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High range resolution (HRR) profiling using stepped-frequency pulse trains suffers from range shift and the attenuation/dispersion of range profiles while the target of interest is moving. To overcome these two drawbacks, a new algorithm based on the maximum likelihood (ML) estimation is proposed in this paper. Without altering the conventional stepped-frequency waveform, this algorithm can estimate the target velocity and thereby compensate the phase errors caused by the target's motion. It is shown that the velocity can be accurately estimated and the range profile can be correctly reconstructed.

High resolution radar is an area of vigorous research and development in recent years. It is known that radar's range resolution is inversely proportional to its bandwidth. Therefore, the increase in bandwidth correspondingly improves the radar's range resolution. However, the wideband radar pulses complicate the design of transmitters and receivers. Also, such radar receivers are subject to potential interference from other sources.

To overcome these drawbacks, Ruttenberg [

As mentioned in [

A new motion compensation algorithm, based on the maximum likelihood (ML) estimation, is provided in this paper. It will be shown that this algorithm can estimate the target's radial velocity accurately and reconstruct the distorted HRRP successfully. Without altering the conventional waveforms, the new algorithm can be implemented on the in-service stepped-frequency radars.

The remainder of this paper is organized as follows. In Section 2, the signal model of moving targets in stepped-frequency radar systems is formulated. In Section 3, the ML estimator of the radial velocity is derived. Then, using the fast Fourier transforms to reduce the computational load, the new algorithm is proposed. In Section 4, some numerical examples are given to demonstrate the performance of the proposed algorithm. Section 6 presents the conclusions drawn from this work.

A stepped-frequency pulse train is a series of pulses modulated with different carrier frequencies. The carrier frequency of the first pulse is _{c}_{c}_{r}_{n}_{0} and _{R}_{R}

The first phase term of the right side of _{R}

Denoting
_{k}_{k}_{k}_{k}^{j}^{(}^{ηk}^{1+}^{μk}^{12)}, ^{j}^{(}^{ηk}^{2+}^{μk}^{22)},…,^{j}^{(}^{ηk}^{(}^{N}^{−1)+}^{μk}^{(}^{N}^{−1)2)}]^{T}^{T}_{0}, _{1},…, _{N}_{−1}], _{0}, _{1},…, _{N}_{−1}], _{0}, _{1},…,_{K}_{−1}]^{T}_{0}, Ω_{1},…, Ω_{K}_{−1}].

The motion compensation and range profiling of a moving target can be seen as the estimation of the scatterers' amplitudes, ranges, and velocities. According to ^{H}_{0} = _{1} = … = _{K}_{−1} =

Supposing that the number of pulses _{0}′, _{1}′,…, _{N}_{−1}′].

With the above-mentioned preparation, the motion compensation algorithm will be introduced in this section. If the number of scatterers

Firstly, the parameter

Secondly, the FFT is adopted to estimate the
_{μ}_{′} = ‖FFT(^{2}. The largest _{μ}_{′} are chosen, and their sum is denoted as _{μ}_{′}. The _{μ}_{′} is a function of

Thirdly, the _{μ}_{′} is maximized with respect to _{R}_{min}, _{R}_{max}] is the search space of the target radial velocity.

Finally, the estimation of target radial velocity _{R}

As the FFT is not exactly equivalent to the ML estimator of

However, if the target is non-cooperative, the number of scatterers

An example is introduced here to validate the MDL criterion in our topic. We consider a moving target including four scatterers. The number of pulses in the stepped-frequency train is 512, and the signal-to-noise ratio (SNR) is 0 dB. The SMLE algorithm is used to yield the ML estimation of parameters and the following values for the MDL criterion (see

The minimum of the MDL is obtained, as expected, for the

With the MDL criterion and the SMLE algorithm proposed above, the scatterer number and the velocity of the target of interest can be estimated iteratively as follows.

Assume that the number of scatterers is

Obtain scatterers' parameters by using SMLE.

Calculate the MDL(

Assume

In this section, some numerical examples are given to show the performance of the proposed new algorithm. The noise is AWGN in all of these simulations, and the parameters of the radar waveform used in this section are shown in

First, we consider a moving target including only one scatterer. The range and the velocity of the target are 128Δ_{r}

Another moving target including seven scatterers is considered. The ranges of the scatterers are 140Δ

The number of scatterers is obtained through the MDL criterion and the radial velocity is estimated by the SMLE. Simulation results are presented in

In this paper, a new algorithm based on the ML estimation is proposed for HRR profiling of moving targets. This algorithm can be implemented on the in-service stepped-frequency radar systems, without changing their waveforms and system structures. The performance of this algorithm is guaranteed by the asymptotical optimality of the ML estimation [

The authors would like to thank the anonymous reviewers for providing them with a large number of detailed suggestions for improving the manuscript.

Root mean square error of the target velocity versus SNR.

Simulation results of the moving target including seven scatterers.

An example of the MDL criterion.

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|---|---|---|---|---|

MDL | 1100 | 924.99 | 748.11 | 572.3 | 574.9 | 576.12 | 577.36 | 578.7 | 580.14 | 581.6 |

Parameters of the simulated stepped-frequency pulse train.

Radar center frequency (_{c} |
9 GHz |

Frequency step size (Δ |
1 MHz |

Pulse number ( |
512 |

Range resolution (Δ |
0.293 m |

Pulse repetition interval (PRI) | 1 ms |

Pulse width ( |
0.5 μs |