Investigating Intra-Pulse Doppler Frequency Coupled in the Radar Echo Signal of a Plasma Sheath-Enveloped Target

Abstract

In detecting hypersonic vehicles, the radar echo signal is coupled with an intra-pulse Doppler frequency (I-D frequency) component caused by relative motion of a plasma sheath (PSh) and the vehicle, which can induce the phenomenon of a ghost target in a one-dimensional range profile. In order to investigate the I-D frequency generated by the relative motion of a PSh, this study transforms a linear frequency modulated (LFM) signal into a single carrier frequency signal based on echo signal equivalent time delay-dechirp processing and realizes high resolution and fast extraction of the I-D frequency coupled with the frequency-domain echo signal. Furthermore, by relying on the computation of the surface flow field of the RAMC-II Blunt Cone Reentry Vehicle, the coupled I-D frequency in the radar echo signal of a PSh-enveloped target under circumstances of typical altitudes and carrier frequencies is extracted and further investigated, revealing the variation law of I-D frequency. The key findings of this study provide a novel approach for suppressing anomalies in radar detection of PSh-enveloped targets as well as effective detecting and as robust target tracking.

1. Introduction

During high-velocity motion of near-space hypersonic vehicles, the high-velocity incoming flow causes energy conversion, during which enormous kinetic energy is transformed into internal energy that causes high temperature. According to the “real effect of high-temperature gas”, the vehicle surface is enveloped in a PSh [1,2], which changes the target electromagnetic (EM) characteristics of hypersonic vehicles. When the EM wave propagates in the PSh, charged particles and various ablation products will cause certain complicated EM effects on EM waves, resulting in a poor EM environment of the target and severe distortion of the EM wave [3,4].

The existing research on PSh-enveloped targets primarily attached great importance to the impact of a exerted on EM wave transmission characteristics, analyzing amplitude attenuation and phase distortion of EM waves induced by the PSh [5,6,7,8,9,10,11]. Researchers have investigated plasma ground generation devices to analyze the impact mechanism of plasma on EM waves based on ground tests. As various hypersonic vehicles emerge, robust radar detection on PSh-enveloped targets has gradually been carried out [2,12,13,14,15]. Ding et al. constructed a time-domain radar echo model of a PSh-enveloped target and jointly analyzed the characteristics of a radar echo signal in multiple domains, revealing the cause of ghost targets [16]. Later, based on parallel compensation processing of I-D frequency, they improved the gain of the energy ratio of real target to ghost target and effectively suppressed the ghost target [17].

Owing to the fact that a linear frequency modulated (LFM) signal belongs to the scope of a large time-bandwidth product signal, the effective extraction of I-D frequency remains under the influence of signal bandwidth. The relative motion of a PSh with fluid characteristics can cause coupling amongst different I-D frequency components in radar echo signals that are relatively close. The existing I-D frequency extraction methods for a LFM echo signal usually adopt time-frequency analytical approaches and use time-frequency processing to extract the frequency components of a coupled radar echo signal [18,19,20,21]. Nevertheless, the existing time-frequency approaches have obvious drawbacks. When the Wigner–Ville distribution (WVD) transform is used to extract I-D frequency, it is susceptible to a multi-component signal, and the cross term is formed in the time-frequency domain, causing errors in I-D frequency extraction results. To solve the above problem, some scholars proposed pseudo WVD transform and smooth WVD transform. Despite the fact that the aforementioned approaches can eliminate the influence of the cross term to some degree, the frequency resolution is significantly reduced; for this reason, the relatively close I-D frequency cannot be effectively extracted. Although the short-time Fourier transform (SFT) is capable of frequency analysis under variable time scales, the frequency resolution is very limited. The fractional Fourier transform (FT) has low frequency resolution and can extract a close I-D frequency. However, it requires large amount of computation while consuming more computing resources, which hinders it from being applied in real-time extraction of I-D frequency. During the process of suppressing a ghost target of an echo signal, the abovementioned deficiencies not only cause the degeneration of the extraction accuracy of the I-D frequency but also affect the suppression effect. Additionally, there are few studies on the coupled I-D frequency component in the echo of a PSh-enveloped target, and the influence law of flight altitude and carrier frequency on I-D frequency remains to be obtained.

In this study, by relying on equivalent echo time delay estimation, we transformed the problem of frequency extraction into one of estimating time-domain signal parameters. Based on equivalent time delay-deviation processing, the influence of bandwidth is eliminated, and the I-D frequency high-resolution extraction of the echo signal can be effectively conducted. Furthermore, by adopting our proposed approach, the variation law regarding I-D frequency is revealed according to the computation with respect to the flow field. The key contributions of this study are summarized as follows.

Firstly, by relying on the technology of estimating pulse parameters, the time delay of the radar echo signal is equivalent. By performing dechirp-Fourier transform, we realized high-resolution extraction of the coupled I-D frequency component in a radar echo signal.

Secondly, according to the computation results of the surface flow field of the RAMC-II Blunt Cone Reentry Vehicle, the spatial distribution characteristics of the PSh velocity field along the region from the stagnation point to the tail end are analyzed. Our proposed method is capable of extracting the I-D frequency under typical altitude and carrier frequency, thereby revealing the variation law of the coupled I-D frequency in the echo signal of a PSh-enveloped target (different flight altitudes and carrier frequencies). The key findings of this study provide a novel approach for suppressing anomalies in radar detection of PSh-enveloped targets, in effective target detection as well as in robust target tracking.

The remainder of this paper is organized as follows. Section 2 outlines the extraction process of the I-D frequency coupled in the echo signal of the PSh-enveloped target. Section 3 verifies the feasibility of our proposal via simulation, revealing the variation law of I-D frequency under circumstances of typical flight altitudes and carrier frequencies. Section 4 summarizes the simulation and theoretical research of this study.

2. I-D Frequency Extraction of PSh-Enveloped Target Echo Signal

2.1. Radar Echo Model of the Target Enveloped in a PSh

When the velocity of a hypersonic vehicle exceeds Mach 10, its surface will be enveloped in a PSh, which is demonstrated as follows.

Assume the radar signal as a far-field plane wave when the PSh-enveloped target locates at far field. The echo signals are generated at different positions of the target reference points (the regions of the stagnation point, the middle part, and the tail end). Figure 1b shows that L reference points are selected on the vehicle surface. Assume the electron density of the i-th layer at the l-th reference point is Ne(l,i). Due to the spatial distribution of the characteristic parameters involving the electron density Ne(l,i), the collision frequency Ve(l,i), and the velocity V(l,i) of the PSh, the plasma’s multi-dimensional modulation effect exerted on the echo signals differs at each position [16]. Therefore, the characteristic frequency f_p(l,i) of this layer can be described by

$$f_p(l,i)=\frac{1}{2\pi }\sqrt{\frac{Ne(l,i)·e^2}{m_e·{\epsilon }_0}},$$

(1)

where e is the unit charge, m_e is the electron mass, and ε₀ is the vacuum permittivity. Correspondingly, the dielectric constant ε(l,i) can be expressed as

$$\epsilon (l,i)=(1-2\pi \frac{f_p^2(l,i)}{{(2\pi f_c)}^2+v_e^2(l,i)}-j4{\pi }^2\frac{f_p^2(l,i)v_e(l,i)/(2\pi f_c)}{{(2\pi f_c)}^2+v_e^2(l,i)})·{\epsilon }_0$$

(2)

where f_c denotes the carrier frequency of EM wave, and j denotes the unit vector of the imaginary part.

By adopting the above Equation (2) and μ₀ (vacuum permeability), the propagation constant k(l,i) and the intrinsic wave impedance Z(l,i) of the i-th layer at the l-th position of plasma can be obtained, for which the specific expression can be described by

$$k(l,i)=2\pi f_c\sqrt{{\mu }_0\epsilon (l,i)}$$

(3)

$$Z(l,i)=\sqrt{{\mu }_0/\epsilon (l,i)}.$$

(4)

By comparing the carrier frequency f_c of the radar signal with the characteristic frequency f_p(l,i) at each position, the reflection coefficient at different positions and the coupled velocity components in the radar echo signal can be determined according to the above Equations (3) and (4).

When both the characteristic frequency f_p(l,i) of the plasma sheath at the l-th reference point of the i-th layer and the carrier frequency f_c of the radar echo signal altogether satisfy the following three conditions

$$f_c>f_p(l,i-1)\hspace{1em}\hspace{1em}{f}_c\approx f_p(l,i)\hspace{1em}\hspace{1em}{f}_c<f_p(l,i+1),$$

(5)

the incident depth of the EM wave at the l-th reference position of the PSh is i, the reflection coefficient of the i-th layer is taken as the reflection coefficient at the reference point position, and the flow-field velocity of the i-th layer is the velocity of the reflected wave coupling at the reference point.

The reflection coefficient R(l) and the coupled velocity v(l) of the echo signal wave at the l-th reference position can be expressed as

$$R(l)=\frac{\left(A(l,i)+B(l,i)/Z(l,i+1)\right)-Z_0\left(C(l,i)+D(l,i)/Z(l,i+1)\right)}{\left(A(l,i)+B(l,i)/Z(l,i+1)\right)+Z_0\left(C(l,i)+D(l,i)/Z(l,i+1)\right)}$$

(6)

$$v(l)=V(l,i),$$

(7)

where Z₀ denotes the wave impedance in vacuum; Z(l, i + 1) denotes the wave impedance at the l-th reference point of the (i + 1)-th layer of the PSh; and A(l), B(l), C(l), and D(l) denote the transmission parameters at the l-th reference point, which are expressed as

$$\left[\begin{array}{cc}A(l)& B(l)\\ C(l)& D(l)\end{array}\right]=\left[\begin{array}{cc}{A}_1(l)& B_1(l)\\ C_1(l)& D_1(l)\end{array}\right]\dots \left[\begin{array}{cc}{A}_{i-1}(l)& B_{i-1}(l)\\ C_{i-1}(l)& D_{i-1}(l)\end{array}\right]\dots \left[\begin{array}{cc}{A}_i(l)& B_i(l)\\ C_i(l)& D_i(l)\end{array}\right].$$

(8)

In the above Equation (8), the symbol “[.]” denotes the transmission matrix of the i-th layer, which can be rewritten as

$$\left[\begin{array}{cc}{A}_i(l)& B_i(l)\\ C_i(l)& D_i(l)\end{array}\right]=\left[\begin{array}{cc}\cosh (jk(l,i)d_i)& Z(l,i)·\sinh (jk(l,i)d_i)\\ \sinh (jk(l,i)d_i)/Z(l,i)& \cosh (jk(l,i)d_i)\end{array}\right],$$

(9)

where d_i denotes the thickness of the i-th layer of the PSh.

When both the characteristic frequency f_p(l,i) of the PSh at the l-th reference point of the i-th layer and the carrier frequency f_c of the radar signal altogether satisfy the following condition

$$f_c>\{f_p(l,i)\left|i=1,2\dots I\right.\},$$

(10)

it means that the EM wave is capable of penetrating into the PSh at this position and will be reflected on the target surface. The above Equations (6) and (7) can be simplified into

$$R(l)=\frac{B(l,i)-Z_{0}D(l,i)}{B(l,i)+Z_{0}D(l,i)}$$

(11)

$$v(l)=V_0,$$

(12)

where V₀ denotes the actual moving velocity of the target (i.e., the radial velocity).

The LFM pulse signal model transmitted by the radar can be described by [22,23,24]

$$S(t)=Arect(\frac{t}{T_p})\exp (j2\pi (f_{c}t+\frac{u}{2}{t}^2)),$$

(13)

where “rect(.)” denotes the gate function, the symbol “A” denotes the radar signal amplitude, u denotes the modulation frequency, and T_p denotes the pulse width.

According to

Table 1. Simulation parameters of flow field and echo.

Parameter Type	Parameter Setting	Values
Parameters of target and flow field	Maximum plasma density	Nemax = 2.67 × 1020/m3
	Height	H = 40 km
	Shape of aircraft	Blunt cone
	Velocity of vehicle	Vtar = 4521 m/s
Radar signal parameters	Carrier frequency	fc = 3.3 GHz
	Bandwidth	B = 5 MHz
	Pulse width	Tp = 500 μs
	Distance of target	R0 = 80 km/100 km/120 km

presented in the following Section 3.1, the reflection coefficient and the I-D frequency at different points of the PSh-enveloped target are computed, and the radar echo model in time domain is established. It is worth noting that since this paper mainly discusses narrowband radar, and the pulse width is not very large, we adopt the go-stop model when building the radar echo model, which means that the time delay is approximately a fixed value in the pulse time. Therefore, the radar echo model in the time domain can be described as

$$R_{pla}(t)={\displaystyle \sum _{l=1}^{L}R(l)·S(f_c+f_d(l),t-{\tau }_0)}.$$

(14)

In the above Equation (14), τ₀ denotes the time delay, and the term f_d(l) = 2v(l) × f_c/c denotes the I-D frequency corresponding to the coupling velocity of the radar echo signal at the position of the l-th reference point (c denotes the speed of light). The above Equation (14) can be rewritten as [19]

$$R_{pla}(t)={\displaystyle \sum _{l=1}^{L}R(l)·rect(\frac{t-{\tau }_0}{T_p})\exp (j\pi u{(t-{\tau }_0)}^2)·{\varphi }_0}(l),$$

(15)

where φ₀(l) represents the phase term that includes the I-D frequency, which can be expressed as

$${\phi }_0(l)=\exp (-j2\pi (f_c+f_d(l)){\tau }_0)·\exp (j2\pi f_d(l)t).$$

(16)

It can be deduced from the above Equations (15) to (16) that the relative motion of the PSh causes the echo signals to couple with different I-D frequency components. The I-D frequency component causes abnormal processing of the matched filter, thus causing a “ghost target” phenomenon on the one-dimensional range profile.

2.2. Equivalent Time Delay Derivation

When the PSh-enveloped target is detected by radar, the target-to-radar range causes a time delay, which further leads to the envelope offset of the radar signal on the time axis.

The transmitted radar signal is demonstrated as follows.

When the transmitted signal is reflected on the target surface, its echo will be coupled with the time delay that is caused by the target-to-radar range, thereby inducing the envelope offset phenomenon. The schematic diagram demonstrating the echo signal is presented in the following figure.

It can be observed that the range causes an envelope offset of the echo signal. In Figure 2, the start time of the transmitting signal is 0, and the end time is the pulse width T_p of the signal. In Figure 3, the envelope of the echo is offset to the right side along the time axis, and its start time correlates with the delay of the echo signal coupling. Combining Equations (13) and (14), we can further conclude that the envelope offset of the echo is τ₀, while its time delay τ₀ can be equivalent by extracting its start time.

The time delay τ₀ corresponding to the echo’s start time can be calculated by

$${\tau }_0=T_{OA}=\frac{2\times R_0}{c},$$

(17)

where R₀ denotes the target-to-radar range.

2.3. Estimating Pulse Parameter of the Echo

It has been discussed in Section 2.2 that the time delay can be equivalently obtained by estimating the echo signal’s start time. In this subsection, we estimate the echo signal’s start/end time according to the difference of the boxes (DOB) filter.

Let the length of the DOB filter h(n) be N (the order is 2N + 1), so that h(n) can be described by

$$h(n)=\left\{\begin{array}{cc}1\hfill & n\in [-N,-1]\hfill \\ 0\hfill & n=0\hfill \\ -1\hfill & n\in [1,N]\hspace{1em}\hfill \end{array}\right..$$

(18)

Based on Equation (18), convolution processing is conducted with respect to the echo signal to obtain the output signal y(n), which can be calculated by

$$y(n)=R_{pla}(t)*h(n)={\displaystyle \sum _x{R}_{pla}(t)\times h(n+1-x)}.$$

(19)

By substituting Equation (15) into Equation (19), the calculation process can be subdivided into the following seven situations:

• ① When $1\le n\le N$, Equation (19) can be rewritten as

  $$y\left(n\right)={\displaystyle \sum _{p=1}^n{R}_{pla}\left(p\right)};$$

  (20)

• ② When $n=N+1$, Equation (19) can be rewritten as

  $$y\left(n\right)={\displaystyle \sum _{p=2}^{N+1}{R}_{pla}\left(p\right)};$$

  (21)

• ③ When $N+2\le n\le 2N+1$, Equation (19) can be rewritten as

  $$y\left(n\right)={\displaystyle \sum _{p=n-N+1}^n{R}_{pla}\left(p\right)}-{\displaystyle \sum _{q=1}^{n-N-1}{R}_{pla}\left(q\right)};$$

  (22)

• ④ When $2N+2\le n\le M-1$, Equation (19) can be converted into

  $$y\left(n\right)={\displaystyle \sum _{p=n-N+1}^n{R}_{pla}\left(p\right)}-{\displaystyle \sum _{q=n-2N}^{n-N-1}{R}_{pla}\left(q\right)};$$

  (23)

• ⑤ When $M\le n\le M+N-1$, Equation (19) can be rewritten as

  $$y\left(n\right)={\displaystyle \sum _{p=n-N+1}^M{R}_{pla}\left(p\right)}-{\displaystyle \sum _{q=n-2N}^{n-N-1}{R}_{pla}\left(q\right)};$$

  (24)

• ⑥ When $n=M+N$, Equation (19) can be rewritten as

  $$y\left(n\right)=-{\displaystyle \sum _{p=M-N}^{M-1}{R}_{pla}\left(p\right)};$$

  (25)

• ⑦ When $M+N+1\le n\le M+2N$, Equation (19) can be rewritten as

  $$y\left(n\right)=-{\displaystyle \sum _{p=n-2N}^M{R}_{pla}\left(p\right)}.$$

  (26)

Combining Equations (20)–(26), the recursive expression of y(n) can be described by

$$y\left(n+1\right)=\left\{\begin{array}{l}y\left(n\right)+R_{pla}\left(n\right), 1\le n\le N-1\hfill \\ y\left(n\right)-R_{pla}\left(n-N\right)+R_{pla}\left(n-N+1\right)+R_{pla}\left(n+1\right), N+1\le n\le 2N\hfill \\ y\left(n\right)+R_{pla}\left(n-2N+1\right)-R_{pla}\left(n-N\right)-R_{pla}\left(n-N+1\right)+R_{pla}\left(n+1\right), 2N+1\le n\le M-2\hfill \\ y\left(n\right)+R_{pla}\left(n-2N\right)-R_{pla}\left(n-N\right)-R_{pla}\left(n-N+1\right), M-1\le n\le M+N-2\hfill \\ y\left(n\right)+R_{pla}\left(n-2N\right), M+N-1\le n\le M+2N-1\hfill \end{array}\right.,$$

(27)

where y(n) is updated by sliding recursion, forming a maximum peak and a minimum peak at the echo’s start time and end time, respectively.

By extracting the peak position, the start time T_OA and the end time T_OE of the echo signal can be estimated by

$$T_{OA}=\frac{n_{\max }-N}{Fs}\hspace{1em}\hspace{1em}{T}_{OE}=\frac{n_{\min }-N}{Fs},$$

(28)

where n_max denotes the coordinates of the maximum value, and n_min denotes the coordinates of the minimum value.

2.4. Extracting I-D Frequency

By adopting the echo signal’s prior parameters (modulation frequency μ, pulse width T_p, and Sampling frequency F_s), the dechirp signal D(t) can be constructed, which is expressed as

$$D(t)=\exp (-j\pi \mu {(0:1/F_s:T_p)}^2).$$

(29)

According to the estimation result of T_OA and T_OE described in Equation (28), when substituting the dechirp Equation (29) to the echo signal’s start time, we obtain

$$DZ(T_{OA}:T_{OE})=D(t).$$

(30)

When the echo signal is processed according to Equation (30), then we obtain

$$RD(t)=R_{pla}(t)\times Dz(t).$$

(31)

The above Equation (31) can further be rewritten as

$$RD(t)={\displaystyle \sum _{l=1}^{L}Ref(n)·rect(\frac{t-{\tau }_0}{T_p})·{\phi }_0}(l).$$

(32)

When we substitute Equation (16) into the above Equation (32), we obtain

$$RD(t)={\displaystyle \sum _{l=1}^{L}R(l)·rect(\frac{t-{\tau }_0}{T_p})\exp (-j2\pi (f_c+f_d(l)){\tau }_0)·\exp (j2\pi f_d(l)t)}.$$

(33)

From Equation (33), we observe that the echo signal after being dechirp-processed does not contain the time quadratic term, and the coefficient with the time term is the I-D frequency.

When we perform FT with respect to Equation (33), we obtain

$$RD(f)={\displaystyle \sum _{t=1}^T[RD(t)]}\exp (j\frac{2\pi f(t-1)}{T}),$$

(34)

where T denotes the total sampling point numbers of the echo signal. By substituting Equation (33) into Equation (34), the echo signal represented in the frequency domain after being dechirp-processed can be rewritten as

$$RD(f)={\displaystyle \sum _{l=1}^{L}R(l)sa(f-f_d(l))\exp (-j2\pi f{\tau }_0)\exp (j2\pi f_d(l){\tau }_0)\exp (j2\pi f_c{\tau }_0)},$$

(35)

where the sa(.) function produces a frequency-domain peak, and different I-D frequencies generate different frequency-domain peaks. The peak coordinates corresponding to f_d(l) can be obtained through the peak search, and the frequency corresponding to the peak coordinates can be solved to realize the extraction of I-D frequencies.

For the abovementioned Equations (17)–(35), the following Figure 4 demonstrates the flow chart of the intra-pulse Doppler frequency extraction method proposed in this study.

2.5. Analyzing Variation Law of I-D Frequency

According to the aforementioned computation results, the I-D frequency of the echo signal of a PSh-enveloped target under circumstances of typical altitudes and carrier frequencies is extracted, for which the variation law regarding the I-D frequency is analyzed.

Assume the attack angle of the vehicle in the flow-field calculation is 0 and the PSh on the vehicle surface is symmetric regarding planes passing through the X-axis. The slice processing of the XOY plane is conducted according to the computation results of the of plasma flow field, for which the schematic diagram is presented in Figure 5.

It can be observed that by extracting the data at the position of the reference points, the numerical computation results (electron density, collision frequency, and velocity component) of the characteristic parameters in different areas of the PSh can be obtained, for which the specific process is demonstrated in Figure 6.

The I-D frequency in the echo signal can be extracted via conducting the specific steps presented in the above Figure 6. By using the obtained computation results of the flow field, we investigated the coupled I-D frequency in the echo signal characterized by different flight altitudes and carrier frequencies and revealed the variation law of the I-D frequency.

3. Simulation Analysis of I-D Frequency Extraction

3.1. Pulse Parameter Extraction of Radar Echo Signal

Based on the plasma flow-field data of the blunt cone aircraft at the height of 40 km, the maximum reflection intensity and coupled Doppler frequency of the radar echo are calculated at each reference point after it is perpendicular to the incident plasma flow field on the surface of the aircraft. The plasma flow-field parameters and radar signal parameters are shown in Table 1. By referring to the aforementioned Equations (13) and (15), the transmitted signal and the time-domain radar echo signal are simulated, and a convolution operation is conducted with respect to the echo signal based on the DOB filter to extract the echo’s start time. Considering the target-to-radar range, we also analyzed the relationship between the start time and the time delay.

As shown in Figure 7, the red dotted line indicates results from the echo signal output through the DOB filter. It can be observed that the three target-to-radar ranges differ, and the offset caused by the time-domain echo signal differs accordingly. The transmitted signal possesses 500 μs pulse width, 0 μs start time, and 500 μs end time, respectively.

When the target-to-radar range is 80 km, the echo’s time delay is computed to be 533.3 μs using Equation (17). The DOB filter processes the echo to form two peaks in the time domain. The start time of the echo measured by the method proposed in this paper is 533.3 μs, which is consistent with the theoretical value. The echo’s end time is 1033.4 μs, and its pulse width is estimated to be 20.01 μs throughout the start and end times, in which the error with the actual pulse width is 0.02%. When the target-to-radar range is 100 km, the time delay generated by the target is 666.7 μs. The output maximum peak position based on the DOB filter is 666.7 μs, which is consistent with the time delay caused by the target, in which the error between the echo’s pulse width and the actual pulse width is 0.02%. When the target-to-radar range is 120 km, the target time delay is 800 μs. The DOB output result of the echo is consistent with the actual delay of the target, and the error between the estimated pulse width of the echo and the actual pulse width is 0.02%.

It can be acknowledged that the start time of the echo can be extracted based on DOB filter processing, and the echo’s time delay can be obtained.

3.2. I-D Frequency Extraction of Echo Signals

The existence of the difference in spatial distribution of the PSh suggests that the echo and the multi-I-D frequency components will be coupled. In this section, based on the simulated radar echo signal in Section 3.1, we analyze the inadequate conventional time-frequency processing method (WVD, fractional Fourier transform (FRFT)) to extract the I-D frequency. According to Equations (14), (29), and (35), the simulated radar echo signal is processed, and the frequency extraction method was verified. In this section, the target distance of the simulated echo is 100 km, and other simulation parameters are consistent with those in Section 3.1.

It can be observed from Figure 8a that since the I-D frequencies of the echo (the target echo signal at the same range) differ, the time-domain echo signal is distorted, causing a “periodic dispersion” phenomenon. Additionally, we observe from Figure 8b that a significant modulation phenomenon occurs in the frequency-domain echo signal, and a serious “oscillation” phenomenon appears in the bandwidth.

We observe from Figure 8c that the energy of the FRFT transformation results distributes on a straight line. The FRFT transform of the echo is limited by the frequency resolution. When the I-D frequencies are closely coupled in the echo, energy aliasing will occur, under which circumstance the peak position will expand, preventing the I-D frequency from being extracted effectively.

Furthermore, Figure 8d demonstrates that some slant lines occur in a two-dimensional time-frequency plane of Wigner–Ville transform results of the echo signals, which have the same slope and intercepts (the same modulation frequency and different carrier frequencies). Because of the existence of multiple I-D frequency components in the echo, the time-frequency curve contains many cross terms, causing errors in the extraction results of the I-D frequency.

Figure 9a suggests that when the echo is dechirp-processed, its time-frequency curve is a straight line perpendicular to the frequency axis. According to Equation (32), the dechirp processing can effectively eliminate the modulation frequency u of the echo, for which reason the signal only contains the carrier frequency information. Figure 9b demonstrates that the frequency-domain characteristics of the dechirp-processed echo can be obtained by Fourier transform. Since the modulation frequency of the echo is eliminated, no influence will be exerted on the bandwidth in the frequency domain (the echo signal is transformed from a LFM signal to a single carrier frequency signal). There are several significant peaks in the frequency spectrum, and the peak positions correspond to the signal carrier frequency.

Table 2. Performance of the proposed method.

	FRFT	WVD	Proposed Method
Minimum frequency interval	Δf = 0.14 MHz	Δf = 0.183 MHz	Δf = 0.0657 MHz
Computation time	T = 6.5 s	T = 7.8 s	T = 1.4 s
Critical SNR	SNR = −12 dB	SNR = −10 dB	SNR = −9 dB

shows that under the circumstance of the same parameters, our proposed method is capable of extracting the minimum frequency interval, which is only 0.0657 MHz. Since the I-D frequency is usually low and the intra-pulse frequency components are relatively close to each other, our proposed method can effectively extract the I-D frequency of the coupled echo signal. Additionally, the method exhibits very low computational complexity, and its operation time is much less than the other two methods under the same circumstances. The critical SNR, for which this method can effectively extract the minimum frequency interval, is −9 dB, whereas the critical SNRs of the other two methods are −12 dB and −10 dB, respectively. The critical SNR of our proposed method in this study is a bit higher than the other two methods. One reasonable explanation is that our proposed method needs to estimate the pulse parameters. A SNR that is too low reduces the estimation accuracy of the pulse parameters and causes similar I-D frequencies to be easily aliased due to noise.

3.3. Velocity Distribution Characteristics of the PSh

Based on the simulated flow-field data of the conical target at different heights, this subsection analyzes the distribution characteristics of the velocity of the plasma flow field under circumstances of typical aircraft flight velocity (V = 24 Mach, 1 Mach = 340 m/s) and different flight altitudes (H = 30 km~70 km). The electron density distribution of the simulated flow field of the cone aircraft at different altitudes is shown in Figure 10.

Figure 11a–d respectively represent the velocity distribution characteristics of the plasma flow field at four altitudes. The red dashed line represents the vehicle surface. The plasma is produced within the stagnation point area of the vehicle. Influenced by the action of high-velocity incoming flow, the plasma moves along the stagnation point area to the tail-end area and forms a PSh enveloping the vehicle. As a result, the spatial distribution of the plasma flow velocity varies significantly within different reference areas.

At 30 km altitude, the innermost flow-field velocity of the PSh is similar to the actual velocity of the vehicle. Vertically outward along the vehicle surface, the flow-field velocity declines mildly. The velocity of the outer flow field decreases faster than that of the inner flow field, and the velocity of the outer flow field is far lower than the vehicle velocity. Within the area of the stagnation point, the inner flow-field velocity has an approximately uniform distribution, but the outer flow-field velocity sharply decreases. From the area of the stagnation point to that of the tail end, the velocity distribution interval of the plasma flow field first increases and then decreases, while the velocity descending gradient decreases accordingly.

As shown in Figure 10, with the increase in flight altitude, the electron density of the flow field decreases, but the thickness of the plasma sheath also increases accordingly. At the position close to the vehicle surface, the range of the flow field whose velocity is close to the target velocity gradually increases. The descending gradient of the flow-field velocity within the stagnation point area is approximately constant, the descending gradient of the flow-field velocity in the middle area and the tail-end area decreases.

3.4. Variation Law of the Coupled I-D Frequency in an Echo Signal under Typical Parameters

According to the aforementioned computation results, this subsection analyzes the coupled I-D frequency in the echo signal of the PSh-enveloped target and reveals the variation law of the I-D frequency through different carrier frequencies (f_c = 3.3 GHz, 5.8 GHz, 9.5 GHz, and 18 GHz) and different flight altitudes (H = 30 km, 50 km, 60 km, and 70 km).

At the flight altitude of 30 km:

Figure 12. Reflection coefficient of plasma sheath to electromagnetic wave at an altitude of 30 km: (a) f_c = 3.3 GHz; (b) f_c = 5.8 GHz; (c) f_c = 9.5 GHz; (d) f_c = 18 GHz.

Figure 12. Reflection coefficient of plasma sheath to electromagnetic wave at an altitude of 30 km: (a) f_c = 3.3 GHz; (b) f_c = 5.8 GHz; (c) f_c = 9.5 GHz; (d) f_c = 18 GHz.

Figure 13. The Doppler spectrum of the echo at an altitude of 30 km: (a) f_c = 3.3 GHz; (b) f_c = 5.8 GHz; (c) f_c = 9.5 GHz; (d) f_c = 18 GHz.

Figure 13. The Doppler spectrum of the echo at an altitude of 30 km: (a) f_c = 3.3 GHz; (b) f_c = 5.8 GHz; (c) f_c = 9.5 GHz; (d) f_c = 18 GHz.

It can be observed that the relative motion of the PSh causes the echo signal to couple with different I-D frequencies. As can be seen from Figure 12, at an altitude of 30 km, with the increase in carrier frequency, the strongest reflection positions of electromagnetic waves in the plasma sheath are all in the outer layer of the PSh. At this time, the plasma sheath has a strong shielding effect on electromagnetic waves. As can be seen from Figure 11a, when the flight altitude of the aircraft is 30 km, the velocity outside the PSh is more than 10 Mach lower than the target flight velocity. Therefore, the Doppler frequency coupled with the outside of the PSh is much lower than the Doppler frequency corresponding to the target velocity. Figure 13a suggests that when the carrier frequency is 3.3 GHz, the maximum I-D frequency coupled in the echo signal is 0.1244 MHz, under which circumstance the corresponding velocity is 16.6 Mach. Because of the extremely high electron density at this flight altitude (the peak characteristic frequency of the plasma flow field within the tail-end area is 23 GHz, which exceeds the radar signal’s carrier frequency), the wave incident depth remains approximately unchanged as the carrier frequency (5.8 GHz, 9.5 GHz, and 18 GHz) increases, under which circumstance the velocity corresponding to the maximum I-D frequency coupled in the echo signal approximates 17 Mach.

At the flight altitude of 50 km:

Figure 14. The Doppler spectrum of the echo at an altitude of 50 km: (a) f_c = 3.3 GHz; (b) f_c = 5.8 GHz; (c) f_c = 9.5 GHz; (d) f_c = 18 GHz.

Figure 14. The Doppler spectrum of the echo at an altitude of 50 km: (a) f_c = 3.3 GHz; (b) f_c = 5.8 GHz; (c) f_c = 9.5 GHz; (d) f_c = 18 GHz.

At 50 km altitude, the electron density decreases, the PSh shielding effect imposed on the wave degrades, and the incident depth increases. As shown in Figure 14, when the carrier frequency is 3.3 GHz, the maximum I-D frequency coupled in the echo signal is 0.1595 MHz, for which the corresponding velocity is 21.3 Mach, and the maximum peak energy is 91 dB. When the carrier frequency increases to 5.8 GHz and 9.5 GHz, the maximum I-D frequency coupled in the echo signal increases. This is because the incident depth increases, while the velocity corresponding to the maximum I-D frequency is 22.5 Mach. As the incident depth continues to increase, the secondary energy absorption effect of the PSh exerted on the echo signal intensifies, and the maximum peak energy is reduced by more than 5 dB. As the carrier frequency increases further to 18 GHz, the incident depth continues to increase, and the maximum I-D frequency coupled in the echo signal is 0.9316 MHz, to which the corresponding velocity is 23 Mach. The secondary energy absorption effect reduces the maximum peak energy to 75.6 dB.

At the flight altitude of 60 km:

Figure 15. The Doppler spectrum of the echo at an altitude of 60 km: (a) f_c = 3.3 GHz; (b) f_c = 5.8 GHz; (c) f_c = 9.5 GHz; (d) f_c = 18 GHz.

Figure 15. The Doppler spectrum of the echo at an altitude of 60 km: (a) f_c = 3.3 GHz; (b) f_c = 5.8 GHz; (c) f_c = 9.5 GHz; (d) f_c = 18 GHz.

When the altitude is 60 km, the electron density of the PSh is further reduced, under which circumstance the PSh shielding effect exerted on the wave is alleviated.

It can be observed from Figure 15a,b that the carrier frequencies are 3.3 GHz and 5.8 GHz, respectively, for which the velocity corresponding to the maximum I-D frequency coupled in the echo signal approximates 21.7 Mach. Furthermore, the secondary energy absorption effect of the PSh imposed on the echo signal lowers the maximum peak energy from 91.5 dB to 82.8 dB. As the carrier frequency increases to 9.5 GHz, the incident depth increases, penetrating through the PSh in some areas (the wave is reflected on the vehicle surface), and the I-D frequency component coupled in the echo signal increases from two to three. An increase in the number of Doppler peaks means an increase in the number of major Doppler components of the coupling, which is due to an increase in strong reflection locations. From Figure 15c, we observe that the maximum I-D frequency coupled in the echo signal is 0.518 MHz, which approximates the Doppler frequency corresponding to the target velocity, and the maximum peak energy is 79.5 dB. As the carrier frequency increases to 18 GHz, the EM wave component penetrating the PSh gradually increases, under which circumstance the energy of the I-D frequency corresponding to the target velocity reaches its maximum value of 86.25 dB.

At a flight altitude of 70 km:

Figure 16. The Doppler spectrum of the echo at an altitude of 70 km: (a) f_c = 3.3 GHz; (b) f_c = 5.8 GHz; (c) f_c = 9.5 GHz; (d) f_c = 18 GHz.

Figure 16. The Doppler spectrum of the echo at an altitude of 70 km: (a) f_c = 3.3 GHz; (b) f_c = 5.8 GHz; (c) f_c = 9.5 GHz; (d) f_c = 18 GHz.

At the altitude of 70 km, the electron density is far less than that at 30 km, 50 km, and 60 km. The PSh in the vehicle’s middle and tail-end areas exert no shielding effect on the wave, for which reason the wave can penetrate the PSh and then be reflected on the vehicle surface. In the above scenario, the carrier frequency is 3.3 GHz, and the maximum I-D frequency component coupled in the echo signal is 0.1808 MHz, approximating the Doppler frequency corresponding to the target velocity. Under the circumstance of 3.3 GHz carrier frequency, despite the fact that the EM wave penetrates through the PSh, the penetration area is limited, so the secondary absorption effect of the flow field causes the minimum peak energy of the maximum I-D frequency. As the carrier frequency increases to 5.8 GHz and further to 9.5 GHz, the I-D frequency energy corresponding to the target velocity increases from 85 dB to 90 dB. As the carrier frequency increases to 18 GHz, the incident depth increases within the areas of the stagnation point and the near-stagnation point, under which circumstance the echo signal couples more frequency components. At 18 GHz carrier frequency, the wave intensity of the echo signal of the target surface increases, and the I-D frequency energy corresponding to the target velocity in the echo signal increases to 92 dB, which is much higher than other I-D frequency energies.

Table 3. The velocity corresponding to the maximum Doppler frequency.

Maximum Velocity	fc = 3.3 GHz	fc = 5.8 GHz	fc = 9.5 GHz	fc = 18 GHz
H = 30 km	16.63 Ma	16.77 Ma	16.68 Ma	16.77 Ma
H = 50 km	22.32 Ma	22.8 Ma	22.71 Ma	22.83 Ma
H = 60 km	21.72 Ma	21.53 Ma	24.06 Ma	23.99 Ma
H = 70 km	24.17 Ma	23.85 Ma	23.92 Ma	23.99 Ma

Table 3. The velocity corresponding to the maximum Doppler frequency.

Maximum Velocity	fc = 3.3 GHz	fc = 5.8 GHz	fc = 9.5 GHz	fc = 18 GHz
H = 30 km	16.63 Ma	16.77 Ma	16.68 Ma	16.77 Ma
H = 50 km	22.32 Ma	22.8 Ma	22.71 Ma	22.83 Ma
H = 60 km	21.72 Ma	21.53 Ma	24.06 Ma	23.99 Ma
H = 70 km	24.17 Ma	23.85 Ma	23.92 Ma	23.99 Ma

Combining Figure 13, Figure 14, Figure 15 and Figure 16, the simulation results are listed in the Table 3. It can be ascertained that at relatively low flight altitude, due to the extremely strong EM shielding effect of the PSh, the EM wave is reflected outside the PSh, and the I-D frequency coupled in the echo signal is far less than the Doppler frequency corresponding to the target velocity. As the carrier frequency increases, the corresponding velocity of the I-D frequency coupled in the echo signal remains approximately unchanged. As the flight altitude continues to increase, the electron density decreases, the EM shielding effect of the PSh degrades, the incident depth increases, and the I-D frequency component coupled in the echo signal increases. It must be noted that the secondary energy absorption effect of the PSh exerted on the EM wave results in the overall downward trend of the echo signal energy. When the flight altitude increases to 60 km and 70 km, the shielding effect of the PSh sharply degrades. When the carrier frequency remains at no less than 5.8 GHz, the EM wave is capable of penetrating through the PSh and will be reflected on the target surface, and then the echo signal couples with the I-D frequency corresponding to the target velocity. As the carrier frequency increases, the energy absorption effect of the PSh exerted on the reflection wave of the target surface degrades, whereas the energy of the I-D frequency peak corresponding to the target velocity increases slowly.

4. Conclusions

In this study, we proposed an I-D frequency analysis method for the radar echo of a PSh-enveloped target, revealing the variation law of I-D frequency according to the aforementioned numerical computation results cited in the references.

Specifically, we adopted a difference filter to conduct parameter estimation with respect to the echo signal, which is then dealt with based on equivalent time delay theory and dechirp-Fourier transform, thereby realizing high-resolution extraction of I-D frequency. Furthermore, we formulated the spatial distribution characteristics of the velocity field of the PSh along the area of the stagnation point to that of the tail end. By performing our proposed method in this study, we extracted the I-D frequency coupled in the echo signal under circumstances of typical altitudes (H = 30 km, 50 km, 60 km, and 70 km) and typical carrier frequencies (f_c = 3.3 GHz, 5.8 GHz, 9.5 GHz, and 18 GHz). We also analyzed the influence exerted by different altitudes and carrier frequencies on the peak position, peak energy, and peak number of I-D frequency components, through which the variation law of I-D frequency was investigated.

Despite the abovementioned research findings, certain positive factors or potential shortcomings must be pointed out concerning related parameter settings, experimental setup, etc., which may facilitate or hinder the practical application of extracting I-D frequency components. Our future research interests will be focusing on suppressing abnormal phenomena in radar detection of PSh-enveloped targets, robust target detection methods, and reliable target tracking, laying a theoretical foundation for improving the generality and robustness of radar detection approaches.