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Photonic fractional Fourier transformer for chirp radar with ghost target elimination

Guanyu Han, Shangyuan Li, Xiaoxiao Xue, and Xiaoping Zheng

Open Access Open Access

Abstract

For the first time, to the best of our knowledge, we propose a photonic fractional Fourier transformer (PFrFTer), which is used in chirp radar for detecting multiple non-cooperative targets. Based on photonic rotation of the time-frequency plane, the optimal fractional Fourier domain is formed, and the received broadband chirp signals are projected on it, where they behave as impulses. Moreover, through manipulating the fractional Fourier transform spectrum, the PFrFTer contributes to the cancellation of two ghost target sources, so that the ghost targets in multiple-target circumstances are removed. The simulation and experimental results show that the proposed PFrFTer can adapt to multiple non-cooperative targets environments and is immune to ghost targets at optimal working conditions, which agrees well with the theoretical analysis.

© 2020 Optical Society of America

Microwave photonics (MWP) radar, which combines the traditional radar systems and the emerging MWP techniques, has been intensively studied over the past years [1]. By replacing the traditional electronic modules with photonic function units, the radar systems can have a broader bandwidth, reduced size and weight, and faster processing speed [2]. Especially, the applications of optical dechirping in MWP chirp radars [3–8] make the broadband chirp signal directly down-converted to be a single-frequency signal to obtain the time delay information. Therefore, the sampling rate and the computation load can be greatly reduced, enabling the real-time and high-resolution detection and imaging for radar.

Nevertheless, most of the present optical dechirping methods are used to process the received echo from a single cooperative target. Actually, in typical scenarios, the targets being detected are usually non-cooperative, implying that their number and locations are unknown. When more than one target exists in radar’s detection range, the received chirp signals are overlapped in both time and frequency domains and produce undesired ghost targets. To eliminate the ghost target, a new optical processing method is in need to process signals in both time and frequency domains in order to separate the overlapped signals. The fractional Fourier transform (FrFT) deals with signals in both time and frequency domain [9]. In this Letter, we propose a photonic fractional Fourier transformer (PFrFTer) immune to ghost targets.

The PFrFTer is established based on photonic rotation of the time-frequency plane, leading to the formation of the optimal fractional Fourier domain (FrFD), where the broadband chirp signals behave as impulses. Meanwhile, to eliminate the ghost targets, the FrFT spectrum is manipulated by the PFrFTer through optical field processing, so that two ghost target sources have opposite amplitudes and cancel out each other. In the experiment, the echoes of two point targets are projected to the FrFD, and the range profile is obtained according to the FrFT spectrum. The ghost target can be removed easily by applying the proposed PFrFTer which works at the optimal condition. Without any extra optical or RF components, both FrFT operation and ghost target eliminations are achieved by an integrated dual-parallel Mach–Zehnder modulator (DP-MZM), which contributes to a simple and compact structure. Additionally, the optimal working condition of the PFrFTer is fixed and independent of the target number and locations, enabling the PFrFTer to adapt to non-cooperative cases.

The schematic diagram and FrFT principle of the proposed PFrFTer are shown in Fig. 1. The two sub-MZMs of the DP-MZM, respectively, are modulated by the received and transmitted signal ${s_{\!r}}(t)$ and ${s_{\!0}}(t)$, where ${s_{{\!0}}}(t) = \exp [j{\beta _0}(t)] = \exp j(2\pi {f_0}t + \pi k{t^2})$ with ${f_0}$ being the starting frequency and $k$ being the chirp rate. We first consider the single target case, where a target is located at $L$ from the antenna. Accordingly, the received signal has a time delay $\tau {{= 2}}L/c$ with respect to the transmitted signal, and the output optical field of the DP-MZM is

$${E_{\!O}} \propto {E_{\!I}}[1 + {s_{\!r}}(t) + {s_{\!0}}(t) + {\rm c.c.}],$$
where ${E_{\!I}}$ is the optical carrier. Ignoring the DC term, the photocurrent of photodetector (PD) is
$$\begin{split}&i(t) \propto {E_{\!O}}E_O^* \propto s_0^*(t) + s_r^*(t) + {s_{\!r}}(t)s_0^*(t) + {\rm c.c.}\\[-2pt] &{\propto} [1 + \exp (j2\pi k\tau t) + {s_{\!r}}(t)]\exp [- j2\pi ({f_0}t + k{t^2}/2)] + {\rm c.c.}\end{split}$$

As shown in Fig. 1(b), and according to Eq. (2), the photonic rotation of the time-frequency plane is achieved by using the transmitted chirp signal as a rotator to apply the chirp modulation. The rotation angle $\alpha$ is given by $\cot \alpha = - 2\pi k$. The followed Fourier transform (FT) projects the signals onto the new frequency axis, which is the FrFD. We can see that the time-frequency distribution of the received chirp signal with chirp rate $k$ is perpendicular to the rotated frequency axis, leading to its energy concentration in the FrFD. Meanwhile, the spectra of other signals will spread in the FrFD [10] and can be easily removed by threshold detection. Thus, the final output of the PFrFTer is

$$\begin{split}\!\!\!{\rm{PFrFT}}[{s_{\!r}}(t)]& = \int_{- \infty}^\infty {{s_{\!r}}(t)s_0^*(t)\exp (- j2\pi f^\prime\! t){\rm d}t} \\ &\propto \int_{- \infty}^\infty {{s_{\!r}}(t){K_\alpha}(t,f^\prime)} {\rm d}t \propto \delta [2\pi (f^\prime - k\tau)],\!\end{split}$$
where ${K_\alpha}(t,f^\prime) = \exp [- j2\pi ({f_0}t + \pi k{t^2}\!/2 + f^\prime\! t)]$ is the transform kernel that is similar to the conventional FrFT transform kernel: ${K_\alpha}(t,u) = \exp j\{[({t^2} + {u^2})\cot \alpha] /2 - ut\csc \alpha\}$ [3], and $f^\prime $ is the fractional frequency and its relationship with the conventional frequency is $f^\prime = f\sin \alpha$. Note that the phase term $\exp j(\cot \alpha\cdot u^2/2)$ in the traditional FrFT kernel is ignored here, because it is irrelevant to the processed signal and can be discarded in most radar applications to simplify the computation [11]. Besides, in the PFrFT kernel, we introduce an additional frequency term $2\pi\! {f_0}$, which downconverts the signal to the low-frequency band of the FrFD and further reduces the sampling rate.
 figure: Fig. 1.

Fig. 1. (a) Schematic diagram of the proposed PFrFTer. (b) Principle of photonic rotation of the time-frequency plane. (c) Projecting the signals onto the FrFD. OC, optical carrier; DP-MZM, dual-parallel Mach–Zehnder modulator; PD, photodetector; FT, Fourier transform.

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By identifying the impulse location ${f^\prime _{0}}$, we can obtain the range information of the target:

$$L = \tau c/2 = \left| {{f^\prime_0}c/2k} \right|.$$

Usually, the targets are non-cooperative, which means that their number and locations are unknown. For the multiple-target case, each received echo can also serve as the transform kernel to transform other received signals. Thus, the FrFT of the received signals happens and leads to the ghost targets. Fortunately, the DP-MZM offers more flexibility to control the optical field by tuning the three biases, which allows us to manipulate the FrFT spectrum to achieve ghost target elimination. Now we consider a situation in which the echoes come from $n\,(n \ge 2)$ targets. For the $i$th target, we assume it has a distance ${L_i}$ away from the antenna. Thus, the received chirp signals are

$$\begin{split}{s_{\!r}}(t) &= \sum\nolimits_{i = 1}^n {{r_i}{s_{{r_i}}}(t)} = \sum\nolimits_{i = 1}^n {{r_i}\exp [j{\beta _i}(t)]} \\[-2pt] &= \sum\nolimits_{i = 1}^n {{r_i}\exp j[2\pi {f_0}(t - {\tau _i}) + \pi k{{(t - {\tau _i})}^2}]} ,\end{split}$$
where ${\tau _i} = 2{L_i}/c$ is the relative time delay with respect to the transmitted signal, and ${r_i}$ is the amplitude of the received echo from the $i$th target. Thus, the optical field after the DP-MZM is
$${E_{\!O}} \propto {E_{\!I}}\{\cos [\sum\nolimits_{i = 1}^n {{m_i}{s_{{r_i}}}(t)} + {\phi _1}] + {e^{j{\phi _3}}}\cos [{m_0}{s_{\!0}}(t) + {\phi _2}]\} ,$$
where ${m_i}$ and ${\phi _i}$, respectively, represent the modulation index and phase bias of MZM-$i$. Expanding Eq. (6) at the small-signal situation, we have
$$\begin{split}{E_{\!O}} &\propto {E_{\!I}}\left\{a + \sum\limits_{i = 0}^n {{b_i}\exp [\pm j{\beta _i}(t)]}\right. \\[-3pt]&\qquad+ \left.\sum\limits_{i,j = 1}^n {{c_\textit{ij}}\exp j[\pm {\beta _i}(t) \mp {\beta _j}(t)]} \right\},\end{split}$$
where
$$\begin{split}a &= \cos {\phi _1}\prod\nolimits_{p = 1}^n {{J_0}({m_p})} + {e^{j{\phi _3}}}\cos {\phi _2}{J_0}({m_0}),\\[-2pt]{b_0} &= - {e^{j{\phi _3}}}\sin {\phi _2}{J_0}({m_0}),\\[-2pt]{b_i} &= - \sin {\phi _1}{J_1}({m_i})\prod\nolimits_{p = 1,p \ne i}^n {{J_0}({m_p})} \quad (i \ne 0),\\[-2pt]{c_\textit{ij}} &= - \cos {\phi _1}{J_1}({m_i}){J_1}({m_j})\prod\nolimits_{p = 1,p \ne i,j}^n {{J_0}({m_p})} .\end{split}$$

Apart from the OC and the first-order sidebands, there is a third term in Eq. (7), which is from the second-order intermodulation distortion (IMD2) of the MZM-1. Here we ignore the IMD2 term $\exp j[\pm {\beta _i}(t) \pm {\beta _j}(t)]$, because it corresponds to chirp signals with the chirp rate of $2k$, which cannot concentrate on the FrFD and will be removed by threshold detection. Thus, the output of the PFrFTer is

$$\begin{split}&{\rm{PFrFT}}[{s_{\!r}}(t)] = \int_{- \infty}^\infty {{E_{\!O}}E_O^*\exp (- j2\pi {f^\prime}\!t){\rm d}t} {|_{{\rm{threshold}}\;{\rm{detection}}}}\\[-2pt] &\propto \int_{- \infty}^\infty {\left\{\begin{array}{l}\sum\limits_{i = 1}^n {{\Gamma _i}\exp j[{\beta _i}(t) - {\beta _0}(t)]} \\ + \sum\limits_{i,j = 1,i \ne j}^n {{\Gamma _\textit{ij}}\exp j[{\beta _i}(t) - {\beta _j}(t)]} \end{array} \right\}} \exp (- j2\pi {f^\prime}\!t){\rm d}t\\[-2pt] &\propto \sum\limits_{i = 1}^n\! {\int_{- \infty}^\infty\! {{\Gamma _i}{s_i}(t){K_{{\alpha _0}}}(t,f^\prime){\rm d}t}} + \!\sum\limits_{i,j = 1,i \ne j}^n\! {\int_{- \infty}^\infty\! {{\Gamma _\textit{ij}}{s_i}(t){K_{{\alpha _j}}}(t,f^\prime){\rm d}t}} \\[-2pt]& \propto \sum\limits_{i = 1}^n {{\Gamma _i}\delta [2\pi (f^\prime - k{\tau _i})] +} \sum\limits_{i,j = 1,i \ne j}^n {{\Gamma _\textit{ij}}\delta [2\pi (f^\prime - k| {{\tau _i} - {\tau _j}} |)]} ,\end{split}$$
where
$${\Gamma _i} \propto {b_i}b_0^* +{\rm c.c.}\;,\quad \;{\Gamma _\textit{ij}} \propto ac_\textit{ij}^* + {b_i}b_j^* + {\rm c.c.}$$
$$\begin{split}{K_{{\alpha _0}}}(t,f^\prime) &= \exp [- j2\pi (2\pi {f_0}t + k{t^2}\!/2 + f^\prime\! t)],\\{K_{{\alpha _j}}}(t,f^\prime) &= \exp \{- j2\pi [{f_0}(t - {\tau _j}) + k{(t - {\tau _j})^2}\!/2 + f^\prime t]\} .\end{split}$$

The FrFT result in Eq. (9) can be categorized into two types: the first $n$ impulses represent the $n$ real targets which come from projecting the received signals onto the transmitted signal formed FrFD, and the rest $C_n^2 = n(n - 1)/2$ terms are the ghost targets, which originate from projecting one received signal onto the other received signals formed FrFDs. As the target number $n$ increases, the number of ghost targets will be far more than that of the real targets, which is unacceptable for radar systems.

 figure: Fig. 2.

Fig. 2. Principle of ghost target elimination in a multiple-target situation.

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The received signal ${s_{{r_i}}}(t)$ actually experiences two projections onto the received signal ${s_{{r_j}}}(t)$ formed FrFD, corresponding to two ghost target sources, as shown in Fig. 2. One happens at the electro-optic modulation period due to the IMD2 of the MZM-1 and the other happens at the PD end. Equation (10) establishes the relationship between the optical field and the FrFT spectrum, which means that the FrFT spectrum can be manipulated by processing the optical field. To eliminate the ghost target, the two sources should have opposite amplitudes and cancel each other, as shown in Fig. 2; meanwhile, the real targets are preserved, which leads to

$$ac_\textit{ij}^* = - {b_i}b_\textit{ij}^*, \quad {b_i}{b_0} \ne 0.$$

Substituting Eq. (10) into Eq. (12), the biases need to meet

$$\begin{split} \Gamma_{i} &\propto \sin \phi_1 \sin \phi_{2} \cos \phi_{3} J_{1}(m_{0}) J_{1}(m_{i}) \prod_{p=1, p \neq i}^{n} J_{0}(m_{p}) \neq 0 \\[-2pt] \Gamma_{i j} &\propto\left\{\begin{array}{l} \sin ^{2} \phi_{1} f_{0}(m_{i}) J_{0}(m_{j}) \prod_{p=1, p \neq i, j}^{n} J_{0}^{2}(m_{p}) \\[4pt] -\left[\begin{array}{c} \cos \phi_{1} \cos \phi_{2} \cos \phi_{3} J_{0}(m_{0}) \\ +\cos ^{2} \phi_{1} \prod_{p=1}^{n} J_{0}(m_{p})\end{array}\right]\prod\limits_{p=1, p \neq i, j} J_{0}(m_{p})\end{array}\right\}\\[-2pt] &\quad\times J_{1}(m_{i}) J_{1}(m_{j})=0. \end{split}$$

Simplifying Eq. (13), we have

$$\left\{\begin{array}{l}\cos {\phi _1}\cos {\phi _2}\cos {\phi _3}{J_0}({m_0}) = -\! \cos 2{\phi _1}\prod\nolimits_{p = 1}^n {{J_0}({m_p})} \\{\phi _1} \ne {{0}},{\phi _{{2}}} \ne {{0}},{\phi _{{3}}} \ne \pi /2\;\end{array}\right..$$

Equation (14) shows that the optimal bias condition of the DP-MZM is related to the intensities of the transmitted and received signals. However, in particular, when

$${\phi _{{1}}} = \pi {{/4}},\quad {\phi _{{2}}} = \pi {{/2}},\quad{\phi _3} = 0,$$
implying the MZM-1 and MZM-2, respectively, are biased at the quadrature and null point, Eq. (14) can be satisfied permanently; meanwhile, the real targets have maximum amplitudes. It should be noted that this bias setting is irrelevant to the target number and signal intensities, meaning that it can adapt to various non-cooperative detection environments.

Based on the analysis, we simulate two situations wherein two and three targets with different distances are detected. The chirp signal we use covers the full bandwidth of the X-band (8–12 GHz) and has a 10 µs time duration with a 0.5 duty ratio and a 25 GHz sampling rate. We set the modulation index ${m_0} = 0.01$ for the transmitted signal, while the modulation indices of the received signals are set to be ${m_i} = 0.001(i = 1,2,3)$. The FrFT spectrum is obtained by applying fast Fourier transform to the square-law detected optical field, and the threshold detection is followed to remove energy-scattered signals. Finally, the range profile is calculated according to Eq. (4).

We first simulate a double-target case, where two point targets, respectively, are located at 40 and 100 m away from the antenna, corresponding to two received echoes with ${\tau _1} = 0.267\,\,{\rm{\unicode{x00B5} s}}$ and ${\tau _2} = 0.667\,\,{\rm{\unicode{x00B5} s}}$. For the traditional optical dechirping method, both sub-MZMs are biased at null point (${\phi _1} = {\phi _2} = \pi /2,\;{\phi _3} = 0$) to realize the carrier suppressed modulation. As shown in Fig. 3(a), a ghost target situates between the two real targets and is 60 m from the antenna, which is the distance difference of the two real ones. Then, when the DP-MZM is biased at the optimal working points (${\phi _1} = \pi /4,\;{\phi _2} = \pi /2,\;{\phi _3} = 0$), the ghost target is eliminated and the real targets are preserved, as shown in Fig. 3(b).

 figure: Fig. 3.

Fig. 3. Simulated range profiles for (a), (b) double-target and (c), (d) triple-target cases. (a) and (c) without ghost target elimination; (b) and (d) with ghost target elimination.

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Then a triple-target simulation test is performed to further verify the effectiveness of the proposed scheme. We add a third target located at 56.25 m, corresponding to ${\tau _3} = 0.375\,\,{\rm{\unicode{x00B5} s}}$. As shown in Fig. 3(c), apart from the three real targets, we can also see three ghost targets within the detection range before ghost target elimination, and their locations (16.25, 43.75, and 60 m) are exactly the distance differences of any two real targets. When the proposed method is applied, all of the ghost targets are removed, and only real targets exist, as shown in Fig. 3(d), which proves the feasibility of the proposed scheme in non-cooperative cases.

A proof-of-concept experimental setup is demonstrated in Fig. 4. An 8–12 GHz chirp signal with 1µs duration is generated by an arbitrary waveform generator (Tektronix, AWG70002A). Then the chirp signal is split by a 3 dB coupler into two paths. One is sent to the transmitting antenna for target detection, and the other is fed to the DP-MZM (Fujitsu, FTM 7961EX). The receiving antenna, which collects the returning echoes, is put close to the transmitting antenna. After being amplified by an electrical amplifier (EA), the returning echoes are applied to the other microwave port of the DP-MZM. An erbium-doped fiber amplifier (EDFA) is followed to compensate for the insertion loss, and an optical bandpass filter is cascaded after the EDFA to remove the amplified spontaneous emission noise. The photocurrent is sampled by a digital storage oscilloscope (DSO, Agilent DSO81204B) to obtain the FrFT spectrum by digital signal processing (DSP). Finally, the range profile is obtained according to the FrFT spectrum.

 figure: Fig. 4.

Fig. 4. Experimental setup of the proposed PFrFTer-based chirp radar system. AWG, arbitrary waveform generator; EA, electrical amplifier; LD, laser diode; EDFA, erbium-doped fiber amplifier; OBPF, optical bandpass filter; DSO, digital storage oscilloscope; DSP, digital signal processing.

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In the experiment, first two trihedral corner reflectors, respectively, are placed at different distances (about 90 and 175 cm) away from the antenna pair, as shown in Fig. 5(a). We first use the traditional optical dechirping method, and the two sub-MZMs are biased at the null point (${{{V}}_{{1}}}= 3{{.50\,\,\rm V,}}\;{{{V}}_{{2}}}= 3{{.48\,\,\rm V,}}\;{{{V}}_{{3}}}= 0$). A ghost target whose location is the distance difference of the two real targets (about 85 cm) appears, as shown in Fig. 5(b). Then MZM-1, MZM-2, and MZM-3, respectively, are biased at the quadrature, null, and maximum points (${{{V}}_{{1}}}= 1{{.73\,\,\rm V,}}\;{{{V}}_{{2}}}= 3{{.48\,\,\rm V,}}\;{{{V}}_{{3}}}= 0$). In this case, the ghost target is removed, and the real targets survive, as Fig. 5(c) shows.

 figure: Fig. 5.

Fig. 5. Experimental configuration for detecting two trihedral corner reflectors placed at (a), (d) different distances and range profiles (c), (f) with and (b), (e) without ghost target elimination.

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To verify the effectiveness of the PFrFTer in a non-cooperative case, we keep the optimal bias settings and, meanwhile, alter the relative position of the two reflectors. It can be seen that no ghost target appears while the range of target 2 changes according to Fig. 5(f). However, the traditional method still results in the ghost target, as shown in Fig. 5(e).

In summary, a DP-MZM-based PFrFTer for chirp radar is theoretically and experimentally demonstrated. Based on the photonic rotation of the time-frequency plane, the FrFD is formed, where received chirp signal behaves as an impulse. A theoretical analysis shows that a proper optical field processing results in the cancellation of the ghost target sources caused by the FrFT of multiple received signals, which can be achieved by tuning the biases of the DP-MZM. Both simulation and experiment results validate the effectiveness of the proposed PFrFT scheme in multiple non-cooperative targets cases. Besides, the integrated DP-MZM enables the system to have a reduced size and a simple structure, and the optimal working condition is independent of the target number and locations, showing that this PFrFTer can find its applications in complicated detection environments.

Funding

National Natural Science Foundation of China (61690191); National Key Research and Development Program of China (2019YFB2203301, 2018YFA0701902).

Disclosures

The authors declare no conflicts of interest.

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Figures (5)
Fig. 1.
Fig. 1. (a) Schematic diagram of the proposed PFrFTer. (b) Principle of photonic rotation of the time-frequency plane. (c) Projecting the signals onto the FrFD. OC, optical carrier; DP-MZM, dual-parallel Mach–Zehnder modulator; PD, photodetector; FT, Fourier transform.

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Fig. 2.
Fig. 2. Principle of ghost target elimination in a multiple-target situation.

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Fig. 3.
Fig. 3. Simulated range profiles for (a), (b) double-target and (c), (d) triple-target cases. (a) and (c) without ghost target elimination; (b) and (d) with ghost target elimination.

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. Experimental setup of the proposed PFrFTer-based chirp radar system. AWG, arbitrary waveform generator; EA, electrical amplifier; LD, laser diode; EDFA, erbium-doped fiber amplifier; OBPF, optical bandpass filter; DSO, digital storage oscilloscope; DSP, digital signal processing.

View in Article | Download Full Size | PDF

Fig. 5.
Fig. 5. Experimental configuration for detecting two trihedral corner reflectors placed at (a), (d) different distances and range profiles (c), (f) with and (b), (e) without ghost target elimination.

View in Article | Download Full Size | PDF

Equations (15)

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E O ∝ E I [ 1 + s r ( t ) + s 0 ( t ) + c . c . ] ,
i ( t ) ∝ E O E O ∗ ∝ s 0 ∗ ( t ) + s r ∗ ( t ) + s r ( t ) s 0 ∗ ( t ) + c . c . ∝ [ 1 + exp ⁡ ( j 2 π k τ t ) + s r ( t ) ] exp ⁡ [ − j 2 π ( f 0 t + k t 2 / 2 ) ] + c . c .
P F r F T [ s r ( t ) ] = ∫ − ∞ ∞ s r ( t ) s 0 ∗ ( t ) exp ⁡ ( − j 2 π f ′ t ) d t ∝ ∫ − ∞ ∞ s r ( t ) K α ( t , f ′ ) d t ∝ δ [ 2 π ( f ′ − k τ ) ] ,
L = τ c / 2 = | f 0 ′ c / 2 k | .
s r ( t ) = ∑ i = 1 n r i s r i ( t ) = ∑ i = 1 n r i exp ⁡ [ j β i ( t ) ] = ∑ i = 1 n r i exp ⁡ j [ 2 π f 0 ( t − τ i ) + π k ( t − τ i ) 2 ] ,
E O ∝ E I { cos ⁡ [ ∑ i = 1 n m i s r i ( t ) + ϕ 1 ] + e j ϕ 3 cos ⁡ [ m 0 s 0 ( t ) + ϕ 2 ] } ,
E O ∝ E I { a + ∑ i = 0 n b i exp ⁡ [ ± j β i ( t ) ] + ∑ i , j = 1 n c ij exp ⁡ j [ ± β i ( t ) ∓ β j ( t ) ] } ,
a = cos ⁡ ϕ 1 ∏ p = 1 n J 0 ( m p ) + e j ϕ 3 cos ⁡ ϕ 2 J 0 ( m 0 ) , b 0 = − e j ϕ 3 sin ⁡ ϕ 2 J 0 ( m 0 ) , b i = − sin ⁡ ϕ 1 J 1 ( m i ) ∏ p = 1 , p ≠ i n J 0 ( m p ) ( i ≠ 0 ) , c ij = − cos ⁡ ϕ 1 J 1 ( m i ) J 1 ( m j ) ∏ p = 1 , p ≠ i , j n J 0 ( m p ) .
P F r F T [ s r ( t ) ] = ∫ − ∞ ∞ E O E O ∗ exp ⁡ ( − j 2 π f ′ t ) d t | t h r e s h o l d d e t e c t i o n ∝ ∫ − ∞ ∞ { ∑ i = 1 n Γ i exp ⁡ j [ β i ( t ) − β 0 ( t ) ] + ∑ i , j = 1 , i ≠ j n Γ ij exp ⁡ j [ β i ( t ) − β j ( t ) ] } exp ⁡ ( − j 2 π f ′ t ) d t ∝ ∑ i = 1 n ∫ − ∞ ∞ Γ i s i ( t ) K α 0 ( t , f ′ ) d t + ∑ i , j = 1 , i ≠ j n ∫ − ∞ ∞ Γ ij s i ( t ) K α j ( t , f ′ ) d t ∝ ∑ i = 1 n Γ i δ [ 2 π ( f ′ − k τ i ) ] + ∑ i , j = 1 , i ≠ j n Γ ij δ [ 2 π ( f ′ − k | τ i − τ j | ) ] ,
Γ i ∝ b i b 0 ∗ + c . c . , Γ ij ∝ a c ij ∗ + b i b j ∗ + c . c .
K α 0 ( t , f ′ ) = exp ⁡ [ − j 2 π ( 2 π f 0 t + k t 2 / 2 + f ′ t ) ] , K α j ( t , f ′ ) = exp ⁡ { − j 2 π [ f 0 ( t − τ j ) + k ( t − τ j ) 2 / 2 + f ′ t ] } .
a c ij ∗ = − b i b ij ∗ , b i b 0 ≠ 0.
Γ i ∝ sin ⁡ ϕ 1 sin ⁡ ϕ 2 cos ⁡ ϕ 3 J 1 ( m 0 ) J 1 ( m i ) ∏ p = 1 , p ≠ i n J 0 ( m p ) ≠ 0 Γ i j ∝ { sin 2 ⁡ ϕ 1 f 0 ( m i ) J 0 ( m j ) ∏ p = 1 , p ≠ i , j n J 0 2 ( m p ) − [ cos ⁡ ϕ 1 cos ⁡ ϕ 2 cos ⁡ ϕ 3 J 0 ( m 0 ) + cos 2 ⁡ ϕ 1 ∏ p = 1 n J 0 ( m p ) ] ∏ p = 1 , p ≠ i , j J 0 ( m p ) } × J 1 ( m i ) J 1 ( m j ) = 0.
{ cos ⁡ ϕ 1 cos ⁡ ϕ 2 cos ⁡ ϕ 3 J 0 ( m 0 ) = − cos ⁡ 2 ϕ 1 ∏ p = 1 n J 0 ( m p ) ϕ 1 ≠ 0 , ϕ 2 ≠ 0 , ϕ 3 ≠ π / 2 .
ϕ 1 = π / 4 , ϕ 2 = π / 2 , ϕ 3 = 0 ,
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