Abstract

A simplified millimeter-wave (mm-wave) radio-over-fiber (RoF) system employing a combination of optical heterodyning in signal generation and radio frequency (RF) self-homodyning in data recovery process is proposed and demonstrated. Three variants of the system are considered in which two independent uncorrelated lasers with a frequency offset equal to the desired mm-wave carrier frequency are used to generate the transmitted signal. Uncorrelated phase noise in the resulting mm-wave signal after photodetection was overcome by using RF self-homodyning in the data recovery process. Theoretical analyses followed by experimental results and simulated characterizations confirm the system’s performance. A key advantage of the system is that it avoids the need for high-speed electro-optic and electronic devices operating at the RF carrier frequency at both the central station and base stations.

© 2012 OSA

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2011 (2)

Y. Li, A. Maedar, L. Fan, A. Nigam, and J. Chou, “Overview of femtocell support in advanced WiMAX systems,” IEEE Commun. Mag. 49(7), 122–130 (2011).
[CrossRef]

A. H. M. Razibul Islam, M. Bakaul, A. Nirmalathas, and G. Town, “Millimeter-wave radio-over-Fiber system based on heterodyned unlocked light sources and self-homodyned RF receiver,” IEEE Photon. Technol. Lett. 23(8), 459–461 (2011).
[CrossRef]

2010 (3)

2009 (4)

2008 (1)

C.-S. Choi, Y. Shoji, and H. Ogawa, “Millimeter-wave fiber-fed wireless access systems based on dense wavelength-division-multiplexing networks,” IEEE Trans. Microw. Theory Tech. 56(1), 232–241 (2008).
[CrossRef]

2007 (2)

P. Gamage, A. Nirmalathas, C. Lim, M. Bakaul, D. Novak, and R. Waterhouse, “Efficient transmission scheme for AWG-based DWDM millimeter-wave fiber-radio systems,” IEEE Photon. Technol. Lett. 19(4), 206–208 (2007).
[CrossRef]

Z. Jia, J. Yu, G. Ellinas, and G. K. Chang, “Key enabling technologies for optical-wireless networks: optical millimeter-wave generation, wavelength resuse and architecture,” J. Lightwave Technol. 25(11), 3452–3471 (2007).
[CrossRef]

2006 (3)

C. Wu and X. Zhang, “Impact of nonlinear distortion in radio over fiber systems with single-sideband and tandem single-sideband subcarrier modulations,” J. Lightwave Technol. 24(5), 2076–2090 (2006).
[CrossRef]

S. Pradhan, G. E. Town, and K. J. Grant, “Dual wavelength DBR fiber laser,” IEEE Photon. Technol. Lett. 18(16), 1741–1743 (2006).
[CrossRef]

L. Chen, H. Wen, and S. Wen, “A radio-over-fiber system with a novel scheme for millimeter-wave generation and wavelength reuse for up-link connection,” IEEE Photon. Technol. Lett. 18(19), 2056–2058 (2006).
[CrossRef]

2005 (1)

A. Wiberg, P. Millan, M. Andres, P. A. Andrekson, and P. O. Hedevkist, “Fiber-optic 40-GHz mm-wave link with 2.5 Gb/s data transmission,” IEEE Photon. Technol. Lett. 17(9), 1938–1940 (2005).
[CrossRef]

2003 (1)

2001 (1)

T. Kuri and K. Kitayama, “Optical heterodyne detection of millimeter-wave-band radio-on-fiber signals with a remote dual-mode local light source,” IEEE Trans. Microw. Theory Tech. 49(10), 2025–2029 (2001).
[CrossRef]

2000 (1)

L. A. Johansson and A. J. Seeds, “Millimeter-wave modulated optical signal generation with high spectral purity and wide-locking bandwidth using a fiber-integrated optical injection phase-lock loop,” IEEE Photon. Technol. Lett. 12(6), 690–692 (2000).
[CrossRef]

1999 (1)

T. Kuri, K. Kitayama, and Y. Ogawa, “Fiber-optic millimeter-wave uplink system incorporating remotely fed 60-GHz-band optical pilot tone,” IEEE Trans. Microw. Theory Tech. 47, 1332–1337 (1999).
[CrossRef]

1998 (1)

R.-P. Braun, G. Grosskopf, D. Rohde, and F. Schmidt, “Low-phase-noise millimeter-wave generation at 64 GHz and data transmission using optical sideband injection locking,” IEEE Photon. Technol. Lett. 10(5), 728–730 (1998).
[CrossRef]

1994 (1)

J. J. O’Reilly and P. M. Lane, “Remote delivery of video services using mm-wave and optics,” J. Lightwave Technol. 12(2), 369–375 (1994).
[CrossRef]

1990 (2)

I. Garrett, D. J. Bond, J. B. Waite, D. S. L. Littis, and G. Jacobsen, “Impact of phase noise in weakly coherent systems: a new and accurate approach,” J. Lightwave Technol. 8(3), 329–337 (1990).
[CrossRef]

J. R. Barry and E. A. Lee, “Performance of coherent optical receivers,” Proc. IEEE 78(8), 1369–1394 (1990).
[CrossRef]

1988 (1)

G. J. Foschini, L. J. Greenstein, and G. Vannucci, “Noncoherent detection of coherent lightwave signals corrupted by phase noise,” IEEE Trans. Commun. 36(3), 306–314 (1988).
[CrossRef]

1987 (1)

I. Garrett and G. Jacobsen, “The effect of laser linewidth on coherent optical receivers with nonsynchronous demodulation,” J. Lightwave Technol. 5(4), 551–560 (1987).
[CrossRef]

Andrekson, P. A.

A. Wiberg, P. Millan, M. Andres, P. A. Andrekson, and P. O. Hedevkist, “Fiber-optic 40-GHz mm-wave link with 2.5 Gb/s data transmission,” IEEE Photon. Technol. Lett. 17(9), 1938–1940 (2005).
[CrossRef]

Andres, M.

A. Wiberg, P. Millan, M. Andres, P. A. Andrekson, and P. O. Hedevkist, “Fiber-optic 40-GHz mm-wave link with 2.5 Gb/s data transmission,” IEEE Photon. Technol. Lett. 17(9), 1938–1940 (2005).
[CrossRef]

Bakaul, M.

A. H. M. Razibul Islam, M. Bakaul, A. Nirmalathas, and G. Town, “Millimeter-wave radio-over-Fiber system based on heterodyned unlocked light sources and self-homodyned RF receiver,” IEEE Photon. Technol. Lett. 23(8), 459–461 (2011).
[CrossRef]

P. Gamage, A. Nirmalathas, C. Lim, M. Bakaul, D. Novak, and R. Waterhouse, “Efficient transmission scheme for AWG-based DWDM millimeter-wave fiber-radio systems,” IEEE Photon. Technol. Lett. 19(4), 206–208 (2007).
[CrossRef]

Barry, J. R.

J. R. Barry and E. A. Lee, “Performance of coherent optical receivers,” Proc. IEEE 78(8), 1369–1394 (1990).
[CrossRef]

Bond, D. J.

I. Garrett, D. J. Bond, J. B. Waite, D. S. L. Littis, and G. Jacobsen, “Impact of phase noise in weakly coherent systems: a new and accurate approach,” J. Lightwave Technol. 8(3), 329–337 (1990).
[CrossRef]

Braun, R.-P.

R.-P. Braun, G. Grosskopf, D. Rohde, and F. Schmidt, “Low-phase-noise millimeter-wave generation at 64 GHz and data transmission using optical sideband injection locking,” IEEE Photon. Technol. Lett. 10(5), 728–730 (1998).
[CrossRef]

Bucholtz, F.

Chang, G. K.

Chen, L.

L. Chen, H. Wen, and S. Wen, “A radio-over-fiber system with a novel scheme for millimeter-wave generation and wavelength reuse for up-link connection,” IEEE Photon. Technol. Lett. 18(19), 2056–2058 (2006).
[CrossRef]

Chien, H. C.

Choi, C.-S.

C.-S. Choi, Y. Shoji, and H. Ogawa, “Millimeter-wave fiber-fed wireless access systems based on dense wavelength-division-multiplexing networks,” IEEE Trans. Microw. Theory Tech. 56(1), 232–241 (2008).
[CrossRef]

Chou, J.

Y. Li, A. Maedar, L. Fan, A. Nigam, and J. Chou, “Overview of femtocell support in advanced WiMAX systems,” IEEE Commun. Mag. 49(7), 122–130 (2011).
[CrossRef]

Chowdhury, A.

Devgan, P.

Dong, Z.

Ellinas, G.

Fan, L.

Y. Li, A. Maedar, L. Fan, A. Nigam, and J. Chou, “Overview of femtocell support in advanced WiMAX systems,” IEEE Commun. Mag. 49(7), 122–130 (2011).
[CrossRef]

Foschini, G. J.

G. J. Foschini, L. J. Greenstein, and G. Vannucci, “Noncoherent detection of coherent lightwave signals corrupted by phase noise,” IEEE Trans. Commun. 36(3), 306–314 (1988).
[CrossRef]

Gamage, P.

P. Gamage, A. Nirmalathas, C. Lim, M. Bakaul, D. Novak, and R. Waterhouse, “Efficient transmission scheme for AWG-based DWDM millimeter-wave fiber-radio systems,” IEEE Photon. Technol. Lett. 19(4), 206–208 (2007).
[CrossRef]

Garrett, I.

I. Garrett, D. J. Bond, J. B. Waite, D. S. L. Littis, and G. Jacobsen, “Impact of phase noise in weakly coherent systems: a new and accurate approach,” J. Lightwave Technol. 8(3), 329–337 (1990).
[CrossRef]

I. Garrett and G. Jacobsen, “The effect of laser linewidth on coherent optical receivers with nonsynchronous demodulation,” J. Lightwave Technol. 5(4), 551–560 (1987).
[CrossRef]

Godinez, M.

Grant, K. J.

S. Pradhan, G. E. Town, and K. J. Grant, “Dual wavelength DBR fiber laser,” IEEE Photon. Technol. Lett. 18(16), 1741–1743 (2006).
[CrossRef]

Greenstein, L. J.

G. J. Foschini, L. J. Greenstein, and G. Vannucci, “Noncoherent detection of coherent lightwave signals corrupted by phase noise,” IEEE Trans. Commun. 36(3), 306–314 (1988).
[CrossRef]

Grosskopf, G.

R.-P. Braun, G. Grosskopf, D. Rohde, and F. Schmidt, “Low-phase-noise millimeter-wave generation at 64 GHz and data transmission using optical sideband injection locking,” IEEE Photon. Technol. Lett. 10(5), 728–730 (1998).
[CrossRef]

Hedevkist, P. O.

A. Wiberg, P. Millan, M. Andres, P. A. Andrekson, and P. O. Hedevkist, “Fiber-optic 40-GHz mm-wave link with 2.5 Gb/s data transmission,” IEEE Photon. Technol. Lett. 17(9), 1938–1940 (2005).
[CrossRef]

Hoekman, M.

Hraimel, B.

Hsueh, Y. T.

Huang, M. F.

Insua, I. G.

Jacobsen, G.

I. Garrett, D. J. Bond, J. B. Waite, D. S. L. Littis, and G. Jacobsen, “Impact of phase noise in weakly coherent systems: a new and accurate approach,” J. Lightwave Technol. 8(3), 329–337 (1990).
[CrossRef]

I. Garrett and G. Jacobsen, “The effect of laser linewidth on coherent optical receivers with nonsynchronous demodulation,” J. Lightwave Technol. 5(4), 551–560 (1987).
[CrossRef]

Jia, Z.

Jian, W.

Johansson, L. A.

L. A. Johansson and A. J. Seeds, “Generation and transmission of millimeter-wave data-modulated optical signals using an optical injection phase-lock loop,” J. Lightwave Technol. 21(2), 511–520 (2003).
[CrossRef]

L. A. Johansson and A. J. Seeds, “Millimeter-wave modulated optical signal generation with high spectral purity and wide-locking bandwidth using a fiber-integrated optical injection phase-lock loop,” IEEE Photon. Technol. Lett. 12(6), 690–692 (2000).
[CrossRef]

Kitayama, K.

T. Kuri and K. Kitayama, “Optical heterodyne detection of millimeter-wave-band radio-on-fiber signals with a remote dual-mode local light source,” IEEE Trans. Microw. Theory Tech. 49(10), 2025–2029 (2001).
[CrossRef]

T. Kuri, K. Kitayama, and Y. Ogawa, “Fiber-optic millimeter-wave uplink system incorporating remotely fed 60-GHz-band optical pilot tone,” IEEE Trans. Microw. Theory Tech. 47, 1332–1337 (1999).
[CrossRef]

Kuri, T.

T. Kuri and K. Kitayama, “Optical heterodyne detection of millimeter-wave-band radio-on-fiber signals with a remote dual-mode local light source,” IEEE Trans. Microw. Theory Tech. 49(10), 2025–2029 (2001).
[CrossRef]

T. Kuri, K. Kitayama, and Y. Ogawa, “Fiber-optic millimeter-wave uplink system incorporating remotely fed 60-GHz-band optical pilot tone,” IEEE Trans. Microw. Theory Tech. 47, 1332–1337 (1999).
[CrossRef]

Lane, P. M.

J. J. O’Reilly and P. M. Lane, “Remote delivery of video services using mm-wave and optics,” J. Lightwave Technol. 12(2), 369–375 (1994).
[CrossRef]

Lee, E. A.

J. R. Barry and E. A. Lee, “Performance of coherent optical receivers,” Proc. IEEE 78(8), 1369–1394 (1990).
[CrossRef]

Leinse, A.

Li, Y.

Y. Li, A. Maedar, L. Fan, A. Nigam, and J. Chou, “Overview of femtocell support in advanced WiMAX systems,” IEEE Commun. Mag. 49(7), 122–130 (2011).
[CrossRef]

Lieu, C.

Lim, C.

P. Gamage, A. Nirmalathas, C. Lim, M. Bakaul, D. Novak, and R. Waterhouse, “Efficient transmission scheme for AWG-based DWDM millimeter-wave fiber-radio systems,” IEEE Photon. Technol. Lett. 19(4), 206–208 (2007).
[CrossRef]

Littis, D. S. L.

I. Garrett, D. J. Bond, J. B. Waite, D. S. L. Littis, and G. Jacobsen, “Impact of phase noise in weakly coherent systems: a new and accurate approach,” J. Lightwave Technol. 8(3), 329–337 (1990).
[CrossRef]

Liu, T.

Maedar, A.

Y. Li, A. Maedar, L. Fan, A. Nigam, and J. Chou, “Overview of femtocell support in advanced WiMAX systems,” IEEE Commun. Mag. 49(7), 122–130 (2011).
[CrossRef]

Marpaung, D.

McKinney, J.

Millan, P.

A. Wiberg, P. Millan, M. Andres, P. A. Andrekson, and P. O. Hedevkist, “Fiber-optic 40-GHz mm-wave link with 2.5 Gb/s data transmission,” IEEE Photon. Technol. Lett. 17(9), 1938–1940 (2005).
[CrossRef]

Nie, Q.

Nigam, A.

Y. Li, A. Maedar, L. Fan, A. Nigam, and J. Chou, “Overview of femtocell support in advanced WiMAX systems,” IEEE Commun. Mag. 49(7), 122–130 (2011).
[CrossRef]

Nirmalathas, A.

A. H. M. Razibul Islam, M. Bakaul, A. Nirmalathas, and G. Town, “Millimeter-wave radio-over-Fiber system based on heterodyned unlocked light sources and self-homodyned RF receiver,” IEEE Photon. Technol. Lett. 23(8), 459–461 (2011).
[CrossRef]

P. Gamage, A. Nirmalathas, C. Lim, M. Bakaul, D. Novak, and R. Waterhouse, “Efficient transmission scheme for AWG-based DWDM millimeter-wave fiber-radio systems,” IEEE Photon. Technol. Lett. 19(4), 206–208 (2007).
[CrossRef]

Novak, D.

P. Gamage, A. Nirmalathas, C. Lim, M. Bakaul, D. Novak, and R. Waterhouse, “Efficient transmission scheme for AWG-based DWDM millimeter-wave fiber-radio systems,” IEEE Photon. Technol. Lett. 19(4), 206–208 (2007).
[CrossRef]

O’Reilly, J. J.

J. J. O’Reilly and P. M. Lane, “Remote delivery of video services using mm-wave and optics,” J. Lightwave Technol. 12(2), 369–375 (1994).
[CrossRef]

Ogawa, H.

C.-S. Choi, Y. Shoji, and H. Ogawa, “Millimeter-wave fiber-fed wireless access systems based on dense wavelength-division-multiplexing networks,” IEEE Trans. Microw. Theory Tech. 56(1), 232–241 (2008).
[CrossRef]

Ogawa, Y.

T. Kuri, K. Kitayama, and Y. Ogawa, “Fiber-optic millimeter-wave uplink system incorporating remotely fed 60-GHz-band optical pilot tone,” IEEE Trans. Microw. Theory Tech. 47, 1332–1337 (1999).
[CrossRef]

Plettemeier, D.

Pradhan, S.

S. Pradhan, G. E. Town, and K. J. Grant, “Dual wavelength DBR fiber laser,” IEEE Photon. Technol. Lett. 18(16), 1741–1743 (2006).
[CrossRef]

Razibul Islam, A. H. M.

A. H. M. Razibul Islam, M. Bakaul, A. Nirmalathas, and G. Town, “Millimeter-wave radio-over-Fiber system based on heterodyned unlocked light sources and self-homodyned RF receiver,” IEEE Photon. Technol. Lett. 23(8), 459–461 (2011).
[CrossRef]

Roeloffzen, C.

Rohde, D.

R.-P. Braun, G. Grosskopf, D. Rohde, and F. Schmidt, “Low-phase-noise millimeter-wave generation at 64 GHz and data transmission using optical sideband injection locking,” IEEE Photon. Technol. Lett. 10(5), 728–730 (1998).
[CrossRef]

Sakib, M. N.

Schaffer, G.

Schmidt, F.

R.-P. Braun, G. Grosskopf, D. Rohde, and F. Schmidt, “Low-phase-noise millimeter-wave generation at 64 GHz and data transmission using optical sideband injection locking,” IEEE Photon. Technol. Lett. 10(5), 728–730 (1998).
[CrossRef]

Seeds, A. J.

L. A. Johansson and A. J. Seeds, “Generation and transmission of millimeter-wave data-modulated optical signals using an optical injection phase-lock loop,” J. Lightwave Technol. 21(2), 511–520 (2003).
[CrossRef]

L. A. Johansson and A. J. Seeds, “Millimeter-wave modulated optical signal generation with high spectral purity and wide-locking bandwidth using a fiber-integrated optical injection phase-lock loop,” IEEE Photon. Technol. Lett. 12(6), 690–692 (2000).
[CrossRef]

Shoji, Y.

C.-S. Choi, Y. Shoji, and H. Ogawa, “Millimeter-wave fiber-fed wireless access systems based on dense wavelength-division-multiplexing networks,” IEEE Trans. Microw. Theory Tech. 56(1), 232–241 (2008).
[CrossRef]

Town, G.

A. H. M. Razibul Islam, M. Bakaul, A. Nirmalathas, and G. Town, “Millimeter-wave radio-over-Fiber system based on heterodyned unlocked light sources and self-homodyned RF receiver,” IEEE Photon. Technol. Lett. 23(8), 459–461 (2011).
[CrossRef]

Town, G. E.

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Figures (15)

Fig. 1
Fig. 1

(a) Millimeter-wave generation schemes exploiting heterodyning of uncorrelated optical carriers, where (a): data is imposed on two independent optical carriers located at CS and separated by 35.75 GHz, similar to a OCS-DSB modulation format (Scheme A), (b): data is imposed on one optical carrier separated by 35.75 GHz from a second independent optical carrier both located at CS, similar to an OSSB + C modulation format (Scheme B), (c): data is imposed on one optical carrier located at CS and coupled with an optical LO (separated by 35.75 GHz) at the BS to enable local optical heterodyning prior to photodetection (Scheme C); (d): describes RF self-homodyning for phase-insensitive recovery of data at the receiver irrespective of heterodyning schemes.

Fig. 2
Fig. 2

SSB phase noise at mm-wave IF due to uncorrelated lasers.

Fig. 3
Fig. 3

Experimental set up of Scheme A (top); Insets (i)-(iv) show optical and RF spectra at different points of the setup in Scheme A.

Fig. 4
Fig. 4

BER and eye diagrams for Scheme A.

Fig. 5
Fig. 5

Experimental set up of Scheme B (top); Insets (i)-(iv) show optical and RF spectra at different points of the setup in Scheme B.

Fig. 6
Fig. 6

BER and eye diagrams for Scheme B.

Fig. 7
Fig. 7

Experimental set up of Scheme C (top); Insets (i)-(v) show optical and RF spectra at different points of the setup in Scheme C.

Fig. 8
Fig. 8

BER and eye diagrams for Scheme C.

Fig. 9
Fig. 9

Simulation model of Scheme A for system characterization.

Fig. 10
Fig. 10

BER and eye diagrams for Scheme A.

Fig. 11
Fig. 11

Simulated power penalty versus relative intensity noise (RIN) curve.

Fig. 12
Fig. 12

BER curves due to frequency drifts introduced by variation in frequency separation of both the lasers.

Fig. 13
Fig. 13

BER curves at various laser linewidths.

Fig. 14
Fig. 14

Power penalty vs. laser linewidths for both LO-based and self-homodyned data recovery techniques.

Fig. 15
Fig. 15

BER curves at various noise variances in a wireless channel.

Tables (2)

Tables Icon

Table 1 Specifications used for the SSB plot

Tables Icon

Table 2 Specifications used for SNR calculations

Equations (31)

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E 1 expj( 2π f 1 t+ ϕ 1 )
E 2 expj( 2π f 2 t+ ϕ 2 )
A 1 ( t ) =expj( 2π f 1 t+ ϕ 1 )[ 1+jmcos( 2π f m t ) ] =expj( 2π f 1 t+ ϕ 1 )[ 1+ m 2 ( expj2π f m t+expj2π f m t ) ] =[ expj( 2π f 1 t+ ϕ 1 )+ m 2 { expj( 2π f 1 t+ ϕ 1 +2π f m t )+expj( 2π f 1 t+ ϕ 1 2π f m t ) } ]
A 2 ( t ) =expj( 2π f 2 t+ ϕ 2 )[ 1+jmcos( 2π f m t ) ] =expj( 2π f 2 t+ ϕ 2 )[ 1+ m 2 ( expj2π f m t+expj2π f m t ) ] =[ expj( 2π f 2 t+ ϕ 2 )+ m 2 { expj( 2π f 2 t+ ϕ 2 +2π f m t )+expj( 2π f 2 t+ ϕ 2 2π f m t ) } ]
A f 1 ( t )=[ expj( 2π f 1 t+ ϕ 1 + φ 0 )+ m 2 expj{ 2π( f 1 + f m )t+ ϕ 1 + φ 1 } + m 2 expj{ 2π( f 1 f m )t+ ϕ 1 + φ 2 } ]
A f 2 ( t )=[ expj( 2π f 2 t+ ϕ 2 + φ 0 )+ m 2 expj{ 2π( f 2 + f m )t+ ϕ 2 + φ 1 } + m 2 expj{ 2π( f 2 f m )t+ ϕ 2 + φ 2 } ]
A * f 1 ( t )=[ expj( 2π f 1 t+ ϕ 1 + φ 0 )+ m 2 expj{ 2π( f 1 + f m )t+ ϕ 1 + φ 1 } + m 2 expj{ 2π( f 1 f m )t+ ϕ 1 + φ 2 } ]
A * f 2 ( t )=[ expj( 2π f 2 t+ ϕ 2 + φ 0 )+ m 2 expj{ 2π( f 2 + f m )t+ ϕ 2 + φ 1 } + m 2 expj{ 2π( f 2 f m )t+ ϕ 2 + φ 2 } ]
i p ( t ) =×[ ( A f 1 ( t )+ A f 2 ( t ) )×( A f 1 * ( t )+ A f 2 * ( t ) ) ] =×[ 2+ m 2 + m 2 cos{ 2π f m t+( φ 1 φ 0 ) }+ m 2 cos{ 2π f m t+( φ 0 φ 2 ) }+ m 2 cos{ 2π f m t+( φ 0 φ 2 ) } + m 2 cos{ 2π f m t+( φ 1 φ 0 ) }+ m 2 4 cos{ 2π.2 f m t+( φ 1 φ 2 ) }+ m 2 4 cos{ 2π.2 f m t+( φ 1 φ 2 ) } +cos{ 2π( f 1 f 2 )t+( ϕ 1 ϕ 2 )+( φ 0 φ 0 ) }+ m 2 4 cos{ 2π( f 1 f 2 )t+( ϕ 1 ϕ 2 )+( φ 1 φ 1 ) } + m 2 4 cos{ 2π( f 1 f 2 )t+( ϕ 1 ϕ 2 )+( φ 2 φ 2 ) }+ m 2 cos{ 2π[ ( f 1 f 2 ) f m ]t+( ϕ 1 ϕ 2 )+( φ 0 φ 1 ) } + m 2 cos{ 2π[ ( f 1 f 2 ) f m ]t+( ϕ 1 ϕ 2 )+( φ 2 φ 0 ) }+ m 2 cos{ 2π[ ( f 1 f 2 )+ f m ]t+( ϕ 1 ϕ 2 )+( φ 0 φ 2 ) } + m 2 cos{ 2π[ ( f 1 f 2 )+ f m ]t+( ϕ 1 ϕ 2 )+( φ 1 φ 0 ) }+ m 2 4 cos{ 2π[ ( f 1 f 2 )+2 f m ]t+( ϕ 1 ϕ 2 )+( φ 1 φ 2 ) } + m 2 4 cos{ 2π[ ( f 1 f 2 )2 f m ]t+( ϕ 1 ϕ 2 )+( φ 2 φ 1 ) } ]
i p ( t ) =×[ cos{ 2π( f 1 f 2 )t+( ϕ 1 ϕ 2 )+( φ 0 φ 0 ) }+ m 2 4 cos{ 2π( f 1 f 2 )t+( ϕ 1 ϕ 2 )+( φ 1 φ 1 ) } + m 2 4 cos{ 2π( f 1 f 2 )t+( ϕ 1 ϕ 2 )+( φ 2 φ 2 ) }+ m 2 cos{ 2π[ ( f 1 f 2 ) f m ]t+( ϕ 1 ϕ 2 )+( φ 0 φ 1 ) } + m 2 cos{ 2π[ ( f 1 f 2 ) f m ]t+( ϕ 1 ϕ 2 )+( φ 2 φ 0 ) }+ m 2 cos{ 2π[ ( f 1 f 2 )+ f m ]t+( ϕ 1 ϕ 2 )+( φ 0 φ 2 ) } + m 2 cos{ 2π[ ( f 1 f 2 )+ f m ]t+( ϕ 1 ϕ 2 )+( φ 1 φ 0 ) }+ m 2 4 cos{ 2π[ ( f 1 f 2 )+2 f m ]t+( ϕ 1 ϕ 2 )+( φ 1 φ 2 ) } + m 2 4 cos{ 2π[ ( f 1 f 2 )2 f m ]t+( ϕ 1 ϕ 2 )+( φ 2 φ 1 ) } ]
i p ( t )=[ cos{ 2π( f 1 f 2 )t+( ϕ 1 ϕ 2 ) }+ m 2 4 cos{ 2π( f 1 f 2 )t+( ϕ 1 ϕ 2 ) } + m 2 4 cos{ 2π( f 1 f 2 )t+( ϕ 1 ϕ 2 )+ }+ m 2 cos{ 2π[ ( f 1 f 2 ) f m ]t+( ϕ 1 ϕ 2 ) } + m 2 cos{ 2π[ ( f 1 f 2 ) f m ]t+( ϕ 1 ϕ 2 ) }+ m 2 cos{ 2π[ ( f 1 f 2 )+ f m ]t+( ϕ 1 ϕ 2 ) } + m 2 cos{ 2π[ ( f 1 f 2 )+ f m ]t+( ϕ 1 ϕ 2 ) }+ m 2 4 cos{ 2π[ ( f 1 f 2 )+2 f m ]t+( ϕ 1 ϕ 2 ) } + m 2 4 cos{ 2π[ ( f 1 f 2 )2 f m ]t+( ϕ 1 ϕ 2 ) } ]
Mcos[ 2π{ ( f 1 f 2 )+n f m }t+( ϕ 1 ϕ 2 ) ]
cosC=Mcos[ 2π{ ( f 1 f 2 )+A f m }t+( ϕ 1 ϕ 2 ) ] cosD=Mcos[ 2π{ ( f 1 f 2 )+B f m }t+( ϕ 1 ϕ 2 ) ],
cosC×cosD = M 2 { cos[ 2π(AB) f m t ]+cos[ 2π{ 2( f 1 f 2 )+( A+B ) f m }t+2( ϕ 1 ϕ 2 ) ] }
0.5{ cos( 2π f m t )+cos[ 2π{ 2( f 1 f 2 )+ f m }t+2( ϕ 1 ϕ 2 ) ] }
A 1 ( t )=[ expj( 2π f 1 t+ ϕ 1 )+ m 2 { expj( 2π f 1 t+ ϕ 1 +2π f m t )+expj( 2π f 1 t+ ϕ 1 2π f m t ) } ]
A 2 ( t )=expj( 2π f 2 t+ ϕ 2 )
A f 1 ( t )=[ expj( 2π f 1 t+ ϕ 1 + φ 0 )+ m 2 expj{ 2π( f 1 + f m )t+ ϕ 1 + φ 1 } + m 2 expj{ 2π( f 1 f m )t+ ϕ 1 + φ 2 } ]
A f 2 ( t )=expj( 2π f 2 t+ ϕ 2 + φ 0 )
i p ( t )=×[ 2+ m 2 2 + m 2 cos{ 2π f m t+( φ 1 φ 0 ) }+ m 2 cos{ 2π f m t+( φ 0 φ 2 ) } + m 2 4 cos{ 2π.2 f m t+( φ 1 φ 2 ) } +cos{ 2π( f 1 f 2 )t+( ϕ 1 ϕ 2 )+( φ 0 φ 0 ) } + m 2 cos{ 2π[ ( f 1 f 2 ) f m ]t+( ϕ 1 ϕ 2 )+( φ 2 φ 0 ) } + m 2 cos{ 2π[ ( f 1 f 2 )+ f m ]t+( ϕ 1 ϕ 2 )+( φ 1 φ 0 ) } ]
A LO ( t )=expj[ 2π f 2 t+ ϕ 2 ]
i p ( t )=×[ 2+ m 2 2 + m 2 cos{ 2π f m t+( φ 1 φ 0 ) }+ m 2 cos{ 2π f m t+( φ 0 φ 2 ) } + m 2 4 cos{ 2π.2 f m t+( φ 1 φ 2 ) } +cos{ 2π( f 1 f 2 )t+( ϕ 1 ϕ 2 )+ φ 0 } + m 2 cos{ 2π[ ( f 1 f 2 ) f m ]t+( ϕ 1 ϕ 2 )+ φ 2 } + m 2 cos{ 2π[ ( f 1 f 2 )+ f m ]t+( ϕ 1 ϕ 2 )+ φ 1 } ]
i p ( t )=( P 1 + P 2 )+2 P 1 . P 2 cos[ 2π f IF t+ ϕ IF ]
S I ( f )= 2 P 1 P 2 1/πΔ υ IF 1+ [ 2(f f IF ) Δ υ IF ] 2
P IF = I 2 p (t) . R L = 1 2 ( 2 P 1 . P 2 ) 2 . R L
P IF = 1 2 ( 2 P 1 . P 2 ) 2 . R L = 1 2 I av 2 . R L
p N = p th + 1 4 ( p shot + p RINtotal )
P N = B n [ p th + 1 4 ( p shot + p RINtotal ) ]
N F Amp =N F LNA +( N F MPA 1 G LNA )
PL=20 log 10 ( 4.π.d. f RF c )
SNR= 1 2 I av 2 . R L . G LNA . G MPA . G Tx . G Rx P N .N F Amp .PL.IL.KT B n .N F Rx

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