Abstract

The recently-proposed multi-dimensional digital predistortion (DPD) technique is experimentally investigated in terms of nonlinearity order, memory length, oversampling rate, digital-to-analog conversion resolution, carrier frequency dependence and RF input power tolerance, in both directly-modulated and externally-modulated multi-band radio-over-fiber (RoF) systems. Similar characteristics of the multi-dimensional digital predistorter are identified in directly-modulated and externally-modulated RoF systems. The experimental results suggest implementing a memory-free multi-dimensional digital predistorter involving nonlinearity orders up to 5 at 2 × oversampling rate for practical multi-band RoF systems. Using the suggested parameters, the multi-dimensional DPD is able to improve the RF input power tolerance by greater than 3dB for each band in a two-band RoF system, indicating an enhancement of RF power transmitting efficiency.

© 2014 Optical Society of America

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References

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  1. K. Andersson, C. Åhlund, “Optimized access network selection in a combined WLAN/LTE environment,” Wirel. Pers. Commun. 61(4), 739–751 (2011).
    [CrossRef]
  2. M. J. Crisp, S. Li, A. Wonfor, R. V. Penty, and I. H. White, “Demonstration of a radio over fiber distributed antenna network for combined in-building WLAN and 3G coverage,” Optical Fiber Communication Conference 2007, JTh81 (2007).
  3. S. Ghafoor, L. Hanzo, “Radio-over-fiber transmission for distributed antennas radio-over-fiber transmission for distributed antennas,” IEEE Commun. Lett. 15(12), 1368–1371 (2011).
    [CrossRef]
  4. D. Waken, A. Nkansah, N. J. Gomes, “Radio over fiber link design for next generation wireless systems,” J. Lightwave Technol. 28(16), 2456–2464 (2010).
    [CrossRef]
  5. S. Fu, W. D. Zhong, P. Shum, Y. J. Wen, “Simultaneous multichannel photonic up-conversion based on nonlinear polarization rotation of an SOA for radio-over-fiber system,” IEEE Photon. Technol. Lett. 21(9), 563–565 (2009).
    [CrossRef]
  6. J. Zhou, S. Fu, F. Luan, J. H. Wong, S. Aditya, P. Shum, K. E. K. Lee, “Tunable multi-tap bandpass microwave photonic filter using a windowed Fabry-Perot filber-based multi-wavelength tunable laser,” J. Lightwave Technol. 29(22), 3381–3386 (2011).
    [CrossRef]
  7. X. N. Fernando, A. B. Sesay, “Higher order adaptive filter based predistortion for nonlinear distortion compensation of radio over fiber links,” Proceedings of the International Conference on Communications 2000, 367–371 (2000).
    [CrossRef]
  8. X. N. Fernando, A. B. Sesay, “Adaptive asymmetric linearization of microwave fiber optic links for wireless access,” IEEE Trans. Vehicular Technol. 51(6), 1576–1586 (2002).
    [CrossRef]
  9. K. Hayasaka, T. Higashino, K. Tsukamoto, and S. Komaki, “A theoretical estimation of IMD on heterogeneous OFDM service over SCM RoF link,” International Topical Meeting on & Microwave Photonics Conference 2011, 328–330 (2011).
    [CrossRef]
  10. A. Ferreira, T. Silveira, D. Fonseca, R. Ribeiro, P. Monteiro, “Highly linear integrated optical transmitter for subcarrier multiplexed systems,” IEEE Photon. Technol. Lett. 21(7), 438–440 (2009).
    [CrossRef]
  11. Y. Pei, K. Xu, J. Li, A. Zhang, Y. Dai, Y. Ji, J. Lin, “Complexity-reduced digital predistortion for subcarrier multiplexed radio over fiber systems transmitting sparse multi-band RF signals,” Opt. Express 21(3), 3708–3714 (2013).
    [CrossRef] [PubMed]
  12. S. A. Bassam, M. Helaoui, F. M. Ghannouchi, “2-D digital predistortion (2-D-DPD) architecture for concurrent dual-band transmitters,” IEEE Trans. Microw. Theory Tech. 59(10), 2547–2553 (2011).
    [CrossRef]
  13. Y. J. Liu, W. Chen, J. Zhou, B. H. Zhou, F. M. Ghannouchi, “Digital predistortion for concurrent dual-band transmitters using 2-D modified memory polynomials,” IEEE Trans. Microw. Theory Tech. 61(1), 281–290 (2013).
    [CrossRef]
  14. L. Ding, G. T. Zhou, Z. Ma, D. R. Morgan, J. S. Kenney, J. Kim, C. R. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun. 52(1), 159–165 (2004).
    [CrossRef]
  15. D. Guo, “Power amplifier and front end module requirements for IEEE 802.11n applications,” High Frequency Electronics (2011).

2013 (2)

Y. Pei, K. Xu, J. Li, A. Zhang, Y. Dai, Y. Ji, J. Lin, “Complexity-reduced digital predistortion for subcarrier multiplexed radio over fiber systems transmitting sparse multi-band RF signals,” Opt. Express 21(3), 3708–3714 (2013).
[CrossRef] [PubMed]

Y. J. Liu, W. Chen, J. Zhou, B. H. Zhou, F. M. Ghannouchi, “Digital predistortion for concurrent dual-band transmitters using 2-D modified memory polynomials,” IEEE Trans. Microw. Theory Tech. 61(1), 281–290 (2013).
[CrossRef]

2011 (4)

S. A. Bassam, M. Helaoui, F. M. Ghannouchi, “2-D digital predistortion (2-D-DPD) architecture for concurrent dual-band transmitters,” IEEE Trans. Microw. Theory Tech. 59(10), 2547–2553 (2011).
[CrossRef]

J. Zhou, S. Fu, F. Luan, J. H. Wong, S. Aditya, P. Shum, K. E. K. Lee, “Tunable multi-tap bandpass microwave photonic filter using a windowed Fabry-Perot filber-based multi-wavelength tunable laser,” J. Lightwave Technol. 29(22), 3381–3386 (2011).
[CrossRef]

K. Andersson, C. Åhlund, “Optimized access network selection in a combined WLAN/LTE environment,” Wirel. Pers. Commun. 61(4), 739–751 (2011).
[CrossRef]

S. Ghafoor, L. Hanzo, “Radio-over-fiber transmission for distributed antennas radio-over-fiber transmission for distributed antennas,” IEEE Commun. Lett. 15(12), 1368–1371 (2011).
[CrossRef]

2010 (1)

2009 (2)

S. Fu, W. D. Zhong, P. Shum, Y. J. Wen, “Simultaneous multichannel photonic up-conversion based on nonlinear polarization rotation of an SOA for radio-over-fiber system,” IEEE Photon. Technol. Lett. 21(9), 563–565 (2009).
[CrossRef]

A. Ferreira, T. Silveira, D. Fonseca, R. Ribeiro, P. Monteiro, “Highly linear integrated optical transmitter for subcarrier multiplexed systems,” IEEE Photon. Technol. Lett. 21(7), 438–440 (2009).
[CrossRef]

2004 (1)

L. Ding, G. T. Zhou, Z. Ma, D. R. Morgan, J. S. Kenney, J. Kim, C. R. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun. 52(1), 159–165 (2004).
[CrossRef]

2002 (1)

X. N. Fernando, A. B. Sesay, “Adaptive asymmetric linearization of microwave fiber optic links for wireless access,” IEEE Trans. Vehicular Technol. 51(6), 1576–1586 (2002).
[CrossRef]

Aditya, S.

Åhlund, C.

K. Andersson, C. Åhlund, “Optimized access network selection in a combined WLAN/LTE environment,” Wirel. Pers. Commun. 61(4), 739–751 (2011).
[CrossRef]

Andersson, K.

K. Andersson, C. Åhlund, “Optimized access network selection in a combined WLAN/LTE environment,” Wirel. Pers. Commun. 61(4), 739–751 (2011).
[CrossRef]

Bassam, S. A.

S. A. Bassam, M. Helaoui, F. M. Ghannouchi, “2-D digital predistortion (2-D-DPD) architecture for concurrent dual-band transmitters,” IEEE Trans. Microw. Theory Tech. 59(10), 2547–2553 (2011).
[CrossRef]

Chen, W.

Y. J. Liu, W. Chen, J. Zhou, B. H. Zhou, F. M. Ghannouchi, “Digital predistortion for concurrent dual-band transmitters using 2-D modified memory polynomials,” IEEE Trans. Microw. Theory Tech. 61(1), 281–290 (2013).
[CrossRef]

Dai, Y.

Ding, L.

L. Ding, G. T. Zhou, Z. Ma, D. R. Morgan, J. S. Kenney, J. Kim, C. R. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun. 52(1), 159–165 (2004).
[CrossRef]

Fernando, X. N.

X. N. Fernando, A. B. Sesay, “Adaptive asymmetric linearization of microwave fiber optic links for wireless access,” IEEE Trans. Vehicular Technol. 51(6), 1576–1586 (2002).
[CrossRef]

X. N. Fernando, A. B. Sesay, “Higher order adaptive filter based predistortion for nonlinear distortion compensation of radio over fiber links,” Proceedings of the International Conference on Communications 2000, 367–371 (2000).
[CrossRef]

Ferreira, A.

A. Ferreira, T. Silveira, D. Fonseca, R. Ribeiro, P. Monteiro, “Highly linear integrated optical transmitter for subcarrier multiplexed systems,” IEEE Photon. Technol. Lett. 21(7), 438–440 (2009).
[CrossRef]

Fonseca, D.

A. Ferreira, T. Silveira, D. Fonseca, R. Ribeiro, P. Monteiro, “Highly linear integrated optical transmitter for subcarrier multiplexed systems,” IEEE Photon. Technol. Lett. 21(7), 438–440 (2009).
[CrossRef]

Fu, S.

J. Zhou, S. Fu, F. Luan, J. H. Wong, S. Aditya, P. Shum, K. E. K. Lee, “Tunable multi-tap bandpass microwave photonic filter using a windowed Fabry-Perot filber-based multi-wavelength tunable laser,” J. Lightwave Technol. 29(22), 3381–3386 (2011).
[CrossRef]

S. Fu, W. D. Zhong, P. Shum, Y. J. Wen, “Simultaneous multichannel photonic up-conversion based on nonlinear polarization rotation of an SOA for radio-over-fiber system,” IEEE Photon. Technol. Lett. 21(9), 563–565 (2009).
[CrossRef]

Ghafoor, S.

S. Ghafoor, L. Hanzo, “Radio-over-fiber transmission for distributed antennas radio-over-fiber transmission for distributed antennas,” IEEE Commun. Lett. 15(12), 1368–1371 (2011).
[CrossRef]

Ghannouchi, F. M.

Y. J. Liu, W. Chen, J. Zhou, B. H. Zhou, F. M. Ghannouchi, “Digital predistortion for concurrent dual-band transmitters using 2-D modified memory polynomials,” IEEE Trans. Microw. Theory Tech. 61(1), 281–290 (2013).
[CrossRef]

S. A. Bassam, M. Helaoui, F. M. Ghannouchi, “2-D digital predistortion (2-D-DPD) architecture for concurrent dual-band transmitters,” IEEE Trans. Microw. Theory Tech. 59(10), 2547–2553 (2011).
[CrossRef]

Giardina, C. R.

L. Ding, G. T. Zhou, Z. Ma, D. R. Morgan, J. S. Kenney, J. Kim, C. R. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun. 52(1), 159–165 (2004).
[CrossRef]

Gomes, N. J.

Hanzo, L.

S. Ghafoor, L. Hanzo, “Radio-over-fiber transmission for distributed antennas radio-over-fiber transmission for distributed antennas,” IEEE Commun. Lett. 15(12), 1368–1371 (2011).
[CrossRef]

Helaoui, M.

S. A. Bassam, M. Helaoui, F. M. Ghannouchi, “2-D digital predistortion (2-D-DPD) architecture for concurrent dual-band transmitters,” IEEE Trans. Microw. Theory Tech. 59(10), 2547–2553 (2011).
[CrossRef]

Ji, Y.

Kenney, J. S.

L. Ding, G. T. Zhou, Z. Ma, D. R. Morgan, J. S. Kenney, J. Kim, C. R. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun. 52(1), 159–165 (2004).
[CrossRef]

Kim, J.

L. Ding, G. T. Zhou, Z. Ma, D. R. Morgan, J. S. Kenney, J. Kim, C. R. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun. 52(1), 159–165 (2004).
[CrossRef]

Lee, K. E. K.

Li, J.

Lin, J.

Liu, Y. J.

Y. J. Liu, W. Chen, J. Zhou, B. H. Zhou, F. M. Ghannouchi, “Digital predistortion for concurrent dual-band transmitters using 2-D modified memory polynomials,” IEEE Trans. Microw. Theory Tech. 61(1), 281–290 (2013).
[CrossRef]

Luan, F.

Ma, Z.

L. Ding, G. T. Zhou, Z. Ma, D. R. Morgan, J. S. Kenney, J. Kim, C. R. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun. 52(1), 159–165 (2004).
[CrossRef]

Monteiro, P.

A. Ferreira, T. Silveira, D. Fonseca, R. Ribeiro, P. Monteiro, “Highly linear integrated optical transmitter for subcarrier multiplexed systems,” IEEE Photon. Technol. Lett. 21(7), 438–440 (2009).
[CrossRef]

Morgan, D. R.

L. Ding, G. T. Zhou, Z. Ma, D. R. Morgan, J. S. Kenney, J. Kim, C. R. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun. 52(1), 159–165 (2004).
[CrossRef]

Nkansah, A.

Pei, Y.

Ribeiro, R.

A. Ferreira, T. Silveira, D. Fonseca, R. Ribeiro, P. Monteiro, “Highly linear integrated optical transmitter for subcarrier multiplexed systems,” IEEE Photon. Technol. Lett. 21(7), 438–440 (2009).
[CrossRef]

Sesay, A. B.

X. N. Fernando, A. B. Sesay, “Adaptive asymmetric linearization of microwave fiber optic links for wireless access,” IEEE Trans. Vehicular Technol. 51(6), 1576–1586 (2002).
[CrossRef]

X. N. Fernando, A. B. Sesay, “Higher order adaptive filter based predistortion for nonlinear distortion compensation of radio over fiber links,” Proceedings of the International Conference on Communications 2000, 367–371 (2000).
[CrossRef]

Shum, P.

J. Zhou, S. Fu, F. Luan, J. H. Wong, S. Aditya, P. Shum, K. E. K. Lee, “Tunable multi-tap bandpass microwave photonic filter using a windowed Fabry-Perot filber-based multi-wavelength tunable laser,” J. Lightwave Technol. 29(22), 3381–3386 (2011).
[CrossRef]

S. Fu, W. D. Zhong, P. Shum, Y. J. Wen, “Simultaneous multichannel photonic up-conversion based on nonlinear polarization rotation of an SOA for radio-over-fiber system,” IEEE Photon. Technol. Lett. 21(9), 563–565 (2009).
[CrossRef]

Silveira, T.

A. Ferreira, T. Silveira, D. Fonseca, R. Ribeiro, P. Monteiro, “Highly linear integrated optical transmitter for subcarrier multiplexed systems,” IEEE Photon. Technol. Lett. 21(7), 438–440 (2009).
[CrossRef]

Waken, D.

Wen, Y. J.

S. Fu, W. D. Zhong, P. Shum, Y. J. Wen, “Simultaneous multichannel photonic up-conversion based on nonlinear polarization rotation of an SOA for radio-over-fiber system,” IEEE Photon. Technol. Lett. 21(9), 563–565 (2009).
[CrossRef]

Wong, J. H.

Xu, K.

Zhang, A.

Zhong, W. D.

S. Fu, W. D. Zhong, P. Shum, Y. J. Wen, “Simultaneous multichannel photonic up-conversion based on nonlinear polarization rotation of an SOA for radio-over-fiber system,” IEEE Photon. Technol. Lett. 21(9), 563–565 (2009).
[CrossRef]

Zhou, B. H.

Y. J. Liu, W. Chen, J. Zhou, B. H. Zhou, F. M. Ghannouchi, “Digital predistortion for concurrent dual-band transmitters using 2-D modified memory polynomials,” IEEE Trans. Microw. Theory Tech. 61(1), 281–290 (2013).
[CrossRef]

Zhou, G. T.

L. Ding, G. T. Zhou, Z. Ma, D. R. Morgan, J. S. Kenney, J. Kim, C. R. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun. 52(1), 159–165 (2004).
[CrossRef]

Zhou, J.

Y. J. Liu, W. Chen, J. Zhou, B. H. Zhou, F. M. Ghannouchi, “Digital predistortion for concurrent dual-band transmitters using 2-D modified memory polynomials,” IEEE Trans. Microw. Theory Tech. 61(1), 281–290 (2013).
[CrossRef]

J. Zhou, S. Fu, F. Luan, J. H. Wong, S. Aditya, P. Shum, K. E. K. Lee, “Tunable multi-tap bandpass microwave photonic filter using a windowed Fabry-Perot filber-based multi-wavelength tunable laser,” J. Lightwave Technol. 29(22), 3381–3386 (2011).
[CrossRef]

IEEE Commun. Lett. (1)

S. Ghafoor, L. Hanzo, “Radio-over-fiber transmission for distributed antennas radio-over-fiber transmission for distributed antennas,” IEEE Commun. Lett. 15(12), 1368–1371 (2011).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

S. Fu, W. D. Zhong, P. Shum, Y. J. Wen, “Simultaneous multichannel photonic up-conversion based on nonlinear polarization rotation of an SOA for radio-over-fiber system,” IEEE Photon. Technol. Lett. 21(9), 563–565 (2009).
[CrossRef]

A. Ferreira, T. Silveira, D. Fonseca, R. Ribeiro, P. Monteiro, “Highly linear integrated optical transmitter for subcarrier multiplexed systems,” IEEE Photon. Technol. Lett. 21(7), 438–440 (2009).
[CrossRef]

IEEE Trans. Commun. (1)

L. Ding, G. T. Zhou, Z. Ma, D. R. Morgan, J. S. Kenney, J. Kim, C. R. Giardina, “A robust digital baseband predistorter constructed using memory polynomials,” IEEE Trans. Commun. 52(1), 159–165 (2004).
[CrossRef]

IEEE Trans. Microw. Theory Tech. (2)

S. A. Bassam, M. Helaoui, F. M. Ghannouchi, “2-D digital predistortion (2-D-DPD) architecture for concurrent dual-band transmitters,” IEEE Trans. Microw. Theory Tech. 59(10), 2547–2553 (2011).
[CrossRef]

Y. J. Liu, W. Chen, J. Zhou, B. H. Zhou, F. M. Ghannouchi, “Digital predistortion for concurrent dual-band transmitters using 2-D modified memory polynomials,” IEEE Trans. Microw. Theory Tech. 61(1), 281–290 (2013).
[CrossRef]

IEEE Trans. Vehicular Technol. (1)

X. N. Fernando, A. B. Sesay, “Adaptive asymmetric linearization of microwave fiber optic links for wireless access,” IEEE Trans. Vehicular Technol. 51(6), 1576–1586 (2002).
[CrossRef]

J. Lightwave Technol. (2)

Opt. Express (1)

Wirel. Pers. Commun. (1)

K. Andersson, C. Åhlund, “Optimized access network selection in a combined WLAN/LTE environment,” Wirel. Pers. Commun. 61(4), 739–751 (2011).
[CrossRef]

Other (4)

M. J. Crisp, S. Li, A. Wonfor, R. V. Penty, and I. H. White, “Demonstration of a radio over fiber distributed antenna network for combined in-building WLAN and 3G coverage,” Optical Fiber Communication Conference 2007, JTh81 (2007).

X. N. Fernando, A. B. Sesay, “Higher order adaptive filter based predistortion for nonlinear distortion compensation of radio over fiber links,” Proceedings of the International Conference on Communications 2000, 367–371 (2000).
[CrossRef]

K. Hayasaka, T. Higashino, K. Tsukamoto, and S. Komaki, “A theoretical estimation of IMD on heterogeneous OFDM service over SCM RoF link,” International Topical Meeting on & Microwave Photonics Conference 2011, 328–330 (2011).
[CrossRef]

D. Guo, “Power amplifier and front end module requirements for IEEE 802.11n applications,” High Frequency Electronics (2011).

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Figures (13)

Fig. 1
Fig. 1

The indirect learning architecture of the multi-dimensional DPD.

Fig. 2
Fig. 2

Experimental setup for a two-band directly-modulated RoF system.

Fig. 3
Fig. 3

EVM performance of output of directly-modulated RoF link for different sample rates, nonlinearity orders and memory lengths. (a) Band 1, sampling rate = 40 MSa/s; (b) Band 1, sampling rate = 60 MSa/s; (c) Band 1, sampling rate = 80 MSa/s; (d) Band 2, sampling rate = 40 MSa/s; (e) Band 2, sampling rate = 60 MSa/s; (f) Band 2, sampling rate = 80 MSa/s.

Fig. 4
Fig. 4

Power spectra and constellation diagrams after directly-modulated RoF link. (a) power spectra of Band 1 at 2.3 GHz; (b) constellation diagram of Band 1 without DPD; (c) constellation diagram of Band 1 with independent DPD; (d) constellation diagram of Band 1 with multi-dimensional DPD; (e) power spectra of Band 2 at 2.462 GHz; (f) constellation diagram of Band 2 without DPD; (g) constellation diagram of Band 2 with independent DPD; (h) constellation diagram of Band 2 with multi-dimensional DPD.

Fig. 5
Fig. 5

EVM as a function of DAC resolution for directly-modulated RoF systems. (a) Band 1 at 2.3 GHz; (b) Band 2 at 2.462 GHz.

Fig. 6
Fig. 6

EVM performance against RF input power after directly-modulated RoF link. (a) Band 1 at 2.3 GHz; (b) Band 2 at 2.462 GHz.

Fig. 7
Fig. 7

Experimental setup for a two-band externally-modulated RoF system.

Fig. 8
Fig. 8

EVM performance of output of externally-modulated RoF link for different sample rates, nonlinearity orders and memory lengths. (a) Band 1, sampling rate = 40 MSa/s; (b) Band 1, sampling rate = 80 MSa/s; (c) Band 2, sampling rate = 40 MSa/s; (d) Band 2, sampling rate = 80 MSa/s.

Fig. 9
Fig. 9

Power spectra and constellation diagrams after externally-modulated RoF link. (a) power spectra of Band 1 at 2.3 GHz; (b) constellation diagram of Band 1 without DPD; (c) constellation diagram of Band 1 with independent DPD; (d) constellation diagram of Band 1 with multi-dimensional DPD; (e) power spectra of Band 2 at 2.462 GHz; (f) constellation diagram of Band 2 without DPD; (g) constellation diagram of Band 2 with independent DPD; (h) constellation diagram of Band 2 with multi-dimensional DPD.

Fig. 10
Fig. 10

EVM as a function of DAC resolution for externally-modulated RoF systems. (a) Band 1 at 2.3 GHz; (b) Band 2 at 2.462 GHz.

Fig. 11
Fig. 11

EVM performance against RF input power after externally-modulated RoF link. (a) Band 1 at 2.3 GHz; (b) Band 2 at 2.462 GHz.

Fig. 12
Fig. 12

DPD system model for practical RoF link using uplink feedback.

Fig. 13
Fig. 13

DPD system model for practical RoF link using additional PD at central unit.

Tables (6)

Tables Icon

Table 1 Comparison in ACPR and EVM for directly-modulated RoF systems

Tables Icon

Table 2 Tolerance to the RF power difference for directly-modulated RoF systems

Tables Icon

Table 3 The dependency on carrier frequency for directly-modulated RoF systems

Tables Icon

Table 4 Comparison in ACPR and EVM for externaltly-modulated RoF systems

Tables Icon

Table 5 Tolerance to the RF power difference for externally-modulated RoF systems

Tables Icon

Table 6 The dependency on carrier frequency for externally-modulated RoF systems

Equations (3)

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

z i (n)= k=1 K q=0 Q m = k 1 M k a i,k,q ( m k ) x i (nq) | x i (nq) | j=1 f i = j=1 k ± f j k | x j (nq) | .
z i = U i a i .
a i = [ a i,1,0 (1) ,.., a i,1,Q (1) ,.., a i,k,0 (1) ,.., a i,k,Q (1) ,.., a i,k,0 ( M k ) ,.., a i,k,Q ( M k ) ,.., a i,K,Q ( M K ) ] T , U i =[ u i,1,0 (1) ,.., u i,1,Q (1) ,.., u i,k,0 (1) ,.., u i,k,Q (1) ,.., u i,k,0 ( M k ) ,.., u i,k,Q ( M k ) ,.., u i,K,Q ( M K ) ], u i,k,q ( m k ) = [ u i,k,q ( m k ) (0),.., u i,k,q ( m k ) (N1)] T , and u i,k,q ( m k ) (n)= y i (n-q) | y i (n-q) | j=1 f i = j=1 k ± f j k | 1 G j y j (n-q) | .

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