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Wideband reconfigurable signal generation based on recirculating frequency-shifting using an optoelectronic loop

Ziqiang Yin, Xiangpeng Zhang, Chenyu Liu, Henan Zeng, and Wangzhe Li

Open Access Open Access

Abstract

A novel photonic-assisted reconfigurable wideband signal generator for linearly frequency-modulated (LFM) signals generation is proposed and experimentally demonstrated. A frequency-shifting recirculating optoelectronic (FSRO) loop is employed to shift and stitch a seed signal repeatedly in the time and the frequency domains through feedback modulation. After experiencing multiple recirculation, a time duration and bandwidth extended LFM signal with a quadratically varying phase of no phase discontinuity is generated. By simply changing loop parameters, the time duration, the bandwidth and the central frequency of the generated LFM signals are adjustable. Although the phase noise will deteriorate during the recirculation, the generated LFM signals with an increased time bandwidth product (TBWP) are still suitable for practical radar applications. A proof-of-concept experiment is carried out. The generation of LFM signals in C and X bands with a TBWP increased by a maximum factor of 196 are achieved. Microwave imaging of rotational targets based on the generated LFM signal is also demonstrated.

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

1. Introduction

Wideband reconfigurable radar waveforms are highly desired in modern radar systems to achieve ultra-high-resolution, anti-interference and multifunctional integration. As a typical radar signal, linearly frequency-modulated (LFM) signals have been widely employed in modern radar systems [1,2]. Conventionally, LFM signals are generated using voltage-controlled oscillators or direct digital synthesizers [3,4]. The central frequency and bandwidth of the generated LFM signals are limited by the state-of-the-art electronic devices to a several gigahertz, which may not meet requirements in advanced radar systems. Microwave photonic (MWP) technologies present an attractive alternative to electronic approaches thanks to the inherent characteristics of photonics in terms of ultra-wide bandwidth, low transmission loss, wavelength agility and low timing jitter [5–10]. Numerous photonic techniques have been proposed and developed to generate tunable and wideband LFM signals within the past few years. One of typical methods is space or frequency-to-time mapping [11–14]. By mapping the envelope of shaped optical spectrum to the waveform in the time domain using dispersive element, large-bandwidth LFM signal generation is realized. However, the time duration of the signal is rather small, making it unsuitable for radars with long-distance applications. Another way to generate broadband LFM signals is based on external modulation [15–17]. Utilizing the nonlinear effect of electrooptic modulation, a narrow-band electrical signal is converted to the high-order sideband of an optical carrier to achieve a large center frequency and bandwidth. However, the improvement of bandwidth is limited due to the limited order of sidebands. Fourier domain mode locking opto-electronic oscillators (FDML-OEO) are another alternative to generate versatile LFM signals [18,19]. By matching the wavelength of a sweeping laser to the frequency of a tunable microwave photonic filter formed in the oscillator periodically, a wideband LFM signal with ultra-low phase noise is generated. However, the time duration of generated signal is limited to the loop length, and the coherence is limited by the instability of the laser wavelength of each oscillation period. In order to improve the reconfigurability and the bandwidth of LFM signals, photonics-assisted schemes based on optical frequency-shifting (OFS) loop have been proposed and become particularly promising due to the availability of improving the TBWP of LFM signals [20–22]. By periodically shifting an optical carrier at a fixed frequency spacing in the OFS loop, multiple optical wavelengths are first developed. In [20], the multi-wavelengths with a quadratic spectral phase are converted to an optical chirp after shaping its spectral envelope. In [21,22], the multi-wavelengths arranged at equal time intervals are filled by a narrow-band chirped signal and then heterodyning with the optical carrier to generate a TBWP-multiplied LFM signal. The bandwidth of generated LFM signals can reach tens of gigahertz. However, the wavelength drift of the optical carrier would cause the instability of the frequency of the output beat signal. In addition, there is no mechanism to guarantee the phase continuity at any stitching position of two adjacent frequency-shifting signals, which would affect practical application of the method.

In this paper, a novel reconfigurable wideband linear frequency-modulated (LFM) signals generation scheme is proposed based on a frequency-shifting recirculating optoelectronic (FSRO) loop. LFM signals with an increased and variable TBWP are generated. The FSRO loop of proposed scheme is formed by a microwave photonic link and an electrical feedback link. In the microwave photonic link, an LFM pulse seed signal is frequency-upconverted by beating a frequency-shifted optical carrier and one sideband of the optical carrier modulated by the seed signal. Then, through the electrical feedback link, the frequency-upconverting signal is delayed and replaces the seed signal to modulate the optical carrier and undergoes the same frequency-up-conversion process repeatedly. Thus, as long as the loop delay equals to the time duration of the seed signal and a loop gain is properly set, the seed signal would experience multiple time-frequency shifting inside the FSRO loop and all the time-frequency shifted signals would be stitched continuously to form a TBWP-multiplied LFM signal. The time duration, the bandwidth and the central frequency of the generated LFM signal can be adjusted by changing the parameters of the loop and the seed signal. The proposed system is immune to the wavelength drift of the laser. As the two optical sidebands for beating are obtained by simultaneously modulating the same optical carrier, whose wavelength drift is eventually eliminated after photoelectrical detection. Moreover, by matching the phase process of the seed signal with the FSRO loop when adjusting the loop parameters, the phase of the stitched waveform at the splicing position is continuous without any phase jump, which means that the generated signals are suitable for practical radar applications. An experimental realization of the proposed method is developed, realizing the generation of LFM signals with an adjustable period from 15 to over 145 us, and bandwidth of 2 to 2.8 GHz on the basis of a seed signal with a bandwidth of 200MHz. Additionally, a proof-of-concept rotational microwave imaging experiment based on the developed signal generator is conducted, exhibiting a good quality of the signal.

2. Principle

The schematic of the proposed reconfigurable wideband signal generation system using a FSRO loop is shown in Fig. 1. The FSRO loop consists of a microwave photonic link and an electrical feedback link. In the microwave photonic link, a light wave from a continuous wave (CW) laser is split into two branches and guided into two equivalent frequency-shifted modulators, where the split optical carriers are frequency-shifted by a single-tone reference signal and single sideband (SSB) modulated by an LFM pulse seed signal respectively. The frequency shifting in the upper branch is implemented by an acousto-optic frequency shifter (AOFS), and the SSB modulation in the lower branch is realized by a Mach-Zehnder modulator (MZM) and an optical bandpass filter (OBPF). The frequency-shifted optical carrier and the SSB signal are then amplified by an erbium-doped fiber amplifier (EDFA) and heterodyned by a photodetector (PD) after propagating through a variable optical delay line (VODL) to generate a frequency-shifting (FS) microwave signal. Notice that the FS signal can be an up-converted signal or a down-converted signal, depending on whether the frequency-shifted optical carrier and the SSB signal are located at the same sides of the optical carrier. If they are on the same side of the optical carrier, the finally obtained LFM signal can be a counter-chirped LFM signal. The following principle-description takes the situation that they are located on both sides of the optical carrier as an example, as shown in the Fig. 2. After being filtered and amplified, the first generated FS (FS1) signal replaces the seed signal to modulate the optical carrier in the lower branch through the electrical feedback link, thus producing a second SSB (SS2) signal. The SS2 signal and the frequency-shifted optical carrier are heterodyning subsequently to generate a second FS (FS2) signal. Accordingly, the new generated FS2 signal repeat the feedback modulation and up-conversion process in the FSRO loop. Under the condition that the loop delay and the frequency of the reference signal are equal to the time duration and bandwidth of the seed signal respectively, as long as the phase matching condition of the loop is satisfied, a TBWP multiplied LFM signal can be obtained by stitching those FS signals.

 figure: Fig. 1.

Fig. 1. The schematic diagram of the proposed signal generator. AOFS, acousto-optic frequency shifter; MZM, Mach-Zehnder modulator; OBPF, optical bandpass filter; EDFA, erbium-doped fiber amplifier; VODL, variable optical delay line; PD, photodetector; LNA, low noise amplifier; BPF, bandpass filter; PS, power splitter; EC, electronic coupler.

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 figure: Fig. 2.

Fig. 2. The spectrum evolutions of the optical signal and electrical signal. (a) The spectrum of the external drive signal. (b) The spectrum of the optical signal at point b in Fig. 1. (c) The spectrum of the electrical signal at point c in Fig. 1. (FSn: the nth frequency-shifting signal; SS: the single-sideband signal generated by modulating the seed signal to the +1st sideband of the optical carrier; SSn: the nth single-sideband signal generated by modulating the corresponding FSn-1 to the +1st sideband of the optical carrier.)

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Mathematically, the principle of the wideband signal generation based on the FSRO loop can be expressed as follows. Assuming the optical field of the light wave sent to the FSRO loop can be written as:

$${E_{in}}(t )= {E_c}\exp [{j({2\pi {f_c}t + \phi (t )} )} ]$$
where Ec is the amplitude of the optical carrier, fc is the carrier frequency, andφ(t) is the phase jitter of the laser. Assuming the seed signal is a LFM pulse with a time width of τs and a repetition period of Ts, which can be written as:
$${E_s}(t )= \sum\limits_n {rect} \left[ {\frac{{t - n{T_s}}}{{{\tau_s}}}} \right]{V_s}\cos [{2\pi {f_s}({t - n{T_s}} )+ \pi k{{({t - n{T_s}} )}^2}} ]$$
where Es(t) is the electric field of the seed signal, Vs is the amplitude, fs is the start frequency, and n=0,1,2,3, …, N-1, denotes the nth LFM pulse of the seed signal. In Eq. (2), k = Bs/τs is the chirp rate of the seed signal, in which Bs is the bandwidth of the seed signal. Within the pulse duration of one cycle, the seed signal can be rewritten as:
$${E_s}(t )= {V_s}\cos ({2\pi {f_s}t + \pi k{t^2}} )$$
At the same time, the single-tone reference signal applied to the AOFS can be written as:
$${E_{ref}}(t )= {V_{ref}}\cos ({2\pi {f_{ref}}t} )$$
where Eref is the electric field of the reference signal. Vref is amplitude, and fref is the frequency. When the reference signal is applied to the AOM, the acoustic wave will travel in the AOM with the frequency of fref, realizing frequency shift of the optical carrier. Therefore, the output of two equivalent frequency-shifted modulators can be expressed as:
$$\left[ {\begin{array}{l} {{E_x}(t )}\\ {{E_y}(t )} \end{array}} \right] = \left[ {\begin{array}{c} {{E_{ref}}\textrm{exp} \{{j[{2\pi ({{f_c}t - {f_{ref}}t} )+ \phi (t )} ]} \}}\\ {{E_s}\textrm{exp} \left\{ {j\left[ {2\pi \left( {{f_c}t + {f_s}t + \frac{1}{2}k{t^2}} \right) + \phi (t )} \right]} \right\}} \end{array}} \right]$$
where Ex(t) and Ey(t) are the optical fields of the output of two equivalent frequency-shifted modulators in the upper branch and the lower branch respectively, Eref and Es are the amplitude of Ex and Ey respectively. The two light are then sent to a PD to generate a frequency-shifting (FS) microwave signal. After passing through the electrical feedback link, the first generated FS (FS1) signal can be expressed as:
$${E_1}(t )= {E_1}\cos [{2\pi ({{f_s} + {f_{ref}}} )({t - \tau } )+ \pi k{{({t - \tau } )}^2}} ]$$
where E1 is the amplitude of the FS1 signal, τ is the loop delay. When the time duration of the seed signal is equal to the loop delay, the FS1 signal would replace the seed signal to modulate the MZM in the pulse interval time and then heterodyning with the frequency-shifted optical carrier to generate the second frequency-shifting (FS2) microwave signal:
$${E_2}(t )= {E_2}\cos [{2\pi ({{f_s} + 2{f_{ref}}} )({t - 2\tau } )+ \pi k{{({t - 2\tau } )}^2} + 2\pi {f_{ref}}\tau } ]$$
Where E2 is the amplitude of the FS2 signal. When the seed signal has a bandwidth equaling to the frequency of reference signal, FS2 and FS1 is continuous in the time and the frequency domains. When the loop delay and the frequency of reference signal meet the following condition:
$$2\pi {f_s}\tau + \pi {f_{ref}}\tau = 2m\pi $$
where m=0,1,2,3, …, M-1. The phase at the stitching position is also continuous without any phase jump, which make it possible to stitching the frequency-shifting microwave signals to a TBWP multiplied LFM signal. In this way, the stitched LFM signal can be written as:
$${E_{12}}(t )= {E_{12}}\cos [{2\pi ({{f_s} + {f_{ref}}} )t + \pi k{t^2}} ]$$
where E12(t) is the stitching LFM signal of FS1 signal and FS2 signal, E12 is the amplitude. The time duration and bandwidth of E12(t) are 2τs and 2Bs, respectively. Repeating the above frequency up-conversion and feedback modulation in the FSRO loop until the frequency of the nth frequency-shifting microwave signal reaching the upper cut-off frequency of the electric bandpass filter. The output of the FSRO loop could be written as:
$${E_{FSRO}}(t )= \sum\limits_n {rect[\frac{{t - n{T_s}}}{T}] \cdot E\cos [{2\pi ({{f_s} + {f_{ref}}} )(t - n{T_s}) + \pi k{{({t - n{T_s}} )}^2}} ]}$$
where E is the amplitude of the generated LFM signal, T = Pτs is the time duration, in which T ≤ Ts and P is an integer representing the recirculating times. Equation (10) represents an electrical LFM signal with a start frequency of fs+fref, a bandwidth of PBs and a time duration of Pτs. It can be seen that the time duration and the bandwidth of the generated signal are all multiply P times. Thus, by controlling the times of recirculation, the TBWP are significantly increased and adjustable. Due to the losses of the modulators and the PD in the FSRO loop, power amplification is applied in the loop to compensate for the attenuated power per round trip.

3. Experiment and results

The proposed wideband signal generator is experimentally evaluated. Figure 1 shows the experimental setup. A CW light generated by a 1550 nm laser (TeraXion PS-TNL) with a power of 13 dBm is sent into the FRSO loop and equally split into two branches. One of the branches is sent into an AOFS (Extinction ratio: 57dB, Frequency shift: 200MHz) driven by a single-tone reference signal at a frequency of about 200 MHz to generate an optical reference signal. And the other one is sent to a MZM (EOSPACE, AX-DS5-20) driven by a seed signal, which is a LFM pulse signal with a start frequency of 9.2GHz, a bandwidth of about 200 MHz, a pulse width of about 10.385 us, and a pulse repetition interval (PRI) of 150us generating from an arbitrary waveform generator (Tektronix AWG70001A). By biasing the MZM at a minimum transmission point (MITP), the +1st optical sideband of the modulated optical signal can be obtained at the output of an OBPF (EXFO XTM-50), which is located on the other side of the optical carrier comparing to the reference light. The generated LFM optical signal and the optical reference signal are combined by an optical coupler. Then the combined light wave is delayed by a variable optical delay line of around 10 us and beaten at a PD with responsivity of 0.7 A/W and 3 dB bandwidth of 18 GHz to generate a frequency-shifting signal. After being filtered by a BPF with work frequency from 8 GHz to 12.2 GHz, the frequency-shifting signal with a start frequency of 9.4 GHz and a bandwidth of 200 MHz is sent back to modulate the MZM transmission through the electrical feedback link. At this time, the pulse duration of the seed signal has just ended. Therefore, the feedback signal replaces the seed signal to experience the same up-conversion process repeatedly. An EDFA (KEOPSYS, EDFA-C-26G-S-FA) and a LNA with a gain of 31 dB and noise figure of 2.2 dB are applied to enhance the power of the combined lights and electrical frequency-shifting signal respectively.

After experiencing 14 times recirculation, the highest frequency of the frequency-shifting signal will reach the upper cut-off frequency of the electrical BPF (Work frequency: 8-12.2 GHz). Therefore, an LFM signal with a start frequency of 9.4 GHz and a bandwidth of 2.8 GHz is obtained, whose time duration has also increased by 14 times to about 145 us. Figure 3(a) shows the generated microwave waveform in the time domain, which is measured by a high-speed digital oscilloscope (Keysight DSO-X 92004A). The period of the microwave signal is about 150 us, equaling to the PRI of the seed signal. A zoom-in view of the dashed frame shown in Fig. 3(a) with a 20-ns time duration is plotted in Fig. 3(b). It can be seen that the phase at the connection of the fourth sub-period and the fifth sub-period is continuous with no jump. To further evaluate the performance of phase matching, the residual phase varying with time of the generated LFM signal is depicted in Fig. 3(c), which has the same order of magnitude as a 2.8 GHz LFM signal generated by photonic-assisted frequency multiplication. Figure 3(d) shows the frequency spectrum of the generated microwave waveform measured by an electrical spectrum analyzer (ESA, Keysight N9030A). As we can see, a waveform ranges from 9.4GHz to 12.2GHz are observed with a power fluctuation about 3 dB, indicating that the generated LFM signal has a flat power. The bandwidth is increased to 2.8GHz, which is 14 times the bandwidth of the seed signal. Compared with external modulation, the bandwidth multiplication factor is increased by 10 times. The instantaneous frequency in Fig. 3(e) is obtained with short-time Fourier transform (STFT) function, from which we can see that the generated LFM signal with a frequency from 9.4 to 12.2GHz has a good linearity, and the in-band spurs is about 41.5dB. The time duration is around 145 us, corresponding to a TBWP of 407288, which is 196 times the TBWP of the seed signal. Figure 3(f) reports the results of cross-correlation between the 1st and the 10th pulses. As can be seen that the peak-to-sidelobe ratio is around 13.16 dB, and the 0.1 dB power discrepancy between first sidelobes of the compressed pulse, which is mainly caused by the phase matching error. The 3-dB pulse width of the compressed pulse is 0.34 ns, corresponding to a pulse compression ratio (PCR) of about 427625. As for a LFM signal, the PCR is approximately equal to the TBWP. Therefore, the PCR and TBWP of the signals have been improved greatly comparing with existing methods of LFM signal generation whose TBWPs are actually limited. Moreover, if a higher frequency reference signal is applied to the AOM of the FSRO loop, the PCR, and TBWP can be further improved.

 figure: Fig. 3.

Fig. 3. (a) Waveform of the generated LFM signal in the time domain; (b) zoom-in view with a time duration of 20 ns; (c) calculated residual phase varying with time of the generated LFM signal; (d) the electrical spectra of the generated LFM signal; (e) calculated instantaneous frequency-time diagram of the generated LFM signal; (f) the cross-correlation between the 1st and the 10th pulses;

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To evaluate the reconfigurability of the FSRO-based generator, experiments are performed in which the time duration, the bandwidth and the frequency range are altered by changing the recirculation times and the start frequency of seed signals. The time-frequency curve of generated LFM signals with different time duration and frequency range are shown in Fig. 4. As can be seen, the center frequencies of generated LFM signals are tuned from 4.5 GHz to 10.8 GHz, the bandwidths are turned from 2 GHz to 3 GHz, and the time duration are turned from 15.5 us to 145 us, demonstrating that the proposed system has the capability of generating wideband signals with variable period and frequency. The corresponding TBWP is increased by 196, 100, 100 and 121 times respectively. Notice that the bandwidth can potentially reach several tens of gigahertz in our demonstrated platform when the frequency of reference signal increased.

 figure: Fig. 4.

Fig. 4. Instantaneous frequency of generated LFM signals with different frequency range and time duration. (a) 9.4-12.2 GHz, 85.5us; (b) 4.4-6.4 GHz, 103.8us; (c) 4.4-6.4GHz, 15.5us; (d) 5.6-3.4 GHz, 114.2us.

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Finally, a proof-of-concept rotational microwave imaging experiment using the generated LFM signal operating in 9.4-12.2GHz with a PRI of 145 us is conducted to verify the quality of the generated signals in the practical scenario. The imaging radar system and experiment scene are shown in Fig. 5. A photonics-based de-chirp receiver is used in the experiment, which is based on our previous work [23]. As shown in Fig. 5(b), a rotating platform with a speed of 60 degree/s is located at around 5.6 m far from the radar, and three trihedral corner reflectors (TCRs) are placed on it as the imaging targets in the experiment. As shown in Fig. 5(c), the range distances between them are 4cm and 7cm respectively, and the cross-range distances are 24cm and 26 cm respectively. The generated LFM signal is emitted by the FSRO-based transmitter, reflected by the TCRs and de-chirped in the photonics-based receiver.

 figure: Fig. 5.

Fig. 5. (a) Schematic diagram of the ISAR imaging experimental setup; (b) the ISAR imaging experimental scene; (c) the distance of the TCRs.

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After pulse compression, three TCRs can be identified in the one-dimensional range profile, as shown in Fig. 6(a). The measured range resolution is around 4.8 cm, which is close to the theoretical value of 4.75 cm. An ISAR image of the three targets is obtained after two-dimensional Fourier transforming the de-chirped signal, as depicted in Fig. 6(b). The calculated distances of the three TCRs are 15.1 cm and 15.4 cm in the range direction and 19.3cm and 22.0cm in the cross-range direction respectively, which are consistent with the real condition.

 figure: Fig. 6.

Fig. 6. (a) The electrical spectrum of the de-chirped echo of the rotating TCRs; (b) the ISAR image of three TCRs.

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The experimental results show the effectiveness of the generated LFM signals. Compared with existing microwave photonic radar system using the LFM signals produced by other photonics-assisted signal generation methods, the radar system using the FSRO-based LFM signals generator has a potential to achieve excellent reconfigurability and high-resolution imaging at a very small cost. Therefore, the proposed LFM signal generation scheme is a promising alternative for the development of multifunctional and reconfigurable wideband LFM signals generation.

4. Conclusion

A novel wideband reconfigurable LFM signal generator is proposed and experimentally demonstrated. In the proposed system, a frequency-shifting recirculating optoelectronic loop is employed to adjust the time duration and bandwidth of generated LFM signals, extending the TBWP of the LFM signals. A phase matching process is introduced to the generator to maintain a continuous phase for signal stitching. The performances of proposed signal generator are evaluated through a series of experiments.

Funding

National Key Research and Development Program of China (2018YFA0701900, 2018YFA0701901); National Natural Science Foundation of China (61690191, 61701476).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (6)
Fig. 1.
Fig. 1. The schematic diagram of the proposed signal generator. AOFS, acousto-optic frequency shifter; MZM, Mach-Zehnder modulator; OBPF, optical bandpass filter; EDFA, erbium-doped fiber amplifier; VODL, variable optical delay line; PD, photodetector; LNA, low noise amplifier; BPF, bandpass filter; PS, power splitter; EC, electronic coupler.

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Fig. 2.
Fig. 2. The spectrum evolutions of the optical signal and electrical signal. (a) The spectrum of the external drive signal. (b) The spectrum of the optical signal at point b in Fig. 1. (c) The spectrum of the electrical signal at point c in Fig. 1. (FSn: the nth frequency-shifting signal; SS: the single-sideband signal generated by modulating the seed signal to the +1st sideband of the optical carrier; SSn: the nth single-sideband signal generated by modulating the corresponding FSn-1 to the +1st sideband of the optical carrier.)

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Fig. 3.
Fig. 3. (a) Waveform of the generated LFM signal in the time domain; (b) zoom-in view with a time duration of 20 ns; (c) calculated residual phase varying with time of the generated LFM signal; (d) the electrical spectra of the generated LFM signal; (e) calculated instantaneous frequency-time diagram of the generated LFM signal; (f) the cross-correlation between the 1st and the 10th pulses;

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Fig. 4.
Fig. 4. Instantaneous frequency of generated LFM signals with different frequency range and time duration. (a) 9.4-12.2 GHz, 85.5us; (b) 4.4-6.4 GHz, 103.8us; (c) 4.4-6.4GHz, 15.5us; (d) 5.6-3.4 GHz, 114.2us.

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Fig. 5.
Fig. 5. (a) Schematic diagram of the ISAR imaging experimental setup; (b) the ISAR imaging experimental scene; (c) the distance of the TCRs.

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Fig. 6.
Fig. 6. (a) The electrical spectrum of the de-chirped echo of the rotating TCRs; (b) the ISAR image of three TCRs.

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Equations (10)

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E i n ( t ) = E c exp ⁡ [ j ( 2 π f c t + ϕ ( t ) ) ]
E s ( t ) = ∑ n r e c t [ t − n T s τ s ] V s cos ⁡ [ 2 π f s ( t − n T s ) + π k ( t − n T s ) 2 ]
E s ( t ) = V s cos ⁡ ( 2 π f s t + π k t 2 )
E r e f ( t ) = V r e f cos ⁡ ( 2 π f r e f t )
[ E x ( t ) E y ( t ) ] = [ E r e f exp { j [ 2 π ( f c t − f r e f t ) + ϕ ( t ) ] } E s exp { j [ 2 π ( f c t + f s t + 1 2 k t 2 ) + ϕ ( t ) ] } ]
E 1 ( t ) = E 1 cos ⁡ [ 2 π ( f s + f r e f ) ( t − τ ) + π k ( t − τ ) 2 ]
E 2 ( t ) = E 2 cos ⁡ [ 2 π ( f s + 2 f r e f ) ( t − 2 τ ) + π k ( t − 2 τ ) 2 + 2 π f r e f τ ]
2 π f s τ + π f r e f τ = 2 m π
E 12 ( t ) = E 12 cos ⁡ [ 2 π ( f s + f r e f ) t + π k t 2 ]
E F S R O ( t ) = ∑ n r e c t [ t − n T s T ] ⋅ E cos ⁡ [ 2 π ( f s + f r e f ) ( t − n T s ) + π k ( t − n T s ) 2 ]
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