Abstract

We employed photonic radio frequency (RF) arbitrary waveform generation to demonstrate space–time compression of ultrabroadband wireless signals through highly scattering multipath channels. To the best of our knowledge, this is the first experimental report that explores an RF-photonic transmitter to both characterize channel dispersions in real wireless environments and generate predistorted waveforms to achieve focusing through the multipath channels. Our experiments span a three octave frequency range of 2–18 GHz, nearly an order of magnitude beyond the 2GHz instantaneous bandwidth reported in previous spatiotemporal focusing experiments relying on electronic waveform generators.

© 2013 Optical Society of America

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  1. M.-G. Benedetto, T. Kaiser, A. F. Molisch, I. Oppermann, C. Politano, and D. Porcino, UWB Communication Systems: A Comprehensive Overview (Hindawi, 2006).
  2. J. Yao, IEEE Microw. Mag. 10(4), 82 (2009).
    [CrossRef]
  3. Q. Wang and J. Yao, Opt. Express 15, 14667 (2007).
    [CrossRef]
  4. A. F. Molisch, in IEEE International Symposium on Microwave, Antennas, Propagation and EMC Technologies for Wireless Communications (2007), Vols. I and II, paper AK41.
  5. A. Dezfooliyan and A. M. Weiner, Optical Fiber Communication Conference (OFC) (2013), paper OTh3D.4.
  6. A. M. Weiner, Nat. Photonics 5, 332 (2011).
    [CrossRef]
  7. S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, Nat. Commun. 1, 81 (2010).
    [CrossRef]
  8. O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
    [CrossRef]
  9. D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, Nat. Commun. 2, 447 (2011).
    [CrossRef]
  10. A. Dezfooliyan and A. M. Weiner, IEEE Trans. Antennas Propag. 60, 4855 (2012).
    [CrossRef]
  11. A. Dezfooliyan and A. M. Weiner, IEEE Trans. Antennas Propag. 60, 3407 (2012).
    [CrossRef]
  12. A. Dezfooliyan and A. M. Weiner, IET Commun. 7, 1287 (2013).
    [CrossRef]
  13. I. H. Naqvi, P. Besnier, and G. E. Zein, IET Microw. Antennas Propag. 5, 468 (2011).
    [CrossRef]
  14. J. Chou, Y. Han, and B. Jalali, IEEE Photon. Technol. Lett. 15, 581 (2003).
    [CrossRef]
  15. I. S. Lin, J. D. McKinney, and A. M. Weiner, IEEE Microw. Wirel. Compon. Lett. 15, 226 (2005).
    [CrossRef]
  16. C. Wang and J. Yao, IEEE Trans. Microwave Theory Tech. 56, 542 (2008).
    [CrossRef]
  17. A. M. Weiner, Ultrafast Optics (Wiley, 2009).
  18. J. Azana and M. A. Muriel, IEEE J. Quantum Electron. 36, 517 (2000).
    [CrossRef]
  19. V. Torres-Company, D. E. Leaird, and A. M. Weiner, Opt. Express 19, 24718 (2011).
    [CrossRef]
  20. A. Dezfooliyan and A. M. Weiner, Opt. Express 21, 22974 (2013).
    [CrossRef]
  21. J. D. McKinney and A. M. Weiner, IEEE Trans. Microwave Theory Tech. 54, 1681 (2006).
    [CrossRef]
  22. J. D. McKinney, D. Peroulis, and A. M. Weiner, IEEE Trans. Microwave Theory Tech. 56, 710 (2008).
    [CrossRef]
  23. G. Lerosey, J. De Rosny, A. Tourin, and M. Fink, Science 315, 1120 (2007).
    [CrossRef]
  24. P. Blomgren, P. Kyritsi, A. D. Kim, and G. Papanicolaou, IEEE J. Oceanic Eng. 33, 341 (2008).
    [CrossRef]
  25. M. Fink, C. Prada, F. Wu, and D. Cassereau, Proc. IEEE Ultrason. Symp. 682, 681 (1989).
    [CrossRef]
  26. X. Li and M. I. Stockman, Phys. Rev. B 77, 195109 (2008).
    [CrossRef]

2013 (2)

A. Dezfooliyan and A. M. Weiner, IET Commun. 7, 1287 (2013).
[CrossRef]

A. Dezfooliyan and A. M. Weiner, Opt. Express 21, 22974 (2013).
[CrossRef]

2012 (2)

A. Dezfooliyan and A. M. Weiner, IEEE Trans. Antennas Propag. 60, 4855 (2012).
[CrossRef]

A. Dezfooliyan and A. M. Weiner, IEEE Trans. Antennas Propag. 60, 3407 (2012).
[CrossRef]

2011 (5)

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
[CrossRef]

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, Nat. Commun. 2, 447 (2011).
[CrossRef]

A. M. Weiner, Nat. Photonics 5, 332 (2011).
[CrossRef]

V. Torres-Company, D. E. Leaird, and A. M. Weiner, Opt. Express 19, 24718 (2011).
[CrossRef]

I. H. Naqvi, P. Besnier, and G. E. Zein, IET Microw. Antennas Propag. 5, 468 (2011).
[CrossRef]

2010 (1)

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, Nat. Commun. 1, 81 (2010).
[CrossRef]

2009 (1)

J. Yao, IEEE Microw. Mag. 10(4), 82 (2009).
[CrossRef]

2008 (4)

C. Wang and J. Yao, IEEE Trans. Microwave Theory Tech. 56, 542 (2008).
[CrossRef]

J. D. McKinney, D. Peroulis, and A. M. Weiner, IEEE Trans. Microwave Theory Tech. 56, 710 (2008).
[CrossRef]

P. Blomgren, P. Kyritsi, A. D. Kim, and G. Papanicolaou, IEEE J. Oceanic Eng. 33, 341 (2008).
[CrossRef]

X. Li and M. I. Stockman, Phys. Rev. B 77, 195109 (2008).
[CrossRef]

2007 (2)

G. Lerosey, J. De Rosny, A. Tourin, and M. Fink, Science 315, 1120 (2007).
[CrossRef]

Q. Wang and J. Yao, Opt. Express 15, 14667 (2007).
[CrossRef]

2006 (1)

J. D. McKinney and A. M. Weiner, IEEE Trans. Microwave Theory Tech. 54, 1681 (2006).
[CrossRef]

2005 (1)

I. S. Lin, J. D. McKinney, and A. M. Weiner, IEEE Microw. Wirel. Compon. Lett. 15, 226 (2005).
[CrossRef]

2003 (1)

J. Chou, Y. Han, and B. Jalali, IEEE Photon. Technol. Lett. 15, 581 (2003).
[CrossRef]

2000 (1)

J. Azana and M. A. Muriel, IEEE J. Quantum Electron. 36, 517 (2000).
[CrossRef]

1989 (1)

M. Fink, C. Prada, F. Wu, and D. Cassereau, Proc. IEEE Ultrason. Symp. 682, 681 (1989).
[CrossRef]

Austin, D. R.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, Nat. Commun. 2, 447 (2011).
[CrossRef]

Azana, J.

J. Azana and M. A. Muriel, IEEE J. Quantum Electron. 36, 517 (2000).
[CrossRef]

Benedetto, M.-G.

M.-G. Benedetto, T. Kaiser, A. F. Molisch, I. Oppermann, C. Politano, and D. Porcino, UWB Communication Systems: A Comprehensive Overview (Hindawi, 2006).

Besnier, P.

I. H. Naqvi, P. Besnier, and G. E. Zein, IET Microw. Antennas Propag. 5, 468 (2011).
[CrossRef]

Blomgren, P.

P. Blomgren, P. Kyritsi, A. D. Kim, and G. Papanicolaou, IEEE J. Oceanic Eng. 33, 341 (2008).
[CrossRef]

Boccara, A. C.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, Nat. Commun. 1, 81 (2010).
[CrossRef]

Bondareff, P.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, Nat. Commun. 2, 447 (2011).
[CrossRef]

Bromberg, Y.

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
[CrossRef]

Cassereau, D.

M. Fink, C. Prada, F. Wu, and D. Cassereau, Proc. IEEE Ultrason. Symp. 682, 681 (1989).
[CrossRef]

Chatel, B.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, Nat. Commun. 2, 447 (2011).
[CrossRef]

Chou, J.

J. Chou, Y. Han, and B. Jalali, IEEE Photon. Technol. Lett. 15, 581 (2003).
[CrossRef]

De Rosny, J.

G. Lerosey, J. De Rosny, A. Tourin, and M. Fink, Science 315, 1120 (2007).
[CrossRef]

Dezfooliyan, A.

A. Dezfooliyan and A. M. Weiner, Opt. Express 21, 22974 (2013).
[CrossRef]

A. Dezfooliyan and A. M. Weiner, IET Commun. 7, 1287 (2013).
[CrossRef]

A. Dezfooliyan and A. M. Weiner, IEEE Trans. Antennas Propag. 60, 4855 (2012).
[CrossRef]

A. Dezfooliyan and A. M. Weiner, IEEE Trans. Antennas Propag. 60, 3407 (2012).
[CrossRef]

A. Dezfooliyan and A. M. Weiner, Optical Fiber Communication Conference (OFC) (2013), paper OTh3D.4.

Fink, M.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, Nat. Commun. 1, 81 (2010).
[CrossRef]

G. Lerosey, J. De Rosny, A. Tourin, and M. Fink, Science 315, 1120 (2007).
[CrossRef]

M. Fink, C. Prada, F. Wu, and D. Cassereau, Proc. IEEE Ultrason. Symp. 682, 681 (1989).
[CrossRef]

Gigan, S.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, Nat. Commun. 2, 447 (2011).
[CrossRef]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, Nat. Commun. 1, 81 (2010).
[CrossRef]

Han, Y.

J. Chou, Y. Han, and B. Jalali, IEEE Photon. Technol. Lett. 15, 581 (2003).
[CrossRef]

Jalali, B.

J. Chou, Y. Han, and B. Jalali, IEEE Photon. Technol. Lett. 15, 581 (2003).
[CrossRef]

Kaiser, T.

M.-G. Benedetto, T. Kaiser, A. F. Molisch, I. Oppermann, C. Politano, and D. Porcino, UWB Communication Systems: A Comprehensive Overview (Hindawi, 2006).

Katz, O.

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
[CrossRef]

Kim, A. D.

P. Blomgren, P. Kyritsi, A. D. Kim, and G. Papanicolaou, IEEE J. Oceanic Eng. 33, 341 (2008).
[CrossRef]

Kyritsi, P.

P. Blomgren, P. Kyritsi, A. D. Kim, and G. Papanicolaou, IEEE J. Oceanic Eng. 33, 341 (2008).
[CrossRef]

Leaird, D. E.

Lerosey, G.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, Nat. Commun. 1, 81 (2010).
[CrossRef]

G. Lerosey, J. De Rosny, A. Tourin, and M. Fink, Science 315, 1120 (2007).
[CrossRef]

Li, X.

X. Li and M. I. Stockman, Phys. Rev. B 77, 195109 (2008).
[CrossRef]

Lin, I. S.

I. S. Lin, J. D. McKinney, and A. M. Weiner, IEEE Microw. Wirel. Compon. Lett. 15, 226 (2005).
[CrossRef]

McCabe, D. J.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, Nat. Commun. 2, 447 (2011).
[CrossRef]

McKinney, J. D.

J. D. McKinney, D. Peroulis, and A. M. Weiner, IEEE Trans. Microwave Theory Tech. 56, 710 (2008).
[CrossRef]

J. D. McKinney and A. M. Weiner, IEEE Trans. Microwave Theory Tech. 54, 1681 (2006).
[CrossRef]

I. S. Lin, J. D. McKinney, and A. M. Weiner, IEEE Microw. Wirel. Compon. Lett. 15, 226 (2005).
[CrossRef]

Molisch, A. F.

A. F. Molisch, in IEEE International Symposium on Microwave, Antennas, Propagation and EMC Technologies for Wireless Communications (2007), Vols. I and II, paper AK41.

M.-G. Benedetto, T. Kaiser, A. F. Molisch, I. Oppermann, C. Politano, and D. Porcino, UWB Communication Systems: A Comprehensive Overview (Hindawi, 2006).

Muriel, M. A.

J. Azana and M. A. Muriel, IEEE J. Quantum Electron. 36, 517 (2000).
[CrossRef]

Naqvi, I. H.

I. H. Naqvi, P. Besnier, and G. E. Zein, IET Microw. Antennas Propag. 5, 468 (2011).
[CrossRef]

Oppermann, I.

M.-G. Benedetto, T. Kaiser, A. F. Molisch, I. Oppermann, C. Politano, and D. Porcino, UWB Communication Systems: A Comprehensive Overview (Hindawi, 2006).

Papanicolaou, G.

P. Blomgren, P. Kyritsi, A. D. Kim, and G. Papanicolaou, IEEE J. Oceanic Eng. 33, 341 (2008).
[CrossRef]

Peroulis, D.

J. D. McKinney, D. Peroulis, and A. M. Weiner, IEEE Trans. Microwave Theory Tech. 56, 710 (2008).
[CrossRef]

Politano, C.

M.-G. Benedetto, T. Kaiser, A. F. Molisch, I. Oppermann, C. Politano, and D. Porcino, UWB Communication Systems: A Comprehensive Overview (Hindawi, 2006).

Popoff, S.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, Nat. Commun. 1, 81 (2010).
[CrossRef]

Porcino, D.

M.-G. Benedetto, T. Kaiser, A. F. Molisch, I. Oppermann, C. Politano, and D. Porcino, UWB Communication Systems: A Comprehensive Overview (Hindawi, 2006).

Prada, C.

M. Fink, C. Prada, F. Wu, and D. Cassereau, Proc. IEEE Ultrason. Symp. 682, 681 (1989).
[CrossRef]

Silberberg, Y.

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
[CrossRef]

Small, E.

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
[CrossRef]

Stockman, M. I.

X. Li and M. I. Stockman, Phys. Rev. B 77, 195109 (2008).
[CrossRef]

Tajalli, A.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, Nat. Commun. 2, 447 (2011).
[CrossRef]

Torres-Company, V.

Tourin, A.

G. Lerosey, J. De Rosny, A. Tourin, and M. Fink, Science 315, 1120 (2007).
[CrossRef]

Walmsley, I. A.

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, Nat. Commun. 2, 447 (2011).
[CrossRef]

Wang, C.

C. Wang and J. Yao, IEEE Trans. Microwave Theory Tech. 56, 542 (2008).
[CrossRef]

Wang, Q.

Weiner, A. M.

A. Dezfooliyan and A. M. Weiner, IET Commun. 7, 1287 (2013).
[CrossRef]

A. Dezfooliyan and A. M. Weiner, Opt. Express 21, 22974 (2013).
[CrossRef]

A. Dezfooliyan and A. M. Weiner, IEEE Trans. Antennas Propag. 60, 3407 (2012).
[CrossRef]

A. Dezfooliyan and A. M. Weiner, IEEE Trans. Antennas Propag. 60, 4855 (2012).
[CrossRef]

A. M. Weiner, Nat. Photonics 5, 332 (2011).
[CrossRef]

V. Torres-Company, D. E. Leaird, and A. M. Weiner, Opt. Express 19, 24718 (2011).
[CrossRef]

J. D. McKinney, D. Peroulis, and A. M. Weiner, IEEE Trans. Microwave Theory Tech. 56, 710 (2008).
[CrossRef]

J. D. McKinney and A. M. Weiner, IEEE Trans. Microwave Theory Tech. 54, 1681 (2006).
[CrossRef]

I. S. Lin, J. D. McKinney, and A. M. Weiner, IEEE Microw. Wirel. Compon. Lett. 15, 226 (2005).
[CrossRef]

A. M. Weiner, Ultrafast Optics (Wiley, 2009).

A. Dezfooliyan and A. M. Weiner, Optical Fiber Communication Conference (OFC) (2013), paper OTh3D.4.

Wu, F.

M. Fink, C. Prada, F. Wu, and D. Cassereau, Proc. IEEE Ultrason. Symp. 682, 681 (1989).
[CrossRef]

Yao, J.

J. Yao, IEEE Microw. Mag. 10(4), 82 (2009).
[CrossRef]

C. Wang and J. Yao, IEEE Trans. Microwave Theory Tech. 56, 542 (2008).
[CrossRef]

Q. Wang and J. Yao, Opt. Express 15, 14667 (2007).
[CrossRef]

Zein, G. E.

I. H. Naqvi, P. Besnier, and G. E. Zein, IET Microw. Antennas Propag. 5, 468 (2011).
[CrossRef]

IEEE J. Oceanic Eng. (1)

P. Blomgren, P. Kyritsi, A. D. Kim, and G. Papanicolaou, IEEE J. Oceanic Eng. 33, 341 (2008).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. Azana and M. A. Muriel, IEEE J. Quantum Electron. 36, 517 (2000).
[CrossRef]

IEEE Microw. Mag. (1)

J. Yao, IEEE Microw. Mag. 10(4), 82 (2009).
[CrossRef]

IEEE Microw. Wirel. Compon. Lett. (1)

I. S. Lin, J. D. McKinney, and A. M. Weiner, IEEE Microw. Wirel. Compon. Lett. 15, 226 (2005).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

J. Chou, Y. Han, and B. Jalali, IEEE Photon. Technol. Lett. 15, 581 (2003).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

A. Dezfooliyan and A. M. Weiner, IEEE Trans. Antennas Propag. 60, 4855 (2012).
[CrossRef]

A. Dezfooliyan and A. M. Weiner, IEEE Trans. Antennas Propag. 60, 3407 (2012).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (3)

C. Wang and J. Yao, IEEE Trans. Microwave Theory Tech. 56, 542 (2008).
[CrossRef]

J. D. McKinney and A. M. Weiner, IEEE Trans. Microwave Theory Tech. 54, 1681 (2006).
[CrossRef]

J. D. McKinney, D. Peroulis, and A. M. Weiner, IEEE Trans. Microwave Theory Tech. 56, 710 (2008).
[CrossRef]

IET Commun. (1)

A. Dezfooliyan and A. M. Weiner, IET Commun. 7, 1287 (2013).
[CrossRef]

IET Microw. Antennas Propag. (1)

I. H. Naqvi, P. Besnier, and G. E. Zein, IET Microw. Antennas Propag. 5, 468 (2011).
[CrossRef]

Nat. Commun. (2)

D. J. McCabe, A. Tajalli, D. R. Austin, P. Bondareff, I. A. Walmsley, S. Gigan, and B. Chatel, Nat. Commun. 2, 447 (2011).
[CrossRef]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, Nat. Commun. 1, 81 (2010).
[CrossRef]

Nat. Photonics (2)

O. Katz, E. Small, Y. Bromberg, and Y. Silberberg, Nat. Photonics 5, 372 (2011).
[CrossRef]

A. M. Weiner, Nat. Photonics 5, 332 (2011).
[CrossRef]

Opt. Express (3)

Phys. Rev. B (1)

X. Li and M. I. Stockman, Phys. Rev. B 77, 195109 (2008).
[CrossRef]

Proc. IEEE Ultrason. Symp. (1)

M. Fink, C. Prada, F. Wu, and D. Cassereau, Proc. IEEE Ultrason. Symp. 682, 681 (1989).
[CrossRef]

Science (1)

G. Lerosey, J. De Rosny, A. Tourin, and M. Fink, Science 315, 1120 (2007).
[CrossRef]

Other (4)

A. F. Molisch, in IEEE International Symposium on Microwave, Antennas, Propagation and EMC Technologies for Wireless Communications (2007), Vols. I and II, paper AK41.

A. Dezfooliyan and A. M. Weiner, Optical Fiber Communication Conference (OFC) (2013), paper OTh3D.4.

M.-G. Benedetto, T. Kaiser, A. F. Molisch, I. Oppermann, C. Politano, and D. Porcino, UWB Communication Systems: A Comprehensive Overview (Hindawi, 2006).

A. M. Weiner, Ultrafast Optics (Wiley, 2009).

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Figures (5)

Fig. 1.
Fig. 1.

Two receiver antennas are located a few wavelengths apart. Due to strong multipath scattering, the received RF signal comprises multiple components with delays and attenuations specific to the receiver location. For a fixed transmit antenna (Tx), the impulse responses for different receivers are nearly uncorrelated. With knowledge of the channel impulse responses, Tx may transmit a waveform matched to the Tx–Rx1 multipath channel that results in significant peaking at the target receiver (Rx1). The waveform received by Rx2 remains noise-like.

Fig. 2.
Fig. 2.

Experimental setup. PC, polarization controller; IM, intensity modulator; EDFA, erbium-doped fiber amplifier; PD, photodetector; AMP, electrical amplifier; Tx, transmit antenna; Rx, receive antenna; LNA, low noise amplifier.

Fig. 3.
Fig. 3.

Photonically generated RF waveform over frequencies from baseband to 18GHz with a time aperture of 16.2ns. (a) and (b) The sounding waveform generated via the NF-FTM technique. The chirp signal obtained provides an RF spectrum that extends smoothly out to 18GHz. (c) and (d) When the conventional FTM technique was employed, the synthesized waveform was badly distorted and certain frequencies were strongly attenuated.

Fig. 4.
Fig. 4.

(a) and (b) Impulse responses of Tx–Rx1 and Tx–Rx2 channels. In the small subfigures, we zoom in on the dotted rectangles to show more details. (c) and (d) Corresponding power spectra of the Tx–Rx1 and Tx–Rx2 links.

Fig. 5.
Fig. 5.

(a) and (b) When the phase compensated (PC) transmit waveform is designed based on the Tx–Rx1 response, a significant peaking results at the target receiver. Receiver Rx2 located just a couple of wavelengths away from the target receiver gets only a noise like interference. (c) and (d) When the Tx transmits a waveform designed to achieve peaking at Rx2, the received signal by Rx1 now remains noise-like.

Equations (3)

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

HSys(ω)=YRec(ω)XTrans(ω),
XPC(ω)=exp(jarg(HSys(ω))),
YPC(ω)=Hsys(ω)XPC(ω)=|Hsys(ω)|.

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