Di Che, An Li Xi Chen, Hu Qian, and W. Shieh, “Rejuvenating direct modulation and direct detection for modern optical communications,” Opt. Commun. 409, 86–93 (2017).

[Crossref]

H. Khodakarami, Di Che, and W. Shieh, “Information Capacity of Polarization-Modulated and Directly Detected Optical Systems Dominated by Amplified Spontaneous Emission Noise,” J. Lightwave Technol. 35, 2797–2802 (2017).

[Crossref]

J. Bijsterbosch and A. Volgenan, “Solving the Rectangular assignment problem and applications,” Annals of Operations Research 181, 443–462 (2010).

[Crossref]

M. Nazarathy, X. Liu, L. Christen, Y. Lize, and A. Wilner, “Self-coherent optical detection of multisymbol differential phase-shift-keyed transmission,” J. Lightwave Technol. 26, 1921–1934 (2008).

[Crossref]

Liu Xiang, S. Chandrasekhar, and A. Leven, “Digital self-coherent detection,” Opt. Express 16, 792–803 (2008)

[Crossref]

H.-W. Hübers, “Terahertz Heterodyne Receivers,” IEEE Journal of Selected Topics in Quantum Electronics 14, 378–391 (2008).

[Crossref]

H.-W. Hübers, S. G. Pavlov, A. D. Semenov, R. Köhler, L. Mahler, A. Tredicucci, H. E. Beere, D. A. Ritchie, and E. H. Linfield, “Terahertz quantum cascade laser as local oscillator in a heterodyne receiver,” Opt. Express 13, 5890–5896 (2005).

[Crossref]
[PubMed]

S. Betti, G. De Marchis, and E. Iannone, “Polarization modulated direct detection optical transmission systems,” J. Lightwave Technol. 10, 1985–1997 (1992).

[Crossref]

J. Munkres, “Algorithms for the Assignment and Transportation Problems,” Journal of the Society for Industrial and Applied Mathematics 5, 32–38 (1957).

[Crossref]

H. W. Kuhn, “The Hungarian Method for the assignment problem,” Naval Research Logistics Quarterly 2, 83–97 (1955).

[Crossref]

H.-W. Hübers, S. G. Pavlov, A. D. Semenov, R. Köhler, L. Mahler, A. Tredicucci, H. E. Beere, D. A. Ritchie, and E. H. Linfield, “Terahertz quantum cascade laser as local oscillator in a heterodyne receiver,” Opt. Express 13, 5890–5896 (2005).

[Crossref]
[PubMed]

M. Bertolotti, The history of laserInstitute of Physics Publishing, London (2005).

S. Betti, G. De Marchis, and E. Iannone, “Polarization modulated direct detection optical transmission systems,” J. Lightwave Technol. 10, 1985–1997 (1992).

[Crossref]

J. Bijsterbosch and A. Volgenan, “Solving the Rectangular assignment problem and applications,” Annals of Operations Research 181, 443–462 (2010).

[Crossref]

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing(Prentice Hall, 1999).

H. Khodakarami, Di Che, and W. Shieh, “Information Capacity of Polarization-Modulated and Directly Detected Optical Systems Dominated by Amplified Spontaneous Emission Noise,” J. Lightwave Technol. 35, 2797–2802 (2017).

[Crossref]

Di Che, An Li Xi Chen, Hu Qian, and W. Shieh, “Rejuvenating direct modulation and direct detection for modern optical communications,” Opt. Commun. 409, 86–93 (2017).

[Crossref]

J. B. Hough, M. Krishnapur, Y. Peres, and B. Virag, Zeros of Gaussian analytic functions and determinantal point processes (University Lecture Series, 2009) vol. 51.

H.-W. Hübers, “Terahertz Heterodyne Receivers,” IEEE Journal of Selected Topics in Quantum Electronics 14, 378–391 (2008).

[Crossref]

H.-W. Hübers, S. G. Pavlov, A. D. Semenov, R. Köhler, L. Mahler, A. Tredicucci, H. E. Beere, D. A. Ritchie, and E. H. Linfield, “Terahertz quantum cascade laser as local oscillator in a heterodyne receiver,” Opt. Express 13, 5890–5896 (2005).

[Crossref]
[PubMed]

S. Betti, G. De Marchis, and E. Iannone, “Polarization modulated direct detection optical transmission systems,” J. Lightwave Technol. 10, 1985–1997 (1992).

[Crossref]

N. Kikuchi, K. Mandai, S. Sasaki, and K. Sekine, “Proposal and first experimental demonstration of digital incoherent optical field detector for chromatic dispersion compensation,” in Proceedings of European Conference on Optical Communications 2006, Post-deadline Paper Th4.4.4.

H.-W. Hübers, S. G. Pavlov, A. D. Semenov, R. Köhler, L. Mahler, A. Tredicucci, H. E. Beere, D. A. Ritchie, and E. H. Linfield, “Terahertz quantum cascade laser as local oscillator in a heterodyne receiver,” Opt. Express 13, 5890–5896 (2005).

[Crossref]
[PubMed]

J. B. Hough, M. Krishnapur, Y. Peres, and B. Virag, Zeros of Gaussian analytic functions and determinantal point processes (University Lecture Series, 2009) vol. 51.

H. W. Kuhn, “The Hungarian Method for the assignment problem,” Naval Research Logistics Quarterly 2, 83–97 (1955).

[Crossref]

H.-W. Hübers, S. G. Pavlov, A. D. Semenov, R. Köhler, L. Mahler, A. Tredicucci, H. E. Beere, D. A. Ritchie, and E. H. Linfield, “Terahertz quantum cascade laser as local oscillator in a heterodyne receiver,” Opt. Express 13, 5890–5896 (2005).

[Crossref]
[PubMed]

H.-W. Hübers, S. G. Pavlov, A. D. Semenov, R. Köhler, L. Mahler, A. Tredicucci, H. E. Beere, D. A. Ritchie, and E. H. Linfield, “Terahertz quantum cascade laser as local oscillator in a heterodyne receiver,” Opt. Express 13, 5890–5896 (2005).

[Crossref]
[PubMed]

N. Kikuchi, K. Mandai, S. Sasaki, and K. Sekine, “Proposal and first experimental demonstration of digital incoherent optical field detector for chromatic dispersion compensation,” in Proceedings of European Conference on Optical Communications 2006, Post-deadline Paper Th4.4.4.

S. Betti, G. De Marchis, and E. Iannone, “Polarization modulated direct detection optical transmission systems,” J. Lightwave Technol. 10, 1985–1997 (1992).

[Crossref]

J. Munkres, “Algorithms for the Assignment and Transportation Problems,” Journal of the Society for Industrial and Applied Mathematics 5, 32–38 (1957).

[Crossref]

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing(Prentice Hall, 1999).

H.-W. Hübers, S. G. Pavlov, A. D. Semenov, R. Köhler, L. Mahler, A. Tredicucci, H. E. Beere, D. A. Ritchie, and E. H. Linfield, “Terahertz quantum cascade laser as local oscillator in a heterodyne receiver,” Opt. Express 13, 5890–5896 (2005).

[Crossref]
[PubMed]

J. B. Hough, M. Krishnapur, Y. Peres, and B. Virag, Zeros of Gaussian analytic functions and determinantal point processes (University Lecture Series, 2009) vol. 51.

M. Petkovic, “Iterative Methods for Simultaneous Inclusion of Polynomial Zeros,” Lecture Notes in Mathematics Volume 1387, Springer-VerlagBerlin Heidelberg (1989).

[Crossref]

Di Che, An Li Xi Chen, Hu Qian, and W. Shieh, “Rejuvenating direct modulation and direct detection for modern optical communications,” Opt. Commun. 409, 86–93 (2017).

[Crossref]

H.-W. Hübers, S. G. Pavlov, A. D. Semenov, R. Köhler, L. Mahler, A. Tredicucci, H. E. Beere, D. A. Ritchie, and E. H. Linfield, “Terahertz quantum cascade laser as local oscillator in a heterodyne receiver,” Opt. Express 13, 5890–5896 (2005).

[Crossref]
[PubMed]

N. Kikuchi, K. Mandai, S. Sasaki, and K. Sekine, “Proposal and first experimental demonstration of digital incoherent optical field detector for chromatic dispersion compensation,” in Proceedings of European Conference on Optical Communications 2006, Post-deadline Paper Th4.4.4.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing(Prentice Hall, 1999).

N. Kikuchi, K. Mandai, S. Sasaki, and K. Sekine, “Proposal and first experimental demonstration of digital incoherent optical field detector for chromatic dispersion compensation,” in Proceedings of European Conference on Optical Communications 2006, Post-deadline Paper Th4.4.4.

H.-W. Hübers, S. G. Pavlov, A. D. Semenov, R. Köhler, L. Mahler, A. Tredicucci, H. E. Beere, D. A. Ritchie, and E. H. Linfield, “Terahertz quantum cascade laser as local oscillator in a heterodyne receiver,” Opt. Express 13, 5890–5896 (2005).

[Crossref]
[PubMed]

Di Che, An Li Xi Chen, Hu Qian, and W. Shieh, “Rejuvenating direct modulation and direct detection for modern optical communications,” Opt. Commun. 409, 86–93 (2017).

[Crossref]

H. Khodakarami, Di Che, and W. Shieh, “Information Capacity of Polarization-Modulated and Directly Detected Optical Systems Dominated by Amplified Spontaneous Emission Noise,” J. Lightwave Technol. 35, 2797–2802 (2017).

[Crossref]

N. Tesla, Lecture Before the New York Academy of Sciences-April 6, 1897, Leland I. Anderson, ed., Twenty-First Century Books, pp. 73–74 (1994).

H.-W. Hübers, S. G. Pavlov, A. D. Semenov, R. Köhler, L. Mahler, A. Tredicucci, H. E. Beere, D. A. Ritchie, and E. H. Linfield, “Terahertz quantum cascade laser as local oscillator in a heterodyne receiver,” Opt. Express 13, 5890–5896 (2005).

[Crossref]
[PubMed]

J. B. Hough, M. Krishnapur, Y. Peres, and B. Virag, Zeros of Gaussian analytic functions and determinantal point processes (University Lecture Series, 2009) vol. 51.

J. Bijsterbosch and A. Volgenan, “Solving the Rectangular assignment problem and applications,” Annals of Operations Research 181, 443–462 (2010).

[Crossref]

Di Che, An Li Xi Chen, Hu Qian, and W. Shieh, “Rejuvenating direct modulation and direct detection for modern optical communications,” Opt. Commun. 409, 86–93 (2017).

[Crossref]

J. Bijsterbosch and A. Volgenan, “Solving the Rectangular assignment problem and applications,” Annals of Operations Research 181, 443–462 (2010).

[Crossref]

H.-W. Hübers, “Terahertz Heterodyne Receivers,” IEEE Journal of Selected Topics in Quantum Electronics 14, 378–391 (2008).

[Crossref]

K. Kikuchi, “Fundamentals of Coherent Optical Fiber Communications,” J. Lightwave Technol. 34, 157–179 (2016).

[Crossref]

M. Nazarathy, X. Liu, L. Christen, Y. Lize, and A. Wilner, “Self-coherent optical detection of multisymbol differential phase-shift-keyed transmission,” J. Lightwave Technol. 26, 1921–1934 (2008).

[Crossref]

H. Khodakarami, Di Che, and W. Shieh, “Information Capacity of Polarization-Modulated and Directly Detected Optical Systems Dominated by Amplified Spontaneous Emission Noise,” J. Lightwave Technol. 35, 2797–2802 (2017).

[Crossref]

S. Betti, G. De Marchis, and E. Iannone, “Polarization modulated direct detection optical transmission systems,” J. Lightwave Technol. 10, 1985–1997 (1992).

[Crossref]

J. Munkres, “Algorithms for the Assignment and Transportation Problems,” Journal of the Society for Industrial and Applied Mathematics 5, 32–38 (1957).

[Crossref]

H. W. Kuhn, “The Hungarian Method for the assignment problem,” Naval Research Logistics Quarterly 2, 83–97 (1955).

[Crossref]

Di Che, An Li Xi Chen, Hu Qian, and W. Shieh, “Rejuvenating direct modulation and direct detection for modern optical communications,” Opt. Commun. 409, 86–93 (2017).

[Crossref]

Liu Xiang, S. Chandrasekhar, and A. Leven, “Digital self-coherent detection,” Opt. Express 16, 792–803 (2008)

[Crossref]

A. Magen and O. Amrani, “Approaching coherent performance in differential detection via diversity,” Opt. Express 23, 4529–4538 (2015).

[Crossref]
[PubMed]

H.-W. Hübers, S. G. Pavlov, A. D. Semenov, R. Köhler, L. Mahler, A. Tredicucci, H. E. Beere, D. A. Ritchie, and E. H. Linfield, “Terahertz quantum cascade laser as local oscillator in a heterodyne receiver,” Opt. Express 13, 5890–5896 (2005).

[Crossref]
[PubMed]

N. Tesla, Lecture Before the New York Academy of Sciences-April 6, 1897, Leland I. Anderson, ed., Twenty-First Century Books, pp. 73–74 (1994).

The only relevant exception is that of the differential receiver, which we discuss later in the paper.

M. Bertolotti, The history of laserInstitute of Physics Publishing, London (2005).

N. Kikuchi, K. Mandai, S. Sasaki, and K. Sekine, “Proposal and first experimental demonstration of digital incoherent optical field detector for chromatic dispersion compensation,” in Proceedings of European Conference on Optical Communications 2006, Post-deadline Paper Th4.4.4.

This implies that the lowest sampling rate of Bs = (Ba + Bb)/2 is achieved when Δν = (Ba − Bb)/2.

Obviously, the roles of a(t) and b(t) are symmetric, meaning that the two waveforms can also be obtained from the zeros of Iab (t) and Ibb (t) = |b(t)|2.

This procedure will rigorously reproduce the correct waveforms only as long as none of the zeros are common to both a(t) and b(t) (or to a(t) and b* (t)). Yet the probability of this happening with unrelated waveforms is too low to be of any practical relevance.

The reason that it is the inverse and not the direct transform has to do with opposite sign conventions between our definition of the Fourier transform (adopted from the optics literature) and the definition used in digital signal processing applications.

Notice that the inability to reconstruct the constant phase offset is not unique to the proposed reconstruction method. The constant phase is not recoverable also in standard homodyne and heterodyne receivers.

A. Mecozzi and M. Shtaif, “Coherent detection with an incoherent local oscillator: supplementary material,” figshare (2018), http://dx.doi.org/10.6084/m9.figshare.7064774

M. Petkovic, “Iterative Methods for Simultaneous Inclusion of Polynomial Zeros,” Lecture Notes in Mathematics Volume 1387, Springer-VerlagBerlin Heidelberg (1989).

[Crossref]

Some insight can be obtained by considering the example where a(t) is periodic with period T . Then the zeros of a(t − Td) are obtained by rotating the zeros of a(t) by an angle of 2πTd /T , and the pairs of zeros are perfectly distinguishable even when Td /T is a small fraction of unity (e.g. Td ~ 0.1T).

J. B. Hough, M. Krishnapur, Y. Peres, and B. Virag, Zeros of Gaussian analytic functions and determinantal point processes (University Lecture Series, 2009) vol. 51.

A. V. Oppenheim, R. W. Schafer, and J. R. Buck, Discrete-Time Signal Processing(Prentice Hall, 1999).

This is because we define the Z-extension as in Eq. (3), in a way consistent with a definition of Z-transform as Σk ak zk . With the most conventional definition of Z-transform Σk ak z−k the zeros of the minimum phase waveform would be all inside the unit circle.

When a zero of Iaa (t) falls exactly on the unit circle, it must be a double zero because it coincides with its inverse conjugate. One such zero can be included in Z_aa.

It should be stressed that this equivalence is only in terms of the measured quantities and not in terms of the processing that we apply to them.