A Variable-Scale Coherent Integration Method for Moving Target Detection in Wideband Radar

Abstract

Accurate integration of the extended target’s energy is one of the important challenges of moving target detection in wideband radar. In this paper, a coherent integration method for wideband radar, i.e., variable-scale moving target detection (VSMTD), is proposed to resist range migration and Doppler broadening. On the one hand, subband decomposition can effectively integrate the energy of the extended target in range using variable-scale transformation, accomplished by modulating the filter bank. On the other hand, it increases the coherent integration time by mitigating the range migration in a sufficiently narrow subband. The discrete Fourier transform (DFT) modulated filter bank and the fast Fourier transform (FFT) algorithm are also used to achieve fast VSMTD implementation. Finally, the simulation results demonstrate the superior performance of the proposed VSMTD method.

1. Introduction

Narrowband radar is commonly used in conventional moving target detection due to its straightforward processing steps, high signal-to-noise ratio (SNR), and inexpensive hardware requirements. However, since the development of wideband radar and related technologies, wideband radar is increasingly used to detect moving targets due to its target recognition capability and diversity gain [1,2]. Wideband radar can discriminate a target into multiple scattering centers, depending on the range extent of the target and the radar bandwidth. Normally, target detection is performed first, using all of the target’s energy found in the range profile [3]. Thereafter, high resolution range profile (HRRP) can be used to attempt target recognition [4].

When the required probability of detection is high, a “golden rule” is that coherent integration must first be used to obtain a sufficient SNR [5,6]. A widely applied coherent integration method is moving target detection (MTD) [6]. The radar’s range units are relatively small in wideband radar observation; therefore, even a slowly moving target’s range walk will exceed several range units. This type of across range unit (ARU) will limit the integration time for MTD. In this regard, many methods have been proposed to improve moving target detection performance via long-time coherent integration [7,8,9,10,11,12,13,14,15,16,17,18], including the Radon–Fourier transform (RFT) based method introduced by Xu et al. [16,17,18]. The RFT method can successfully resolve coupling between range migration and phase modulations by concurrently searching in the range and velocity directions of moving targets. In general, existing pulse integration methods overcome the ARU problem by fitting the target motion mode. However, because such methods necessitate a considerable amount of computation, the demand for instantaneity may not be reached when wideband radar has a large amount of data. Furthermore, these methods cannot be directly applied to moving extended target detection, because they ignore the target energy dispersed across the range, while still detecting the “point target”.

As there is no coherence across range cells for extended targets, only noncoherent integration can be carried out in the range direction; that is, accumulation can be performed after the envelope detection [19,20,21,22]. Additionally, some wideband detectors were proposed based on noncoherent accumulation of range cells [23,24]. For instance, He et al. [23] proposed a double-threshold (M/N) algorithm based on scattering density. The detection performance of this algorithm is related to the density of scattering centers, and it only works well at a lower scattering density. These methods can provide greater detection performance for many types of extended targets; however, wideband radar targets are more prone to cause the ARU effect in pulse integration. Many researchers ignored the impact of pulse integration with regard to broadband radar detection; there are few studies devoted to this issue [24].

In this paper we propose a variable-scale moving target detection (VSMTD) method based on subband coherent integration for the detection of moving extended targets. The bandwidth of a signal processing system dictates its processing rate (calculation scale), whereas the bandwidth of a radar system determines its resolution (observation scale). As a result, multi-rate processing can be used to achieve the scale transformation of radar targets. VSMTD first converts the wideband signal into suitable subbands through a multi-rate filter bank, which is equivalent to integrating the energy of the scattering center in range direction. The reflection coefficients of the target are assumed to be constant over the processing of coherent integration [16,17,18]. Next, to operate coherent integration in adjacent pulses, Doppler filter banks of subbands were designed and used. As some narrowband assumptions and approximations are not applicable to wideband signals, the Doppler filter bank has different parameters for Doppler frequency in each subband. In contrast to existing integration detection methods, which need to integrate the energy in each scattering center separately, the method we propose can efficiently accumulate the energy of the target in both the pulse dimension and the range dimension at the same time. Furthermore, comparisons between VSMTD and existing integration methods are provided with regard to four aspects: coherent integration time, integration gain, detection performance, and computational complexity. Finally, some numerical experiments are provided to illustrate that VSMTD can obtain a better integration gain in different SNR backgrounds for wideband extended targets. The performance of the proposed method is also compared with that of RFT cascaded range integration. VSMTD is better than the sliding window method in the case of high SNR, and the computational requirement is much lower than the sliding window method.

2. Signal Modeling for Extended Target in a Long-Time Coherent Integration

Considering a radar system which transmits a linear frequency modulation (LFM) signal, the transmit signal model may be given as

$$s_t\left(\tau \right)=rect\left(\frac{\tau }{T_p}\right)\exp \left\{j\pi K{\tau }^2\right\}$$

(1)

where $T_p$ is the pulse duration, $K$ is the rate of frequency modulation, and $B=K{T}_p$ is the signal bandwidth. It is assumed that the radar’s bandwidth is large enough to distinguish all the major scattering centers of the targets in range. Considering a moving target with scattering centers moving along the radial direction of the radar with velocity $V_T$, the initial slant range from the $l$th scattering center to the radar is $R_l$. This paper uses the traditional “stop-go” motion model to simplify analysis. It is worth noting that when the target is moving at high enough speeds (e.g., hypersonic targets), this model may not function [25]. The radial distance of the moving target may be represented as

$$R_l(t_{\eta })=R_l+V_T{t}_{\eta }$$

(2)

where $t_{\eta }\in \left[-CPI/2,CPI/2\right]$ is the slow time (that is, each $t_{\eta }$ radar transmits a pulse for one observation), and $CPI$ represents the time of the coherent processing interval; then, the two-dimensional echo of the moving extended target may be given as

$$s\left(\tau ,t_{\eta }\right)={\displaystyle \sum _{l=1}^L{A}_{l}rect}\left(\frac{\tau -2R_l(t_{\eta })/c}{T_p}\right)\exp \left(j\pi K{\left(\tau -2R_l(t_{\eta })/c\right)}^2\right)\exp \left(-j\frac{4\pi f_c{R}_l(t_{\eta })}{c}\right)$$

(3)

where $\tau$ is the fast time, and $L$ is the distribution range of the extended target. The first exponential term is the quadratic phase of the chirp signal, and the second exponential term is the linear phase after quadrature demodulation [16]. After range compression, the two-dimensional echo of the target may be given as

$$s_{rm}\left(\tau ,t_{\eta }\right)={\displaystyle \sum _{l=1}^L{A}_{r{m}_l}\sin \mathrm{c}}\left(B(\tau -2R_l(t_{\eta })/c)\right)\exp \left(-j4\pi f_c{R}_l(t_{\eta })/c\right)$$

(4)

where $A_{r{m}_l}=T_{p}B{A}_l$ is the amplitude of the compressed signal. Substituting (2) into (4), the time-domain echo model of the extended target is

$$s_{rm}\left(\tau ,t_{\eta }\right)={\displaystyle \sum _{l=1}^L{A}_{T_l}\sin \mathrm{c}}\left(B(\tau -2R_l(t_{\eta })/c)\right)\exp \left(j2\pi f_d{t}_{\eta }\right)$$

(5)

where $A_{T_l}=A_{r{m}_l}\exp (-j4\pi f_c{R}_l/c)$ is the back-scattering coefficient in the complex form. This paper defines $A_{T_l}$ as a constant in the time of coherent integration. The Doppler frequency $f_d$ of the target is defined as

$$f_d=\frac{-2f V_T}{c}$$

(6)

3. Variable-Scale Transformation of the Radar Target

3.1. Variable-Scale Transformation by Subband Decomposition

The observation scale of a radar system is related to the bandwidth of the radar signal. The variable scale transformation of radar targets can be achieved by changing the bandwidth of the received signal. The signal model in the frequency-domain is given as

$$S\left(f_{\tau },t_{\eta }\right)={\displaystyle \sum _{l=1}^L{A}_{T_l}rect}\left\{\frac{f_{\tau }}{B}\right\}\exp \left\{\frac{-j4\pi R_l(t_{\eta })f_{\tau }}{c}\right\}\exp \left(j2\pi f_d{t}_{\eta }\right)$$

(7)

Divide the signal in the frequency domain into M subbands evenly. Ideally, the sub-band decomposed signal is given as

$$S\left(f_{\tau },t_{\eta },m\right)={\displaystyle \sum _{l=1}^L{A}_{T_l}rect}\left\{\frac{M\left(f_{\tau }-f_m\right)}{B}\right\}\exp \left\{\frac{-j4\pi R_l(t_{\eta })f_{\tau }}{c}\right\}\exp \left(j2\pi f_d{t}_{\eta }\right)$$

(8)

where $m=0 1 2 \dots \dots M-1$ is the subband number and $f_m=B\cdot (m/M-1/2)$ is the subband frequency offset. The transformed signal $s_{t_{\eta }}\left(\tau ,m\right)$ is given as

$$s\left(\tau ,t_{\eta },m\right)=\frac{B}{M}{\displaystyle \sum _{l=1}^L{A}_{T_l}\sin \mathrm{c}\left\{\frac{B}{M}\left(\tau -\frac{2R_l(t_{\eta })}{c}\right)\right\}}\exp \left\{j\cdot f_m\tau \right\}\exp \left(j2\pi f_{dm}{t}_{\eta }\right)$$

(9)

It can be seen from (9) that the sinc envelope of each scattering center has an $M$ times broadening. The penultimate exponential term is a modulation term resulting from the uniform subband decomposition. The last exponential term is the Doppler phase. As shown in (9), according to the definition of 3 dB resolution, the resolution of each subband signal increases by $M$ times after subband decomposition. ${\rho }_M$ represents the resolution of each subband after decomposition of the $M$ uniform subband. The Doppler frequency $f_d$ of the wideband signal is related to the signal frequency $f$, as shown in Figure 1. For subband signals, the center frequency $f_{c,sub}$ of the subband is used to approximate the signal frequency. The Doppler frequency in a different subband, $f_{dm}$, may be defined as

$$f_{dm}=-\frac{2V_T}{c}(f_c+(\frac{m}{M}-\frac{1}{2})B)$$

(10)

3.2. Design of Variable-Scale Transformation Filter Bank

Multi-rate filter banks are commonly employed in signal processing to decompose the signal spectrum into numerous adjacent frequency bands. In Figure 2, the decomposition process is completed in the analysis filter bank $H_m\left(z\right)$, and the synthesis process is completed in synthesis filter bank $I_m\left(z\right)$. $\downarrow D$ and $\uparrow D$ signify down-sampling and up-sampling by a factor of $D$. Assuming that the original signal satisfies the Nyquist sampling law, the analysis filter bank reduces the subband signal bandwidth by a factor of $M$; therefore, the sampling rate of the subband can theoretically be reduced by $M$ times at most.

In theory, it is possible to design a complete reconstruction filter bank with $M$ channels separately; however, when $M$ is large (over 50), the complexity of filter design becomes intolerable. Considering that we need to divide the signal into uniform subbands (the width of each subband is the same), a complex modulation filter bank can be used to decompose and synthesize the signal. The complex modulation filter bank only needs to design one or two prototype low-pass filters, which greatly reduces the workload of filter bank design. De Haan et al. [26] proposed a method to simplify the prototype filter design into an optimization problem that minimizes the passband response error and the in-band aliasing error. The frequency response of the prototype filter and the analysis filter bank are shown in the Figure 3. Figure 4 illustrates the result of decomposing the same target with a different number of subbands.

4. Filter Bank Design for Variable-Scale Moving Target Detection (VSMTD)

A radar system with a range sampling rate of $f_s$ and a pulse repetition frequency of $PRF$ is considered. In the coherent processing interval ($CPI$), the discretized form of range-compressed signal (5) is expressed as $s_{rm}(n_r,n_a)$. The number of range cells is $N_r=round(f_s(2r_a/c+T_p))$. The number of coherent processing pulses is $N_a=CPI\cdot PRF$. Since the frequency responses of all discrete-time filters are periodic, there is a blind speed in moving target detection, which is limited by the pulse repetition frequency of the radar system. The calculation of the blind speed may be given as

$$v_b=f_{c}PRF/4c$$

(11)

As shown in Figure 5, the moving target detection (MTD) using a Doppler filter bank may be written as

$$G_v(n_r)={\displaystyle \sum _{n_a=-N_a/2}^{N_a/2-1}s\left(n_r,n_a\right)H_v(n_a)}$$

(12)

The Doppler filter bank $H_v(n_a)$ is defined as

$$H_v(n_a)=\exp \left(j2\pi \frac{f_{c}2V}{c}{n}_a\cdot PRT\right)=\exp \left(j2\pi f_d{n}_a\cdot PRT\right)$$

(13)

where $v$ is the velocity parameter of the filters. As a result of the existence of blind velocity, the velocity parameter of MTD’s Doppler filter bank only needs to cover $v\in [-v_b,v_b]$, and the search interval is

$${\Delta }_{MTD}=f_c/(2c\cdot CPI)$$

(14)

The scale transformation by subband decomposition is performed on the compressed signal (5). The range sampling rate after variable scale processing is $f_s\cdot D/M$. The discretized form of the subband signal (9) is written as $s_{srm}(n_{sr},n_a,m)$. The number of range cells after subband down-sampling is $N_{sr}=round(f_s(2r_a/c+T_p)D/M)$. The subband Doppler filter bank is defined as

$$\begin{array}{l}{H}_{vm}(n_a)=\exp \left(j2\pi f_{dm}{n}_a\cdot PRT\right)\\ f_{dm}=-\frac{2V}{c}(f_c+(\frac{m}{M}-\frac{1}{2})B)\end{array}$$

(15)

The result of coherent integration using the subband Doppler filter bank may be given as

$$G_{sub\_m}(n_{sr},v)={\displaystyle \sum _{n_a=-N_a/2}^{N_a/2-1}{s}_{srm}(n_{sr},n_a,m)H_{vm}(n_a)}$$

(16)

A block diagram of the subband Doppler filter bank is shown in Figure 6. For the same velocity parameter, the Doppler frequencies that need to be compensated in each subband are different. Combined with the variable-scale processing method based on the filter bank proposed in Section 3, the processing block diagram of VMTD is illustrated in Figure 7. First, the range compressed signal is decomposed into multiple subbands by the analysis filter bank, and then the coherent integration and velocity estimate are performed by the subband Doppler filter bank. Since there is no coherence between the subband signals, the noncoherent accumulation of the coherent integration output of subbands can improve the SNR of VSMTD. The output of VSMTD is

$$G_{VSMTD}(n_{sr},v)={\displaystyle \sum _{m=0}^{M-1}{\left|G_{sv}(n_{sr},m)\right|}^2}$$

(17)

Although the signal used for moving target detection loses some characteristics of the target, this does not affect subsequent classification and recognition processing. High resolution signals in the original scale can be obtained using the synthesis filter banks in Figure 2.

5. Performance Analysis of VSMTD

This section discusses the performance of VSMTD. First, we discuss the SNR gain achieved by the noncoherence integration of subbands and the coherent integration of pulses. Next, we discuss the performance of VSMTD in terms of detection metrics and computational complexity.

5.1. Noncoherent Gain of Scale Transformation

The signal-to-noise ratio in the range cell of the extended target is given by

$$SNR=\frac{P_t{G}^2{\lambda }^2\sigma }{{\left(4\pi \right)}^{3}k{T}_{e}BFL{R}^4}$$

(18)

where $P_t$ is the radar transmit peak power; $G$ is the antenna gain; $\lambda$ is the wavelength; $\sigma$ is the RCS of the scattering center in the range cell; $k$ is the Boltzmann constant; $T_e$ is the effective noise temperature; $F$ is the radar receiver noise figure; $L$ is the loss of radar system; and $R$ is the range of scattering center. The bandwidth of the subband signal after scale transformation is reduced to $1/M$ of the original bandwidth. The SNR gain of the matched filter becomes $1/M^2$ of the original signal because the time-bandwidth product (TBP) reduced to $1/M^2$ of the original signal. Since the range unit is enlarged by $M$ times, the RCS in the range unit is $M\sigma$. The SNR of the radar range cell in subband is given by

$$SN{R}_{sub}=\frac{1}{M^2}\frac{P_t{G}^2{\lambda }^2(M\sigma )}{{\left(4\pi \right)}^{3}k{T}_e(\frac{1}{M}B)FL{L}_{VS}{R}^4}=\frac{P_t{G}^2{\lambda }^2\sigma }{{\left(4\pi \right)}^{3}k{T}_{e}BFL{L}_{VS}{R}^4}$$

(19)

where $L_{VS}$ represents the processing loss of the nonideal variable-scale filter bank. There is no coherence between the $M$ subbands, so noncoherent integration can be used between subbands to improve the SNR. The integration gain is always less than $M$, which is a type of noncoherent integration loss. Marcum [27] thought this loss is between $\sqrt{M}$ and $M$. The SNR of the radar range cell after scale transformation is given by

$$SN{R}_{VS}=\frac{P_t{G}^2{\lambda }^{2}I\left(M\right)\sigma }{{\left(4\pi \right)}^{3}k{T}_{e}BFL{L}_{VS}{R}^4}$$

(20)

where $I\left(M\right)$ is a factor of noncoherent integration gain [28].

5.2. Gain of Coherent Integration between Pulses

Coherent processing time (CPT) is the processing interval required to achieve a full coherent integration during the total coherent integration in slow time dimension, i.e., the pulses dimension. For an extended target moving at high speed, the across range unit between adjacent echoes of each scattering center will limit the coherent processing time, as shown in Figure 8. The coherent processing time of a moving target can be calculated as

$$CPT=round(\rho /\left|v\right|)PRT$$

(21)

where $\rho$ is the length of range cells, i.e., the resolution scale.

It can be seen from Equation (21) that when other parameters of the radar system are fixed, the coherent processing time is inversely proportional to the magnitude of the velocity. When the moving velocity of the target scattering center is larger, the movement across range unit is larger, and the coherent processing time is correspondingly shorter. The MTD results of single scattering points with different velocities at the same resolution scale are shown in Figure 9. As a result of the different degrees of scatter centers moving across the range unit at different speeds, the faster the speed, the shorter the coherent processing time. At the same time, due to the short coherent processing time, the observation window in the time domain is shortened. From the perspective of time-frequency analysis, shortening of the time domain window corresponds to broadening of the frequency domain (velocity domain), i.e., Doppler broadening. This Doppler broadening can be clearly seen in Figure 9. The faster the target, the more severe the broadening, and the smaller the integration gain.

Considering the problem of short coherent processing time, Xu et al. [16,17,18] proposed a Radon–Fourier transform method to improve the coherent integration gain. This method searches for the optimal integration path on all velocity parameters. Theoretically, the RFT can achieve coherent integration of all echoes without limitation of the target speed. However, the RFT cannot effectively gather the energy of the extended target in the range, and it requires extensive calculation. The VSMTD method proposed in this paper adopts another idea, which is to transform the broadband signal into a subband sufficient to ignore the range walk for coherent integration. Furthermore, if nonideal target motion (such as high-order motion) is taken into account, it greatly affects the computational efficiency of the RFT; however, the method proposed here is not sensitive to this change. As shown in Figure 10, VSMTD can accumulate more echo energy through scale transformation. It is assumed that VSMTD can accumulate all $N_a$ echoes by choosing appropriate scale parameters $M$. Expansion of the coherent processing time will increase the number of echoes accumulated during the coherent integration, and the coherent integration gain will increase accordingly. The number of integration echoes by different integration methods is given by

$$N_{a,MTD}=round(\rho /\left|v\right|)$$

(22)

$$N_{a,RFT}=N_a$$

(23)

$$N_{a,VSMTD}=M{N}_{a,MTD}=N_a$$

(24)

5.3. Comparison of SNR Gain

As shown in Figure 10, VSMTD can not only increase the CPT, but also accumulate the target energy in the range direction. This method performs scale transformation and long-term coherent integration at the same time. For a moving extended target with range $R$ and RCS $\sigma$, the SNR gain for different coherent integration methods is given by

$$SN{R}_{MTD}=\frac{P_t{G}^2{\lambda }^2{N}_{a,MTD}\sigma }{{\left(4\pi \right)}^{3}k{T}_{e}BFL{R}^4}$$

(25)

$$SN{R}_{RFT}=\frac{P_t{G}^2{\lambda }^2{N}_a\sigma }{{\left(4\pi \right)}^{3}k{T}_{e}BFL{R}^4}$$

(26)

$$SN{R}_{VSMTD}=\frac{P_t{G}^2{\lambda }^{2}I\left(M\right)N_a\sigma }{{\left(4\pi \right)}^{3}k{T}_{e}BFL{L}_{VS}{R}^4}$$

(27)

VSMTD increases the coherent integration gain $N_a$ by processing in subbands. Compared with RFT, this method can also obtain a gain factor $I\left(M\right)$ by noncoherent integration in the range direction. The coherent integration in the pulse dimension and the noncoherent integration in the range dimension constitute the overall gain of the VSMTD.

5.4. Comparison of Computational Complexity

Although VSMTD implemented by the filter bank adds many filters, it can achieve efficient parallel filtering with some equivalent relationships and the polyphase structure of the filter banks. As shown in Figure 11, if the filter before the D-fold decimator is moved after the decimator, the power of variable $z$ of the filter is reduced by a factor of D.

The theory of polyphase structure is used to reduce the computational complexity of the filter bank [29]. $L_n$-order filter: the filter $H(z)$ group is evenly divided into M groups, and $L_n/M=Q$ ($Q$ is an integer); then $H(z)$ can be written as

$$\begin{array}{l}H(z)={\displaystyle \sum _{k=0}^{M-1}{E}_k(z^M)z^{-k}}\\ E_k(z^M)={\displaystyle \sum _{n=0}^{Q-1}h(nM+k)}{(z^M)}^{-n} ,k=0,1,\dots ,M-1\end{array}$$

(28)

where $E_k(z^M)$ is the polyphase component of $H(z)$. Equation (28) is the polyphase representation of $H(z)$. An efficient form of the DFT filter bank is shown in Figure 12, and the efficient implementation of VSMTD is shown in Figure 13. The down-sampling process is moved before filtering, and the polyphase representation of the filter bank is used to simplify the filter structure.

MTD integrated along the column (or the row) can be greatly improved by the Fast Fourier Transform (FFT). However, the RFT cannot be accelerated by the FFT due to integration along different lines, so it can only search for velocity parameters in the velocity dimension. VSMTD integrates along the column (or the row) in subbands, so it can be realized by the FFT in subbands; calculation complexity is also greatly reduced. Floating-Pointing operation is used to compare the computational complexity of the MTD, RFT, and VSMTD. Each FLOP can be either a real multiplication or a real addition; the computational complexity of these three methods are provided in

Table 1. Computational Complexity of All Methods.

Methods	FLOP
MTD	$N_r(5N_a{\log }_2(N_a))$
RFT VSMTD	$Nr(6N_a{N}_a+N_a(N_a-1))$ $6N_r{L}_n/M+N_r(5M{\log }_2(M))/M+N_r(5N_a{\log }_2(N_a))+N_r$

.

Figure 14a shows the computational requirements of three methods. The range cell number $N_r$ is 1024. The number of subbands $M$ is 128, and the subband filter order $L_n$ is 256. It is evident that the computational complexity of the RFT is much larger than that of MTD and VSMTD. VSMTD has a greater computational requirement than MTD due to the use of the DFT modulation filter bank; however, this part of the increment does not increase as the number of integration pulses increases. For $N_a=128$, the FLOP of VSMTD is 1.0714% greater than that of MTD. And for $N_a=512$, VSMTD is 0.2083% greater than MTD.

6. Simulation and Experiments

To verify the proposed VSMTD, this section provides simulations and experiments. The simulation parameters are set as in

Table 2. Simulation experiment parameters.

Parameters	Symbol	Value
Carrier frequency	$f_c$	10 GHz
Pulse duration	$T_p$	$10 \mathsf{\mu }\mathrm{s}$
Bandwidth	$B$	500 MHz
Coherent processing interval	$CPI$	25.6 ms
Number of subbands	$M$	128
Number of pulses	$N_a$	128
Filter order	$L_n$	256
Down-sampling multiple	$D$	64
Number of scattering center	$N_l$	8
Velocity of the target	$V_T$	270 m/s

; all experiments are carried out in accordance with Nyquist sampling theorem. The RCS of each scattering center is the same in the simulations presented in this section.

A wideband surveillance radar was designed for the simulation. Given that a target observed by the radar is actually extended to eight scatter centers, a 128-channel subbands analysis filter bank is used to process the simulated echo signal generated by the parameters in Table 2. The output of the variable-scale transformation filter bank is shown in Figure 15. The target energy distributed in range becomes distributed in subbands. The VSMTD filter bank proposed is used to coherently integrate the target signal in different subbands. The results of VSMTD are shown in Figure 16. VSMTD can have a significant integration effect using subband Doppler filter banks; the velocity of the target is accurately estimated.

To verify the SNR gain of VSMTD methods in different SNR backgrounds, comparative experiments were designed. When the SNR of the single echo is 5 dB, other parameters of the radar are shown in Table 2. The integration result is shown in Figure 17. Figure 17a shows the MTD result. The speed estimation result broadens due to the ARU of the extended target, and the integration gain of MTD deteriorates at the same time as shown in Figure 17b. In this regard, the RFT method has none of these problems. By compensating for the trajectory of the target, the RFT method can precisely integrate the energy of each scattering center, as shown in Figure 17d. The integration result and the range profile of VSMTD is illustrated in Figure 17e,f. A decrease in bandwidth results in a corresponding decrease in range cells. Thus, the target’s energy is concentrated in a few range cells. This type of integration in range can improve the SNR of the result. It is noted that although both RFT and VSMTD can achieve better coherent integration in the 5 dB SNR background, it is obvious that VSMTD has a better suppression effect on the background noise.

The effect of this comparison becomes more pronounced when the SNR is reduced. The different methods’ outputs in −5 dB SNR are illustrated in Figure 18. MTD was completely disabled in the −5 dB SNR background, and the RFT output of the target was also almost drowned-out by the noise. However, VSMTD can still achieve a better integration effect in a −5 dB SNR background. The simulation experiments fully verify that the VSMTD method proposed has a better coherent integration performance than MTD and the RFT method for an extended target with high-speed.

The processing gain for different methods is illustrated in Figure 19. It is noted that in the case of a SNR input of −6 dB to 4 dB, the VSMTD processing effect is the best. In the simulation conditions of this section, the VSMTD SNR gain is, on average, 8.85 dB higher than that of the RFT method, and 14.62 dB higher than that of MTD.

Although the RFT method and VSMTD eliminate the influence of range walking, VSMTD simultaneously concentrates the energy of multiple scattering centers into a single range cell, which improves the SNR of the cell being tested. Similar results can be obtained using a sliding window to integrate the RFT output. The ideal sliding window integration is to slide one range cell at a time from the first range cell with a window the same length as the target. This sliding window strategy ensures that the maximum energy of the target is extracted. As wideband radar generates such a large amount of data, it is impossible to slide only one range cell at a time. The “half sliding window” strategy is usually adopted in practice [30]. However, this strategy may fragment the real extended target signal, resulting in a decrease in detection performance. This paper adopts the “half sliding window” strategy as the practical sliding window method. The output SNR with different strategies of range integration is shown in Figure 20. As a result of the use of a nonideal filter, the output SNR of VSMTD is 1.93 dB lower on average than that of an ideal sliding window. In the case of low SNR, VSMTD’s performance is worse than that of a practical sliding window; in the case of high SNR, VSMTD’s performance is better than that of a practical sliding window. However, the design idea of performing range integration after the RFT obviously wastes considerable calculation effort, which is not necessary. The FLOP of VSMTD is 15% of the RFT method with the practical sliding window under the simulation parameters used in this paper.

The cost of using VSMTD to improve the detection of SNR is the loss of the high resolution range profile (HRRP). The high resolution information is not lost, but contained in the subbands; the HRRP can be restored through the synthesis filter bank. The reconstructed results of the measured data of a large aircraft processed by VSMTD are shown in Figure 21. With the reconstruction of the synthesis filter bank, signal scale is restored to the level of the original signal. As the subband coherent integration overcomes the influence of the range migration, the HRRP with a high SNR can be finally obtained.

7. Conclusions

For moving extended target detection via coherent integration, a novel method, i.e., VSMTD, is proposed in this paper to realize long-time coherent integration. The VSMTD filter bank accomplishes the scale transformation of the radar target with subband decomposition. Next, the subband Doppler filter bank compensates the center frequency offset used for successive long-time coherent integration. Furthermore, the gain of VSMTD is analyzed for comparison with existing integration methods. VSMTD can integrate pulses without the restriction of ARU (as with the RFT method), and it has benefits due to integration in range. The integration performance of VSMTD in range is related to the number of scattering points and filter loss.

Additionally, this paper compares the computational complexity of MTD, the RFT method, and VSMTD. The integration method realized by VSMTD can be simplified using the polyphase structure of the DFT filter bank and the FFT algorithm. In theory, the computational complexity of VSMTD is similar to that of MTD, and much less than that of the RFT method. Under the typical parameters, VSMTD has less than 2% more computational complexity than MTD, which is acceptable considering the gain provided by VSMTD. Finally, numerical experiments are provided to show the processing effects of different integration methods. The VSMTD output is, on average, approximately 8 dB higher than the RFT method, and approximately 15 dB higher than MTD, in the case of −6 dB to 4 dB SNR. The experiments verify that the VSMTD method proposed in this paper can provide a better processing gain in the same SNR background. In general, integration in subbands allows VSMTD to efficiently integrate the target energy from two dimensions. After VSMTD processing, the HRRP of the target can be reconstructed via the synthesis filter bank.