In this paper, we investigate the code design problem of improving the detection performance of a moving target in the presence of nonhomogeneous signal-dependent clutter for moving target-detecting (MTD) radar systems. The optimization metric is constructed based on the signal to clutter and noise ratio (SCNR) of interpulse matched filtering. Under the frameworks of cyclic and majorization-minimization algorithms, we propose a novel algorithm, named CMMCODE, to tackle the code design optimization problem in the case of unknown precise target Doppler information and nonhomogeneous clutter. In the white-noise case, the simplified algorithm is also given based on CMMCODE algorithm. The presented algorithm is computationally efficient and convergent. Numerical examples show the effectiveness of the proposed algorithms.

Radars are required to deal with the simultaneous effects of receiver noise, signal-dependent clutter, and signal-independent interference, when detecting targets since echoes are received from the natural environment such as land, sea, and weather. The signal-dependent clutter can be seen as the convolution result of the transmitted signal and unwanted obstacles, whose scattering magnitude can be many orders larger than that of the target. Additionally, the signal-independent interference consists of various types of noise and jamming signals. Moving target detecting (MTD) is a mode of operation of a radar to discriminate a moving target against clutter. By taking advantage of the fact that Doppler frequency shift of a moving target is different from that of clutter, MTD technology plays an important role in modern radar systems [

The transmit waveform design has been paid much attention to in radar research area in the last decades. The optimal radar waveform is synthesized based on different performance objectives. In the early 1960s, the problem of designing radar signals and receivers that are optimum for detecting a point target masked by a background of clutter returns and thermal noise has been investigated. The problem of choosing an optimum signal has been discussed in [

A lot of work has also been carried out to radar code design by intrapulse coding [

The clutter model in these researches can be classified into two kinds, i.e., homogeneous and nonhomogeneous clutter, according to the properties of the clutter covariance matrix. Homogeneous clutter has an identical clutter covariance matrix for all resolution cells, and code design for homogeneous clutter can improve the detection performance for all cells. However, the clutter covariance matrix of one resolution cell in nonhomogeneous clutter may be different from that of the other cells. Therefore, code design for a certain range cell may deteriorate the others in nonhomogeneous clutter. In this paper, we will investigate the code design problem of detecting a moving target in the presence of nonhomogeneous clutter for MTD radar systems. Two different scenarios including general signal-independent interference and white noise are considered with the problem of detecting a moving target without precise Doppler frequency shift information. The optimization problems in this work are highly nonconvex and, therefore, firstly, made convex via relaxation of the original problems. Under the frameworks of cyclic algorithms [

The effect of nonhomogeneous clutter and a moving target is considered. To our knowledge, code design for improving detection performance in nonhomogeneous clutter has not been addressed in the literature.

The optimization metrics are constructed based on interpulse matched filtering, which is computationally efficient for coherent processing in a CPI. The corresponding optimization metric is given, and the connections between the SCNR and optimization metric are analytically addressed.

Under the frameworks of majorization-minimization and cyclic algorithms, we propose new algorithms to tackle the code design optimization problem in the case of unknown precise target Doppler information and nonhomogeneous clutter. The presented algorithm is computationally efficient and convergent.

The rest of this paper is organized as follows. In Section

We assume that a pulsed-Doppler radar transmits a coherent burst of

The radar scene can be split into different cells in range and velocity domains by a pulsed-Doppler radar at the same time. Range cells are obtained by transmitting a waveform with a certain bandwidth, whereas velocity cells are obtained by utilizing a burst of pulses that have a high Doppler frequency resolution. Suppose that there exists

We further assume that the clutter, distributed across the Doppler frequency domain, is composed of many individual scattering points, which are statistically independent. Consequently,

Note that, after downconverted and demodulated in the receiver, the received baseband signals are first compressed in the range domain to separate echo reflected by different range cells, and then, Doppler processing is performed to distinguish moving targets from stationary clutter. In radar engineering, the received signal is sampled at the time delays corresponding to all the range cells under test, whose interval is determined by the range resolution of the radar system. The discrete-time received signal

According to (

Suppose that the covariance matrices of

Suppose that the Doppler frequency shift of the target

If the target Doppler frequency shift is unknown, but the considered interval

Suppose that the target Doppler frequency shift is uniformly distributed over the interval

Until now, we have only considered the design metric for only one range cell under test, and for homogeneous clutter scene, the detection performance will be improved for all range cells. However, for the nonhomogeneous clutter scene, the covariance matrices

Consider that the signal-independent noise or interferences usually have the same statistical characteristic, which implies that

In a nonhomogeneous clutter scene, the SCNR expression is different from each other, and code design for a certain range cell may not optimize the SCNR in other range cells. Therefore, we should use the abovementioned design metric, which is the sum of clutter and signal-independent noise output of matched filters of all range cells, to evaluate the optimization performance.

The abovementioned design metric can also be transformed into the following expression, which is expressed as

To optimize the detection performance in a nonhomogeneous clutter scene, the optimization problem can be formulated under an energy constraint:

The objective function of the abovementioned optimization problem consists of a quadratic form

Let

Let

Lemma

Let

Let

The majorization-minimization (MM) method can be used to solve optimization problems that are difficult to tackle directly.

Let

By ignoring the constant items in (

This minimization problem can be transformed into a simple form:

The closed-form solution of (

Now, we summarize the algorithm, and the corresponding steps are given in Table

The proposed algorithm for code design in nonhomogeneous clutter.

Step 1: set |

Step 2: calculate |

Step 3: |

Step 4: |

Step 5: |

Step 6: repeat steps 3–5 until convergence |

In Table

According to the majorization-minimization scheme, the sequence of the objective function values

The optimization problem in (

Since the added item is just a constant, it will not affect the optimal solution of (

The limit point

Proof of Theorem

A special case for code design in a MTD radar system is that the signal-dependent interference only contains receiver noise, which is white and statistically independent with each other. In this case, let

We can obtain a lemma similar to Lemma

Let

By ignoring the constant item in (

Let

Consequently, the optimization problem is also equivalent to

The closed-form solution of (

Now, we summarize the algorithm, and the corresponding steps are given in Table

The proposed algorithm for code design in nonhomogeneous clutter and white noise.

Step 1: set |

Step 2: calculate |

Step 3: |

Step 4: |

Step 5: |

Step 5: |

Step 6: repeat steps 3–5 until convergence, |

In Table

Numerical examples are provided to demonstrate the detection performance improvement of the proposed algorithms. Two code design examples including the general signal-independent interference and white noise are given. In the general signal-independent interference example, the entries of the corresponding covariance matrix

In the white-noise example, (

To compare the performance improvement between the coded and uncoded cases, we define the transmit code as

This metric also indicates the algorithm performance in terms of merit (

Although CMMCODE algorithm is designed for the radar code for a nonhomogeneous clutter case, this algorithm can also be applied for obtaining optimal codes for the homogeneous clutter case. The goodness of the resultant codes is investigated by comparing with uncoded system, CoRe, and CADCODE algorithms in [

It is no surprise that the coded system, exploring the resultant codes of CoRe, CADCODE, and CMMCODE algorithms, outperform the uncoded system. Additionally, we note that average metric of the codes of CoRe, CADCODE, and CMMCODE are almost the same in Figure

(a) Average metric vs. transmit energy

We also investigate the detection probability [

In Figure

Convergence of CADCODE and CMMCODE.

In this example, we assume there exist 3 different range cells under the detection test. To demonstrate the effectiveness of code design, the spectrum model of the clutter is assumed to be Gaussian, and its corresponding covariance coefficient is given by

Clutter spectrum shape of different range cells.

Considering that CoRe and CADCODE algorithms can only synthesis code according to a certain clutter covariance matrix, we choose CADCODE algorithm for comparing the average metric and average detection probability with CMMCODE algorithm in a nonhomogeneous clutter case. Average detection probability (ADP) is defined to be the mean value of the detection probability in a different range cell and expressed as

In Figures

(a) Average metric of the resultant codes of CoRe, CADCODE, and CMMCODE algorithms and the uncoded system for 3 range cells; (b) average detection probability of an adaptive matched filter detector; and (c) IMP factor for different range cells.

To show the performance improvement of SCNR, the metric IMP is defined as

Figure

Figures

Probability of detection vs. probability of false alarm for 3 range cells with the resultant codes of CADCODE and CMMCODE algorithms and the uncoded system. (a) _{d} vs. _{fa} for clutter _{1}; (b) _{d} vs. _{fa} for clutter _{2}; and (c) _{d} vs. _{fa} for clutter _{3}.

In this example, the signal-independent interference is assumed to be receiver noise, which is white and statistically independent with each other. The noise power

Figures

(a) Average metric of the resultant codes of CoRe, CADCODE, and CMMCODE algorithms and the uncoded system for 3 range cells; (b) average detection probability of an adaptive matched filter detector; and (c) IMP factor for different range cells.

Figures

Probability of detection vs. probability of false alarm for 3 range cells with the resultant codes of CADCODE and CMMCODE algorithms and the uncoded system. (a) _{d} vs. _{fa} for clutter _{1}; (b) _{d} vs. _{fa} for clutter _{2}; and (c) _{d} vs. _{fa} for clutter _{3}.

In this paper, code design problem for a MTD radar system in a nonhomogeneous clutter scenario is investigated. Considering the SCNR metric of an interpulse matched filtering output, we construct the corresponding optimization problem for optimizing a radar waveform in a CPI and transform the optimization problem by majorization-minimization. For solving this optimization problem, we propose an algorithm, i.e., CMMCODE, based on cyclic and majorization-minimization algorithms, which is computationally efficient. In the white-noise case, the simplified algorithm is also given based on CMMCODE algorithm.

CMMCODE algorithm demonstrates computational efficiency and fast convergence. In the homogeneous clutter case, its performance is as good as CoRe and CADCODE algorithms. In the nonhomogeneous clutter case, it has better balanced performances for different range cells.

Because of the (semi) negative definiteness of

With Lemma

With Lemma

Hence, we can easily obtain

Note that

It is easy to show that

Due to the positive definiteness of

The data used to support this study are available within the article.

The authors declare that they have no conflicts of interest.

This work was supported by Fundamental Research Funds for Central Universities of China under grant no. NS2019024.