- Email: [email protected]
The effect of flicker noise on the phase noise of opto-electronic oscillator
Optik 125 (2014) 1572–1574
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
The effect of flicker noise on the phase noise of opto-electronic oscillator Jun Hong a,∗ , Sheng-Xing Yao a , Zhu-lin Li a , Xiao-Yong Fang a , Jian Guo b a b
Department of Electrical and Information Engineering, Hunan Institute of Technology, Hengyang, Hunan, China School of Information Science and Engineering, Southeast University, Nanjing, Jiangsu, China
a r t i c l e
i n f o
Article history: Received 13 April 2013 Accepted 1 September 2013
Keywords: Opto-electronic oscillator Flick noise Phase noise
a b s t r a c t A new analytic theory model of opto-electronic oscillator (OEO) is derived and verified by experiments in this paper, where the flick and white noise are both considered. Based on this model, the effect of flick noise on the phase noise is analyzed and results show that our model can describe the phase noise characteristics of OEO more accurately than traditional model. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction
2. Theory
Opto-electronic oscillator (OEO) is a new kind of microwave source introduced by Yao in 1995 [1,2], and it has been getting more and more attentions due to its super-low phase noise characteristics in microwave and even higher band [3–10]. There are two problems needing to be solved after the appearance of super-low phase noise OEO with long fiber delay: one is the spurs introduced by the small free spectral range (FSR) accompanying long fiber delay, which has been effectively depressed by both schemes of paratactic multi-loop OEO [3] and injection locked dual OEO [6]; the other problem is the difficulty that theory model can not accurately describe OEO’s near-dc phase noise determined by colored noise, since only the white noise is considered in traditional analytic models. Flick noise, also called 1/f noise, is an important part of color noise and it is the transition band between white noise and other colored noise [11]. Numerical model containing flick noise has been reported in some articles [12–14], while the analytic model is proposed for the first time and verified by experiments with the considerations of not only white but also flick noise. In following sections, we first build the phase noise theory for microwave link, and then based on this theory, new analytic phase noise model for single-loop OEO is derived, containing both flick and white noise. Finally, we compare our model with traditional analytic model by experiments.
2.1. Phase noise of microwave link Microwave link consists of kinds of devices, whose noise would be coupled into carrier, and that is the so-called device’s phase noise [15,16]. The noise of device is mainly consists of white noise and 1/f noise. The white phase noise results from white noise and is inversely proportional to the input power, which can be expressed by [17] b0 =
FkT0 P0
(1)
where F is the noise figure, k is the Boltzmann constant, T0 is the temperature and P0 is the power of input signal. The link white phase noise is derived as
b0,t =
F1 +
F2 − 1 A21
0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.09.006
F3 − 1 A22 A21
+ ···
kT0 P0
(2)
1/f noise is another type of noise, which results in 1/f phase noise of device. Generally speaking, 1/f noise only affects the low frequency signal, but if carrier passes through the nonlinear device, the 1/f noise could modulate the carrier and up-converse to higher frequency. Considering white noise and 1/f noise, the phase noise of device can be written as Sϕ (f ) = b0 + b−1
∗ Corresponding author. E-mail address: [email protected] (J. Hong).
+
1 f
(3)
where b0 is power spectrum density of white phase noise and b−1 is the 1/f phase noise constant. As for cascaded microwave link,
J. Hong et al. / Optik 125 (2014) 1572–1574
since the total 1/f phase noise is independent of carrier’s power, the whole link’s 1/f phase noise can be expressed by b−1,i
-60
(4)
i=1
Based on Eqs. (1)–(4), we can get the cascaded link’s phase noise as
Sϕ,t (f ) =
F1 +
F2 − 1 A21
+
F3 − 1 A22 A21
+ ···
kT0 + P0
n
b−1,i
i=1
1 f
(5)
2.2. Phase noise of single-loop OEO The configuration of single-loop OEO is shown in Fig. 1. The setup consists of a laser diode (LD), a variable optical attenuator (VOA), a Lithium Niobate Mach-Zehnder intensity modulator (MZM), a Single-mode optical fiber (SMF), a Pin-photodiode (PD), a microwave amplifier, a tunable microwave attenuator, a narrowband filter and a microwave coupler. The light wave (black line) from the LD is sent to the MZM, modulated by oscillating signal originating from noise at the microwave input port of the MZM, and then sent to the SMF fiber. After transmission through the optical delay line, the optical signals turn into the electrical signals (dashed line) through the PD, after being amplified and filtered, and then feed back to the electric port of the MZM. Signal, whose loop-gain is greater than one and phase-shift is multiples of 2, is able to oscillate, and detailed theoretical analysis can be found in references [1,2]. Based on the quasi-linear theory of OEO, the oscillation power at ω can be derived by [1,2] ∗ /2R GA2 Vin Vin |Vout |2 = 2 2R 1 + |G(V0 )| − 2G(V0 )cos(ω − ϕ(ω))
(6)
where Vin is the oscillation noise source, GA is the amplitude gain of amplifier, V0 is the amplitude of Vin , G(V0 ) is the closed-loop gain and near 1, is the delay time caused by fiber and ϕ(ω) is the phase shifter of open-loop link. Considering both of white and flick noise, Eq. (6) can be expressed as N0 + N−1 |Vout |2 = 2R 1 + |G(V0 )|2 − 2G(V0 )cos(ω − ϕ(ω))
b-1 = -100 dBc/Hz b-1 = -90 dBc/Hz
-90 -100 -110 -120 -130 -140 -150
10000
100000
Offset Frequency(Hz) Fig. 2. SSB phase noise of OEO as a function of offset frequency in different value of b−1,t .
where N0 and N−1 are the power spectrum density of systematic white and flick noise respectively. Above equation can be further simplified as P(ω) =
N0 N−1 + 2 − 2cos(ω − ϕ(ω)) 2 − 2cos(ω − ϕ(ω))
(8)
At last, the single side band (SSB) phase noise of OEO can be expressed as L(f )
=
b−1,t f −1 N0 /POSC + 2 − 2cos(2f − ϕ(f )) 2 − 2cos(2f − ϕ(f ))
=
b0,t b−1,t f −1 + 2 − 2cos(2f − ϕ(f )) 2 − 2cos(2f − ϕ(f ))
=
Sϕ,t (f ) 2 − 2cos(2f − ϕ(f ))
(7) -80 -90 -100
Our Model
-110 -120 -130 Traditional Model -140 -150 100
1000
10000
100000
Offset Frequency (Hz) Fig. 1. Schematic diagram of a typical single-loop OEO.
(9)
where Sϕ,t (f) is link’s phase noise, composed of white and flick phase noise and expressed by Eq. (5); b0,t is total white phase noise and b−1,t is total flick phase noise coefficient of open-loop link. Assuming that 6 km SMF-28 fiber is applied and the value of b0,t is about −140 dBc/Hz, the SSB phase noise of OEO as a function of offset frequency in different value of b−1,t is shown in Fig. 2. Four colored curves represent the SSB phase noise data of single-loop OEO with 6 km fiber at b−1,t of −120, −110, −100 and −90 dBc/Hz. The curves own high spurs due to small FSR caused by long fiber and the SSB phase noise increases with b−1,t . In the case of low b−1,t , the flick noise makes influence on the near-dc phase noise, but if the flick noise is high enough, the far-dc phase noise could be enhanced evidently. The comparison of our model and traditional model is also implemented, and the theoretical data is shown in Fig. 3. In this case that the flick noise is relatively low, we can see that in the far-dc phase noise range, two theoretical curves agree well with
SSB Phase Noise(dBc/Hz)
P(ω) =
-80
1000
Summarizing above analysis: device’s phase noise consists of white and 1/f phase noise; white phase noise is inversely proportional to carrier’s power but 1/f phase noise is not influenced by input power; Link’s phase noise is the total of every device’s contribution.
P(ω) =
b-1 = -120 dBc/Hz b-1 = -110 dBc/Hz
-70
SSB Phase Noise(dBc/Hz)
b−1,t =
n
1573
Fig. 3. Theoretical comparation of our model and traditional model.
1574
J. Hong et al. / Optik 125 (2014) 1572–1574
SSB phase noise (dBc/Hz)
-60
From above equation, we can find that the SSB phase noise consists of two parts: Part one is proportional to f−2 and determined by link’s white noise; Part two is proportional to f−3 and determined by link’s flick noise. Analysis based on Eq. (10) agrees well with the experimental data shown in Fig. 4.
-70 -80
f
-3
Our Model
f
-90
-2
Instrument's Floor Noise
-100
4. Conclusions
-2
f Traditional Model
-110 -120 -130 100
1000
Phase noise theory for microwave link is derived in this paper, and based on this theory, new model for single-loop OEO has also been built. It is an analytic model where link’s white noise and flick noise are both considered. Compared with traditional model, our model agrees better with experimental data, especially for near-dc phase noise.
10000
Offset frequency (Hz) Fig. 4. Experimental and theoretical phase noise data of OEO.
each other, but in the near-dc range, the theoretical phase noise in our model is obviously higher than the phase noise in traditional model due to the effect of flick noise considered in our model. 3. Experiments In order to prove the validity of our model, experiments of single-loop OEO has also been finished. In order to make measurement convenient, 50 m SMF-28 fiber is used in experiments, because short fiber results high phase noise which can be measured by frequently-used spectrum analyzer. Experiments based on the schematic in Fig. 1 are performed. The LD used in experiments is a distributed feedback semiconductor laser (Ortel 1772), and we operated it at the full rated power of 70 mW. A variable optical attenuator is employed to adjust the input optical power of the MZmodulator, placed between the LD and the MZ-modulator. Since high output of the LD makes its relative intensity noise (RIN) relatively lower, varying optical output power through the VOA can not only change the output optical power but also make its amplitude noise as a constant. The electro-optic modulator in this experiment is a low-V MZM (Convega LN058), and its V is about 1.8 V at 10 GHz. The PD is a high-saturation (35 mW) and high-speed (30 GHz) PIN-photodiode (Optilab PD-30). The amplifier’s gain and NF are 50 dB and 1.8 dB respectively, and we can adjust the gain through the rear adjustable attenuator. The −3 dB bandwidth of the MICROWAVE filter (K&L 3C60) is 20 MHz and the center frequency is 10 GHz. Experimental data is shown in Fig. 4, and black and red curves are theoretical data of traditional model and our model respectively. It is obvious that the theoretical data of our model agrees better with experimental data than traditional model due to not considering the flick noise in it. As for experimental data, we find that measured phase noise consists of two sections: f−3 part and f−2 part, f−n meaning that the SSB phase noise decreases by 10 × n dB per decade frequency. The closer near dc the more n is, and this situation has well been explained with Lesson model [11]. In following section, it would be analyzed with our theoretical model of Eq. (9). If ω 1, Eq. (9) can be simplified by L(f ) =
b0,t (2)
2
f −2 +
b−1,t (2)2
f −3
(10)
Acknowledgment This work is supported by the Scientific Research Youth Fund of Hunan Provincial Education Department (Grant No. 13B158). References [1] X.S. Yao, L. Maleki, Optoelectronic microwave oscillator, J. Opt. Soc. Am. B 13 (August) (1996) 1725–1735. [2] X.S. Yao, L. Maleki, Optoelectronic oscillator for photonic systems, J. Quantum. Lett. 32 (July) (1996) 1141–1149. [3] X.S. Yao, L. Maleki, Multiloop optoelectronic oscillator, J. Quantum. Lett. 36 (January) (2000) 79–84. [4] Y. Jiang, J.L. Yu, Y.T. Wang, L.T.E. Zhang, Z. Yang, An optical domain combined dual-loop optoelectronic oscillator, Photon. Technol. Lett. 19 (June) (2007) 807–809. [5] E. Shumakher, G. Eisenstein, A novel multiloop optoelectronic oscillator, Photon. Technol. Lett. 20 (November) (2008) 1181–1183. [6] W.M. Zhou, G. Blasche, Injection-locked dual opto-electronic oscillator with ultra-low phase noise and ultra-low spurious level, Trans. Microwave Theory Techn. 53 (March) (2005) 929–933. [7] K.H. Lee, J.Y. Kim, W.Y. Choi, Injection-locked hybrid optoelectronic oscillators for single-mode oscillation, Photon. Technol. Lett. 20 (October) (2008) 1645–1647. [8] I. Ozdur, M. Akbulut, N. Hoghooghi, D. Mandridis, M.U. Piracha, P.J. Delfyett, Optoelectronic loop design with 1000 finesse Fabry–Perot etalon, Opt. Express 35 (March) (2010) 799–801. [9] J. Hong, C. Yang, Effects of the biasing voltage of modulator on the phase noise of opto-electronic oscillator, Microwave Opt. Technol. Lett. 54 (2012) 689–692. [10] J. Hong, C.X. Yang, B. Zhang, et al., Oscillation power of opto-electronic oscillator limited by nonlinearities of Mach-Zehnder modulator and microwave amplifier, in: IEEE The International Topic Meeting on Microwave Photonics, 2011, pp. 77–80. [11] D.B. Lesson, A simple model of feedback oscillator noise spectrum[J], Proc. IEEE 54 (2) (1966) 329–330. [12] E. Levy, M. Horowitz, O. Okusaga, et al., Study of dual-loop optoelectronic oscillators[C], in: Frequency Control Symposium, 2009, pp. 505–507. [13] C.R. Menyuk, E.C. Levy, O. Okusaga, et al., An analytical model of the dualinjection-locked opto-electronic oscillator (DIL-OEO)[C], in: Frequency Control Symposium, 2009, pp. 870–874. [14] O. Okusaga, W.M. Zhou, E. Levy, et al., Experimental and simulation study of dual injection-locked OEOs[C], in: Frequency Control Symposium, 2009, pp. 875–879. [15] J. Mukherjee, P. Roblin, S. Akhtar, An analytic circuit-based model for white and flicker phase noise in LC oscillators, IEEE Trans. Circuits Syst. I, Reg. Papers 54 (2007) 1584–1598. [16] Z.F. Zhang, J. Lau, Experimental study on MOSFET’s flicker noise under switching conditions and modelling in RF applications, in: IEEE Conference on Custom Integrated Circuits, 2001, pp. 393–396. [17] E. Rubiola, Phase Noise and Frequency Stability in Oscillator, Cambridge University Press, New York, 2009.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
The effect of flicker noise on the phase noise of opto-electronic oscillator Jun Hong a,∗ , Sheng-Xing Yao a , Zhu-lin Li a , Xiao-Yong Fang a , Jian Guo b a b
Department of Electrical and Information Engineering, Hunan Institute of Technology, Hengyang, Hunan, China School of Information Science and Engineering, Southeast University, Nanjing, Jiangsu, China
a r t i c l e
i n f o
Article history: Received 13 April 2013 Accepted 1 September 2013
Keywords: Opto-electronic oscillator Flick noise Phase noise
a b s t r a c t A new analytic theory model of opto-electronic oscillator (OEO) is derived and verified by experiments in this paper, where the flick and white noise are both considered. Based on this model, the effect of flick noise on the phase noise is analyzed and results show that our model can describe the phase noise characteristics of OEO more accurately than traditional model. © 2013 Elsevier GmbH. All rights reserved.
1. Introduction
2. Theory
Opto-electronic oscillator (OEO) is a new kind of microwave source introduced by Yao in 1995 [1,2], and it has been getting more and more attentions due to its super-low phase noise characteristics in microwave and even higher band [3–10]. There are two problems needing to be solved after the appearance of super-low phase noise OEO with long fiber delay: one is the spurs introduced by the small free spectral range (FSR) accompanying long fiber delay, which has been effectively depressed by both schemes of paratactic multi-loop OEO [3] and injection locked dual OEO [6]; the other problem is the difficulty that theory model can not accurately describe OEO’s near-dc phase noise determined by colored noise, since only the white noise is considered in traditional analytic models. Flick noise, also called 1/f noise, is an important part of color noise and it is the transition band between white noise and other colored noise [11]. Numerical model containing flick noise has been reported in some articles [12–14], while the analytic model is proposed for the first time and verified by experiments with the considerations of not only white but also flick noise. In following sections, we first build the phase noise theory for microwave link, and then based on this theory, new analytic phase noise model for single-loop OEO is derived, containing both flick and white noise. Finally, we compare our model with traditional analytic model by experiments.
2.1. Phase noise of microwave link Microwave link consists of kinds of devices, whose noise would be coupled into carrier, and that is the so-called device’s phase noise [15,16]. The noise of device is mainly consists of white noise and 1/f noise. The white phase noise results from white noise and is inversely proportional to the input power, which can be expressed by [17] b0 =
FkT0 P0
(1)
where F is the noise figure, k is the Boltzmann constant, T0 is the temperature and P0 is the power of input signal. The link white phase noise is derived as
b0,t =
F1 +
F2 − 1 A21
0030-4026/$ – see front matter © 2013 Elsevier GmbH. All rights reserved. http://dx.doi.org/10.1016/j.ijleo.2013.09.006
F3 − 1 A22 A21
+ ···
kT0 P0
(2)
1/f noise is another type of noise, which results in 1/f phase noise of device. Generally speaking, 1/f noise only affects the low frequency signal, but if carrier passes through the nonlinear device, the 1/f noise could modulate the carrier and up-converse to higher frequency. Considering white noise and 1/f noise, the phase noise of device can be written as Sϕ (f ) = b0 + b−1
∗ Corresponding author. E-mail address: [email protected] (J. Hong).
+
1 f
(3)
where b0 is power spectrum density of white phase noise and b−1 is the 1/f phase noise constant. As for cascaded microwave link,
J. Hong et al. / Optik 125 (2014) 1572–1574
since the total 1/f phase noise is independent of carrier’s power, the whole link’s 1/f phase noise can be expressed by b−1,i
-60
(4)
i=1
Based on Eqs. (1)–(4), we can get the cascaded link’s phase noise as
Sϕ,t (f ) =
F1 +
F2 − 1 A21
+
F3 − 1 A22 A21
+ ···
kT0 + P0
n
b−1,i
i=1
1 f
(5)
2.2. Phase noise of single-loop OEO The configuration of single-loop OEO is shown in Fig. 1. The setup consists of a laser diode (LD), a variable optical attenuator (VOA), a Lithium Niobate Mach-Zehnder intensity modulator (MZM), a Single-mode optical fiber (SMF), a Pin-photodiode (PD), a microwave amplifier, a tunable microwave attenuator, a narrowband filter and a microwave coupler. The light wave (black line) from the LD is sent to the MZM, modulated by oscillating signal originating from noise at the microwave input port of the MZM, and then sent to the SMF fiber. After transmission through the optical delay line, the optical signals turn into the electrical signals (dashed line) through the PD, after being amplified and filtered, and then feed back to the electric port of the MZM. Signal, whose loop-gain is greater than one and phase-shift is multiples of 2, is able to oscillate, and detailed theoretical analysis can be found in references [1,2]. Based on the quasi-linear theory of OEO, the oscillation power at ω can be derived by [1,2] ∗ /2R GA2 Vin Vin |Vout |2 = 2 2R 1 + |G(V0 )| − 2G(V0 )cos(ω − ϕ(ω))
(6)
where Vin is the oscillation noise source, GA is the amplitude gain of amplifier, V0 is the amplitude of Vin , G(V0 ) is the closed-loop gain and near 1, is the delay time caused by fiber and ϕ(ω) is the phase shifter of open-loop link. Considering both of white and flick noise, Eq. (6) can be expressed as N0 + N−1 |Vout |2 = 2R 1 + |G(V0 )|2 − 2G(V0 )cos(ω − ϕ(ω))
b-1 = -100 dBc/Hz b-1 = -90 dBc/Hz
-90 -100 -110 -120 -130 -140 -150
10000
100000
Offset Frequency(Hz) Fig. 2. SSB phase noise of OEO as a function of offset frequency in different value of b−1,t .
where N0 and N−1 are the power spectrum density of systematic white and flick noise respectively. Above equation can be further simplified as P(ω) =
N0 N−1 + 2 − 2cos(ω − ϕ(ω)) 2 − 2cos(ω − ϕ(ω))
(8)
At last, the single side band (SSB) phase noise of OEO can be expressed as L(f )
=
b−1,t f −1 N0 /POSC + 2 − 2cos(2f − ϕ(f )) 2 − 2cos(2f − ϕ(f ))
=
b0,t b−1,t f −1 + 2 − 2cos(2f − ϕ(f )) 2 − 2cos(2f − ϕ(f ))
=
Sϕ,t (f ) 2 − 2cos(2f − ϕ(f ))
(7) -80 -90 -100
Our Model
-110 -120 -130 Traditional Model -140 -150 100
1000
10000
100000
Offset Frequency (Hz) Fig. 1. Schematic diagram of a typical single-loop OEO.
(9)
where Sϕ,t (f) is link’s phase noise, composed of white and flick phase noise and expressed by Eq. (5); b0,t is total white phase noise and b−1,t is total flick phase noise coefficient of open-loop link. Assuming that 6 km SMF-28 fiber is applied and the value of b0,t is about −140 dBc/Hz, the SSB phase noise of OEO as a function of offset frequency in different value of b−1,t is shown in Fig. 2. Four colored curves represent the SSB phase noise data of single-loop OEO with 6 km fiber at b−1,t of −120, −110, −100 and −90 dBc/Hz. The curves own high spurs due to small FSR caused by long fiber and the SSB phase noise increases with b−1,t . In the case of low b−1,t , the flick noise makes influence on the near-dc phase noise, but if the flick noise is high enough, the far-dc phase noise could be enhanced evidently. The comparison of our model and traditional model is also implemented, and the theoretical data is shown in Fig. 3. In this case that the flick noise is relatively low, we can see that in the far-dc phase noise range, two theoretical curves agree well with
SSB Phase Noise(dBc/Hz)
P(ω) =
-80
1000
Summarizing above analysis: device’s phase noise consists of white and 1/f phase noise; white phase noise is inversely proportional to carrier’s power but 1/f phase noise is not influenced by input power; Link’s phase noise is the total of every device’s contribution.
P(ω) =
b-1 = -120 dBc/Hz b-1 = -110 dBc/Hz
-70
SSB Phase Noise(dBc/Hz)
b−1,t =
n
1573
Fig. 3. Theoretical comparation of our model and traditional model.
1574
J. Hong et al. / Optik 125 (2014) 1572–1574
SSB phase noise (dBc/Hz)
-60
From above equation, we can find that the SSB phase noise consists of two parts: Part one is proportional to f−2 and determined by link’s white noise; Part two is proportional to f−3 and determined by link’s flick noise. Analysis based on Eq. (10) agrees well with the experimental data shown in Fig. 4.
-70 -80
f
-3
Our Model
f
-90
-2
Instrument's Floor Noise
-100
4. Conclusions
-2
f Traditional Model
-110 -120 -130 100
1000
Phase noise theory for microwave link is derived in this paper, and based on this theory, new model for single-loop OEO has also been built. It is an analytic model where link’s white noise and flick noise are both considered. Compared with traditional model, our model agrees better with experimental data, especially for near-dc phase noise.
10000
Offset frequency (Hz) Fig. 4. Experimental and theoretical phase noise data of OEO.
each other, but in the near-dc range, the theoretical phase noise in our model is obviously higher than the phase noise in traditional model due to the effect of flick noise considered in our model. 3. Experiments In order to prove the validity of our model, experiments of single-loop OEO has also been finished. In order to make measurement convenient, 50 m SMF-28 fiber is used in experiments, because short fiber results high phase noise which can be measured by frequently-used spectrum analyzer. Experiments based on the schematic in Fig. 1 are performed. The LD used in experiments is a distributed feedback semiconductor laser (Ortel 1772), and we operated it at the full rated power of 70 mW. A variable optical attenuator is employed to adjust the input optical power of the MZmodulator, placed between the LD and the MZ-modulator. Since high output of the LD makes its relative intensity noise (RIN) relatively lower, varying optical output power through the VOA can not only change the output optical power but also make its amplitude noise as a constant. The electro-optic modulator in this experiment is a low-V MZM (Convega LN058), and its V is about 1.8 V at 10 GHz. The PD is a high-saturation (35 mW) and high-speed (30 GHz) PIN-photodiode (Optilab PD-30). The amplifier’s gain and NF are 50 dB and 1.8 dB respectively, and we can adjust the gain through the rear adjustable attenuator. The −3 dB bandwidth of the MICROWAVE filter (K&L 3C60) is 20 MHz and the center frequency is 10 GHz. Experimental data is shown in Fig. 4, and black and red curves are theoretical data of traditional model and our model respectively. It is obvious that the theoretical data of our model agrees better with experimental data than traditional model due to not considering the flick noise in it. As for experimental data, we find that measured phase noise consists of two sections: f−3 part and f−2 part, f−n meaning that the SSB phase noise decreases by 10 × n dB per decade frequency. The closer near dc the more n is, and this situation has well been explained with Lesson model [11]. In following section, it would be analyzed with our theoretical model of Eq. (9). If ω 1, Eq. (9) can be simplified by L(f ) =
b0,t (2)
2
f −2 +
b−1,t (2)2
f −3
(10)
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