Abstract

A Multi-format receiver for both non-return-to-zero binary-phase-shift-keyed (NRZ-BPSK) signal and non-return-to-zero amplitude-shift-keyed (NRZ-ASK) signal is demonstrated. Multi-format signal detection is based on incoherent BPSK demodulation and ASK-BPSK format conversion. Incoherent BPSK demodulation is realized by a Mach-Zehnder delay interferometer (MZDI) and a feedback decoder. Transmission experiments validate the feasibility of multi-format receiver. This receiver has potential to serve as a useful terminal block for all-optical wavelength division-multiplexed (WDM) networks.

© 2009 Optical Society of America

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References

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  1. A. H. Gnauck, S. Chandrasekhar, J. Leuthold, and L. Stulz, "Demonstration of 42.7-Gb/s DPSK receiver with 45 photons/bit sensitivity," IEEE Photon. Technol. Lett. 15, 99-101 (2003).
    [CrossRef]
  2. A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, and E. Burrows, "25×40-Gb/s copolarized DPSK transmission over 12×100-km NZDF with 50-GHz channel spacing," IEEE Photon. Technol. Lett. 15, 467-469 (2003).
    [CrossRef]
  3. T. Mizuochi, K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, K. Motoshima, and K. Kasahara, "A comparative study of DPSK and OOK WDM transmission over transoceanic distances and their performance degradations due to nonlinear phase noise," J. Lightwave Technol. 21, 1933-1943 (2003).
    [CrossRef]
  4. K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K. Kitayama, "NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter," J. Lightwave Technol. 24, 3751-3758 (2006).
    [CrossRef]
  5. C.  Yan, Y. Su, L. Yi, L. Feng, X. Tian, X. Xu, and Y. Tian, "All-optical format conversion from NRZ to BPSK using a single saturated SOA," IEEE Photon. Technol. Lett. 18, 2368-2370 (2006).
    [CrossRef]
  6. W. Astar and G. M. Carter, "10 Gb/s RZ-OOK to BPSK format conversion by cross-phase modulation in single semiconductor optical amplifier," Electron. Lett. 42, 1472-1474 (2006).
    [CrossRef]
  7. S. H. Lee, K. K. Chow, C. Shu, and C. Lin, "All optical ASK to DPSK format conversion using crossphase modulation in a nonlinear photonic crystal fiber," in Conference on Lasers and Electro-Optics (2005), Paper CFJ2-5.
  8. Huan Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, "All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch," IEEE Photon. Technol. Lett. 19, 1985-1987 (2007).
    [CrossRef]
  9. Y. Lu, F. Liu, M. Qiu, and Y. Su, "All-optical format conversions from NRZ to BPSK and QPSK based on nonlinear responses in silicon microring resonators," Opt. Express 15, 14275-14282 (2007)
    [CrossRef] [PubMed]
  10. D. Norte, E. Park, and A. E. Willner, "All-optical TDM-to-WDM data format conversion in a dynamically reconfigurable WDM network," IEEE Photon. Technol. Lett. 7, 920-922 (1995).
    [CrossRef]
  11. K. Inoue, "Experimental study on channel crosstalk due to fiber four-wave mixing around the zero-dispersion wavelength," J. Lightwave Technol. 12, 1023-1028, Jun. 1994.
    [CrossRef]
  12. A. A. M. Saleh and J. M. Simmons, "Architectural Principles of Optical Regional and Metropolitan Access Networks," J. Lightwave Technol. 17, 2431- 2448 (1999).
    [CrossRef]
  13. H. Leib, "Data-aided incoherent demodulation of DPSK," IEEE Trans. Commun. 43, 722-725 (1995).
    [CrossRef]
  14. A. H. Gnauck and P. J. Winzer, "Optical Phase-Shift-Keyed Transmission," J. Lightwave Technol. 23, 115-130 (2005).
    [CrossRef]
  15. X. Wei,Y. Su, X. Liu, J. Leuthold, and S. Chandrasekhar,"10-Gb/s RZ-DPSK Transmitter Using a Saturated SOA as a Power Booster and Limiting Amplifier," IEEE Photon. Technol. Lett. 16, 1582-1584 (2004).
    [CrossRef]
  16. IEEE 802.3-2005, Section Five, &58.4.1 Transmitter optical specifications, 63-64.
  17. L. Cai, S. L. Xiao, Z. X. Liu, R. Y. Li, M. Zhu and W. S. Hu "Cost-effective WDM-PON for simultaneously transmitting unicast and broadcast/multicast data by superimposing IRZ signal onto NRZ signal" in 34th European Conference on Optical Communication (ECOC), (2008), Paper Th.1.F.4.

2007 (2)

Huan Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, "All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch," IEEE Photon. Technol. Lett. 19, 1985-1987 (2007).
[CrossRef]

Y. Lu, F. Liu, M. Qiu, and Y. Su, "All-optical format conversions from NRZ to BPSK and QPSK based on nonlinear responses in silicon microring resonators," Opt. Express 15, 14275-14282 (2007)
[CrossRef] [PubMed]

2006 (3)

K. Mishina, A. Maruta, S. Mitani, T. Miyahara, K. Ishida, K. Shimizu, T. Hatta, K. Motoshima, and K. Kitayama, "NRZ-OOK-to-RZ-BPSK modulation-format conversion using SOA-MZI wavelength converter," J. Lightwave Technol. 24, 3751-3758 (2006).
[CrossRef]

C.  Yan, Y. Su, L. Yi, L. Feng, X. Tian, X. Xu, and Y. Tian, "All-optical format conversion from NRZ to BPSK using a single saturated SOA," IEEE Photon. Technol. Lett. 18, 2368-2370 (2006).
[CrossRef]

W. Astar and G. M. Carter, "10 Gb/s RZ-OOK to BPSK format conversion by cross-phase modulation in single semiconductor optical amplifier," Electron. Lett. 42, 1472-1474 (2006).
[CrossRef]

2005 (1)

2004 (1)

X. Wei,Y. Su, X. Liu, J. Leuthold, and S. Chandrasekhar,"10-Gb/s RZ-DPSK Transmitter Using a Saturated SOA as a Power Booster and Limiting Amplifier," IEEE Photon. Technol. Lett. 16, 1582-1584 (2004).
[CrossRef]

2003 (3)

A. H. Gnauck, S. Chandrasekhar, J. Leuthold, and L. Stulz, "Demonstration of 42.7-Gb/s DPSK receiver with 45 photons/bit sensitivity," IEEE Photon. Technol. Lett. 15, 99-101 (2003).
[CrossRef]

A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, and E. Burrows, "25×40-Gb/s copolarized DPSK transmission over 12×100-km NZDF with 50-GHz channel spacing," IEEE Photon. Technol. Lett. 15, 467-469 (2003).
[CrossRef]

T. Mizuochi, K. Ishida, T. Kobayashi, J. Abe, K. Kinjo, K. Motoshima, and K. Kasahara, "A comparative study of DPSK and OOK WDM transmission over transoceanic distances and their performance degradations due to nonlinear phase noise," J. Lightwave Technol. 21, 1933-1943 (2003).
[CrossRef]

1999 (1)

1995 (2)

H. Leib, "Data-aided incoherent demodulation of DPSK," IEEE Trans. Commun. 43, 722-725 (1995).
[CrossRef]

D. Norte, E. Park, and A. E. Willner, "All-optical TDM-to-WDM data format conversion in a dynamically reconfigurable WDM network," IEEE Photon. Technol. Lett. 7, 920-922 (1995).
[CrossRef]

1994 (1)

K. Inoue, "Experimental study on channel crosstalk due to fiber four-wave mixing around the zero-dispersion wavelength," J. Lightwave Technol. 12, 1023-1028, Jun. 1994.
[CrossRef]

Abe, J.

Astar, W.

W. Astar and G. M. Carter, "10 Gb/s RZ-OOK to BPSK format conversion by cross-phase modulation in single semiconductor optical amplifier," Electron. Lett. 42, 1472-1474 (2006).
[CrossRef]

Burrows, E.

A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, and E. Burrows, "25×40-Gb/s copolarized DPSK transmission over 12×100-km NZDF with 50-GHz channel spacing," IEEE Photon. Technol. Lett. 15, 467-469 (2003).
[CrossRef]

Carter, G. M.

W. Astar and G. M. Carter, "10 Gb/s RZ-OOK to BPSK format conversion by cross-phase modulation in single semiconductor optical amplifier," Electron. Lett. 42, 1472-1474 (2006).
[CrossRef]

Chandrasekhar, S.

X. Wei,Y. Su, X. Liu, J. Leuthold, and S. Chandrasekhar,"10-Gb/s RZ-DPSK Transmitter Using a Saturated SOA as a Power Booster and Limiting Amplifier," IEEE Photon. Technol. Lett. 16, 1582-1584 (2004).
[CrossRef]

A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, and E. Burrows, "25×40-Gb/s copolarized DPSK transmission over 12×100-km NZDF with 50-GHz channel spacing," IEEE Photon. Technol. Lett. 15, 467-469 (2003).
[CrossRef]

A. H. Gnauck, S. Chandrasekhar, J. Leuthold, and L. Stulz, "Demonstration of 42.7-Gb/s DPSK receiver with 45 photons/bit sensitivity," IEEE Photon. Technol. Lett. 15, 99-101 (2003).
[CrossRef]

Doerr, C.

A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, and E. Burrows, "25×40-Gb/s copolarized DPSK transmission over 12×100-km NZDF with 50-GHz channel spacing," IEEE Photon. Technol. Lett. 15, 467-469 (2003).
[CrossRef]

Feng, L.

C.  Yan, Y. Su, L. Yi, L. Feng, X. Tian, X. Xu, and Y. Tian, "All-optical format conversion from NRZ to BPSK using a single saturated SOA," IEEE Photon. Technol. Lett. 18, 2368-2370 (2006).
[CrossRef]

Gnauck, A. H.

A. H. Gnauck and P. J. Winzer, "Optical Phase-Shift-Keyed Transmission," J. Lightwave Technol. 23, 115-130 (2005).
[CrossRef]

A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, and E. Burrows, "25×40-Gb/s copolarized DPSK transmission over 12×100-km NZDF with 50-GHz channel spacing," IEEE Photon. Technol. Lett. 15, 467-469 (2003).
[CrossRef]

A. H. Gnauck, S. Chandrasekhar, J. Leuthold, and L. Stulz, "Demonstration of 42.7-Gb/s DPSK receiver with 45 photons/bit sensitivity," IEEE Photon. Technol. Lett. 15, 99-101 (2003).
[CrossRef]

Guo, Yili

Huan Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, "All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch," IEEE Photon. Technol. Lett. 19, 1985-1987 (2007).
[CrossRef]

Han, Liuyan

Huan Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, "All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch," IEEE Photon. Technol. Lett. 19, 1985-1987 (2007).
[CrossRef]

Hatta, T.

Inoue, K.

K. Inoue, "Experimental study on channel crosstalk due to fiber four-wave mixing around the zero-dispersion wavelength," J. Lightwave Technol. 12, 1023-1028, Jun. 1994.
[CrossRef]

Ishida, K.

Jiang, Huan

Huan Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, "All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch," IEEE Photon. Technol. Lett. 19, 1985-1987 (2007).
[CrossRef]

Kasahara, K.

Kinjo, K.

Kitayama, K.

Kobayashi, T.

Leib, H.

H. Leib, "Data-aided incoherent demodulation of DPSK," IEEE Trans. Commun. 43, 722-725 (1995).
[CrossRef]

Leuthold, J.

X. Wei,Y. Su, X. Liu, J. Leuthold, and S. Chandrasekhar,"10-Gb/s RZ-DPSK Transmitter Using a Saturated SOA as a Power Booster and Limiting Amplifier," IEEE Photon. Technol. Lett. 16, 1582-1584 (2004).
[CrossRef]

A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, and E. Burrows, "25×40-Gb/s copolarized DPSK transmission over 12×100-km NZDF with 50-GHz channel spacing," IEEE Photon. Technol. Lett. 15, 467-469 (2003).
[CrossRef]

A. H. Gnauck, S. Chandrasekhar, J. Leuthold, and L. Stulz, "Demonstration of 42.7-Gb/s DPSK receiver with 45 photons/bit sensitivity," IEEE Photon. Technol. Lett. 15, 99-101 (2003).
[CrossRef]

Liu, F.

Liu, X.

X. Wei,Y. Su, X. Liu, J. Leuthold, and S. Chandrasekhar,"10-Gb/s RZ-DPSK Transmitter Using a Saturated SOA as a Power Booster and Limiting Amplifier," IEEE Photon. Technol. Lett. 16, 1582-1584 (2004).
[CrossRef]

Lu, Y.

Maruta, A.

Mishina, K.

Mitani, S.

Miyahara, T.

Mizuochi, T.

Motoshima, K.

Norte, D.

D. Norte, E. Park, and A. E. Willner, "All-optical TDM-to-WDM data format conversion in a dynamically reconfigurable WDM network," IEEE Photon. Technol. Lett. 7, 920-922 (1995).
[CrossRef]

Park, E.

D. Norte, E. Park, and A. E. Willner, "All-optical TDM-to-WDM data format conversion in a dynamically reconfigurable WDM network," IEEE Photon. Technol. Lett. 7, 920-922 (1995).
[CrossRef]

Qiu, M.

Raybon, G.

A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, and E. Burrows, "25×40-Gb/s copolarized DPSK transmission over 12×100-km NZDF with 50-GHz channel spacing," IEEE Photon. Technol. Lett. 15, 467-469 (2003).
[CrossRef]

Saleh, A. A. M.

Shimizu, K.

Simmons, J. M.

Stulz, L.

A. H. Gnauck, S. Chandrasekhar, J. Leuthold, and L. Stulz, "Demonstration of 42.7-Gb/s DPSK receiver with 45 photons/bit sensitivity," IEEE Photon. Technol. Lett. 15, 99-101 (2003).
[CrossRef]

A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, and E. Burrows, "25×40-Gb/s copolarized DPSK transmission over 12×100-km NZDF with 50-GHz channel spacing," IEEE Photon. Technol. Lett. 15, 467-469 (2003).
[CrossRef]

Su, Y.

Y. Lu, F. Liu, M. Qiu, and Y. Su, "All-optical format conversions from NRZ to BPSK and QPSK based on nonlinear responses in silicon microring resonators," Opt. Express 15, 14275-14282 (2007)
[CrossRef] [PubMed]

C.  Yan, Y. Su, L. Yi, L. Feng, X. Tian, X. Xu, and Y. Tian, "All-optical format conversion from NRZ to BPSK using a single saturated SOA," IEEE Photon. Technol. Lett. 18, 2368-2370 (2006).
[CrossRef]

X. Wei,Y. Su, X. Liu, J. Leuthold, and S. Chandrasekhar,"10-Gb/s RZ-DPSK Transmitter Using a Saturated SOA as a Power Booster and Limiting Amplifier," IEEE Photon. Technol. Lett. 16, 1582-1584 (2004).
[CrossRef]

Tian, X.

C.  Yan, Y. Su, L. Yi, L. Feng, X. Tian, X. Xu, and Y. Tian, "All-optical format conversion from NRZ to BPSK using a single saturated SOA," IEEE Photon. Technol. Lett. 18, 2368-2370 (2006).
[CrossRef]

Tian, Y.

C.  Yan, Y. Su, L. Yi, L. Feng, X. Tian, X. Xu, and Y. Tian, "All-optical format conversion from NRZ to BPSK using a single saturated SOA," IEEE Photon. Technol. Lett. 18, 2368-2370 (2006).
[CrossRef]

Wei, X.

X. Wei,Y. Su, X. Liu, J. Leuthold, and S. Chandrasekhar,"10-Gb/s RZ-DPSK Transmitter Using a Saturated SOA as a Power Booster and Limiting Amplifier," IEEE Photon. Technol. Lett. 16, 1582-1584 (2004).
[CrossRef]

Wen, He

Huan Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, "All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch," IEEE Photon. Technol. Lett. 19, 1985-1987 (2007).
[CrossRef]

Willner, A. E.

D. Norte, E. Park, and A. E. Willner, "All-optical TDM-to-WDM data format conversion in a dynamically reconfigurable WDM network," IEEE Photon. Technol. Lett. 7, 920-922 (1995).
[CrossRef]

Winzer, P. J.

Xu, X.

C.  Yan, Y. Su, L. Yi, L. Feng, X. Tian, X. Xu, and Y. Tian, "All-optical format conversion from NRZ to BPSK using a single saturated SOA," IEEE Photon. Technol. Lett. 18, 2368-2370 (2006).
[CrossRef]

Yan, C.

C.  Yan, Y. Su, L. Yi, L. Feng, X. Tian, X. Xu, and Y. Tian, "All-optical format conversion from NRZ to BPSK using a single saturated SOA," IEEE Photon. Technol. Lett. 18, 2368-2370 (2006).
[CrossRef]

Yi, L.

C.  Yan, Y. Su, L. Yi, L. Feng, X. Tian, X. Xu, and Y. Tian, "All-optical format conversion from NRZ to BPSK using a single saturated SOA," IEEE Photon. Technol. Lett. 18, 2368-2370 (2006).
[CrossRef]

Zhang, Hanyi

Huan Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, "All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch," IEEE Photon. Technol. Lett. 19, 1985-1987 (2007).
[CrossRef]

Electron. Lett. (1)

W. Astar and G. M. Carter, "10 Gb/s RZ-OOK to BPSK format conversion by cross-phase modulation in single semiconductor optical amplifier," Electron. Lett. 42, 1472-1474 (2006).
[CrossRef]

IEEE Photon. Technol. Lett. (6)

A. H. Gnauck, S. Chandrasekhar, J. Leuthold, and L. Stulz, "Demonstration of 42.7-Gb/s DPSK receiver with 45 photons/bit sensitivity," IEEE Photon. Technol. Lett. 15, 99-101 (2003).
[CrossRef]

A. H. Gnauck, G. Raybon, S. Chandrasekhar, J. Leuthold, C. Doerr, L. Stulz, and E. Burrows, "25×40-Gb/s copolarized DPSK transmission over 12×100-km NZDF with 50-GHz channel spacing," IEEE Photon. Technol. Lett. 15, 467-469 (2003).
[CrossRef]

C.  Yan, Y. Su, L. Yi, L. Feng, X. Tian, X. Xu, and Y. Tian, "All-optical format conversion from NRZ to BPSK using a single saturated SOA," IEEE Photon. Technol. Lett. 18, 2368-2370 (2006).
[CrossRef]

Huan Jiang, He Wen, Liuyan Han, Yili Guo, and Hanyi Zhang, "All-optical NRZ-OOK to BPSK format conversion in an SOA-based nonlinear polarization switch," IEEE Photon. Technol. Lett. 19, 1985-1987 (2007).
[CrossRef]

D. Norte, E. Park, and A. E. Willner, "All-optical TDM-to-WDM data format conversion in a dynamically reconfigurable WDM network," IEEE Photon. Technol. Lett. 7, 920-922 (1995).
[CrossRef]

X. Wei,Y. Su, X. Liu, J. Leuthold, and S. Chandrasekhar,"10-Gb/s RZ-DPSK Transmitter Using a Saturated SOA as a Power Booster and Limiting Amplifier," IEEE Photon. Technol. Lett. 16, 1582-1584 (2004).
[CrossRef]

IEEE Trans. Commun. (1)

H. Leib, "Data-aided incoherent demodulation of DPSK," IEEE Trans. Commun. 43, 722-725 (1995).
[CrossRef]

J. Lightwave Technol. (5)

Opt. Express (1)

Other (3)

IEEE 802.3-2005, Section Five, &58.4.1 Transmitter optical specifications, 63-64.

L. Cai, S. L. Xiao, Z. X. Liu, R. Y. Li, M. Zhu and W. S. Hu "Cost-effective WDM-PON for simultaneously transmitting unicast and broadcast/multicast data by superimposing IRZ signal onto NRZ signal" in 34th European Conference on Optical Communication (ECOC), (2008), Paper Th.1.F.4.

S. H. Lee, K. K. Chow, C. Shu, and C. Lin, "All optical ASK to DPSK format conversion using crossphase modulation in a nonlinear photonic crystal fiber," in Conference on Lasers and Electro-Optics (2005), Paper CFJ2-5.

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Figures (7)

Fig. 1.
Fig. 1.

WDM networks with different signal format. Multi-format transmitters and receivers are used in the aggregation nodes. ASK signal is used in access and metro networks. Signal transmitted in core networks is BPSK format.

Fig. 2.
Fig. 2.

(a) Structure of multi-format receiver, (b) Illustration of the demodulation of NRZ-ASK signal using multi-format receiver.

Fig. 3.
Fig. 3.

Experimental setup. MZDI is 1-bit-delay Mach-Zehnder delay interferometer; PC is polarizer controller; OBPF is optical bandpass filter; MZM is Mach–Zehnder modulator; EDFA is erbium-doped fiber amplifier; VOA is variable optical attenuator; PD is photodetector; con is constructive port; des is destructive port; SOA is semiconductor optical amplifier; DCF is dispersion compensation fiber

Fig. 4.
Fig. 4.

Experimental results of multi-format detection. (a), (b), and (c) Optical eye diagrams at positions A, B, and C in Fig. 3 for NRZ-BPSK demodulation. (d) Electrical eye diagram of the demodulated NZR-BPSK signal; (e), (f), and (g) Optical eye diagrams at positions A, B, and C in Fig. 3 for NRZ-ASK demodulation. (h) Electrical eye diagram of the demodulated NRZ-ASK signal.

Fig. 5.
Fig. 5.

Waveforms of a typical pattern. (a), (b), and (c) Waveforms at positions A, B, and C in Fig.3 for NRZ-BPSK demodulation. (d) Waveform of the demodulated NRZ-BPSK signal; (e), (f), and (g) Waveforms at positions A, B, and C in Fig.3 for NRZ-ASK demodulation. (h) Waveform of the demodulated NRZ-ASK signal. A.U.: arbitrary unit.

Fig. 6.
Fig. 6.

(a) BER results for two different PRBS pattern lengths, (b) BER results before and after transmissions.

Fig. 7.
Fig. 7.

BER results for traditional receivers and multi-format receiver

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

b ( k ) = a ( k )
c ( k ) des = b ( k ) b ( k 1 ) output at destructive port
and c ( k ) con = b ( k ) b ( k 1 ) output at constructive port
d ( k ) = c ( k ) d ( k 1 ) = b ( k ) b ( k 1 ) d ( k 1 )
d ( k 1 ) = c ( k 1 ) d ( k 2 ) = b ( k 1 ) b ( k 2 ) d ( k 2 )
d ( k ) = b ( k ) b ( k 2 ) d ( k 2 )
d ( k ) = b ( k ) b ( 0 ) d ( 0 )
= b ( k ) a ( 0 ) d ( 0 )
d ( k ) = b ( k ) a ( 0 ) d ( 0 )

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