When a stepped-frequency radar is used to obtain the high-resolution range profile (HRRP) of high-speed target, accurate speed estimation and motion compensation must be considered. Therefore, in this paper, a novel mathematical method is presented for estimating the target speed. Firstly, the pulse Doppler method is used to calculate the initial estimation value. Secondly, based on the initial estimation value, the minimum entropy method is used to calculate the coarse estimation value. Finally, based on the coarse estimation value, the minimum
Stepped-frequency (SF) signal is widely used in the high-resolution radar systems [
This paper is organized as follows. Section
The SF radar system transmits a series of frames of pulses. Each frame contains a sequence of
Illustration of
Suppose that the moving target is composed of a set of
In general, during each frame period, the change of envelope with the target motion can be neglected. Therefore, after the down conversion and sampling of the SF received-signal, the baseband signal is shown in
In this paper, the number of points for inverse discrete Fourier transform (IDFT) is
Note that the HRRP of target is calculated only by every single frame; therefore, the influence of target motion on HRRP is only embodied in every single frame. In order to simplify the analytic process, suppose
Equation (
The third term
The fourth term “Spread’’ in (
According to the discussion above, the compensation accuracy of LPT is much higher than that of QPT. Therefore, the LPT needs an accurate speed estimation and the QPT only needs a coarse speed estimation. When the target speed is estimated, according to (
In this paper, the main parameters of SF signal are shown in Table
The main parameters of SF radar signal.
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94 GHz | 128 | 4 MHz | 100 ns | 20 us | 96 |
The main performance of SF radar signal.
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0.3 m | 37.5 m | 0.308 m/s | 28.610 m/s |
In this paper, let us suppose that the target speed is constant and takes no account of its own rotation. Meanwhile, the maximum radial length of the target is less than
The model of target.
According to the discussion above, the influence of target motion on HRRP is embodied in the additional LPT and QPT. In order to get better HRRP, firstly, the target speed must be estimated, and then the additional phase terms need to be eliminated. In this paper, based on some traditional mathematic model, a new accurate and fast speed estimation algorithm is proposed.
This method is based on pulse Doppler effect; firstly, through IDFT with SF received signal, the Doppler frequency can be calculated, and then the target speed can be calculated by
According to the IDFT theory, the speed estimation accuracy of PDA is shown in
In this paper, the accuracy of speed estimation
Figure
The estimation error of PDA.
It is well known that the overall target response in the HRRP is changing with the target motion, such as the reduction of peak response, the divergence of scatters, and the blurring effects. According to the theory of image processing, the entropy is a measure of the randomness of an image. The focused images will have a low entropy values. Thus, the mathematical model of entropy can be used as a quality indicator for HRRP. It is defined by
According to Formula (
The entropy feature of MEA.
Figure
The estimation error of MEA.
This algorithm needs to transmit two different frames and each frame contains
The model of the up frame and results in frame.
According to (
Compensating with a initial speed
According to (
Supposing that the speed of target is When the initial estimation value is chosen, wrong searching direction will prolong searching time and increase computation amount. In order to get global minimum value as accurate as possible, search step size should be less than the width of the peak. Therefore, the computation amount may be significantly increased. Even if the SNR of received signal is high, the curve of evaluation function may also have some local fluctuation nearby the true value. Therefore, if the search algorithm is not very effective, the evaluation function may converge to the local minimum which will cause much higher estimation error.
The frame error feature of MNA.
Figure
The estimation error of MNA.
According to the discussion above, in order to overcome the defects of PDA, MEA, and MNA, an improved speed estimation algorithm is presented. It contains three steps including initial estimation, coarse estimation and accurate estimation.
The SF radar system transmits alternately the up frame and the down frame and the received signals are shown in
PDA is used for initial estimation with the first column in (
According to the discussion in Section
In this paper, according to the main parameters of SF signal in Table
After compensating with initial estimation value, MEA is used for coarse estimation with the first and second row in (
According to the discussion in Sections
According to the discussion of MEA method, the possible value of target speed can be determined. However, this coarse estimation value is only enough to compensate for QPT. Therefore, finally, the MNA is used to increase the estimation accuracy of target speed.
When the coarse estimation value is obtained, the true value of target speed will surely consist in
Figure
The estimation error of CEA.
Figures
The HRRP of SF signal based on PDA.
The HRRP of SF signal based on MEA.
The HRRP of SF signal based on MNA.
The HRRP of SF signal based on CEA.
In this paper, three different speed estimation models are analyzed in detail. The PDA has an accurate estimation value in low SNR. However, it has a large limitation to estimation interval of target speed, which will make a great estimation error for a high-speed moving target. The MNA has a coarse estimation value in a much larger speed estimation interval. However, it has a large limitation to SNR, and the estimation accuracy is not enough for well compensating the influence of target speed on HRRP. The MNA has the best estimation accuracy. However, the local minimum value of frame error function will produce worse estimation accuracy and make a large computation amount.
According to the advantages and defects of PDA, MEA, and MNA, the CEA is presented in this paper. Firstly, according to the main parameters of SF signal, some important performance parameters can be calculated. Secondly, the initial estimation value is obtained by PDA. Thirdly, based on the initial estimation value and the estimation interval of target speed, the MEA is used for the coarse estimation value. Finally, based on the coarse estimation value and the final estimation accuracy, the MNA is used for the accurate estimation. The theory analysis and simulation results confirm that this new method is effective and predominant, which has a much higher speed estimation accuracy in real-time and a much larger estimation interval of target speed. Furthermore, the influence of super-high speed motion of target on HRRP can be perfectly compensated, and the HRRP can be used to improve the detection, recognition, and distance measurement performance of moving target.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the Nation Nature Science Foundation of China no. 61301095 and 61201237, Nature Science Foundation of Heilongjiang Province of China no. QC2012C069, and the Fundamental Research Funds for the Central Universities no. HEUCFZ1129, no. HEUCF130810, and no. HEUCF130817.