A novel method for solving Doppler ambiguous problem based on compressed sensing (CS) theory is proposed in this paper. A pulse train with the random and sparse transmitting time is transmitted. The received signals after matched filtering can be viewed as randomly sparse sampling from the traditional fixedpulse repetition frequency (PRF) echo signals. The whole target echo could be reconstructed via CS recovery algorithms. Through refining the sensing matrix, which is equivalent to increase the sampling frequency of target characteristic, the Doppler unambiguous range is enlarged. In particular, Complex Approximate Message Passing (CAMP) algorithm is developed to estimate the unambiguity Doppler frequency. CramerRao lower bound expressions are derived for the frequency. Numerical simulations validate the effectiveness of the proposed method. Finally, compared with traditional methods, the proposed method only requires transmitting a few sparse probing pulses to achieve a larger Doppler frequency unambiguous range and can also reduce the consumption of the radar time resources.
Long range radars (LRRs) usually work in low Pulse Repetition Frequency (PRF). Although there is no or low range ambiguity, it suffers from serious Doppler or velocity ambiguity. Therefore, Doppler ambiguity resolution is necessarily required to ensure accurate measurement of the radial velocity of a target. One resolution introduced from pulse Doppler technology is adopted by LRRs, which measure the velocity by transmitting high PRF impulse train. However, the high PRF mode requires more radar time resources, which are very precious to phased array radars when tracking multiple targets at the same time. On the other hand, the low signal pulse width may lead to a weak corresponding echo signal and hence a small operational range of the LRR, due to the high repetition frequency.
Many Doppler ambiguity resolution methods have been proposed in [
In this paper, we proposed a novel method to solve Doppler ambiguous problem, which tremendously increases fixedPRF and randomly determines the small amount of measurement corresponding to the time sequence of transmitted probing pulses. Explicitly, a sparse pulse train is randomly selected from traditional fixed repetition frequency pulses, according to the Restricted Isometry Property (RIP) of the CS sensing matrix, and a random perturbation item is added to the transmitting time of each selected pulse before transmitting. The received signals after Matched Filtering (MF) can be considered as randomly sparse sampling of Doppler. By using the CS recovery algorithms, the sensing matrix can be built based on the transmitting time sequence and recover the whole Doppler spectrum. As a result, the number of columns of sensing matrix becomes quite large such that many reconstruction algorithms of CS are not adequate, namely,
To address this problem, the proposed method for enlarging the Doppler unambiguous range in this paper is based on Complex Approximate Message Passing (CAMP) algorithm [
The performance of the CAMP algorithm can be accurately predicted by State Evolution (SE) formalism introduced in [
The rest of the paper is organized as follows. In Section
Let the center frequency to be denoted by
As can be seen from (
The CS theory is able to reconstruct the original signal from a set of nonadaptive measurements sampled at a much lower rate than required by the Nyquist sampling theorem by simultaneously sensing and compressing sparse or compressible signals. The original signal denoted by
The CAMP is a fast and efficient iterative algorithm for solving the problem with complex vectors and matrices involved. In the real number field, the algorithm is referred to as AMP algorithm, which only requires the transpose operation on the sensing matrix
In the ideal scenario, the complex original signal
Clearly, in Algorithm
Algorithm
Fixed point
To construct the measurement matrix
Since
The PRF of LRR is usually low and the typical value of
Clearly,
In practical, the term
After determining the dwell time
The definition of SNR for the CAMPbased CS radar system is provided by [
Consider the case of
The conditional Probability Density Function (PDF) of
In this section, the performance of proposed method is studied with the aid of numerical simulations. We commence the section by investigating the noisefree scenario, where the unambiguous Doppler frequencies are 345.482 Hz and 347.158 Hz. Since radar transmits a fixedPRF (PRF = 100 Hz) pulse train. It is worth pointing out that the pulse train can be used in some LRRs for the velocity measurement or the normal waveform for tracking targets.
Spectrum of recovered slowtime echo signal.
Since the tradeoff between success rate of reconstruction and calculation amount is determined by the value of the refining factor, the influence of refining factor for reconstruction performance is investigated as follows. Monte Carlo (MC) simulation results under different refining factors are depicted in Figure
The success rate of reconstruction with different refining factor.
The success rate of reconstruction estimated at different SNR is shown in Figure
The success rate of reconstruction at different SNR.
As shown in Figure
Comparison of simulation times versus different refining factors.
Refining factor  10  100  500  1000 


Time (s)  0.044040  0.417513  2.064184  3.909073 
The comparison between the Root Mean Square Error (RMSE) and the CRLB for frequency estimation of a sinusoid with the presence of noise is investigated, as shown in Figure
Performance of the proposed method.
The relationship between
In the following, our proposed method is compared with traditional methods, including CRT, Staggered PRF, and nonuniform sample [
Compared with CRT, Staggered PRF, and nonuniform sample.
Method  CS  CRT  Staggered frequency  Nonuniform sampling 

Sample property  Randomly probing pulses  Relatively prime  Close to each other 



Pulse number 






Unambiguous range 




Relied on sparse property of radar echo and CS theory, a novel Doppler ambiguous resolution utilizing sparse probing pulses is proposed. The whole slowtime domain echo signals can be recovered via the adaptive CAMP reconstruction algorithm. By refining the basis matrix, the equivalent high PRF echoes can be obtained and therefore, the Doppler unambiguous range can be enlarged. Using the SE formalism, we have derived the CRLB of the proposed method. The accuracy of estimation is demonstrated by theoretical inference and simulations. Compared with traditional methods, our novel method only transmits a few sparse probing pulses and reduces more than 80% of the time resource consumption of radar. Furthermore, due to the fact that the pulse transmitting time can be controlled by the radar clock, which generally has a frequency of 10 MHz or integral multiple of 10 MHz, the precision of pulse transmitting time adjustment needed in the paper is achievable. Therefore, our proposed method is adequate to be implemented in practical, since only slight adjustment is required on the radar dwell scheduling.
The authors declare that there is no conflict of interests regarding the publication of this paper.
The research was supported by the National Hightech R&D Program of China and the OpenEnd Fund National Laboratory of Automatic Target Recognition (ATR). The authors would like to thank Laura Anitori and Zai Yang for sharing the CAMP Matlab package.