Abstract

Optical communication systems, which operate at very high rates, are often limited by the sampling rate bottleneck. The optical wideband regime may exceed analog to digital converters (ADCs) front-end bandwidth. Multi-channel sampling approaches, such as interleaved ADCs, also known as multicoset sampling in some contexts, have been proposed to sample the wideband signal using several channels. Each channel samples below the Nyquist rate such that the overall sampling rate is preserved. However, this scheme suffers from two practical limitations that make its implementation difficult. First, the inherent anti-aliasing filter of the samplers distorts the wideband signal. Second, it requires accurate time shifts on the order of the signal’s Nyquist rate, which are challenging to maintain. In this work, we propose an alternative multi-channel sampling scheme, the wideband demodulator for optical waveforms (WINDOW), based on analog RF demodulation, where each channel aliases the spectrum using a periodic mixing function before integration and sampling. We show that intentionally using the inherent ADC filter to perform integration increases the signal to noise ratio (SNR). We demonstrate both theoretically and through numerical experiments that our system outperforms interleaved sampling in terms of signal recovery and symbol estimation in the presence of both thermal and quantization noise but is slightly less robust to timing jitter. The main contribution of this work is the application of RF demodulation concepts proposed in the context of sub-Nyquist sampling, e.g. random demodulator and modulated wideband converter, to optical communication signals in the Nyquist regime. We develop a sampling scheme that presents an alternative for optical links where thermal noise in the receiver is the bottleneck.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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  1. C. Laperle and M. O’Sullivan, “Advances in high-speed DACs, ADCs, and DSP for optical coherent transceivers,” J. Lightwave Technol 32, 629–643 (2014).
    [Crossref]
  2. K. Azadet, E. F. Haratsch, H. Kim, F. Saibi, J. H. Saunders, M. Schaffer, L. Song, and M.-L. Yu, “Equalization and FEC techniques for optical transceivers,” IEEE J. Solid-State Circ. 37, 317–327 (2002).
    [Crossref]
  3. R. Venkataramani and Y. Bresler, “Perfect reconstruction formulas and bounds on aliasing error in sub-Nyquist nonuniform sampling of multiband signals,” IEEE Trans. Inf. Theory 46, 2173–2183 (2000).
    [Crossref]
  4. M. Mishali and Y. C. Eldar, “Blind multi-band signal reconstruction: compressed sensing for analog signals,” IEEE Trans. Signal Process. 57, 993–1009 (2009).
    [Crossref]
  5. Y. C. Eldar, Sampling theory: Beyond Bandlimited Systems, (Cambridge University, 2015).
  6. Analog Devices Corp., “A/D converters [Online],” Available: http://www.analog.com/en/analog-to-digital-converters/ad-converters/products/index.html (2009).
  7. M. Mishali and Y. C. Eldar, “From theory to practice: sub-Nyquist sampling of sparse wideband analog signals,” IEEE J. Sel. Top. Signal Process. 4, 375–391 (2010).
    [Crossref]
  8. M. Mishali, Y. C. Eldar, O. Dounaevsky, and E. Shoshan, “Xampling: Analog to digital at sub-Nyquist rates,” IET Circ. Dev. Syst. 46, 2173–2183 (2000).
  9. D. Cohen, S. Tsiper, and Y. C. Eldar, “Analog to Digital Cognitive Radio,” in Handbook of Cognitive Radio (chapter 11) (Springer, 2017), to appear.
  10. B. Razavi, “Problem of timing mismatch in interleaved ADCs,” in Custom Integrated Circuits Conference (2012), pp. 1–8.
  11. M. Mishali and Y. C. Eldar, “Wideband spectrum sensing at sub-Nyquist rates,” IEEE Mag. Sign. Process. 28, 102–135 (2011).
    [Crossref]
  12. J. N. Laska, S. Kirolos, M. F. Duarte, T. S. Ragheb, R. G. Baraniuk, and Y. Massoud, “Theory and Implementation of an analog-to-information converter using random demodulation,” in IEEE International Symposium on Circuits Systems (2007), pp. 1959–1962.
  13. A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, “Beyond Nyquist: efficient sampling of sparse bandlimited signals,” IEEE Trans. Inf. Theory 56, 520–544 (2010).
    [Crossref]
  14. J. R. Barry, E. A. Lee, and D. G. Messerschmitt, Digital Communication, (Springer, 2003).
  15. G. P. Agrawal, Fiber-Optic Communication Systems, (John Wiley & Sons, 2002).
    [Crossref]
  16. I. Djordjevic, W. Ryan, and B. Vasic, Coding for Optical Channels, (Springer Science & Business Media, 2010).
    [Crossref]
  17. G. Katz, D. Sadot, and J. Tabrikian, “Electrical dispersion compensation equalizers in optical direct-and coherent-detection systems,” IEEE Trans. Commun. 54, 2045–2050 (2006).
    [Crossref]
  18. W. Shieh and I. Djordjevic, OFDM for Optical Communications (Academic Press, 2009).
  19. A. C.-C. Yeh, “Minimum-error-probability equalization and multi-user detection,” Georgia Institute of Technology, PhD thesis (1998).
  20. J. Choi, Optimal Combining and Detection, (Cambridge University Press, 2010).
    [Crossref]
  21. J Seberry, B. J Wysocki, and T. A Wysocki, “On some applications of Hadamard matrices,” Metrika 62, 221–239 (2005).
    [Crossref]
  22. D. C. Lee, “Modeling timing jitter in oscillators,” in Proceedings of Forum Design Languages (2001), pp. 3–7.

2014 (1)

C. Laperle and M. O’Sullivan, “Advances in high-speed DACs, ADCs, and DSP for optical coherent transceivers,” J. Lightwave Technol 32, 629–643 (2014).
[Crossref]

2011 (1)

M. Mishali and Y. C. Eldar, “Wideband spectrum sensing at sub-Nyquist rates,” IEEE Mag. Sign. Process. 28, 102–135 (2011).
[Crossref]

2010 (2)

A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, “Beyond Nyquist: efficient sampling of sparse bandlimited signals,” IEEE Trans. Inf. Theory 56, 520–544 (2010).
[Crossref]

M. Mishali and Y. C. Eldar, “From theory to practice: sub-Nyquist sampling of sparse wideband analog signals,” IEEE J. Sel. Top. Signal Process. 4, 375–391 (2010).
[Crossref]

2009 (1)

M. Mishali and Y. C. Eldar, “Blind multi-band signal reconstruction: compressed sensing for analog signals,” IEEE Trans. Signal Process. 57, 993–1009 (2009).
[Crossref]

2006 (1)

G. Katz, D. Sadot, and J. Tabrikian, “Electrical dispersion compensation equalizers in optical direct-and coherent-detection systems,” IEEE Trans. Commun. 54, 2045–2050 (2006).
[Crossref]

2005 (1)

J Seberry, B. J Wysocki, and T. A Wysocki, “On some applications of Hadamard matrices,” Metrika 62, 221–239 (2005).
[Crossref]

2002 (1)

K. Azadet, E. F. Haratsch, H. Kim, F. Saibi, J. H. Saunders, M. Schaffer, L. Song, and M.-L. Yu, “Equalization and FEC techniques for optical transceivers,” IEEE J. Solid-State Circ. 37, 317–327 (2002).
[Crossref]

2000 (2)

R. Venkataramani and Y. Bresler, “Perfect reconstruction formulas and bounds on aliasing error in sub-Nyquist nonuniform sampling of multiband signals,” IEEE Trans. Inf. Theory 46, 2173–2183 (2000).
[Crossref]

M. Mishali, Y. C. Eldar, O. Dounaevsky, and E. Shoshan, “Xampling: Analog to digital at sub-Nyquist rates,” IET Circ. Dev. Syst. 46, 2173–2183 (2000).

Agrawal, G. P.

G. P. Agrawal, Fiber-Optic Communication Systems, (John Wiley & Sons, 2002).
[Crossref]

Azadet, K.

K. Azadet, E. F. Haratsch, H. Kim, F. Saibi, J. H. Saunders, M. Schaffer, L. Song, and M.-L. Yu, “Equalization and FEC techniques for optical transceivers,” IEEE J. Solid-State Circ. 37, 317–327 (2002).
[Crossref]

Baraniuk, R. G.

A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, “Beyond Nyquist: efficient sampling of sparse bandlimited signals,” IEEE Trans. Inf. Theory 56, 520–544 (2010).
[Crossref]

J. N. Laska, S. Kirolos, M. F. Duarte, T. S. Ragheb, R. G. Baraniuk, and Y. Massoud, “Theory and Implementation of an analog-to-information converter using random demodulation,” in IEEE International Symposium on Circuits Systems (2007), pp. 1959–1962.

Barry, J. R.

J. R. Barry, E. A. Lee, and D. G. Messerschmitt, Digital Communication, (Springer, 2003).

Bresler, Y.

R. Venkataramani and Y. Bresler, “Perfect reconstruction formulas and bounds on aliasing error in sub-Nyquist nonuniform sampling of multiband signals,” IEEE Trans. Inf. Theory 46, 2173–2183 (2000).
[Crossref]

Choi, J.

J. Choi, Optimal Combining and Detection, (Cambridge University Press, 2010).
[Crossref]

Cohen, D.

D. Cohen, S. Tsiper, and Y. C. Eldar, “Analog to Digital Cognitive Radio,” in Handbook of Cognitive Radio (chapter 11) (Springer, 2017), to appear.

Djordjevic, I.

I. Djordjevic, W. Ryan, and B. Vasic, Coding for Optical Channels, (Springer Science & Business Media, 2010).
[Crossref]

W. Shieh and I. Djordjevic, OFDM for Optical Communications (Academic Press, 2009).

Dounaevsky, O.

M. Mishali, Y. C. Eldar, O. Dounaevsky, and E. Shoshan, “Xampling: Analog to digital at sub-Nyquist rates,” IET Circ. Dev. Syst. 46, 2173–2183 (2000).

Duarte, M. F.

A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, “Beyond Nyquist: efficient sampling of sparse bandlimited signals,” IEEE Trans. Inf. Theory 56, 520–544 (2010).
[Crossref]

J. N. Laska, S. Kirolos, M. F. Duarte, T. S. Ragheb, R. G. Baraniuk, and Y. Massoud, “Theory and Implementation of an analog-to-information converter using random demodulation,” in IEEE International Symposium on Circuits Systems (2007), pp. 1959–1962.

Eldar, Y. C.

M. Mishali and Y. C. Eldar, “Wideband spectrum sensing at sub-Nyquist rates,” IEEE Mag. Sign. Process. 28, 102–135 (2011).
[Crossref]

M. Mishali and Y. C. Eldar, “From theory to practice: sub-Nyquist sampling of sparse wideband analog signals,” IEEE J. Sel. Top. Signal Process. 4, 375–391 (2010).
[Crossref]

M. Mishali and Y. C. Eldar, “Blind multi-band signal reconstruction: compressed sensing for analog signals,” IEEE Trans. Signal Process. 57, 993–1009 (2009).
[Crossref]

M. Mishali, Y. C. Eldar, O. Dounaevsky, and E. Shoshan, “Xampling: Analog to digital at sub-Nyquist rates,” IET Circ. Dev. Syst. 46, 2173–2183 (2000).

D. Cohen, S. Tsiper, and Y. C. Eldar, “Analog to Digital Cognitive Radio,” in Handbook of Cognitive Radio (chapter 11) (Springer, 2017), to appear.

Y. C. Eldar, Sampling theory: Beyond Bandlimited Systems, (Cambridge University, 2015).

Haratsch, E. F.

K. Azadet, E. F. Haratsch, H. Kim, F. Saibi, J. H. Saunders, M. Schaffer, L. Song, and M.-L. Yu, “Equalization and FEC techniques for optical transceivers,” IEEE J. Solid-State Circ. 37, 317–327 (2002).
[Crossref]

Katz, G.

G. Katz, D. Sadot, and J. Tabrikian, “Electrical dispersion compensation equalizers in optical direct-and coherent-detection systems,” IEEE Trans. Commun. 54, 2045–2050 (2006).
[Crossref]

Kim, H.

K. Azadet, E. F. Haratsch, H. Kim, F. Saibi, J. H. Saunders, M. Schaffer, L. Song, and M.-L. Yu, “Equalization and FEC techniques for optical transceivers,” IEEE J. Solid-State Circ. 37, 317–327 (2002).
[Crossref]

Kirolos, S.

J. N. Laska, S. Kirolos, M. F. Duarte, T. S. Ragheb, R. G. Baraniuk, and Y. Massoud, “Theory and Implementation of an analog-to-information converter using random demodulation,” in IEEE International Symposium on Circuits Systems (2007), pp. 1959–1962.

Laperle, C.

C. Laperle and M. O’Sullivan, “Advances in high-speed DACs, ADCs, and DSP for optical coherent transceivers,” J. Lightwave Technol 32, 629–643 (2014).
[Crossref]

Laska, J. N.

A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, “Beyond Nyquist: efficient sampling of sparse bandlimited signals,” IEEE Trans. Inf. Theory 56, 520–544 (2010).
[Crossref]

J. N. Laska, S. Kirolos, M. F. Duarte, T. S. Ragheb, R. G. Baraniuk, and Y. Massoud, “Theory and Implementation of an analog-to-information converter using random demodulation,” in IEEE International Symposium on Circuits Systems (2007), pp. 1959–1962.

Lee, D. C.

D. C. Lee, “Modeling timing jitter in oscillators,” in Proceedings of Forum Design Languages (2001), pp. 3–7.

Lee, E. A.

J. R. Barry, E. A. Lee, and D. G. Messerschmitt, Digital Communication, (Springer, 2003).

Massoud, Y.

J. N. Laska, S. Kirolos, M. F. Duarte, T. S. Ragheb, R. G. Baraniuk, and Y. Massoud, “Theory and Implementation of an analog-to-information converter using random demodulation,” in IEEE International Symposium on Circuits Systems (2007), pp. 1959–1962.

Messerschmitt, D. G.

J. R. Barry, E. A. Lee, and D. G. Messerschmitt, Digital Communication, (Springer, 2003).

Mishali, M.

M. Mishali and Y. C. Eldar, “Wideband spectrum sensing at sub-Nyquist rates,” IEEE Mag. Sign. Process. 28, 102–135 (2011).
[Crossref]

M. Mishali and Y. C. Eldar, “From theory to practice: sub-Nyquist sampling of sparse wideband analog signals,” IEEE J. Sel. Top. Signal Process. 4, 375–391 (2010).
[Crossref]

M. Mishali and Y. C. Eldar, “Blind multi-band signal reconstruction: compressed sensing for analog signals,” IEEE Trans. Signal Process. 57, 993–1009 (2009).
[Crossref]

M. Mishali, Y. C. Eldar, O. Dounaevsky, and E. Shoshan, “Xampling: Analog to digital at sub-Nyquist rates,” IET Circ. Dev. Syst. 46, 2173–2183 (2000).

O’Sullivan, M.

C. Laperle and M. O’Sullivan, “Advances in high-speed DACs, ADCs, and DSP for optical coherent transceivers,” J. Lightwave Technol 32, 629–643 (2014).
[Crossref]

Ragheb, T. S.

J. N. Laska, S. Kirolos, M. F. Duarte, T. S. Ragheb, R. G. Baraniuk, and Y. Massoud, “Theory and Implementation of an analog-to-information converter using random demodulation,” in IEEE International Symposium on Circuits Systems (2007), pp. 1959–1962.

Razavi, B.

B. Razavi, “Problem of timing mismatch in interleaved ADCs,” in Custom Integrated Circuits Conference (2012), pp. 1–8.

Romberg, J. K.

A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, “Beyond Nyquist: efficient sampling of sparse bandlimited signals,” IEEE Trans. Inf. Theory 56, 520–544 (2010).
[Crossref]

Ryan, W.

I. Djordjevic, W. Ryan, and B. Vasic, Coding for Optical Channels, (Springer Science & Business Media, 2010).
[Crossref]

Sadot, D.

G. Katz, D. Sadot, and J. Tabrikian, “Electrical dispersion compensation equalizers in optical direct-and coherent-detection systems,” IEEE Trans. Commun. 54, 2045–2050 (2006).
[Crossref]

Saibi, F.

K. Azadet, E. F. Haratsch, H. Kim, F. Saibi, J. H. Saunders, M. Schaffer, L. Song, and M.-L. Yu, “Equalization and FEC techniques for optical transceivers,” IEEE J. Solid-State Circ. 37, 317–327 (2002).
[Crossref]

Saunders, J. H.

K. Azadet, E. F. Haratsch, H. Kim, F. Saibi, J. H. Saunders, M. Schaffer, L. Song, and M.-L. Yu, “Equalization and FEC techniques for optical transceivers,” IEEE J. Solid-State Circ. 37, 317–327 (2002).
[Crossref]

Schaffer, M.

K. Azadet, E. F. Haratsch, H. Kim, F. Saibi, J. H. Saunders, M. Schaffer, L. Song, and M.-L. Yu, “Equalization and FEC techniques for optical transceivers,” IEEE J. Solid-State Circ. 37, 317–327 (2002).
[Crossref]

Seberry, J

J Seberry, B. J Wysocki, and T. A Wysocki, “On some applications of Hadamard matrices,” Metrika 62, 221–239 (2005).
[Crossref]

Shieh, W.

W. Shieh and I. Djordjevic, OFDM for Optical Communications (Academic Press, 2009).

Shoshan, E.

M. Mishali, Y. C. Eldar, O. Dounaevsky, and E. Shoshan, “Xampling: Analog to digital at sub-Nyquist rates,” IET Circ. Dev. Syst. 46, 2173–2183 (2000).

Song, L.

K. Azadet, E. F. Haratsch, H. Kim, F. Saibi, J. H. Saunders, M. Schaffer, L. Song, and M.-L. Yu, “Equalization and FEC techniques for optical transceivers,” IEEE J. Solid-State Circ. 37, 317–327 (2002).
[Crossref]

Tabrikian, J.

G. Katz, D. Sadot, and J. Tabrikian, “Electrical dispersion compensation equalizers in optical direct-and coherent-detection systems,” IEEE Trans. Commun. 54, 2045–2050 (2006).
[Crossref]

Tropp, A.

A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, “Beyond Nyquist: efficient sampling of sparse bandlimited signals,” IEEE Trans. Inf. Theory 56, 520–544 (2010).
[Crossref]

Tsiper, S.

D. Cohen, S. Tsiper, and Y. C. Eldar, “Analog to Digital Cognitive Radio,” in Handbook of Cognitive Radio (chapter 11) (Springer, 2017), to appear.

Vasic, B.

I. Djordjevic, W. Ryan, and B. Vasic, Coding for Optical Channels, (Springer Science & Business Media, 2010).
[Crossref]

Venkataramani, R.

R. Venkataramani and Y. Bresler, “Perfect reconstruction formulas and bounds on aliasing error in sub-Nyquist nonuniform sampling of multiband signals,” IEEE Trans. Inf. Theory 46, 2173–2183 (2000).
[Crossref]

Wysocki, B. J

J Seberry, B. J Wysocki, and T. A Wysocki, “On some applications of Hadamard matrices,” Metrika 62, 221–239 (2005).
[Crossref]

Wysocki, T. A

J Seberry, B. J Wysocki, and T. A Wysocki, “On some applications of Hadamard matrices,” Metrika 62, 221–239 (2005).
[Crossref]

Yeh, A. C.-C.

A. C.-C. Yeh, “Minimum-error-probability equalization and multi-user detection,” Georgia Institute of Technology, PhD thesis (1998).

Yu, M.-L.

K. Azadet, E. F. Haratsch, H. Kim, F. Saibi, J. H. Saunders, M. Schaffer, L. Song, and M.-L. Yu, “Equalization and FEC techniques for optical transceivers,” IEEE J. Solid-State Circ. 37, 317–327 (2002).
[Crossref]

IEEE J. Sel. Top. Signal Process. (1)

M. Mishali and Y. C. Eldar, “From theory to practice: sub-Nyquist sampling of sparse wideband analog signals,” IEEE J. Sel. Top. Signal Process. 4, 375–391 (2010).
[Crossref]

IEEE J. Solid-State Circ. (1)

K. Azadet, E. F. Haratsch, H. Kim, F. Saibi, J. H. Saunders, M. Schaffer, L. Song, and M.-L. Yu, “Equalization and FEC techniques for optical transceivers,” IEEE J. Solid-State Circ. 37, 317–327 (2002).
[Crossref]

IEEE Mag. Sign. Process. (1)

M. Mishali and Y. C. Eldar, “Wideband spectrum sensing at sub-Nyquist rates,” IEEE Mag. Sign. Process. 28, 102–135 (2011).
[Crossref]

IEEE Trans. Commun. (1)

G. Katz, D. Sadot, and J. Tabrikian, “Electrical dispersion compensation equalizers in optical direct-and coherent-detection systems,” IEEE Trans. Commun. 54, 2045–2050 (2006).
[Crossref]

IEEE Trans. Inf. Theory (2)

A. Tropp, J. N. Laska, M. F. Duarte, J. K. Romberg, and R. G. Baraniuk, “Beyond Nyquist: efficient sampling of sparse bandlimited signals,” IEEE Trans. Inf. Theory 56, 520–544 (2010).
[Crossref]

R. Venkataramani and Y. Bresler, “Perfect reconstruction formulas and bounds on aliasing error in sub-Nyquist nonuniform sampling of multiband signals,” IEEE Trans. Inf. Theory 46, 2173–2183 (2000).
[Crossref]

IEEE Trans. Signal Process. (1)

M. Mishali and Y. C. Eldar, “Blind multi-band signal reconstruction: compressed sensing for analog signals,” IEEE Trans. Signal Process. 57, 993–1009 (2009).
[Crossref]

IET Circ. Dev. Syst. (1)

M. Mishali, Y. C. Eldar, O. Dounaevsky, and E. Shoshan, “Xampling: Analog to digital at sub-Nyquist rates,” IET Circ. Dev. Syst. 46, 2173–2183 (2000).

J. Lightwave Technol (1)

C. Laperle and M. O’Sullivan, “Advances in high-speed DACs, ADCs, and DSP for optical coherent transceivers,” J. Lightwave Technol 32, 629–643 (2014).
[Crossref]

Metrika (1)

J Seberry, B. J Wysocki, and T. A Wysocki, “On some applications of Hadamard matrices,” Metrika 62, 221–239 (2005).
[Crossref]

Other (12)

D. C. Lee, “Modeling timing jitter in oscillators,” in Proceedings of Forum Design Languages (2001), pp. 3–7.

J. N. Laska, S. Kirolos, M. F. Duarte, T. S. Ragheb, R. G. Baraniuk, and Y. Massoud, “Theory and Implementation of an analog-to-information converter using random demodulation,” in IEEE International Symposium on Circuits Systems (2007), pp. 1959–1962.

D. Cohen, S. Tsiper, and Y. C. Eldar, “Analog to Digital Cognitive Radio,” in Handbook of Cognitive Radio (chapter 11) (Springer, 2017), to appear.

B. Razavi, “Problem of timing mismatch in interleaved ADCs,” in Custom Integrated Circuits Conference (2012), pp. 1–8.

Y. C. Eldar, Sampling theory: Beyond Bandlimited Systems, (Cambridge University, 2015).

Analog Devices Corp., “A/D converters [Online],” Available: http://www.analog.com/en/analog-to-digital-converters/ad-converters/products/index.html (2009).

J. R. Barry, E. A. Lee, and D. G. Messerschmitt, Digital Communication, (Springer, 2003).

G. P. Agrawal, Fiber-Optic Communication Systems, (John Wiley & Sons, 2002).
[Crossref]

I. Djordjevic, W. Ryan, and B. Vasic, Coding for Optical Channels, (Springer Science & Business Media, 2010).
[Crossref]

W. Shieh and I. Djordjevic, OFDM for Optical Communications (Academic Press, 2009).

A. C.-C. Yeh, “Minimum-error-probability equalization and multi-user detection,” Georgia Institute of Technology, PhD thesis (1998).

J. Choi, Optimal Combining and Detection, (Cambridge University Press, 2010).
[Crossref]

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Figures (6)

Fig. 1
Fig. 1

On-off keying (2 PAM) with different pulse shapes: (a) ideal rectangular pulse, (b) LED and (c) laser.

Fig. 2
Fig. 2

(a) Schematic implementation of interleaved sampling, (b) Practical ADC front-end modeled as a LPF with bandwidth b preceding the uniform sampling at rate r samples/s [4].

Fig. 3
Fig. 3

Schematic implementation of the WINDOW analog sampling front-end and digital signal recovery from low rate samples. In each channel, the input signal x(t) is mixed with a periodic function pm (t) with period Ts, integrated over the period Ts and sampled at the low rate 1/Ts. The Nyquist samples x[n] are then recovered using (11).

Fig. 4
Fig. 4

Signal recovery error using on-off keying PAM for WINDOW and interleaved sampling. In the legend, (p) denotes practical experiments and (t) refers to the theoretical bounds.

Fig. 5
Fig. 5

Signal recovery error using 4 PAM and fiber length L0 = 140[km] for WINDOW and interleaved sampling. In the legend, (p) denotes practical experiments and (t) refers to the theoretical bounds.

Fig. 6
Fig. 6

Recovery error using on-off keying and fiber length L0 = 80[km] for WINDOW and interleaved sampling.

Equations (33)

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

x ( t ) = k a k g T ( t k T ) ,
g T ( t ) = 1 e t k T τ ,
g T ( t ) = e 1 2 ( t T 0 ) 2 ,
I o u t = I i n e α L 0
X o u t ( f ) = X i n ( f ) e j β 0 e j L β 1 f e j L 0 β 2 2 f 2 ,
x i [ n ] = x ( n M T + i T ) , n ,
p m ( t ) = l = 0 L 1 p m l u ( t l T ) ,
x ˜ m ( t ) = x ( t ) p m ( t ) ,
y m [ n ] = ( n 1 ) T s n T s x ˜ m ( t ) d t = ( n 1 ) T s n T s x ( t ) l = 0 L 1 p m l u ( t l T ( n 1 ) T s ) d t = l = 0 L 1 p m l ( n 1 ) T s + l T ( n 1 ) T s + ( l + 1 ) T x ( t ) d t T l = 1 L x l [ n ] P m l .
y [ n ] = T Px [ n ] , n ,
x ^ [ n ] = 1 T P y [ n ] , n .
z ^ [ n ] = Ha [ n ] + n [ n ] ,
H i j = { g ( ( j i ) T κ T ) , 0 j i 2 κ 0 , e l s e ,
a ^ n = c M M S E T z ^ [ n ] ,
c M M S E = ( H H T + σ n 2 I ) 1 h .
z ( t ) = x ( t ) + v ( t ) .
y m [ n ] = ( n 1 ) T s n T s ( x ( τ ) + v ( τ ) ) p m ( τ ) d τ = T l = 1 L x l [ n ] p m l + v m [ n ] ,
σ v 2 = ( n 1 ) T s n T s N 0 p m 2 ( τ ) d τ = N 0 T s .
q = Δ 2 B s .
y m [ n ] = Q ( ( n 1 ) T s n T s ( x ( τ ) + v ( τ ) ) p m ( τ ) d τ ) T l = 1 L x l [ n ] p m   l + v m [ n ] + w m [ n ] ,
Δ 2 B i n T s = 2 B i n L T .
y [ n ] = T Px [ n ] + v [ n ] + w [ n ] , n ,
x ^ [ n ] = x [ n ] + r [ n ] ,
r [ n ] 1 T P ( v [ n ] + w [ n ] ) .
S N R = E ( x m 2 [ n ] ) E ( r m 2 [ n ] ) = σ x 2 L T 2 L 2 ( σ v 2 + σ w 2 ) = L T 2 σ x 2 N 0 T s + q 2 / 12 .
E v i 2 = L β 2 [ E u l 2 + E w l 2 ] = L β 2 [ N 0 + q 2 12 ] .
S N R = σ x 2 N 0 / T + 2 2 ( B i n B s ) L / 12 .
y m [ n ] = x m [ n ] + v m c [ n ] + w m c [ n ] .
S N R = E ( x m 2 [ n ] ) E ( ( v m c ) 2 [ n ] ) + E ( ( w m c ) 2 [ n ] ) = σ x 2 N 0 + q 2 / 12 .
S N R = σ x 2 N 0 + 2 2 ( B i n B s ) / 12 .
t k + 1 = ( k + 1 ) T + n = 0 k C w n ,
R M S = v a r ( n = 0 T s / T 1 C w n ) = C T s T .
C = T T s ( p U I ) 2 = T 3 T s p 2 .

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