Abstract

We study analytically and experimentally the performance limits of a Si-photonic (SiP) balanced coherent receiver (CRx) co-packaged with transimpedance amplifiers (TIAs) in a colorless WDM scheme. Firstly, the CRx architecture is depicted and characterization results are presented. Secondly, an analytical expression for the signal-to-noise ratio (SNR) at the CRx output is rigorously developed and various noise sources in the context of colorless reception are outlined. Thirdly, we study experimentally the system-level CRx performance in colorless reception of 16 × 112 Gbps PDM-QPSK WDM channels. Using a 15.5 dBm local oscillator (LO) power, error free transmissions over 4800 and 4160 km at received powers of −3 and −21 dBm per channel, respectively, were achieved in a fully colorless and preamplifierless reception. Next, a set of measurements on one of the center WDM channels is performed where the LO power, received signal power, distance, and number of channels presented to the CRx are swept to evaluate the performance limits of colorless reception. Results reveal that the LO beating with optical noise incoming with the signal is a dominant noise source regardless of received signal power. In the high received signal power regime (~0 dBm/channel), the self-beat noise from out-of-band (OOB) channels is an additional major noise source especially for small LO-to-signal power ratio, short reach and large number of OOB channels. For example, at a received signal power of 0 dBm/channel after 1600 km transmission, the SNR difference between the fully filtered and colorless scenarios, where 1 and 16 channels are passed to the CRx respectively, grows from 0.5 to 3.3 dB as the LO power changes from 12 to 0 dBm. For low received power (~-12 dBm/channel), the effect of OOB channels becomes minor while the receiver shot and thermal noises become more significant. We identify the common mode rejection ratio (CMRR) and sensitivity as the two important CRx specifications that impact the performance at high and low received signal power regimes, respectively. Finally, an excellent match between experimental and analytical SNRs is proven after the derived SNR model is fitted to the experimental data in a least-squares sense. The model is then used to predict that the CRx can operate colorlessly for a fully populated WDM spectrum with 80 channels provided that the LO-to-signal power ratio is properly set.

© 2014 Optical Society of America

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2013 (2)

R. Essiambre, R. Ryf, N. Fontaine, and S. Randel, “Breakthroughs in photonics 2012: Space-division multiplexing in multimode and multicore fibers for high-capacity optical communication,” IEEE Photonics J. 5(2), 0701307 (2013).
[Crossref]

M. Morsy-Osman, M. Chagnon, X. Xu, Q. Zhuge, M. Poulin, Y. Painchaud, M. Pelletier, C. Paquet, and D. V. Plant, “Colorless and preamplifierless reception using an integrated Si-photonic coherent receiver,” IEEE Photonics Technol. Lett. 25, 1027–1030 (2013).

2012 (4)

2011 (3)

C. R. Doerr, L. L. Buhl, Y. Baeyens, R. Aroca, S. Chandrasekhar, X. Liu, L. Chen, and Y. K. Chen, “Packaged monolithic silicon 112-Gb/s coherent receiver,” IEEE Photonics Technol. Lett. 23, 762–764 (2011).

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photonics Technol. Lett. 23, 742–744 (2011).

C. Doerr, L. Zhang, P. Winzer, N. Weimann, V. Houtsma, T. Hu, N. Sauer, L. Buhl, D. Neilson, S. Chandrasekhar, and Y. Chen, “Monolithic InP dual-polarization and dual-quadrature coherent receiver,” IEEE Photonics Technol. Lett. 23, 694–696 (2011).

2010 (5)

2009 (3)

2008 (1)

2006 (1)

P. Ciblat and M. Ghogho, “Blind NLLS carrier frequency-offset estimation for QAM, PSK, and PAM modulations: Performance at low SNR,” IEEE Trans. Commun. 54, 1725–1730 (2006).

1989 (1)

M. Nazarathy, W. Sorin, D. Baney, and S. Newton, “Spectral analysis of optical mixing measurements,” J. Lightwave Technol. 7, 1083–1096 (1989).

1987 (1)

L. Kazovsky, “Multichannel coherent optical communications systems,” J. Lightwave Technol. 5, 1095–1102 (1987).

Ang, K.

Aroca, R.

C. R. Doerr, L. L. Buhl, Y. Baeyens, R. Aroca, S. Chandrasekhar, X. Liu, L. Chen, and Y. K. Chen, “Packaged monolithic silicon 112-Gb/s coherent receiver,” IEEE Photonics Technol. Lett. 23, 762–764 (2011).

Baeyens, Y.

C. R. Doerr, L. L. Buhl, Y. Baeyens, R. Aroca, S. Chandrasekhar, X. Liu, L. Chen, and Y. K. Chen, “Packaged monolithic silicon 112-Gb/s coherent receiver,” IEEE Photonics Technol. Lett. 23, 762–764 (2011).

Baney, D.

M. Nazarathy, W. Sorin, D. Baney, and S. Newton, “Spectral analysis of optical mixing measurements,” J. Lightwave Technol. 7, 1083–1096 (1989).

Beckett, D.

Berthold, J.

K. Roberts, D. Beckett, D. Boertjes, J. Berthold, and C. Laperle, “100G and beyond with digital coherent signal processing,” IEEE Commun. Mag. 48(7), 62–69 (2010).
[Crossref]

Boertjes, D.

K. Roberts, D. Beckett, D. Boertjes, J. Berthold, and C. Laperle, “100G and beyond with digital coherent signal processing,” IEEE Commun. Mag. 48(7), 62–69 (2010).
[Crossref]

Bosco, G.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012).

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photonics Technol. Lett. 23, 742–744 (2011).

Bowers, J. E.

D. Liang and J. E. Bowers, “Photonic integration: Si or InP substrates?” Electron. Lett. 45(12), 578–581 (2009).
[Crossref]

Buhl, L.

C. Doerr, L. Zhang, P. Winzer, N. Weimann, V. Houtsma, T. Hu, N. Sauer, L. Buhl, D. Neilson, S. Chandrasekhar, and Y. Chen, “Monolithic InP dual-polarization and dual-quadrature coherent receiver,” IEEE Photonics Technol. Lett. 23, 694–696 (2011).

Buhl, L. L.

C. R. Doerr, L. L. Buhl, Y. Baeyens, R. Aroca, S. Chandrasekhar, X. Liu, L. Chen, and Y. K. Chen, “Packaged monolithic silicon 112-Gb/s coherent receiver,” IEEE Photonics Technol. Lett. 23, 762–764 (2011).

Carena, A.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012).

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photonics Technol. Lett. 23, 742–744 (2011).

Chagnon, M.

M. Morsy-Osman, M. Chagnon, X. Xu, Q. Zhuge, M. Poulin, Y. Painchaud, M. Pelletier, C. Paquet, and D. V. Plant, “Colorless and preamplifierless reception using an integrated Si-photonic coherent receiver,” IEEE Photonics Technol. Lett. 25, 1027–1030 (2013).

Chandrasekhar, S.

C. R. Doerr, L. L. Buhl, Y. Baeyens, R. Aroca, S. Chandrasekhar, X. Liu, L. Chen, and Y. K. Chen, “Packaged monolithic silicon 112-Gb/s coherent receiver,” IEEE Photonics Technol. Lett. 23, 762–764 (2011).

C. Doerr, L. Zhang, P. Winzer, N. Weimann, V. Houtsma, T. Hu, N. Sauer, L. Buhl, D. Neilson, S. Chandrasekhar, and Y. Chen, “Monolithic InP dual-polarization and dual-quadrature coherent receiver,” IEEE Photonics Technol. Lett. 23, 694–696 (2011).

C. Doerr, P. Winzer, Y. Chen, S. Chandrasekhar, M. Rasras, L. Chen, T. Liow, K. Ang, and G. Lo, “Monolithic polarization and phase diversity coherent receiver in silicon,” J. Lightwave Technol. 28, 520–525 (2010).

Chen, L.

C. R. Doerr, L. L. Buhl, Y. Baeyens, R. Aroca, S. Chandrasekhar, X. Liu, L. Chen, and Y. K. Chen, “Packaged monolithic silicon 112-Gb/s coherent receiver,” IEEE Photonics Technol. Lett. 23, 762–764 (2011).

C. Doerr, P. Winzer, Y. Chen, S. Chandrasekhar, M. Rasras, L. Chen, T. Liow, K. Ang, and G. Lo, “Monolithic polarization and phase diversity coherent receiver in silicon,” J. Lightwave Technol. 28, 520–525 (2010).

Chen, Y.

C. Doerr, L. Zhang, P. Winzer, N. Weimann, V. Houtsma, T. Hu, N. Sauer, L. Buhl, D. Neilson, S. Chandrasekhar, and Y. Chen, “Monolithic InP dual-polarization and dual-quadrature coherent receiver,” IEEE Photonics Technol. Lett. 23, 694–696 (2011).

C. Doerr, P. Winzer, Y. Chen, S. Chandrasekhar, M. Rasras, L. Chen, T. Liow, K. Ang, and G. Lo, “Monolithic polarization and phase diversity coherent receiver in silicon,” J. Lightwave Technol. 28, 520–525 (2010).

Chen, Y. K.

C. R. Doerr, L. L. Buhl, Y. Baeyens, R. Aroca, S. Chandrasekhar, X. Liu, L. Chen, and Y. K. Chen, “Packaged monolithic silicon 112-Gb/s coherent receiver,” IEEE Photonics Technol. Lett. 23, 762–764 (2011).

Ciblat, P.

P. Ciblat and M. Ghogho, “Blind NLLS carrier frequency-offset estimation for QAM, PSK, and PAM modulations: Performance at low SNR,” IEEE Trans. Commun. 54, 1725–1730 (2006).

Curri, V.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012).

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photonics Technol. Lett. 23, 742–744 (2011).

Doerr, C.

C. Doerr, L. Zhang, P. Winzer, N. Weimann, V. Houtsma, T. Hu, N. Sauer, L. Buhl, D. Neilson, S. Chandrasekhar, and Y. Chen, “Monolithic InP dual-polarization and dual-quadrature coherent receiver,” IEEE Photonics Technol. Lett. 23, 694–696 (2011).

C. Doerr, P. Winzer, Y. Chen, S. Chandrasekhar, M. Rasras, L. Chen, T. Liow, K. Ang, and G. Lo, “Monolithic polarization and phase diversity coherent receiver in silicon,” J. Lightwave Technol. 28, 520–525 (2010).

Doerr, C. R.

C. R. Doerr, L. L. Buhl, Y. Baeyens, R. Aroca, S. Chandrasekhar, X. Liu, L. Chen, and Y. K. Chen, “Packaged monolithic silicon 112-Gb/s coherent receiver,” IEEE Photonics Technol. Lett. 23, 762–764 (2011).

Edvold, B.

Essiambre, R.

R. Essiambre, R. Ryf, N. Fontaine, and S. Randel, “Breakthroughs in photonics 2012: Space-division multiplexing in multimode and multicore fibers for high-capacity optical communication,” IEEE Photonics J. 5(2), 0701307 (2013).
[Crossref]

Fatadin, I.

Fontaine, N.

R. Essiambre, R. Ryf, N. Fontaine, and S. Randel, “Breakthroughs in photonics 2012: Space-division multiplexing in multimode and multicore fibers for high-capacity optical communication,” IEEE Photonics J. 5(2), 0701307 (2013).
[Crossref]

Foo, S.

Forghieri, F.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30, 1524–1539 (2012).

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of nonlinear propagation in uncompensated optical transmission links,” IEEE Photonics Technol. Lett. 23, 742–744 (2011).

Geisler, T.

Ghogho, M.

P. Ciblat and M. Ghogho, “Blind NLLS carrier frequency-offset estimation for QAM, PSK, and PAM modulations: Performance at low SNR,” IEEE Trans. Commun. 54, 1725–1730 (2006).

Gnauck, A. H.

Houtsma, V.

C. Doerr, L. Zhang, P. Winzer, N. Weimann, V. Houtsma, T. Hu, N. Sauer, L. Buhl, D. Neilson, S. Chandrasekhar, and Y. Chen, “Monolithic InP dual-polarization and dual-quadrature coherent receiver,” IEEE Photonics Technol. Lett. 23, 694–696 (2011).

Hu, T.

C. Doerr, L. Zhang, P. Winzer, N. Weimann, V. Houtsma, T. Hu, N. Sauer, L. Buhl, D. Neilson, S. Chandrasekhar, and Y. Chen, “Monolithic InP dual-polarization and dual-quadrature coherent receiver,” IEEE Photonics Technol. Lett. 23, 694–696 (2011).

Ip, E.

Ives, D.

Kahn, J. M.

Kazovsky, L.

L. Kazovsky, “Multichannel coherent optical communications systems,” J. Lightwave Technol. 5, 1095–1102 (1987).

Laperle, C.

K. Roberts, D. Beckett, D. Boertjes, J. Berthold, and C. Laperle, “100G and beyond with digital coherent signal processing,” IEEE Commun. Mag. 48(7), 62–69 (2010).
[Crossref]

Liang, D.

D. Liang and J. E. Bowers, “Photonic integration: Si or InP substrates?” Electron. Lett. 45(12), 578–581 (2009).
[Crossref]

Liow, T.

Liu, X.

C. R. Doerr, L. L. Buhl, Y. Baeyens, R. Aroca, S. Chandrasekhar, X. Liu, L. Chen, and Y. K. Chen, “Packaged monolithic silicon 112-Gb/s coherent receiver,” IEEE Photonics Technol. Lett. 23, 762–764 (2011).

Lo, G.

Magill, P.

Malouin, C.

Morin, M.

Morsy-Osman, M.

M. Morsy-Osman, M. Chagnon, X. Xu, Q. Zhuge, M. Poulin, Y. Painchaud, M. Pelletier, C. Paquet, and D. V. Plant, “Colorless and preamplifierless reception using an integrated Si-photonic coherent receiver,” IEEE Photonics Technol. Lett. 25, 1027–1030 (2013).

Moyer, M.

Nazarathy, M.

M. Nazarathy, W. Sorin, D. Baney, and S. Newton, “Spectral analysis of optical mixing measurements,” J. Lightwave Technol. 7, 1083–1096 (1989).

Neilson, D.

C. Doerr, L. Zhang, P. Winzer, N. Weimann, V. Houtsma, T. Hu, N. Sauer, L. Buhl, D. Neilson, S. Chandrasekhar, and Y. Chen, “Monolithic InP dual-polarization and dual-quadrature coherent receiver,” IEEE Photonics Technol. Lett. 23, 694–696 (2011).

Nelson, L.

Newton, S.

M. Nazarathy, W. Sorin, D. Baney, and S. Newton, “Spectral analysis of optical mixing measurements,” J. Lightwave Technol. 7, 1083–1096 (1989).

OSullivan, M.

Painchaud, Y.

M. Morsy-Osman, M. Chagnon, X. Xu, Q. Zhuge, M. Poulin, Y. Painchaud, M. Pelletier, C. Paquet, and D. V. Plant, “Colorless and preamplifierless reception using an integrated Si-photonic coherent receiver,” IEEE Photonics Technol. Lett. 25, 1027–1030 (2013).

Y. Painchaud, M. Poulin, M. Morin, and M. Têtu, “Performance of balanced detection in a coherent receiver,” Opt. Express 17(5), 3659–3672 (2009).
[Crossref] [PubMed]

Paquet, C.

M. Morsy-Osman, M. Chagnon, X. Xu, Q. Zhuge, M. Poulin, Y. Painchaud, M. Pelletier, C. Paquet, and D. V. Plant, “Colorless and preamplifierless reception using an integrated Si-photonic coherent receiver,” IEEE Photonics Technol. Lett. 25, 1027–1030 (2013).

Pelletier, M.

M. Morsy-Osman, M. Chagnon, X. Xu, Q. Zhuge, M. Poulin, Y. Painchaud, M. Pelletier, C. Paquet, and D. V. Plant, “Colorless and preamplifierless reception using an integrated Si-photonic coherent receiver,” IEEE Photonics Technol. Lett. 25, 1027–1030 (2013).

Plant, D. V.

M. Morsy-Osman, M. Chagnon, X. Xu, Q. Zhuge, M. Poulin, Y. Painchaud, M. Pelletier, C. Paquet, and D. V. Plant, “Colorless and preamplifierless reception using an integrated Si-photonic coherent receiver,” IEEE Photonics Technol. Lett. 25, 1027–1030 (2013).

Poggiolini, P.

Poulin, M.

M. Morsy-Osman, M. Chagnon, X. Xu, Q. Zhuge, M. Poulin, Y. Painchaud, M. Pelletier, C. Paquet, and D. V. Plant, “Colorless and preamplifierless reception using an integrated Si-photonic coherent receiver,” IEEE Photonics Technol. Lett. 25, 1027–1030 (2013).

Y. Painchaud, M. Poulin, M. Morin, and M. Têtu, “Performance of balanced detection in a coherent receiver,” Opt. Express 17(5), 3659–3672 (2009).
[Crossref] [PubMed]

Randel, S.

R. Essiambre, R. Ryf, N. Fontaine, and S. Randel, “Breakthroughs in photonics 2012: Space-division multiplexing in multimode and multicore fibers for high-capacity optical communication,” IEEE Photonics J. 5(2), 0701307 (2013).
[Crossref]

Rasras, M.

Raybon, G.

Roberts, K.

K. Roberts, D. Beckett, D. Boertjes, J. Berthold, and C. Laperle, “100G and beyond with digital coherent signal processing,” IEEE Commun. Mag. 48(7), 62–69 (2010).
[Crossref]

Ryf, R.

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Although both LO and signal fields are stationary in their CW form, a modulated signal field is non-stationary rendering the final photocurrent difference Δi non-stationary. Hence, we apply both ensemble and time averaging to obtain a single argument time-averaged ACF from which the PSD is obtained using the Wiener–Khintchine theorem.

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Figures (13)

Fig. 1
Fig. 1 (a) Picture of the PIC of the CRx showing the main buliding blocks, (b) Overall frequency response of the Si-photonic CRx assembly.
Fig. 2
Fig. 2 CMRR versus frequency curves measured at 192.9 THz at the four output RF ports of the CRx when light is applied at the (a) signal port, (b) LO port.
Fig. 3
Fig. 3 Experimental setup for WDM experiments, (IL: Interleaver, DFB laser: Distributed Feedback laser, AWG: Arrayed Waveguide Grating, PBS: Polarization Beam Splitter, PBC: Polarization Beam Combiner, VODL: Variable Optical Delay Line, ECL: External Cavity Laser, OSA: Optical Spectrum Analyzer, VOA: Variable Optical Attenuator).
Fig. 4
Fig. 4 BER for all WDM channels using fully colorless and preamplifierless reception at different transmission distances and at per channel received signal power of −3 and −21 dBm. Insets show the received WDM spectra in back-to-back and after 3200 km transmission where we observe a fairly flat spectrum due to the gain flattening waveshaper in the loop.
Fig. 5
Fig. 5 Measurement results of channel 9: (a) BER versus transmission distance, (b) SNR evaluated after offline DSP versus distance, (c) BER versus SNR using both experimental results and theory where the experimental SNR corresponding to FEC threshold is marked.
Fig. 6
Fig. 6 SNR of channel 9 versus received signal power per channel PSIG after 0, 1600, 3200 and 4800 km transmission with varying number of channels presented to the CRx at an LO power of: (a) 15.5 dBm, (b) 12 dBm, (c) 9 dBm, (d) 6 dBm, (e) 3 dBm, (f) 0 dBm. In each subfigure, different colors represent different number of channels Nch according to the legend. Also, SNR evaluated after 0, 1600, 3200 and 4800 km transmission are shown in each subfigure where the top set of four curves in each subfigure is evaluated in back-to-back and the bottom set is after 4800 km transmission.
Fig. 7
Fig. 7 SNR of channel 9 versus Nch at PSIG = 0 dBm after 0, 1600, 3200 and 4800 km transmission at PLO of (a) 15.5 dBm, (b) 12 dBm, (c) 9 dBm, (d) 6 dBm, (e) 3 dBm, (f) 0 dBm.
Fig. 8
Fig. 8 SNR of channel 9 versus Nch at PSIG = −12 dBm after 0, 1600, 3200 and 4800 km transmission at PLO of (a) 15.5 dBm, (b) 12 dBm, (c) 9 dBm, (d) 6 dBm, (e) 3 dBm, (f) 0 dBm.
Fig. 9
Fig. 9 SNR of channel 9 versus PLO after 0, 1600, 3200 and 4800 km transmission at various Nch and at PSIG of: (a) 0 dBm, (b) −15 dBm.
Fig. 10
Fig. 10 Color coded 2-D plots of SNR versus both PSIG and PLO for (a) back-to-back and Nch = 1, (b) back-to-back and Nch = 16, (c) 1600 km transmission and Nch = 1, (d) 1600 km transmission and Nch = 16, (e) 3200 km transmission and Nch = 1, (f) 3200 km transmission and Nch = 16, (g) 4800 km transmission and Nch = 1, and (h) 4800 km transmission and Nch = 16.
Fig. 11
Fig. 11 SNR versus PSIG comparing both experimental data points with the points evaluated from the fitted SNR analytical model after 0, 1600, 3200 and 4800 km at the following operating conditions: (a) PLO = 12 dBm and Nch = 1, (b) PLO = 12 dBm and Nch = 16, (c) PLO = 3 dBm and Nch = 1, and (d) PLO = 3 dBm and Nch = 16.
Fig. 12
Fig. 12 SNR versus Nch comparing the experimental data points with the points evaluated from the fitted SNR analytical model after 0, 1600, 3200 and 4800 km at the following operating conditions: (a) PLO = 12 dBm and PSIG = 0 dBm, (b) PLO = 12 dBm and PSIG = −12 dBm, (c) PLO = 3 dBm and PSIG = 0 dBm, and (d) PLO = 3 dBm and PSIG = −12 dBm.
Fig. 13
Fig. 13 (a) Simulated intensity eyediagram of the single polarization QPSK signal using NRZ pulse shape, (b) experimental intensity eyediagram obtained after the single polarization IQ modulator, (c) scaling factor (β) versus the amount of residual CD for various polarization rotation angles (θ) obtained from simulations and theory on a 28 Gbaud PDM-QPSK signal.

Tables (1)

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Table 1 List of Symbols and Notation

Equations (69)

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i P ( t ) = [ i = 1 N c h R S I G , i P ( E S I G , i ( t ) + E O N , i ( t ) ) + R L O , s P ( E L O ( t ) ) ] 2 + i s h P + i t h P 2 R S I G , s P R L O , s P [ E S I G , s ( t ) E L O ( t ) + E O N , s ( t ) E L O ( t ) ] + i = 1 N c h R S I G , i P [ E S I G , i 2 ( t ) + E O N , i 2 ( t ) + 2 E S I G , i ( t ) E O N , i ( t ) ] + R L O , s P E L O 2 ( t ) + i s h P + i t h P ,
i N ( t ) = [ i = 1 N c h R S I G , i N ( E S I G , i ( t δ ) + E O N , i ( t δ ) ) R L O , s N ( E L O ( t δ ) ) ] 2 + i s h N + i t h N 2 R S I G , s N R L O , s N [ E S I G , s ( t δ ) E L O ( t δ ) + E O N , s ( t δ ) E L O ( t δ ) ] + i = 1 N c h R S I G , i N [ E S I G , i 2 ( t δ ) + E O N , i 2 ( t δ ) + 2 E S I G , i ( t δ ) E O N , i ( t δ ) ] + R L O , s N E L O 2 ( t δ ) + i s h N + i t h N ,
E S I G , i ( t ) = R e { E ˜ S I G , i ( t ) e j 2 π ν S I G , i t } , E L O ( t ) = R e { E ˜ L O ( t ) e j 2 π ν L O t } , E O N , i ( t ) = R e { E ˜ O N , i ( t ) e j 2 π ν S I G , i t } ,
E 1 ( t ) E 2 ( t ) = R e { E ˜ 1 ( t ) e j 2 π ν 1 t } R e { E ˜ 2 ( t ) e j 2 π ν 2 t } = 1 2 R e { E ˜ 1 ( t ) E ˜ 2 * ( t ) e j 2 π ( ν 1 ν 2 ) t } + 1 2 R e { E ˜ 1 ( t ) E ˜ 2 ( t ) e j 2 π ( ν 1 + ν 2 ) t } ,
i P ( t ) = 2 R S I G , s P R L O , s P 1 2 R e { E ˜ S I G , s ( t ) E ˜ L O * ( t ) e j 2 π Δ ν t + E ˜ O N , s ( t ) E ˜ L O * ( t ) e j 2 π Δ ν t } + 1 2 i = 1 N c h R S I G , i P [ | E ˜ S I G , i ( t ) | 2 + | E ˜ O N , i ( t ) | 2 + 2 R e { E ˜ S I G , i ( t ) E ˜ O N , i * ( t ) } ] + R L O , s P 1 2 | E ˜ L O ( t ) | 2 + i s h P + i t h P ,
i N ( t ) = 2 R S I G , s N R L O , s N 1 2 R e { E ˜ S I G , s ( t δ ) E ˜ L O * ( t δ ) e j 2 π Δ ν ( t δ ) + E ˜ O N , s ( t δ ) E ˜ L O * ( t δ ) e j 2 π Δ ν ( t δ ) } + 1 2 i = 1 N c h R S I G , i N [ | E ˜ S I G , i ( t δ ) | 2 + | E ˜ O N , i ( t δ ) | 2 + 2 R e { E ˜ S I G , i ( t δ ) E ˜ O N , i * ( t δ ) } ] + R L O , s N 1 2 | E ˜ L O ( t δ ) | 2 + i s h N + i t h N ,
Δ i ( t ) = i P ( t ) h P ( t ) i N ( t ) h N ( t ) = Δ i L O S I G ( t ) + Δ i L O O N ( t ) + Δ i S I G S I G ( t ) + Δ i O N O N ( t ) + Δ i S I G O N ( t ) + Δ i L O L O ( t ) + Δ i s h ( t ) + Δ i t h ( t ) ,
Γ ¯ Δ i ( τ ) = Δ i ( t ) Δ i ( t τ ) ¯ = Γ ¯ Δ i L O S I G ( τ ) + Γ ¯ Δ i L O O N ( τ ) + Γ ¯ Δ i S I G S I G ( τ ) + Γ ¯ Δ i O N O N ( τ ) + Γ ¯ Δ i S I G O N ( τ ) + Γ ¯ Δ i L O L O ( τ ) + Γ ¯ Δ i s h ( τ ) + Γ ¯ Δ i t h ( τ ) ,
S Δ i ( f ) = FT  { Γ ¯ Δ i ( τ ) } = S Δ i L O S I G ( f ) + S Δ i L O O N ( f ) + S Δ i S I G S I G ( f ) + S Δ i O N O N ( f ) + S Δ i S I G O N ( f ) + S Δ i L O L O ( f ) + S Δ i s h ( f ) + S Δ i t h ( f ) S Δ i L O S I G ( f ) + S Δ i L O O N ( f ) + S Δ i S I G S I G ( f ) + S Δ i s h ( f ) + S Δ i t h ( f ) ,
S N R = S Δ i L O S I G ( f ) | H r x D S P ( f ) | 2 d f [ S Δ i L O O N ( f ) + S Δ i S I G S I G ( f ) + S Δ i s h ( f ) + S Δ i t h ( f ) ] A C | H r x D S P ( f ) | 2 d f = P Δ i L O S I G σ Δ i L O O N 2 + σ Δ i S I G S I G 2 + σ Δ i s h 2 + σ Δ i t h 2 ,
S N R = 2 P L O P S I G 2 c 1 P L O P O N + c 2 ( σ Δ i s h 2 + σ Δ i t h 2 ) + C M R R ¯ S I G N c h β P S I G 2 ,
c 1 = B c h / 2 B c h / 2 | H P ( f ) R S I G , s P R L O , s P + H N ( f ) e j 2 π f δ R S I G , s N R L O , s N | 2 | H r x D S P ( f ) | 2 d f / B c h | H P ( f ) R S I G , s P R L O , s P + H N ( f ) e j 2 π f δ R S I G , s N R L O , s N | 2 | G ( f ) | 2 | H r x D S P ( f ) | 2 d f / | G ( f ) | 2 d f ,
c 2 = 1 | H P ( f ) R S I G , s P R L O , s P + H N ( f ) e j 2 π f δ R S I G , s N R L O , s N | 2 | G ( f ) | 2 | H r x D S P ( f ) | 2 d f / | G ( f ) | 2 d f ,
C M R R ¯ S I G = | H P ( f ) R S I G P H N ( f ) e j 2 π f δ R S I G N | 2 S | E ˜ S I G | 2 A C ( f ) | H r x D S P ( f ) | 2 d f / S | E ˜ S I G | 2 A C ( f ) | H a v ( f ) | 2 | H r x D S P ( f ) | 2 d f | H P ( f ) R S I G P R L O P + H N ( f ) e j 2 π f δ R S I G N R L O N | 2 | G ( f ) | 2 | H r x D S P ( f ) | 2 d f / | G ( f ) | 2 d f ,
S N R = 2 2 c 1 O S N R 1 + C M R R S I G ¯ N c h β L S R 1 ,
S N R = 2 2 c 1 O S N R 1 + c 2 4 e R L O a v Δ f P S I G + c 2 i T I A 2 Δ f P L O P S I G ,
I D C = R L O P L O + N c h R S I G P S I G ,
I A C p p d = 8 R L O P L O P A P R R S I G P S I G ,
S N R = P L O P S I G α 1 P L O P S I G + α 2 N l o o p s P L O P S I G + α 3 + α 4 ( P L O + N c h P S I G ) + α 5 N c h P S I G 2 ,
α = [ α 1 α 5 ] ,     y = [ P L O 1 P S I G 1 / S N R 1 P L O m P S I G m / S N R m ] ,     X = [ P L O 1 P S I G 1 P L O m P S I G m N l o o p s 1 P L O 1 P S I G 1 N l o o p s m P L O m P S I G m 1 1 P L O 1 + N c h 1 P S I G 1 P L O m + N c h m P S I G m N c h 1 P S I G 1 2 N c h m P S I G m 2 ] ,
α ^ = ( X T X ) 1 X T y ,
Γ ¯ Δ i L O S I G ( τ ) = Δ i L O S I G ( t ) Δ i L O S I G ( t τ ) ¯ = R S I G , s P R L O , s P Re { E ˜ S I G , s ( t ) E ˜ L O * ( t ) e j 2 π Δ ν t } Re { E ˜ S I G , s ( t τ ) E ˜ L O * ( t τ ) e j 2 π Δ ν ( t τ ) } ¯ + R S I G , s N R L O , s N Re { E ˜ S I G , s ( t δ ) E ˜ L O * ( t δ ) e j 2 π Δ ν ( t δ ) } Re { E ˜ S I G , s ( t τ δ ) E ˜ L O * ( t τ δ ) e j 2 π Δ ν ( t τ δ ) } ¯ + R S I G , s P R L O , s P R S I G , s N R L O , s N Re { E ˜ S I G , s ( t ) E ˜ L O * ( t ) e j 2 π Δ ν t } Re { E ˜ S I G , s ( t τ δ ) E ˜ L O * ( t τ δ ) e j 2 π Δ ν ( t τ δ ) } ¯ + R S I G , s P R L O , s P R S I G , s N R L O , s N Re { E ˜ S I G , s ( t δ ) E ˜ L O * ( t δ ) e j 2 π Δ ν ( t δ ) } Re { E ˜ S I G , s ( t τ ) E ˜ L O * ( t τ ) e j 2 π Δ ν ( t τ ) } ¯ .
Γ ¯ Δ i L O S I G ( τ ) = 1 2 R S I G , s P R L O , s P Re { E ˜ S I G , s ( t ) E ˜ S I G , s * ( t τ ) E ˜ L O * ( t ) E ˜ L O ( t τ ) ¯ e j 2 π Δ ν τ } + 1 2 R S I G , s N R L O , s N Re { E ˜ S I G , s ( t δ ) E ˜ S I G , s * ( t τ δ ) E ˜ L O * ( t δ ) E ˜ L O ( t τ δ ) ¯ e j 2 π Δ ν τ } + 1 2 R S I G , s P R L O , s P R S I G , s N R L O , s N Re { E ˜ S I G , s ( t ) E ˜ S I G , s * ( t τ δ ) E ˜ L O * ( t ) E ˜ L O ( t τ δ ) ¯ e j 2 π Δ ν ( τ + δ ) } + 1 2 R S I G , s P R L O , s P R S I G , s N R L O , s N Re { E ˜ S I G , s ( t δ ) E ˜ S I G , s * ( t τ ) E ˜ L O * ( t δ ) E ˜ L O ( t τ ) ¯ e j 2 π Δ ν ( τ δ ) } .
Γ ¯ Δ i L O S I G ( τ ) = 1 2 R S I G , s P R L O , s P R e { Γ ¯ E ˜ S I G ( τ ) Γ E ˜ L O * ( τ ) e j 2 π Δ ν τ } + 1 2 R S I G , s N R L O , s N R e { Γ ¯ E ˜ S I G ( τ ) Γ E ˜ L O * ( τ ) e j 2 π Δ ν τ } + 1 2 R S I G , s P R L O , s P R S I G , s N R L O , s N R e { Γ ¯ E ˜ S I G ( τ + δ ) Γ E ˜ L O * ( τ + δ ) e j 2 π Δ ν ( τ + δ ) } + 1 2 R S I G , s P R L O , s P R S I G , s N R L O , s N R e { Γ ¯ E ˜ S I G ( τ δ ) Γ E ˜ L O * ( τ δ ) e j 2 π Δ ν ( τ δ ) } .
E ˜ S I G , s ( t δ ) E ˜ S I G , s * ( t τ δ ) E ˜ L O * ( t δ ) E ˜ L O ( t τ δ ) ¯ = E ˜ S I G , s ( t δ ) E ˜ S I G , s * ( t τ δ ) ¯ E ˜ L O * ( t δ ) E ˜ L O ( t τ δ ) = Γ E ˜ S I G ( t δ , τ ) ¯ Γ E ˜ L O * ( τ ) = Γ ¯ E ˜ S I G ( τ ) Γ E ˜ L O * ( τ ) ,
E ˜ S I G , s ( t ) E ˜ S I G , s * ( t τ δ ) E ˜ L O * ( t ) E ˜ L O ( t τ δ ) ¯ = E ˜ S I G , s ( t ) E ˜ S I G , s * ( t τ δ ) ¯ E ˜ L O * ( t ) E ˜ L O ( t τ δ ) = Γ E ˜ S I G ( t , τ + δ ) ¯ Γ E ˜ L O * ( τ + δ ) = Γ ¯ E ˜ S I G ( τ + δ ) Γ E ˜ L O * ( τ + δ ) ,
E ˜ S I G , s ( t δ ) E ˜ S I G , s * ( t τ ) E ˜ L O * ( t δ ) E ˜ L O ( t τ ) ¯ = E ˜ S I G , s ( t δ ) E ˜ S I G , s * ( t τ ) ¯ E ˜ L O * ( t δ ) E ˜ L O ( t τ ) = Γ E ˜ S I G ( t δ , τ δ ) ¯ Γ E ˜ L O * ( τ δ ) = Γ ¯ E ˜ S I G ( τ δ ) Γ E ˜ L O * ( τ δ ) ,
Γ ¯ Δ i L O S I G ( τ ) = P L O ( R S I G , s P R L O , s P + R S I G , s N R L O , s N ) R e { Γ ¯ E ˜ S I G ( τ ) e j 2 π Δ ν τ } + P L O R S I G , s P R L O , s P R S I G , s N R L O , s N . [ R e { Γ ¯ E ˜ S I G ( τ + δ ) e j 2 π Δ ν ( τ + δ ) + Γ ¯ E ˜ S I G ( τ δ ) e j 2 π Δ ν ( τ δ ) } ] .
S Δ i L O S I G ( f ) = FT { Γ ¯ Δ i L O S I G ( τ ) } = P L O | R S I G , s P R L O , s P + e j 2 π f δ R S I G , s N R L O , s N | 2 1 2 [ S E ˜ S I G ( f Δ ν ) + S E ˜ S I G ( ( f + Δ ν ) ) ] .
Γ ¯ Δ i L O S I G ( τ ) = P L O R S I G , s P R L O , s P ( R e { Γ ¯ E ˜ S I G ( τ ) e j 2 π Δ ν τ } Γ h P ( τ ) ) + P L O R S I G , s N R L O , s N ( R e { Γ ¯ E ˜ S I G ( τ ) e j 2 π Δ ν τ } Γ h N ( τ ) ) + P L O R S I G , s P R L O , s P R S I G , s N R L O , s N ( R e { Γ ¯ E ˜ S I G ( τ + δ ) e j 2 π Δ ν ( τ + δ ) } Γ h P h N ( τ ) ) + P L O R S I G , s P R L O , s P R S I G , s N R L O , s N ( R e { Γ ¯ E ˜ S I G ( τ δ ) e j 2 π Δ ν ( τ δ ) } Γ h N h P ( τ ) ) ,
Γ h P ( τ ) = h P ( τ ) h P * ( τ ) ,    Γ h N ( τ ) = h N ( τ ) h N * ( τ ) , Γ h P h N ( τ ) = h P ( τ ) h N * ( τ ) ,    Γ h N h P ( τ ) = h N ( τ ) h P * ( τ ) .
S Δ i L O S I G ( f ) = P L O | H P ( f ) R S I G , s P R L O , s P + e j 2 π f δ H N ( f ) R S I G , s N R L O , s N | 2 1 2 [ S E ˜ S I G ( f Δ ν ) + S E ˜ S I G ( ( f + Δ ν ) ) ] ,
E ˜ S I G ( t ) = n = ( a X n + b Y n ) h ( t n T ) ,
J = [ a b b * a * ] .
Γ ¯ E ˜ S I G ( τ ) = E ˜ S I G ( t ) E ˜ S I G * ( t τ ) ¯ = | a | 2 n = m = X n X m * h ( t n T ) h * ( t τ m T ) ¯ + | b | 2 n = m = Y n Y m * h ( t n T ) h * ( t τ m T ) ¯ + a b * n = m = X n Y m * h ( t n T ) h * ( t τ m T ) ¯ + a * b n = m = X n * Y m h ( t n T ) h * ( t τ m T ) ¯ .
X n X m * = Y n Y m * = { 1             n = m 0            n m ,
X n Y m * = X n * Y m = 0.
Γ ¯ E ˜ S I G ( τ ) = ( | a | 2 + | b | 2 ) n = h ( t n T ) h * ( t τ n T ) ¯ = 1 T n = T / 2 T / 2 h ( t n T ) h * ( t τ n T ) d t = 1 T h ( t ) h * ( t τ ) d t = 1 T Γ h ( τ ) ,
S E ˜ S I G ( f ) = 1 T | H ( f ) | 2 = 1 T | G ( f ) | 2 ,
P Δ i L O S I G = P L O | H P ( f ) R S I G , s P R L O , s P + H N ( f ) e j 2 π f δ R S I G , s N R L O , s N | 2 1 T | G ( f ) | 2 | H r x D S P ( f ) | 2 d f = 2 P L O P S I G | H P ( f ) R S I G , s P R L O , s P + H N ( f ) e j 2 π f δ R S I G , s N R L O , s N | 2 | G ( f ) | 2 | H r x D S P ( f ) | 2 d f | G ( f ) | 2 d f ,
S Δ i L O O N ( f ) = P L O | H P ( f ) R S I G , s P R L O , s P + H N ( f ) e j 2 π f δ R S I G , s N R L O , s N | 2 1 2 [ S E ˜ O N ( f Δ ν ) + S E ˜ O N ( ( f + Δ ν ) ) ] ,
S E ˜ O N ( f ) = { 2 N o           | f | B c h / 2 0                | f | > B c h / 2 ,
σ Δ i L O O N 2 = 2 P L O P O N B c h / 2 B c h / 2 | H P ( f ) R S I G , s P R L O , s P + H N ( f ) e j 2 π f δ R S I G , s N R L O , s N | 2 | H r x D S P ( f ) | 2 d f B c h .
Γ ¯ Δ i S I G S I G ( τ ) = Δ i S I G S I G ( t ) Δ i S I G S I G ( t τ ) ¯ = 1 4 [ i = 1 N c h j = 1 N c h R S I G , i P R S I G , j P | E ˜ S I G , i ( t ) | 2 | E ˜ S I G , j ( t τ ) | 2 ¯ + i = 1 N c h j = 1 N c h R S I G , i N R S I G , j N | E ˜ S I G , i ( t δ ) | 2 | E ˜ S I G , j ( t τ δ ) | 2 ¯ i = 1 N c h j = 1 N c h R S I G , i P R S I G , j N | E ˜ S I G , i ( t ) | 2 | E ˜ S I G , j ( t τ δ ) | 2 ¯ i = 1 N c h j = 1 N c h R S I G , i N R S I G , j P | E ˜ S I G , i ( t δ ) | 2 | E ˜ S I G , j ( t τ ) | 2 ¯ ] .
Γ ¯ Δ i S I G S I G ( τ ) = i = 1 N c h j = 1 j i N c h ( R S I G , i P R S I G , j P + R S I G , i N R S I G , j N 2 R S I G , i P R S I G , j N ) P S I G , i P S I G , j + 1 4 i = 1 N c h R S I G , i P 2 | E ˜ S I G , i ( t ) | 2 | E ˜ S I G , i ( t τ ) | 2 ¯ + 1 4 i = 1 N c h R S I G , i N 2 | E ˜ S I G , i ( t δ ) | 2 | E ˜ S I G , i ( t τ δ ) | 2 ¯ 1 4 i = 1 N c h R S I G , i P R S I G , i N | E ˜ S I G , i ( t ) | 2 | E ˜ S I G , i ( t τ δ ) | 2 ¯ 1 4 i = 1 N c h R S I G , i P R S I G , i N | E ˜ S I G , i ( t δ ) | 2 | E ˜ S I G , i ( t τ ) | 2 ¯ = i = 1 N c h j = 1 j i N c h ( R S I G , i P R S I G , j P + R S I G , i N R S I G , j N 2 R S I G , i P R S I G , j N ) P S I G , i P S I G , j + 1 4 i = 1 N c h R S I G , i P 2 Γ ¯ | E ˜ S I G , i | 2 ( τ ) + 1 4 i = 1 N c h R S I G , i N 2 Γ ¯ | E ˜ S I G , i | 2 ( τ ) 1 4 i = 1 N c h R S I G , i P R S I G , i N Γ ¯ | E ˜ S I G , i | 2 ( τ + δ ) 1 4 i = 1 N c h R S I G , i P R S I G , i N Γ ¯ | E ˜ S I G , i | 2 ( τ δ ) ,
S Δ i S I G S I G ( f ) = δ ( f ) i = 1 N c h j = 1 j i N c h ( R S I G , i P R S I G , j P + R S I G , i N R S I G , j N 2 R S I G , i P R S I G , j N ) P S I G , i P S I G , j + 1 4 i = 1 N c h | R S I G , i P e j 2 π f δ R S I G , i N | 2 S | E ˜ S I G , i | 2 ( f ) .
S Δ i S I G S I G ( f ) = δ ( f ) i = 1 N c h j = 1 j i N c h ( R S I G , i P R S I G , j P + R S I G , i N R S I G , j N 2 R S I G , i P R S I G , j N ) P S I G , i P S I G , j + 1 4 i = 1 N c h | H P ( f ) R S I G , i P H N ( f ) e j 2 π f δ R S I G , i N | 2 S | E ˜ S I G , i | 2 ( f ) ,
Γ ¯ | E ˜ S I G | 2 ( τ ) = E ˜ S I G ( t ) E ˜ S I G * ( t ) E ˜ S I G ( t τ ) E ˜ S I G * ( t τ ) ¯ = n = m = l = k = ( a X n + b Y n ) ( a X m + b Y m ) * ( a X l + b Y l ) ( a X k + b Y k ) * h ( t n T ) h * ( t m T ) h ( t τ l T ) h * ( t τ k T ) ¯ ,
( a X n + b Y n ) ( a X m + b Y m ) * ( a X l + b Y l ) ( a X k + b Y k ) * = | a | 4 X n X m * X l X k * + | b | 4 Y n Y m * Y l Y k * + | a | 2 | b | 2 X n X m * Y l Y k * + | a | 2 | b | 2 Y n Y m * X l X k * + | a | 2 | b | 2 Y n X m * X l Y k * + | a | 2 | b | 2 X n Y m * Y l X k * + terms with vanishing statistical averages,
X n X m * X l X k * = Y n Y m * Y l Y k * = { 1             n = m & l = k 1            n = k & m = l & n m 0           otherwise ,
X n X m * Y l Y k * = Y n Y m * X l X k * = { 1             n = m & l = k 0           otherwise ,
Y n X m * X l Y k * = X n Y m * Y l X k * = { 1             n = k & l = m 0           otherwise .
Γ ¯ | E ˜ S I G | 2 ( τ ) = ( | a | 4 + | b | 4 ) [ n = l = | h ( t n T ) | 2 | h ( t τ l T ) | 2 ¯ + n = m = n m h ( t n T ) h * ( t m T ) h ( t τ m T ) h * ( t τ n T ) ¯ ] + 2 | a | 2 | b | 2 [ n = l = | h ( t n T ) | 2 | h ( t τ l T ) | 2 ¯ + n = m = h ( t n T ) h * ( t m T ) h ( t τ m T ) h * ( t τ n T ) ¯ ] = 1 T ( | a | 4 + | b | 4 + 2 | a | 2 | b | 2 ) l = Γ | h | 2 ( τ l T ) + 1 T ( | a | 4 + | b | 4 ) k = k 0 C h ( τ , k ) + 1 T 2 | a | 2 | b | 2 k = C h ( τ , k ) ,
n = l = | h ( t n T ) | 2 | h ( t τ l T ) | 2 ¯ = 1 T n = l = T / 2 T / 2 | h ( t n T ) | 2 | h ( t τ l T ) | 2 d t = 1 T n = T / 2 T / 2 | h ( t n T ) | 2 l = | h ( t τ l T ) | 2 d t      ,let t - n T = z = 1 T n = n T T / 2 n T + T / 2 | h ( z ) | 2 l = | h ( z τ ( l n ) T ) | 2 d z = 1 T l = | h ( z ) | 2 | h ( z τ l T ) | 2 d z = 1 T l = Γ | h | 2 ( τ l T ) ,           where      Γ | h | 2 ( τ ) = | h ( t ) | 2 | h ( t τ ) | 2 d t
n = m = n m h ( t n T ) h * ( t m T ) h ( t τ m T ) h * ( t τ n T ) ¯ = 1 T n = m = n m T / 2 T / 2 h ( t n T ) h * ( t m T ) h ( t τ m T ) h * ( t τ n T ) d t = 1 T n = T / 2 T / 2 h ( t n T ) h * ( t τ n T ) [ m = n m h * ( t m T ) h ( t τ m T ) ] d t    ,let   t - n T = z = 1 T n = n T T / 2 n T + T / 2 h ( z ) h * ( z τ ) [ m = n m h * ( z ( m n ) T ) h ( z τ ( m n ) T ) ] d z = 1 T k = k 0 h ( t ) h * ( t τ ) h * ( t k T ) h ( t τ k T ) d t = 1 T k = k 0 C h ( τ , k ) ,          where   C h ( τ , k ) =   h ( t ) h * ( t τ ) h * ( t k T ) h ( t τ k T ) d t    
S | E ˜ S I G | 2 ( f ) = 1 T 2 | F T { h ( t ) h * ( t ) } | 2 l = δ ( f l T ) + 1 T k = | F T { h ( t ) h * ( t k T ) } | 2 1 T ( | a | 4 + | b | 4 ) | F T { h ( t ) h * ( t ) } | 2 = 1 T 2 | H ( f ) H * ( f ) | 2 l = δ ( f l T ) + 1 T k = | H ( f ) ( e j 2 π f k T H * ( f ) ) | 2 1 T ( | a | 4 + | b | 4 ) | H ( f ) H * ( f ) | 2 ,
FT { Γ | h | 2 ( τ ) } = | FT { h ( t ) h * ( t ) } | 2 = | H ( f ) H * ( f ) | 2 ,
FT { C h ( τ , k ) } =   | FT { h ( t ) h * ( t k T ) } | 2 = | H ( f ) ( e j 2 π f k T H * ( f ) ) | 2 .
S Δ i S I G S I G A C ( f ) = 1 4 i = 1 N c h | H P ( f ) R S I G , i P H N ( f ) e j 2 π f δ R S I G , i N | 2 S | E ˜ S I G , i | 2 A C ( f ) ,
S | E ˜ S I G , i | 2 A C ( f ) = 1 T k = | F T { h i ( t ) h i * ( t k T ) } | 2 1 T ( | a i | 4 + | b i | 4 ) | F T { h i ( t ) h i * ( t ) } | 2 ,
σ Δ i S I G S I G 2 = 1 4 i = 1 N c h | H P ( f ) R S I G , i P H N ( f ) e j 2 π f δ R S I G , i N | 2 S | E ˜ S I G , i | 2 A C ( f ) | H r x D S P ( f ) | 2 d f = 1 4 N c h σ | E ˜ S I G , s e | 2 2 | H P ( f ) R S I G P H N ( f ) e j 2 π f δ R S I G N | 2 S | E ˜ S I G | 2 A C ( f ) | H r x D S P ( f ) | 2 d f S | E ˜ S I G | 2 A C ( f ) | H a v ( f ) | 2 | H r x D S P ( f ) | 2 d f ,
σ Δ i s h 2 = 2 e [ ( R L O N + R L O P ) P L O + N c h ( R S I G N + R S I G P ) P S I G ] Δ f 4 e [ R L O a v P L O + N c h R S I G a v P S I G ] Δ f ,
σ Δ i t h 2 = ( i T I A ) 2 Δ f ,
S Δ i L O L O ( f ) = | H P ( f ) R L O P H N ( f ) e j 2 π f δ R L O N | 2 P L O 2 ( δ ( f ) + R I N ( f ) ) ,
S Δ i S I G O N ( f ) = 1 4 | H P ( f ) R S I G P H N ( f ) e j 2 π f δ R S I G N | 2 N c h ( S E ˜ S I G ( f ) S E ˜ O N ( f ) + S E ˜ S I G ( f ) S E ˜ O N ( f ) ) ,
S Δ i O N O N ( f ) = | H P ( f ) R S I G P H N ( f ) e j 2 π f δ R S I G N | 2 ( ( N o B o ) 2 δ ( f ) + N o 2 B o t r i ( f B o ) ) ,
R I N ( f ) = FT { δ P L O ( t ) δ P L O ( t τ ) / P L O 2 } ,
t r i ( f B o ) = { 1 | f | B o              | f | B o 0                      | f | > B o ,
β = 1 4 S | E ˜ S I G | 2 A C ( f ) | H a v ( f ) | 2 | H r x D S P ( f ) | 2 d f P S I G 2 ,

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