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High-SFDR 100.8km ROF link with optical homodyne detection and genetic-algorithm-assisted digital demodulation

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Abstract

Radio-over-fiber (ROF) link based on phase modulation and coherent detection has been widely proposed for linear transmission. Nowadays, there are increasing demands for long-distance analog radio-frequency (RF) signal transmission, as radars and broadcast systems. In this paper, a high spurious-free-dynamic-range (SFDR) analog coherent ROF link based on optical homodyne detection and genetic-algorithm-assisted digital demodulation is proposed and experimentally investigated. The ROF link is designed for transmitting RF signals ranging from 500 kHz to 100 MHz over a long-distance fiber under the environment of wide temperature. We test the link performance by transmitting different groups of two-tone signals (580 kHz and 600 kHz, 9 MHz and 10 MHz, 49 MHz and 50 MHz, 99 MHz and 100 MHz) over a 100.8-km single-mode fiber (SMF) under the temperature varying from –40°C to 70°C, the shot-noise-limited SFDR of the link are measured to be greater than 122 dB·Hz2/3.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Radio-over-fiber (ROF) links with the advantages of low transmission loss, light weight, and immunity against electromagnetic interference have great application potentials in long-distance transmission civilian and military situations, such as the broadcast-signal transmission in broadcast systems (operating at 530 kHz∼1610 kHz), the detection-signal transmission in multi-static over-the-horizon (OTH) radars (operating at 3 MHz∼30 MHz) and the transmission of radar crystal-oscillation signals (operated at 10 MHz∼100 MHz) [15]. The long-distance ROF links, restricted by the spurious-free dynamic range (SFDR), are mainly used for digital codes transmitting [6,7]. Thus, the key point for realizing the long-distance analog ROF links is to improve the SFDR. The SFDR represents the radio-frequency (RF) power range to which the ROF link can accommodate, considering the effects of noise and nonlinear distortions. In the analog ROF links, the second-order intermodulation distortion (IMD2) and the second-order harmonic distortion (SHD) can be easily filtered out, third-order intermodulation distortion (IMD3) suppression is the primary concern for SFDR improvement [8,9]. The low-biased Mach-Zehnder modulator (MZM) or dual parallel MZM are frequently implemented to suppress the intermodulation distortion in the intensity-modulated ROF links for improving the SFDR [10,11]. However, in these cases, the bias voltage of MZM needs to be precisely controlled and the output noise in long-distance link will be seriously deteriorated caused by the optical amplifier. Thus, the phase-modulation with coherent-detection (PMCD) links have been widely proposed to solve these restrictions. It is known that the optical phase-locked loop (OPLL) could be used in the PMCD links to track and adjust optical phase for improving the SFDR [12,13]. In the approaches, the demodulated signal is fed back to the phase modulator to remain the phase difference in the linear region of the response curve. However, in long-distance transmission links, it is difficult to compensate for the fast power jitter caused by the mismatch between transmission and local-oscillation paths, and the gain of amplifier needs to be precisely designed according to the power of the received signal. So, the approach is more compatible for integrated links rather than long-distance transmission links. Recently, digital demodulation in PMCD links has emerged as an effective method to improve the SFDR, in which the outputs of two paths are used as I and Q components to form a complex quantity and the phase of complex quantity are extracted as the modulated RF signal by the digital signal processing (DSP) [1417]. Hence the RF signal can be demodulated linearly without small signal approximation and nonlinear elimination.

As the approaches in [14,15], the optical wave is divided by two paths to make up the in-phase and quadrature-phase (I/Q) coherent structure, the DSP technique is utilized to recover the modulate signals. The main limitation for long-distance transmission is the mismatch of the delay time between two paths which deteriorate the phase noise of demodulated signals, and the phase jitter of optical fiber that result in the power jitter of received photocurrent. Subsequently, the polarization multiplexing PMCD link has been proposed to reduce the matching demands of the conventional I/Q coherent link [16,17]. for the method in [16], the optical wave from single laser is phase modulated by the polarization modulator and transmitted to the parallel polarization beam splitters to realize the I/Q coherent demodulation. But the polarization state of light in long fiber varies randomly, so the polarization controller is difficult to be accurate adjusted to control the polarized angle of light wave. In PMCD links based on two free-running lasers [17], the remote optical wave is separated as two orthogonally polarized optical signals with one being phase modulated and the other with no modulation. After transmission through single-mode fiber (SMF), the optical signals are coherent detected with a free-running local laser and digital processed at the receiver. However, the phase noise deterioration of receive signals from the lasers would bring a difficulty for low-frequency analog signals receiving.

In this paper, we propose and demonstrate a PMCD scheme for linearized analog signals transmission over a 100.8 km SMF based on the optical phase locking and genetic-algorithm (GA)-assisted I/Q digital demodulating [18]. Owning to the OPLL which synchronous the phase between the local laser diode (LLD) and the remote laser diode (RLD), we overcome the phase noise deterioration of the demodulated RF signal caused by two-path mismatch and the phase jitter of the photocurrent caused by the environmental disturbance in long fiber. Because of the ultra-narrow linewidth lasers, the LLD can be easily phase-locked with the RLD without affecting the RF-signal transmission. In the receiving end, the RF signal is demodulated by the I/Q digital demodulating technique. The amplitude and phase imbalance between the I/Q pathways are precisely calculated by using the GA. Hence, the IMD3 of the received signals can be effectively suppressed. In addition, the relative-intensity-noise (RIN) from the lasers and erbium-doped fiber amplifiers (EDFA) can also be suppressed by the balanced photodiodes. The typical link gain, IMD3 and the shot-noise-limited SFDR are respectively measured to be –11.5 dB, 27.34 dBm and 122.02 dB·Hz2/3 by transmitting two-tone signals (99 MHz and 100 MHz). In order to verify that the system could work under broadband and wide temperature situation, we also measured the SFDRs of the link while transmitting multiple groups of two-tone signals (580 kHz and 600 kHz, 9 MHz and 10 MHz, 49 MHz and 50 MHz, 99 MHz and 100 MHz) under different temperatures (–40°C ∼70°C), which are all proved to be greater than 122 dB·Hz2/3. The demonstrated ROF link with such a high SFDR possess great application potentials in the field of radars and broadcasts systems. It should be noted that the bandwidth of the system could be further broaden by using high-speed analog-to-digital converters (ADCs) for meeting more application scenarios requiring large bandwidth.

2. Principle of operation

The schematic diagram of our scheme is shown in Fig. 1. The optical wave from RLD is sent to the electro-optic phase modulator (PM) and transmitted through the SMF. In the local receiver end, the receiving optical power is amplified by the Raman fiber amplifier (RFA) and EDFA. Note that the use of RFA is to compensate for long-fiber loss without introducing excess amplifier-spontaneous-emission (ASE) noise. Then the optical wave and the local optical wave, which is also amplified by EDFA to improve link gain, are injected into the 90° hybrid coupler. The input optical amplitudes of the 90° hybrid coupler can be written as

$$\begin{array}{l} {E_{PM}}\textrm{ = }\sqrt {{P_s}} \exp \left[ {j{\omega_1}t\textrm{ + }j\frac{{\pi V(t )}}{{{V_\pi }}}\textrm{ + }j{n_1}(t )\textrm{ + }j\Delta \varphi } \right],\\ {E_{LO}}\textrm{ = }\sqrt {{P_{LO}}} \exp [{j{\omega_2}t\textrm{ + }j{n_2}(t )} ], \end{array}$$
where Ps and PLO are the input power of the hybrid coupler, ω1 and ω2 are the input angular frequencies to the hybrid coupler, n1(t) and n2(t) are the input optical phase noise of the hybrid coupler. Δφ is optical phase difference between the lasers, which is introduced by the jitter and optical transmitting delay in the temperature-varying long fiber. Because the amplitude imbalance between the paths of optical hybrid coupler is relatively small, we only consider the phase imbalance. Thus, the output optical wave of the hybrid coupler can be written as
$$\begin{array}{l} {E_1}\textrm{ = }\frac{1}{2}{E_{PM}}\textrm{ + }\frac{1}{2}{E_{LO}},\\ {E_2}\textrm{ = }\frac{1}{2}{E_{PM}} - \frac{1}{2}{E_{LO}} \cdot \exp ({j{\beta_1}} ),\\ {E_3}\textrm{ = }\frac{1}{2}{E_{PM}}\textrm{ + }\frac{1}{2}j{E_{LO}},\\ {E_4}\textrm{ = }\frac{1}{2}{E_{PM}} - \frac{1}{2}j{E_{LO}} \cdot \exp ({j{\beta_2}} ). \end{array}$$

 figure: Fig. 1.

Fig. 1. The proposed link with optical homodyne detection and GA-assisted digital demodulation. PM: phase modulator; ISO: optical isolator; RFA: Raman fiber amplifier; BPD: balanced photodetector; EPD: electrical power divider; LPF: low-pass filter; PID: proportional-integral-differential controller.

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In the equations, β1 and β2 are the phase-imbalance coefficients in the hybrid coupler. After photoelectric conversion by the balanced photodetectors (BPDs), the I/Q photocurrent can be expressed as

$$\begin{array}{l} {I_I}(t )= {R_1}\cos \left( {\frac{{{\beta_1}}}{2}} \right)\sqrt {{P_s}{P_{LO}}} \cos \left[ {\Delta wt\textrm{ + }\frac{{\pi V(t )}}{{{V_\pi }}} - \frac{{{\beta_1}}}{2}\textrm{ + }n(t )\textrm{ + }\Delta \varphi } \right],\\ {I_Q}(t )= m{R_2}\cos \left( {\frac{{{\beta_2}}}{2}} \right)\sqrt {{P_s}{P_{LO}}} \sin \left[ {\Delta w({t - \tau } )\textrm{ + }\frac{{\pi V({t - \tau } )}}{{{V_\pi }}} - \frac{{{\beta_2}}}{2} + n(t )\textrm{ + }\Delta \varphi } \right], \end{array}$$
where R1 and R2 are responsivity of the BPDs. m is the amplitude attenuation caused by the electrical power divider (EPD) and cables. Δw = w1–w2, which is the frequency deviation between optical waves emitted from two lasers. V(t) is the time-varying voltage of the RF signal. τ is the delay difference caused by structures difference between I and Q paths. Vπ is half-wave voltage of PM. n(t)=n1(t)–n2(t), which represents the photocurrent phase noise within the phase-locked bandwidth. After filtering by low-pass filter (LPF) and controlling by proportional-integral-differential controller (PID), the error signal is fed back to regulate the oscillation frequency and phase of LLD. Consequently, the frequency deviation Δw, the phase noise n(t) within the phase-locked bandwidth and the phase difference Δφ can be eliminated. Therefore, the RF signal can be demodulated at the receiving end. Because of the RFA and EDFAs amplification, the gain of link is increased. Due to the balanced detection of BPD, the relative intensity noise including the ASE noise is suppressed, so we only consider the shot noise. To further improve SFDR of the link, the receiving I/Q signals are sampled and saved by the ADC, that can be expressed as
$$\begin{array}{l} {I_I}(n) = {\alpha _1}\cos \left[ {\frac{{\pi V(n )}}{{{V_\pi }}} - \frac{{{\beta_1}}}{2}} \right],\\ {I_Q}(n) = {\alpha _2}\sin \left[ {\frac{{\pi V({n - N} )}}{{{V_\pi }}} - \frac{{{\beta_2}}}{2} - \Delta w\tau } \right], \end{array}$$
where α1 and α2 are the peak voltage of the I/Q signals. The delay difference τ is converted to points number N after ADC. The delay mismatch can be corrected by shifting the sampled points of Q-path in the computer. Afterwards, in order to equal the amplitudes and orthogonalize the phases of the I/Q signals, we estimate the imbalance coefficients by the GA and alter the amplitude and phase of Q-signal. The ideal sampled signals II(n) and IQ(n) can be derived from the original one by
$$\begin{aligned} {I_I}^\prime \left( n \right) &= {I_I}\left( n \right),\\ {{I'}_Q}\left( n \right) &= \left[ {\left( {{\alpha _\textrm{1}}\textrm{/}{\alpha _2}} \right){I_Q}\left( n \right) + {I_I}\left( n \right)\sin \left( {\frac{{{\beta _2} - {\beta _1}}}{2}\textrm{ + }{\Delta }w\tau } \right)} \right]\textrm{/}\cos \left( {\frac{{{\beta _2} - {\beta _1}}}{2}\textrm{ + }{\Delta }w\tau } \right)\\ &\textrm{ = }\left[ {\alpha {I_Q}\left( n \right) + {I_I}\left( n \right)\sin \left( {\Delta \beta } \right)} \right]/\cos \left( {\Delta \beta } \right). \end{aligned}$$
When the IQ(n) is altered to IQ(n) by Eq. (5), the I/Q signals are synthesized by
$$V \propto {\mathop{\rm Im}\nolimits} \{{\ln [{{I_I}^\prime (n )\textrm{ + }j{{I^{\prime}}_Q}(n )} ]} \},$$
where Im (·) and ln (·) are the operations to extract imaginary part and calculate logarithm. In the ideal case, the modulated RF signal can be linearly demodulated without any distortions. However, the imbalance coefficients α and Δβ of the receiver are unknown and difficult to calculate in the scheme. Therefore, the GA is used to estimate α and Δβ. We primarily set α and Δβ as initial population, and approach the optimum values that maximized IMD3 suppression ratio through the evaluation, selection, crossover and mutation. During several generations, the IMD3 suppression ratio is close to maximum and the optimum imbalance coefficients in our link is found out. Thus, the SFDR of our link can be greatly improved by the GA-assisted DSP technique.

3. Experimental results

An experiment based on the setup shown in Fig. 1 is demonstrated and performed. In the transmitting end, an optical wave from a tunable CW laser source (OEwaves, OE4028) operating at 1550.07 nm with an output power of 11 dBm is sent to a phase modulator (Photline, MPZ-LN-10). Then the optical wave is phase modulated by the two-tone RF signals with frequency from 500 kHz to 100 MHz and transmitted in the 100.8 km long fiber. Owing to the long-distance transmission system, we cascaded a Raman fiber amplifier (Amonics) to compensate the fiber loss without introducing excess ASE noise. The power of the optical wave is amplified to 6.6 dBm by the RFA and then amplified to 17.1 dBm by EDFA. Because of the usage of RFA, an optical isolator with 40 dB isolation in receiving end is cascaded to prevent backward pumping light of RFA from entering PM. In another path, the light wave from local laser (OE4030) operating at 1550.07 nm with an output power of 10 dBm is amplified to 17.2 dBm by another EDFA. After that, the light waves in two paths are phase-shifted and combined by an integrated 2×4 90° hybrid coupler (Kylia COH24). I/Q signals are detected by the balanced photodetectors (Discovery DSC705), with about 0.75 A/W responsivity. The Q-path current separated by EPD is fed back to tune the laser oscillation frequency via an integrated circuit that includes LPF and PID controller. Therefore, the frequency deviation and initial phase difference of two laser output, the phase noise within phase-locked bandwidth (∼50 kHz) is eliminated. Note that the phase-locked bandwidth is set between the laser linewidth and the lowest frequency of our modulated signals. The purpose is to ensure stable phase-locking while not affecting the transmission of signals in the high-frequency band. The detected I/Q products are digitized by a multiple-port ADC (Picoscope6000E) of 5 GHz sampling rate and 500 MHz bandwidth, which is capable to save the received signals completely and will not lose the higher harmonic components. Finally, the saved signals are processed by DSP technology, which includes the delay-matching method and the GA-assisted I/Q-demodulating method, to suppress the IMD3 and improve the SFDR of the proposed link.

In order to test the performance of our system at practice temperatures, we put the long fiber into a high-and-low temperature test box to change the temperature from –40°C to 70°C. A test signal with the frequency from 500 kHz to 100 MHz is introduced to the phase modulator, the output signal power of Q-path before ADC is measured by a spectrograph (Rigol RSA3015E-TG). Therefore, the link gains at different frequencies under varying temperatures are shown in Fig. 2. As expected, the link gain remains stable even if the transmit frequency and temperature of the link is changed, so the practicality of the link is first proved.

 figure: Fig. 2.

Fig. 2. Gain of the link at different frequencies under varying temperatures.

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The proposed DSP technique mainly includes I/Q-path delay-match, GA-based imbalance-coefficient estimation and I/Q signals synthesis derived in Eq. (6). At first, the two-tone signals at frequencies of 99 MHz and 100 MHz are transmitted at room temperature. As can be seen from Fig. 3(a) and 3(b), the I/Q waveforms possess a delay difference which is caused by the structure difference between I and Q paths. After correcting the mismatch by shifting the sampled points of Q-path, the imbalance coefficient of the amplitude and phase is estimated by GA while the RF input power is 0 dBm per-tone. The GA flow chart in our scheme is shown in Fig. 4. According to Eq. (5) and Eq. (6), correct α and Δβ can completely suppress IMD3. Therefore, in the GA, we set α and Δβ as the initial population, and set the IMD3-suppression ratio in the output FFT spectrum as the solution. Through the evaluate, selection, crossover and mutation during several generations, α and Δβ are gradually closer to correct value and IMD3-suppression ratio is gradually closer to maximum. In our scheme, the interval values of α and Δβ are initially set as 1.7∼2.0 and –2π∼0, as shown in Fig. 5(a). Through the evaluate, selection, crossover and mutation, the IMD3-suppression ratio is gradually approaching to maximum after 70 generations, as it is shown in Fig. 5(b). The corresponding imbalance coefficient α and Δβ are respectively 1.89086 and –6.10035. Accordingly, the IMD3-suppression ratio achieves to 70.08 dBc. To illustrate the SFDR improvement after DSP, the fast Fourier transform (FFT) power spectra of the output signals without and with our proposed DSP technique are compared in Fig. 6(a) and 6(b), in which the input power of RF signals is 7 dBm per-tone. As can be seen, the IMD3 in the output signal is suppressed by more than 30 dB via using our proposed DSP technique.

 figure: Fig. 3.

Fig. 3. (a) Time series of I and Q signals, (b) zoom-in waveforms of (a) (gray line: I product, red line: Q product before delay matching; blue line: Q product after delay matching).

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 figure: Fig. 4.

Fig. 4. The GA flow chart in our scheme.

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 figure: Fig. 5.

Fig. 5. The maximum IMD3-suppression ratio calculated by the GA, (a) as a function of amplitude and phase imbalance coefficient, (b) as a function as generation.

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 figure: Fig. 6.

Fig. 6. The output FFT power spectra (a) without and (b) with the DSP technique, when the input power is 7 dBm per-tone.

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The output power of fundamental and IMD3 is functional of the two-tone input power applied to the phase modulator, which is shown in Fig. 7. The received photocurrent was 16.8 mA, so the corresponding receiver shot noise after EPD was −155.69 dBm/Hz. It can be seen that the output third-order intermodulation intercept point (OIP3) is increased from 12.7 dBm to 27.34 dBm, and the shot-noise-limited SFDR is improved from 112.26 dB·Hz2/3 to 122.02 dB·Hz2/3 in the proposed scheme.

 figure: Fig. 7.

Fig. 7. Fundamental and IMD3 output power as a function of input power, when the frequencies of the input two-tone signals are 99 MHz and 100 MHz.

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Furthermore, when we change the input frequencies of the two-tone signals to 49 MHz and 50 MHz, 9 MHz and 10 MHz, 580 kHz and 600 kHz, the output power of fundamental and IMD3 signal as the function of the two-tone input power applied to the phase modulator is shown in Fig. 8. As can be seen, the OIP3 are respectively increased to 27.45 dBm, 27.35 dBm and 27.7 dBm, and the shot-noise-limited SFDR is improved to 122.09 dB·Hz2/3, 122.03 dB·Hz2/3 and 122.26 dB·Hz2/3. Therefore, it can be proved that the frequency change has little influence on the SFDR of the proposed scheme. Besides, when the input frequencies of the two-tone signals are set as 580 kHz and 600 kHz, 9 MHz and 10 MHz, 49 MHz and 50 MHz, 99 MHz and 100 MHz, the SFDRs under various temperatures from –40°C to 70°C are measured and plotted in Fig. 9. It can be seen that the SFDRs fluctuate above 122 dB·Hz2/3, which illustrates that our system has the advantages of wide temperature applicability from –40°C to 70°C as well as broadband practicality from 500 kHz to 100 MHz.

 figure: Fig. 8.

Fig. 8. Fundamental and IMD3 signal output power as a function of input power, when the input two-tone frequencies are (a) 49 MHz and 50 MHz, (b) 9 MHz and 10 MHz, (c) 580 kHz and 600 kHz.

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 figure: Fig. 9.

Fig. 9. SFDR under the chosen temperatures, when frequencies of the two-tone signals are set as 580 kHz and 600 kHz, 9 MHz and 10 MHz, 49 MHz and 50 MHz, 99 MHz and 100 MHz.

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4. Conclusion

In conclusion, we have experimentally investigated a high-SFDR 100.8-km ROF coherent link with optical homodyne detection and GA-assisted digital demodulation. By using the OPLL-based homodyne detection, the output phase noise deterioration caused by two lasers beat and environmental fluctuation within the phase-locked bandwidth are eliminated. The SFDR of our proposed link is greatly improved through the advanced DSP technique assisted by the GA. The gains of transmit signals with frequency from 500 kHz to 100 MHz over a 100.8-km SMF at various temperatures (–40°C∼70°C) are not less than –12 dB. While transmitting two-tone signals at frequencies of 99 MHz and 100 MHz, the IMD3 is measured to be 27.34 dBm, which leads to a projected shot-noise-limited SFDR of 122.02 dB·Hz2/3. The SFDRs of the link at various frequencies (580 kHz and 600 kHz, 9 MHz and 10 MHz, 49 MHz and 50 MHz, 99 MHz and 100 MHz) under different temperatures (–40°C ∼70°C) are all greater than 122 dB·Hz2/3. The capability of the proposed link is proved for practically implementation in the long-distance signal transmission at the frequency from 500 kHz to 100 MHz, which can be used in radars and broadcast systems. With the usage of high-speed ADC, the bandwidth of the link could be extended and the link can be potentially used for long-distance analog RF transmission in other frequency bands.

Funding

National Natural Science Foundation of China (61971065, 61625104); State Key Laboratory of Information Photonics and Optical Communications (IPOC2020ZT03); ZTE Research Fund.

Disclosures

The authors declare no conflicts of interest.

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Figures (9)

Fig. 1.
Fig. 1. The proposed link with optical homodyne detection and GA-assisted digital demodulation. PM: phase modulator; ISO: optical isolator; RFA: Raman fiber amplifier; BPD: balanced photodetector; EPD: electrical power divider; LPF: low-pass filter; PID: proportional-integral-differential controller.
Fig. 2.
Fig. 2. Gain of the link at different frequencies under varying temperatures.
Fig. 3.
Fig. 3. (a) Time series of I and Q signals, (b) zoom-in waveforms of (a) (gray line: I product, red line: Q product before delay matching; blue line: Q product after delay matching).
Fig. 4.
Fig. 4. The GA flow chart in our scheme.
Fig. 5.
Fig. 5. The maximum IMD3-suppression ratio calculated by the GA, (a) as a function of amplitude and phase imbalance coefficient, (b) as a function as generation.
Fig. 6.
Fig. 6. The output FFT power spectra (a) without and (b) with the DSP technique, when the input power is 7 dBm per-tone.
Fig. 7.
Fig. 7. Fundamental and IMD3 output power as a function of input power, when the frequencies of the input two-tone signals are 99 MHz and 100 MHz.
Fig. 8.
Fig. 8. Fundamental and IMD3 signal output power as a function of input power, when the input two-tone frequencies are (a) 49 MHz and 50 MHz, (b) 9 MHz and 10 MHz, (c) 580 kHz and 600 kHz.
Fig. 9.
Fig. 9. SFDR under the chosen temperatures, when frequencies of the two-tone signals are set as 580 kHz and 600 kHz, 9 MHz and 10 MHz, 49 MHz and 50 MHz, 99 MHz and 100 MHz.

Equations (6)

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E P M  =  P s exp [ j ω 1 t  +  j π V ( t ) V π  +  j n 1 ( t )  +  j Δ φ ] , E L O  =  P L O exp [ j ω 2 t  +  j n 2 ( t ) ] ,
E 1  =  1 2 E P M  +  1 2 E L O , E 2  =  1 2 E P M 1 2 E L O exp ( j β 1 ) , E 3  =  1 2 E P M  +  1 2 j E L O , E 4  =  1 2 E P M 1 2 j E L O exp ( j β 2 ) .
I I ( t ) = R 1 cos ( β 1 2 ) P s P L O cos [ Δ w t  +  π V ( t ) V π β 1 2  +  n ( t )  +  Δ φ ] , I Q ( t ) = m R 2 cos ( β 2 2 ) P s P L O sin [ Δ w ( t τ )  +  π V ( t τ ) V π β 2 2 + n ( t )  +  Δ φ ] ,
I I ( n ) = α 1 cos [ π V ( n ) V π β 1 2 ] , I Q ( n ) = α 2 sin [ π V ( n N ) V π β 2 2 Δ w τ ] ,
I I ( n ) = I I ( n ) , I Q ( n ) = [ ( α 1 / α 2 ) I Q ( n ) + I I ( n ) sin ( β 2 β 1 2  +  Δ w τ ) ] / cos ( β 2 β 1 2  +  Δ w τ )  =  [ α I Q ( n ) + I I ( n ) sin ( Δ β ) ] / cos ( Δ β ) .
V Im { ln [ I I ( n )  +  j I Q ( n ) ] } ,
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