When using a long range radar (LRR) to track a target with micromotion, the micro-Doppler embodied in the radar echoes may suffer from ambiguity problem. In this paper, we propose a novel method based on compressed sensing (CS) to solve micro-Doppler ambiguity. According to the RIP requirement, a sparse probing pulse train with its transmitting time random is designed. After matched filtering, the slow-time echo signals of the micromotion target can be viewed as randomly sparse sampling of Doppler spectrum. Select several successive pulses to form a short-time window and the CS sensing matrix can be built according to the time stamps of these pulses. Then performing Orthogonal Matching Pursuit (OMP), the unambiguous micro-Doppler spectrum can be obtained. The proposed algorithm is verified using the echo signals generated according to the theoretical model and the signals with micro-Doppler signature produced using the commercial electromagnetic simulation software FEKO.

Estimating and extracting micromotion information have attracted much attention in recent years [

In Synthetic Aperture Radar (SAR)/Inverse Synthetic Aperture Radar (ISAR) imagery, micro-Doppler effect will introduce nonstationary phase modulation into returned signals, which will significantly decrease the readability of images [

Most of the previous researches assume that the probing frequency to a micromotion target is large enough and thus there is no Doppler ambiguity and micro-Doppler ambiguity. However, a long range radar usually works in low PRF, which causes serious Doppler ambiguity and micro-Doppler ambiguity. In [

To solve micro-Doppler ambiguity problem, we propose a novel method based on CS in this paper. According to the RIP requirement of the sensing matrix, a sparse pulse train with random time stamps is designed based on the fixed-PRF pulses. The echo signals after matched filtering can be viewed as randomly sparse sampling of the micro-Doppler spectrum. To reconstruct the micro-Doppler signature from the sparse samples, we propose a short-time-compressed-sensing time-frequency analysis method. A short-time window slides along the slow-time domain echo signals and the reconstruction algorithm OMP is applied within this window to reconstruct the micro-Doppler spectrum. Two kinds of echo signals, one generated according to the theoretical model and one produced by the commercial electromagnetic simulation software FEKO, are used to verify the proposed algorithm.

The transmitted chirp signal can be modeled as

According to the Nyquist-Shannon sampling theorem, for a band-limited baseband signal the sampling rate must be greater than or equal to two times the highest frequency of the signal. For a pulsed radar, the Doppler frequency

In a short-time interval,

The theory of compressed sensing shows that when the signal is sparse or compressible, the signal can be reconstructed accurately or approximately by gathering very few projective values of the signal. Suppose

The dimension of

Signal reconstruction is the process of recovering

According to the description of Section

Generally, the value of

The classical recovery algorithms of compressed sensing are Basis Pursuit and Greedy Matching Pursuit. BP algorithm is the global optimization algorithm which has several advantages including superresolution and stability, but it is also accompanied with high computational complexity. Greedy Matching Pursuit algorithm, such as Orthogonal Matching Pursuit (OMP), is a local optimization algorithm and has the low computational complexity and high level of localization accuracy. The characteristics of OMP, such as easy implementation and fast speed, make it a better choice than BP algorithm [

Doppler frequency shifts generally have the time-varying characteristic and should be analyzed via the joint time-frequency analysis technique. However, when the PRF is lower than the Nyquist, the traditional time-frequency analysis methods are invalid. According to the above theoretical analysis results and the time-variant properties of micro-Doppler, we design a window-weighted compressed sensing method. According to the RIP of the CS sensing matrix, a sparse pulse train is randomly extracted from traditional fixed repetition frequency pulses. By adding a random perturbation item to the transmitting time of each selected pulse, we can obtain a new transmitting time sequence, which is equivalent to those extracted from a high PRF pulse transmitting time set. Then the corresponding measurement matrix can be obtained according to the transmitting time sequence. After matched filtering to the radar echoes, a short-time window slides along the slow-time domain echo signals and the CS method is applied within this window to reconstruct the micro-Doppler spectrum. Similar to the short-time Fourier transform (STFT), we call this method as short-time compressed sensing.

The PRF of a LRR is usually low. For a LRR working in fixed PRF mode, the slow-time domain echoes suffer from serious micro-Doppler ambiguity. By modifying the

When the value of

Illustration of radar dwell scheduling.

Since

The signal within the window is written as

To satisfy the RIP, the perturbation

The process of the short-time compressed sensing on micro-Doppler spectrum reconstruction is summarized as follows:

(i) A new transmitting time sequence is generated by extracting randomly from the fixed-PRF transmitting time sequence. Then a perturbation is added to each element of the new transmitting time sequence. The radar transmits probing pulses according to the scheduled time sequence. Performing matched filtering to the echo pulses, the slow-time domain signals are obtained.

(ii) Slide a short-time window along the slow-time domain signals and construct the corresponding sensing matrix

(iii) Performing the Orthogonal Matching Pursuit algorithm to the signals in the sliding window, the instantaneous micro-Doppler frequency is obtained.

The flowchart for the implementation of the algorithm is shown in Figure

Flowchart of the proposed method.

In this section, simulations are performed to verify the proposed method. The proposed method can be applied to different kinds of micromotion. In the following, two micromotions, that is, spin and precession, are simulated.

The simulation parameters are shown in the list below. One scatter case is as follows.

According to the simulated parameters, the maximum micro-Doppler frequency shift is

In order to investigate the effect of the number of measurements

Micro-Doppler time-frequency spectrum: ambiguous time-frequency spectrum via STFT (a); unambiguous time-frequency spectrum of adding-perturbation fixed-PRF via CS (b); unambiguous time-frequency spectrum of adding-perturbation sparse-PRF via CS (c).

CS reconstruction quality with different measurement number. (a)

Two spin scatterers are simulated and the simulation parameters are shown in the list below. To demonstrate the ambiguity resolving ability of our method, we change the rotating radius

Micro-Doppler time-frequency spectrum: ambiguous time-frequency spectrum via STFT (a); unambiguous time-frequency spectrum of adding-perturbation fixed-PRF via CS (b); unambiguous time-frequency spectrum of adding-perturbation sparse-PRF via CS (c).

Two precession scatterers are simulated and the simulation parameters are shown in the list below (

The simulation results for the precession case are similar to those for the spin case. The sparse rate of (c) of Figure

Micro-Doppler time-frequency spectrum: ambiguous time-frequency spectrum via STFT (a); unambiguous time-frequency spectrum of adding-perturbation fixed-PRF via CS (b); unambiguous time-frequency spectrum of adding-perturbation sparse-PRF via CS (c).

In this experiment, we use the electromagnetic simulation software FEKO to generate sparse-sampled slow-time echo signals and reconstruct the unambiguous micro-Doppler spectrum via CS. A scatterer is rotating with a radius of 30 cm. The rate of angular motion is

According to the configuration, the maximum micro-Doppler frequency is

Micro-Doppler time-frequency spectrum: ambiguous time-frequency spectrum via STFT (a); unambiguous time-frequency spectrum of adding-perturbation sparse-PRF via CS (b).

The proposed short-time compressed sensing method can solve the micro-Doppler ambiguity problem. The sensing matrix built based on the sparse probing time stamps can meet the requirement of RIP property. Performing the OMP algorithm to the signals within the sliding short-time window, the ambiguous micro-Doppler spectrum can be reconstructed. The proposed method can work in low PRF circumstances and requires transmitting fewer probing pulses while at the same time it can achieve larger unambiguous micro-Doppler range.

The authors declare that there is no conflict of interests regarding the publication of this paper.

The research was supported by the National High-Tech R&D Program of China, the Open-End Fund National Laboratory of Automatic Target Recognition (ATR), the Open-End Fund of BITTT Key Laboratory of Space Object Measurement, and the National Natural Science Foundation of China (Grant no. 62101196).