Abstract

A novel OFDM-PON structure based on channel characteristic division is proposed to reduce the sampling and computation requirement at the ONUs. In this method, the preprocessed downstream signal propagated to the ONUs is diversely aliased on spectrum by the sub-Nyquist sampling. With the subcarriers in OFDM symbols distorted according to the channel characteristics and overlaid by sections, users can recover the expected original data sent to the specific ONU lossless. Based on this method, the receiving capability of one of the 32 ONUs in a 40-Gb/s 32-QAM channel characteristic division OFDM-PON experiment is tested. The experiment confirms that the sampling rates and FFT sizes can be reduced to 1/32 on average compared to the conventional method. This new method also supports dynamic bandwidth allocations and improves the system efficiency and security by realizing the addressing process in the physical layer.

© 2011 OSA

1. Introduction

As bit rates increase to higher than 10 Gb/s, time-division multiplexed (TDM) passive optical networks (PON), the current market leader, can hardly continue its dominance due to its limited capabilities [1]. What engineers expect for next-generation optical access networks after the TDM-PON era is low building cost, high operation efficiency, friendly structure and adaptability to fast growing service demands [2]. Among all the potential directions, orthogonal frequency-division multiplexing (OFDM) PON is viewed as an outstanding candidate because of its superior performance, such as its high spectral efficiency, linear dispersion tolerance and equalization simplicity [3]. It is possible to further classify two more specific branches of OFDM-PON: OFDM-PON based on a wavelength-division multiplexing (WDM) architecture [4,5], and OFDM-PON based on sub-carrier allocations [6]. The latter is preferred to the former because of its profound advantages in dynamic bandwidth allocations and low cost on optical devices.

However, the most notable trade-off for OFDM-PON’s agility is the insecurity and the redundancy of data receptions (users must recover all the broadcast downstream subcarriers as a whole and then pick up its own section out of them) [7]. In this paper, we aim to overcome the problems above and introduce a novel scheme called channel characteristic division OFDM-PON (CCD-OFDM-PON) in which the optical network unit (ONU) cost on electrical sampling and computation can be highly reduced, and the addressing is achieved directly in the physical layer based on the channel characteristics, and hence the security is further strengthened. This approach is based on the sub-Nyquist sampling theory that has been utilized in our previous works to support OFDM transmissions for different methods and applications [8,9]. Thanks to the nature of this theory, both the agility and security of OFDM-PON are ensured simultaneously without any extra hardware.

2. Channel characteristic division OFDM-PON

To illustrate the principle of CCD-OFDM-PON and demonstrate the downstream system, we first give a simple example. We then mathematically represent the key step–the channel characteristic based preprocessing.

2.1 A typical CCD-OFDM-PON downstream system

Figure 1 shows an example of a CCD-OFDM-PON downstream with four ONUs and eight OFDM subcarriers. At the optical line terminal (OLT), the downstream data for different ONUs are first serial-parallel converted and buffered. Within the buffered block, a certain slice of the dynamically allocated data is mapped into QAM symbols. These symbols modulate the eight subcarriers out of which two carry data for ONU-1, two for ONU-2, four for ONU-3 and none for ONU-4, according to the proportions at this data slot. The subcarriers are preprocessed according to the feedback and estimation of the channel characteristics, before the 8-point IFFT and the digital-to-analog converter (DAC). The preprocessing changes the subcarrier structure and hence rearranges the information carried by the original subcarriers.

 

Fig. 1 Scheme of CCD-OFDM-PON.

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Once the DAC output waveform is loaded onto light waves, it is launched to the fiber link and propagated to different ONUs through diverse channels. Accordingly, the launched waveform evolves to different shapes after the splitter. At the ONUs, the analog-to-digital converters (ADCs) sample the detected electrical signals in different sub-Nyquist rates to produce spectral aliasing. The sampling rate as well as the FFT size at each ONU is proportional to its preassigned bandwidth. In this example, the sampling rate of ONU-1 and 2 is 1/4 of the DAC, ONU-3 is 1/2 and ONU-4 stays idle, according to the bandwidth allocation shown on the very left side. 2, 2, and 4-point FFTs at ONU-1 to 3, respectively, are implemented, and as such sections of the original subcarriers with different lengths (or nothing at all) are recovered. After compensation and de-mapping, the ONUs retrieve their expected data from the recovered subcarriers directly without addressing. Generally, the propagation and aliasing processes act together as an inversion of the preprocessing, and they naturally distribute the information carried by the subcarrier sections to each target ONU.

2.2 Mathematical description of the preprocessing

In a system with N ONUs, for each OFDM symbol containing nc subcarriers, the propagation and aliasing process can be mathematically written as

HSpr=S,

where H is a fixed nc × nc matrix whose element, Hm(fk), 1≤mN, 1≤knc, describes the frequency response at the k-th subcarrier's frequency, fk, through the link from the DAC at the OLT to the ADC at ONU-m, S pr is an nc × 1 vector composed by the preprocessed subcarriers, and S is an nc × 1 vector composed by the original subcarriers allocated to different ONUs. More specifically, Eq. (1) can be expressed as follows for the example given in Subsection 2.1:

(H1(f1)0H1(f3)0H1(f5)0H1(f7)00H1(f2)0H1(f4)0H1(f6)0H1(f8)H2(f1)0H2(f3)0H2(f5)0H2(f7)00H2(f2)0H2(f4)0H2(f6)0H2(f8)H3(f1)000H3(f5)0000H3(f2)000H3(f6)0000H3(f3)000H3(f7)0000H3(f4)000H3(f8))(Spr1Spr2Spr3Spr4Spr5Spr6Spr7Spr8)=(S1S2S3S4S5S6S7S8).

The preprocessing, as an inversion of the propagation and aliasing process, is to solve for the vector S pr inside the equation. S pr can be computed in a matrix multiplication,

Spr=H1S,

where H −1 is the inverse of H.

After H is obtained by sending training symbols and retrieving the feedback signal through the upstream link, H −1 is computed and saved in the transmitter module. As we can see, the matrix H representing the channel characteristics and the aliasing is block-diagonal, and its inverse, H −1, retains this and is easy to compute. Preprocessed subcarriers for more than one data slot can be computed according to (3) by using the same H −1.

3. Experimental setup and results

Figure 2 shows the experimental setup composed of an OLT, a physical ONU and 31 virtual ONUs to realize a 40-Gb/s CCD-OFDM-PON downstream transmission over a 25 km distance. The 32 ONUs are indexed from ONU-1 to 32, and one of them, ONU-1, is evaluated experimentally. At the offline part, the original pseudo-random data are generated in MatLab without FEC applied. 3/64 of the data are downstream to ONU-1 while the rest of the data are intended for the 31 virtual ONUs. Before the transmission, a test signal is transmitted through the link from the OLT to ONU-1 to estimate the channel characteristics whose amplitude and phase transfer functions are shown as insertion (a) in Fig. 2. One of the main reasons that cause the amplitude fading shown in (a) is the dispersion-induced penalty attributed to the double-sideband and the intensity modulation with direct detection (IM/DD). The greatest dispersion-induced penalty is computed around 3 dB at 8.6 GHz. Likewise, similar but distinct characteristics are factitiously given to those links to ONU-2 to 32 so as to imitate a complete PON structure.

 

Fig. 2 Experimental setup of 40-Gb/s CCD-OFDM-PON downstream, and (a) frequency response of channel to ONU-1 and (b) original and (c) preprocessed subcarriers in one OFDM symbol.

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After the information is loaded into the OLT offline modules, the data are mapped into a 32-QAM format and modulate the 256 subcarriers within the OFDM signals. The upper 128 subcarriers retain conjugate symmetry to the lower 128 in order to guarantee the signal real-valued in the time domain as needed for the IM/DD reception. Among the 128 independent subcarriers, as the ones in a random OFDM symbol sketched in insertion (b) in Fig. 2, six subcarriers are carrying 30 bits of data for ONU-1. In the channel characteristic based preprocessing module, these 256 original subcarriers are then preprocessed according to the sampling rates at the ONUs. Since the processing is real-valued, the newly derived 256 subcarriers retain conjugate symmetry, which is shown in insertion (c). After the 256-point IFFT, the time-domain signal is up-sampled with an over-sampling factor of 1.4 to satisfy the DAC requirement. In the symbol stream, 3% of cycle prefix and 3% of training symbols are inserted for the succeeding compensation. A waveform with an 8.6-GHz bandwidth is output from a Tektronix 7122B arbitrary waveform generator (AWG) operating at 24 GSa/s. As an IM/DD application, the waveform is loaded onto light waves in a Mach-Zehnder modulator (MZM) from Avanex modulating continuous wave light from a 100-kHz-linewidth tunable laser (TL). At the output of the OLT, the optical signal is amplified to 3.77 dBm by an EDFA before launched into a 25 km standard single mode fiber (SSMF) link.

The output power from the 25 km fiber varies at around −1.4 dBm. To emulate a splitting loss, a tunable attenuator with at least 15-dB attenuation is placed before ONU-1. At the ONU front end, the weak signal is first amplified to meet the photodiode (PD) sensitivity and then filtered by an optical band-pass filter (OBPF). The directly detected signal is sampled by a Tektronix DPO72004B oscilloscope operating at a sampling rate of 50 GSa/s. According to the principle, the sampling rate here is required to be 536 (24/1.4/32) MSa/s, and as such we implemented a down-sampling to 536-MSa/s as soon as we saved the sampled data from the oscilloscope to MatLab. After the 12-point FFT and the equalization, six 32-QAM symbols contained in each OFDM symbol are recovered at ONU-1.

The bit error rate (BER) performance curves and the 32-QAM constellation diagrams are plotted in Fig. 3 with two different received powers (−16 dBm with 15-dB attenuation as mentioned above, and −21 dBm with 20-dB attenuation to imitate a higher splitting loss). As a reference, a system transmitting conventional non-preprocessed OFDM signals with ONUs receiving the full-bandwidth broadcast signal at the same sampling rate as the DAC’s is also realized, and its BER performance is also shown in Fig. 3. Compared to this conventional baseline, no obvious optical signal to noise ratio (OSNR) penalty can be observed when using CCD-OFDM-PON. To examine the operation of ONU-1, Fig. 4 plots the BER curve versus the received power in comparison to the conventional OFDM-PON, and a 0.3-dB power penalty on average is measured.

 

Fig. 3 BER performance of CCD-OFDM-PON and conventional OFDM-PON with different received powers.

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Fig. 4 BER-received power performance of CCD-OFDM-PON and conventional OFDM-PON.

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4. Conclusion

By virtue of its efficient structure and tolerant performance, CCD-OFDM-PON is a promising candidate for future access networks in which its simplicity and security are highlighted. With the realization of the OLT and one of the 32 ONUs in a 40-Gb/s CCD-OFDM-PON downstream experiment, we verified that ONUs can recover data with a low sampling rate and computation cost. Since the ADC rates and FFT sizes do not need to be the same as the ones at the OLT and only fractional values are required, the rates and sizes can, on average, be reduced to 1/N of those in the conventional case, where N is the amount of ONUs. Furthermore, thanks to the channel-characteristic division between each ONU, users’ security can be guaranteed in the physical layer. Lastly, the sub-Nyquist sampling not only simplifies users’ hardware but also cuts the redundancy of data for reception. As such the downstream broadcast addressing process can be realized directly in the physical layer.

Acknowledgments

This work was supported in part by the National Nature Science Foundation of China (NSFC) under grant Nos. 60736003, 60932004, 61025004, and 61032005; National 863 Program under grant Nos. 2009AA01Z222, 2009AA01Z223; Foundation of the Key State Lab of Integrated Optoelectronics under grant No. 2010KFB007; and the Ph.D. Programs Foundation of Ministry of Education of China under grant No. 20100002110039.

References

1. P. P. Iannone and K. C. Reichmann, “Optical access beyond 10 Gb/s PON”, in 2010 36th European Conference and Exhibition on Optical Communication (ECOC), (2010), pp. 1–5.

2. D. Breuer, R. Hülsermann, C. Lange, T. Monath, and E. Weis, “Architectural options and challenges for next generation optical access,” in 2010 36th European Conference and Exhibition on Optical Communication (ECOC), (2010), pp. 1–5.

3. N. Cvijetic, “OFDM in optical access networks,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper OMG3.

4. J. Yu, M. Huang, D. Qian, L. Chen, and G. Chang, “Centralized lightwave WDM-PON employing 16-QAM intensity modulated OFDM downstream and OOK modulated upstream signals,” IEEE Photon. Technol. Lett. 20(18), 1545–1547 (2008). [CrossRef]  

5. C. W. Chow, C. H. Yeh, C. H. Wang, F. Y. Shih, C. L. Pan, and S. Chi, “WDM extended reach passive optical networks using OFDM-QAM,” Opt. Express 16(16), 12096–12101 (2008). [CrossRef]   [PubMed]  

6. D. Qian, N. Cvijetic, J. Hu, and T. Wang, “108Gb/s OFDMA-PON with polarization multiplexing and direct-detection,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2009), paper PDPD5.

7. L. Zhang, X. Xin, B. Liu, J. Yu, and Q. Zhang, “A novel ECDM-OFDM-PON architecture for next-generation optical access network,” Opt. Express 18(17), 18347–18353 (2010). [CrossRef]   [PubMed]  

8. L. Cheng, H. Wen, X. Zheng, H. Zhang, Y. Guo, and B. Zhou, “A low-speed receiving method in broadband optical OFDM system,” IEEE Photon. Technol. Lett. 22(15), 1165–1167 (2010). [CrossRef]  

9. L. Cheng, H. Wen, X. Zheng, H. Zhang, and Y. Guo, “Predistortion of high speed optical OFDM signal for aliasing-free receiving in multiple low-bandwidth receiver system,” Chin. Opt. Lett. 8, 377–380 (2010).

References

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  1. P. P. Iannone and K. C. Reichmann, “Optical access beyond 10 Gb/s PON”, in 2010 36th European Conference and Exhibition on Optical Communication (ECOC), (2010), pp. 1–5.
  2. D. Breuer, R. Hülsermann, C. Lange, T. Monath, and E. Weis, “Architectural options and challenges for next generation optical access,” in 2010 36th European Conference and Exhibition on Optical Communication (ECOC), (2010), pp. 1–5.
  3. N. Cvijetic, “OFDM in optical access networks,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper OMG3.
  4. J. Yu, M. Huang, D. Qian, L. Chen, and G. Chang, “Centralized lightwave WDM-PON employing 16-QAM intensity modulated OFDM downstream and OOK modulated upstream signals,” IEEE Photon. Technol. Lett. 20(18), 1545–1547 (2008).
    [CrossRef]
  5. C. W. Chow, C. H. Yeh, C. H. Wang, F. Y. Shih, C. L. Pan, and S. Chi, “WDM extended reach passive optical networks using OFDM-QAM,” Opt. Express 16(16), 12096–12101 (2008).
    [CrossRef] [PubMed]
  6. D. Qian, N. Cvijetic, J. Hu, and T. Wang, “108Gb/s OFDMA-PON with polarization multiplexing and direct-detection,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2009), paper PDPD5.
  7. L. Zhang, X. Xin, B. Liu, J. Yu, and Q. Zhang, “A novel ECDM-OFDM-PON architecture for next-generation optical access network,” Opt. Express 18(17), 18347–18353 (2010).
    [CrossRef] [PubMed]
  8. L. Cheng, H. Wen, X. Zheng, H. Zhang, Y. Guo, and B. Zhou, “A low-speed receiving method in broadband optical OFDM system,” IEEE Photon. Technol. Lett. 22(15), 1165–1167 (2010).
    [CrossRef]
  9. L. Cheng, H. Wen, X. Zheng, H. Zhang, and Y. Guo, “Predistortion of high speed optical OFDM signal for aliasing-free receiving in multiple low-bandwidth receiver system,” Chin. Opt. Lett. 8, 377–380 (2010).

2010 (3)

2008 (2)

J. Yu, M. Huang, D. Qian, L. Chen, and G. Chang, “Centralized lightwave WDM-PON employing 16-QAM intensity modulated OFDM downstream and OOK modulated upstream signals,” IEEE Photon. Technol. Lett. 20(18), 1545–1547 (2008).
[CrossRef]

C. W. Chow, C. H. Yeh, C. H. Wang, F. Y. Shih, C. L. Pan, and S. Chi, “WDM extended reach passive optical networks using OFDM-QAM,” Opt. Express 16(16), 12096–12101 (2008).
[CrossRef] [PubMed]

Chang, G.

J. Yu, M. Huang, D. Qian, L. Chen, and G. Chang, “Centralized lightwave WDM-PON employing 16-QAM intensity modulated OFDM downstream and OOK modulated upstream signals,” IEEE Photon. Technol. Lett. 20(18), 1545–1547 (2008).
[CrossRef]

Chen, L.

J. Yu, M. Huang, D. Qian, L. Chen, and G. Chang, “Centralized lightwave WDM-PON employing 16-QAM intensity modulated OFDM downstream and OOK modulated upstream signals,” IEEE Photon. Technol. Lett. 20(18), 1545–1547 (2008).
[CrossRef]

Cheng, L.

L. Cheng, H. Wen, X. Zheng, H. Zhang, Y. Guo, and B. Zhou, “A low-speed receiving method in broadband optical OFDM system,” IEEE Photon. Technol. Lett. 22(15), 1165–1167 (2010).
[CrossRef]

L. Cheng, H. Wen, X. Zheng, H. Zhang, and Y. Guo, “Predistortion of high speed optical OFDM signal for aliasing-free receiving in multiple low-bandwidth receiver system,” Chin. Opt. Lett. 8, 377–380 (2010).

Chi, S.

Chow, C. W.

Guo, Y.

L. Cheng, H. Wen, X. Zheng, H. Zhang, Y. Guo, and B. Zhou, “A low-speed receiving method in broadband optical OFDM system,” IEEE Photon. Technol. Lett. 22(15), 1165–1167 (2010).
[CrossRef]

L. Cheng, H. Wen, X. Zheng, H. Zhang, and Y. Guo, “Predistortion of high speed optical OFDM signal for aliasing-free receiving in multiple low-bandwidth receiver system,” Chin. Opt. Lett. 8, 377–380 (2010).

Huang, M.

J. Yu, M. Huang, D. Qian, L. Chen, and G. Chang, “Centralized lightwave WDM-PON employing 16-QAM intensity modulated OFDM downstream and OOK modulated upstream signals,” IEEE Photon. Technol. Lett. 20(18), 1545–1547 (2008).
[CrossRef]

Liu, B.

Pan, C. L.

Qian, D.

J. Yu, M. Huang, D. Qian, L. Chen, and G. Chang, “Centralized lightwave WDM-PON employing 16-QAM intensity modulated OFDM downstream and OOK modulated upstream signals,” IEEE Photon. Technol. Lett. 20(18), 1545–1547 (2008).
[CrossRef]

Shih, F. Y.

Wang, C. H.

Wen, H.

L. Cheng, H. Wen, X. Zheng, H. Zhang, Y. Guo, and B. Zhou, “A low-speed receiving method in broadband optical OFDM system,” IEEE Photon. Technol. Lett. 22(15), 1165–1167 (2010).
[CrossRef]

L. Cheng, H. Wen, X. Zheng, H. Zhang, and Y. Guo, “Predistortion of high speed optical OFDM signal for aliasing-free receiving in multiple low-bandwidth receiver system,” Chin. Opt. Lett. 8, 377–380 (2010).

Xin, X.

Yeh, C. H.

Yu, J.

L. Zhang, X. Xin, B. Liu, J. Yu, and Q. Zhang, “A novel ECDM-OFDM-PON architecture for next-generation optical access network,” Opt. Express 18(17), 18347–18353 (2010).
[CrossRef] [PubMed]

J. Yu, M. Huang, D. Qian, L. Chen, and G. Chang, “Centralized lightwave WDM-PON employing 16-QAM intensity modulated OFDM downstream and OOK modulated upstream signals,” IEEE Photon. Technol. Lett. 20(18), 1545–1547 (2008).
[CrossRef]

Zhang, H.

L. Cheng, H. Wen, X. Zheng, H. Zhang, Y. Guo, and B. Zhou, “A low-speed receiving method in broadband optical OFDM system,” IEEE Photon. Technol. Lett. 22(15), 1165–1167 (2010).
[CrossRef]

L. Cheng, H. Wen, X. Zheng, H. Zhang, and Y. Guo, “Predistortion of high speed optical OFDM signal for aliasing-free receiving in multiple low-bandwidth receiver system,” Chin. Opt. Lett. 8, 377–380 (2010).

Zhang, L.

Zhang, Q.

Zheng, X.

L. Cheng, H. Wen, X. Zheng, H. Zhang, Y. Guo, and B. Zhou, “A low-speed receiving method in broadband optical OFDM system,” IEEE Photon. Technol. Lett. 22(15), 1165–1167 (2010).
[CrossRef]

L. Cheng, H. Wen, X. Zheng, H. Zhang, and Y. Guo, “Predistortion of high speed optical OFDM signal for aliasing-free receiving in multiple low-bandwidth receiver system,” Chin. Opt. Lett. 8, 377–380 (2010).

Zhou, B.

L. Cheng, H. Wen, X. Zheng, H. Zhang, Y. Guo, and B. Zhou, “A low-speed receiving method in broadband optical OFDM system,” IEEE Photon. Technol. Lett. 22(15), 1165–1167 (2010).
[CrossRef]

Chin. Opt. Lett. (1)

IEEE Photon. Technol. Lett. (2)

L. Cheng, H. Wen, X. Zheng, H. Zhang, Y. Guo, and B. Zhou, “A low-speed receiving method in broadband optical OFDM system,” IEEE Photon. Technol. Lett. 22(15), 1165–1167 (2010).
[CrossRef]

J. Yu, M. Huang, D. Qian, L. Chen, and G. Chang, “Centralized lightwave WDM-PON employing 16-QAM intensity modulated OFDM downstream and OOK modulated upstream signals,” IEEE Photon. Technol. Lett. 20(18), 1545–1547 (2008).
[CrossRef]

Opt. Express (2)

Other (4)

D. Qian, N. Cvijetic, J. Hu, and T. Wang, “108Gb/s OFDMA-PON with polarization multiplexing and direct-detection,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2009), paper PDPD5.

P. P. Iannone and K. C. Reichmann, “Optical access beyond 10 Gb/s PON”, in 2010 36th European Conference and Exhibition on Optical Communication (ECOC), (2010), pp. 1–5.

D. Breuer, R. Hülsermann, C. Lange, T. Monath, and E. Weis, “Architectural options and challenges for next generation optical access,” in 2010 36th European Conference and Exhibition on Optical Communication (ECOC), (2010), pp. 1–5.

N. Cvijetic, “OFDM in optical access networks,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper OMG3.

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Figures (4)

Fig. 1
Fig. 1

Scheme of CCD-OFDM-PON.

Fig. 2
Fig. 2

Experimental setup of 40-Gb/s CCD-OFDM-PON downstream, and (a) frequency response of channel to ONU-1 and (b) original and (c) preprocessed subcarriers in one OFDM symbol.

Fig. 3
Fig. 3

BER performance of CCD-OFDM-PON and conventional OFDM-PON with different received powers.

Fig. 4
Fig. 4

BER-received power performance of CCD-OFDM-PON and conventional OFDM-PON.

Equations (3)

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H S p r = S ,
( H 1 ( f 1 ) 0 H 1 ( f 3 ) 0 H 1 ( f 5 ) 0 H 1 ( f 7 ) 0 0 H 1 ( f 2 ) 0 H 1 ( f 4 ) 0 H 1 ( f 6 ) 0 H 1 ( f 8 ) H 2 ( f 1 ) 0 H 2 ( f 3 ) 0 H 2 ( f 5 ) 0 H 2 ( f 7 ) 0 0 H 2 ( f 2 ) 0 H 2 ( f 4 ) 0 H 2 ( f 6 ) 0 H 2 ( f 8 ) H 3 ( f 1 ) 0 0 0 H 3 ( f 5 ) 0 0 0 0 H 3 ( f 2 ) 0 0 0 H 3 ( f 6 ) 0 0 0 0 H 3 ( f 3 ) 0 0 0 H 3 ( f 7 ) 0 0 0 0 H 3 ( f 4 ) 0 0 0 H 3 ( f 8 ) ) ( S p r 1 S p r 2 S p r 3 S p r 4 S p r 5 S p r 6 S p r 7 S p r 8 ) = ( S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 )
S p r = H 1 S ,

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