induce a little high-frequency component as shown in Fig. 13(b) yellow and green solid line. Yet, such a high-frequency distortion has a negligible impact on the system performance as the power of it is indeed low, thus, the shape factor α has a negligible impact on the BER performance when the normalized cut-off frequency fco is larger than the ratio of the bandwidth to the digital sampling rate mentioned above.

 figure: Fig. 13.

Fig. 13. Electric spectrum of the signals after, with and before Kaiser LPF with (a) cut-off frequency of 0.23 and 0.28 when shape factor α is 3 (b) shape factor α is 3 and 9 when cut-off frequency is 0.28 with a digital sampling rate of 112-GHz. DN: digital down-sampling.

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In summary, when the tap number and the cut-off frequency of the Kaiser-window-shaped FIR LPF are larger than 7 and the ratio of the bandwidth to the digital sampling rate, respectively, the performance will be acceptable. In this case, the impact of the shape factor can also be ignored.

4.4 Impact of equal interval decimation in down-sampling

Especially, when the digital up-sampling rate is 4, 6, or 8 samples per symbol, that is, 112-GHz (4-SPS), 168-GHz (6-SPS), and 224-GHz (8-SPS) in our system, the digital down-sampling operation can also be implemented by equal interval decimation. For the equal interval decimation, supposing the decimation factor is M, the initial decimation points are first, second, …, M-th, whose corresponding initial decimation phases are 0, 1, …, M-1, respectively. Figure 14 shows the BER performance versus the initial phase of the equal interval decimation when different sampling rates and interpolation schemes are considered. It should be noted that the digital up-sampling with different interpolation schemes is adopted while the equal interval decimation is uniformly employed in the digital down-sampling operation.

 figure: Fig. 14.

Fig. 14. The BER performance versus initial decimation phase of the equal interval decimation in down-sampling with different interpolation schemes when different digital sampling rates of 112-GHz (4-SPS), 168-GHz (6-SPS), and 224-GHz (8-SPS) are used.

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It can be observed that the BER performance is almost constant when p changes. We derive it in Eq. (11) as follow:

$$y(l) = f[i]|{_{i = SPS\cdot l/2 + p}} = m[i]\cdot \exp [ - j2\pi {f_c}\cdot \frac{i}{{SPS\cdot {R_s}}}]$$
where p = [0, 1, …,SPS/2-1] is the initial decimation phase, m[i] is the signal after KK phase retrieval, f[i] is the signal after down-conversion. Thus, Eq. (11) can be further expanded:
$$\begin{aligned} y(l) &= f[i]|{_{i = SPS\cdot l/2 + p}} = m[i]\cdot \exp [ - j2\pi {f_c}\frac{i}{{SPS\cdot {R_s}}}]\\ & = m[i]|{_{i = SPS\cdot l/2 + p}} \cdot \exp [ - j2\pi {f_c}\frac{{SPS\cdot l/2 + p}}{{SPS\cdot {R_s}}}]\\ &= m[i]\cdot \exp [ - j2\pi {f_c}\frac{{SPS}}{{SPS\cdot {R_s}}}\cdot \frac{l}{2}]\cdot \exp [ - j2\pi {f_c}\frac{p}{{SPS\cdot {R_s}}}]\\ & = m[i]\cdot \exp [ - j\pi {f_c}\frac{l}{{{R_s}}}]\cdot \exp [ - j2\pi {f_c}\frac{p}{{SPS\cdot {R_s}}}] \end{aligned}$$

It should be noted that when p is an integer, the right term of Eq. (12) can be regarded as fixed phase noise, which can be eliminated by subsequent DSP algorithms like FOE and VVPE algorithms as described in the experimental setup [11]. In addition, due to the adoption of the clock recovery (CR) algorithm, the impact of the initial phase of m(i) can also be ignored [35].

4.5 Fiber transmission case

To investigate the transmission performance of different interpolation schemes-based KK reception, a 1920-km cascaded Raman amplifier (RFA)-based fiber link is employed as Fig. 15 shows. The digital sampling rate is 112-GHz (4-SPS) for all interpolation schemes and the BER performance versus transmission distance is validated. We have optimized the optical launch power and CSPR every 240-km in the experiments as Fig. 15 shows. It can be observed that when the transmission distance is up to 1200-km, the BER performance of LI-ITP and LC-ITP is the worst as the KK phase retrieval is indeed impaired by the joint large ISI and other linear, nonlinear distortions. We take the 7% HD-FEC threshold of 3.8 × 10−3 as a reference, the LI-ITP, and LC-ITP only reach 480-km while the SC-ITP, TD-FRM, and FD-FRM can double the transmission distance up to 960-km. Similarly, when the SD-FEC threshold of 2 × 10−2 is considered, the maximum transmission distance of the LI-ITP and LC-ITP is 720-km, while the SC-ITP, TD-FRM, and FD-FRM can reach 1440-km with 33-tap HT-FIR filters.

 figure: Fig. 15.

Fig. 15. The BER performance versus transmission distance with different interpolation schemes when tap number of HT-FIR is 33 are used. 7% HD-FEC: hard-decision forward error correction threshold of 3.8 × 10−3, 20% SD-FEC: soft-decision forward error correction threshold of 2 × 10−2.

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5. Conclusions

In conclusion, we have experimentally investigated the computational complexity and the practical performance of different interpolation schemes in the Kramers-Kronig receiver through a 112-Gbit/s SSB DD 16-QAM system over 1920-km Raman amplification (RFA)-based standard single-mode fiber (SSMF). The interpolation schemes are linear interpolation (LI-ITP), nonlinear Lagrange cubic interpolation (LC-ITP), nonlinear spline cubic interpolation (SC-ITP), the time-domain anti-aliasing finite impulse response (FIR) low-pass filter (LPF)-based resampling method (TD-FRM), and the frequency-domain fast Fourier transform (FFT) resampling method (FD-FRM), respectively.

In the optical back-to-back (B2B) case, we show that the TD-FRM outperforms other interpolation schemes and the computational complexity is at least 49.6% and 54.2% lower than the SC-ITP and FD-FRM schemes when a similar BER performance is obtained. Besides, we have also investigated the performance of the tap number, the normalized cut-off frequency, and the shape factor of the Kaiser anti-aliasing FIR LPF in the TD-FRM scheme, and the results show that when the tap number and the normalized cut-off frequency are larger than 7, and the ratio of the bandwidth to the digital sampling rate, superior performance of the TD-FRM scheme can be obtained. Furthermore, we elaborate on a special case when the sampling rate is an integral multiple of the symbol rate, the digital down-sampling can be implemented by equal interval decimation. The result shows that the impact of the initial decimation phase has a negligible impact on the performance due to the adoption of the subsequent clock recovery and phase estimation digital signal processing (DSP) algorithm. Finally, a 1920-km RFA-based transmission link is employed to investigate the transmission performance of different interpolation schemes-based KK reception. The results show that the performance of LI-ITP and LC-ITP is the worst as the KK phase retrieval is indeed impaired by the joint large ISI and other linear, nonlinear distortions, while the TD-FRM with 33-tap HT-FIR filter and 4-SPS digital sampling rate can realize an error-free transmission over 1440-km when SD-FEC of 2 × 10−2 is considered.

We provide an effective reference for the implementation and performance analysis of interpolation schemes in the practical KK receiver or even more extensive DSP techniques in future short-reach, intermediate-haul, and long-haul metropolitan network applications.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but maybe obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but maybe obtained from the authors upon reasonable request.

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Figures (15)
Fig. 1.
Fig. 1. Schematic diagram of (a) up-sampled signals, (b) each component of the 2-order auto convolution and (c) the 2-order auto convolution.

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Fig. 2.
Fig. 2. Basic structure of the time-domain FIR filter-based resample method (TD-FRM). R1 and R2 are the denominator and numerator of the fractional sampling rate ratio, respectively.

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Fig. 3.
Fig. 3. (a) The electrical spectrums of the received signals x[n] (blue dotted line) and the up-sampled signals s[n] (orange solid line), respectively. (b) The electrical spectrums of the filtered signal u[n] (blue solid line), the Kaiser-window-shaped anti-aliasing LPF hk[n] (orange dotted line) and the down-sampled signal y[n] (red solid line).

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Fig. 4.
Fig. 4. Basic structure of the frequency-domain FFT-based resample method (FD-FRM). ZP: zero-padding. SC: samples cutting. FFT: fast Fourier transform. IFFT: inverse fast Fourier transform.

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Fig. 5.
Fig. 5. (a) Experimental setup. (b) Transmitter-side DSP. (c) Different interpolation schemes-based KK reception. (d) Receiver-side DSP. PRBS: pseudo-random bit sequence, AWG: arbitrary waveform generator, IQ-MZM: I/Q Mach-Zehnder modulator, EDFA: erbium-doped fiber amplifier, OBPF: optical band-pass filter, PD: photodetector, DSO: digital sampling oscilloscope, DSP: digital signal processing.

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Fig. 6.
Fig. 6. The BER performance versus CSPR with different interpolation schemes when digital sampling rate is 112-GHz (4-SPS) in B2B case.

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Fig. 7.
Fig. 7. Electrical spectrum of the up-sampled signals, after logarithm operation, aliasing noise, and after root operation with a digital up-sampling rate of 168-GHz (6-SPS) are used, original received intensity signal with an analog sampling rate of 80-GHz (black line) and TD-FRM with a digital up-sampling rate of 168-GHz (blue line and green line) are used as references: (a) LI-ITP and TD-FRM, (b) LC-ITP and TD-FRM, (c) SC-ITP, (d) FD-FRM.

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Fig. 8.
Fig. 8. The BER performance versus samples per symbol (SPS) with different interpolation schemes when tap number of the HT-FIR filter is 33.

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Fig. 9.
Fig. 9. Computational complexity (Nm) versus parallel block length (Np) with different sampling rate and interpolation schemes.

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Fig. 10.
Fig. 10. BER performance versus the tap number of the Kaiser-window-shaped low-pass FIR LPF with different sampling rate.

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Fig. 11.
Fig. 11. Electric spectrum of the signals after Kaiser LPF u[n], the frequency transfer function of the Kaiser LPF hk[n] and the signals after down-sampling y[n], the signals before Kaiser LPF s[n] is used as a reference.

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Fig. 12.
Fig. 12. The BER performance versus shape factor α and cut-off frequency fco.

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Fig. 13.
Fig. 13. Electric spectrum of the signals after, with and before Kaiser LPF with (a) cut-off frequency of 0.23 and 0.28 when shape factor α is 3 (b) shape factor α is 3 and 9 when cut-off frequency is 0.28 with a digital sampling rate of 112-GHz. DN: digital down-sampling.

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Fig. 14.
Fig. 14. The BER performance versus initial decimation phase of the equal interval decimation in down-sampling with different interpolation schemes when different digital sampling rates of 112-GHz (4-SPS), 168-GHz (6-SPS), and 224-GHz (8-SPS) are used.

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Fig. 15.
Fig. 15. The BER performance versus transmission distance with different interpolation schemes when tap number of HT-FIR is 33 are used. 7% HD-FEC: hard-decision forward error correction threshold of 3.8 × 10−3, 20% SD-FEC: soft-decision forward error correction threshold of 2 × 10−2.

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Tables (1)

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Table 1. Complexity comparison of different interpolation schemes [6,14]a

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Equations (12)

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ln ⁡ I ( t ) = 1 2 ln ⁡ [ I ( t ) ] = 1 2 ln ⁡ | A + s ( t ) | 2  =  1 2 ln ⁡ { [ A + s ( t ) ] [ A + s ∗ ( t ) ] } = ln ⁡ A + 1 2 [ s ( t ) A + s ∗ ( t ) A ] ⏟ f i r s t − o r d e r + 1 2 [ s 2 ( t ) 2 A 2 + s ∗ 2 ( t ) 2 A 2 ] ⏟ s e c o n d − o r d e r + 1 2 [ s 3 ( t ) 3 A 3 + s ∗ 3 ( t ) 3 A 3 ] ⏟ t h i r d − o r d e r + ⋯
F [ ln ⁡ I ( t ) ] = ln ⁡ A ⋅ δ ( f ) + 1 2 { S ( f ) A + [ S ( − f ) ] ∗ A } ⏟ f i r s t − o r d e r + 1 2 { S ( f ) ∗ S ( f ) 2 A 2 + [ S ( − f ) ] ∗ ∗ [ S ( − f ) ] ∗ 2 A 2 } ⏟ s e c o n d − o r d e r + 1 2 { S ( f ) ∗ S ( f ) ∗ S ( f ) 3 A 3 + [ S ( − f ) ] ∗ ∗ [ S ( − f ) ] ∗ ∗ [ S ( − f ) ] ∗ 3 A 3 } ⏟ t h i r d − o r d e r + ⋯
S ( f ) ∗ S ( f ) = [ M ( f ) + N ( f ) ] ∗ [ M ( f ) + N ( f ) ] = M ( f ) ∗ M ( f ) ⏟ useful part + 2 M ( f ) ∗ N ( f ) + N ( f ) ∗ N ( f ) ⏟ aliasing part
R u = R 2 R 1 = 120  -  G H z 80  -  G H z = 3 2
s [ n ] = { x [ ( n − 1 ) / R 2 + 1 ] , n ∈ [ 1 , R 2 + 1 , ⋯ , N p ⋅ R 2 + 1 ] 0 , o t h e r w i s e
f c o = π max ( R 1 , R 2 )
I ( t ) = exp ⁡ [ ln ⁡ I ( t ) ] = exp ⁡ { ln ⁡ [ A + s ( t ) ] [ A + s ∗ ( t ) ] } = exp ⁡ { ln ⁡ A + 1 2 [ s ( t ) A + s ∗ ( t ) A ] ⏟ f i r s t − o r d e r + 1 2 [ s 2 ( t ) 2 A 2 + s ∗ 2 ( t ) 2 A 2 ] ⏟ s e c o n d − o r d e r + ⋯ }
w k [ n ] = I 0 ( β ) I 0 ( α ) = I 0 ( α 1 − ( n − N k / 2 N k / 2 ) 2 ) I 0 ( α )
α = { 0.112 ( α s − 8.7 ) α s > 50 d B 0.5842 ( α s − 8.7 ) 0.4 + 0.07886 ( α s − 21 ) 21 d B < α s < 50 d B 0 α s < 21 d B
h k [ n ] = w k [ n ] h L P F [ n ]
y ( l ) = f [ i ] | i = S P S ⋅ l / 2 + p = m [ i ] ⋅ exp ⁡ [ − j 2 π f c ⋅ i S P S ⋅ R s ]
y ( l ) = f [ i ] | i = S P S ⋅ l / 2 + p = m [ i ] ⋅ exp ⁡ [ − j 2 π f c i S P S ⋅ R s ] = m [ i ] | i = S P S ⋅ l / 2 + p ⋅ exp ⁡ [ − j 2 π f c S P S ⋅ l / 2 + p S P S ⋅ R s ] = m [ i ] ⋅ exp ⁡ [ − j 2 π f c S P S S P S ⋅ R s ⋅ l 2 ] ⋅ exp ⁡ [ − j 2 π f c p S P S ⋅ R s ] = m [ i ] ⋅ exp ⁡ [ − j π f c l R s ] ⋅ exp ⁡ [ − j 2 π f c p S P S ⋅ R s ]
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