The complex environments of the littoral zones prevent the radar from operating efficiently. We propose a waveform and filter design approach to help the radar improve the performance in littoral zones. The approach includes a phase-only nonlinear programming method for suppressing correlation sidelobes in specified Doppler and range intervals, and an alternating projection based algorithm for designing receive filters. Several numerical examples are provided to demonstrate the usage and effectiveness of the proposed methods.

Nowadays, radar has been used in many different applications. One of the most intricate working environments for maritime radar is the littoral zone [

Today, many researchers have proposed concepts and methods for developing flexible and multifunctional radar. Considering that the environment is tangled and constantly changing in a littoral zone, the radar systems should be flexible enough to confront all of the possible threats. As mentioned earlier, interested littoral zones are always full of inartificial and man-made objects. The targets with large radar cross-section (RCS) such as wind turbines, ocean liners, beacons, and cliffs will make it very difficult to detect a target with small RCS because of the range sidelobe masking [

In this study, we develop several waveforms and filter design methods for suppressing spectral magnitude in desired bands and correlation sidelobes in specified range and Doppler intervals. The paper is organized as follows. The signal model is provided in Section

We denote vectors and matrices with boldface lower and upper case letters, respectively. The imaginary unit is denoted by

An important operator used in this paper is

We introduce the lag shifting matrix

For many topics in signal processing, the optimization problem is usually related to the minimization of errors, such as parameter estimation, filter design, and signal synthesis. A powerful tool for solving this kind of problem is projection onto sets. The projection operation is defined as follows:

Based on the discussion in Section

the civil and military wireless devices have significant influence on radar performance;

the radar usually has to detect targets with low RCS from close range; thus, the echo from sea surface, that is, the sea clutter, can degrade the radar performance significantly [

numerals natural and artificial objects such as cliffs, reefs, wind turbines, and lighthouses make it difficult to find the interested targets.

These features make the littoral zone a harsh environment for traditional radar. Moreover, the radar scenes are usually quite different when the antenna is aiming in different directions. For example, the direction facing the land and the direction facing the open seas need totally different signal processing methods. Fortunately, the adaptive transmit technologies can be a cure for intricate environments as mentioned in Section

The baseband discrete form of a radar waveform can be expressed as a complex vector

The synthesis of the probe waveform is also the synthesis of the matched filter. First introduced several decades ago [

The output from of the matched filter of the

The output from the matched filter without noise. The solid line corresponds to a large target, and the dash line corresponds to a small target. (a) Linear frequency modulation waveform (LFM) is used as the probe waveform and (b) a specifically designed sequence as the proved waveform.

From Figure

From the discussion above, we can find that if there are strong scatterers near the cell under test (CUT), the sidelobe interference will degrade the performance of the detector. To simplify the notation, here we define the descriptor of strong scatterers. A descriptor is an ordered sequence which is used to describe the vital properties of an interested object. The scatterer descriptor (SD) is defined as

The bandwidth resources in the littoral zones are insufficient due to the existence of a large number of wireless devices. A traditional radar system will suffer performance loss inevitably. Nowadays, the concepts such as “cognitive” and “adaptive waveform design” have been accepted by the radar community, and many researchers have shown that by designing waveforms with specified spectral shapes, the active interference from other devices can be weakened.

To analyze the spectrum of the signal and interference, we introduce the unit discrete Fourier transform (DFT) matrix. The matrix on

The radar systems working in littoral zones usually have to detect objects from close range; consequently, the clutter return becomes the dominant interference. Literature [

The compound-Gaussian model assumes that the clutter vector has the following form:

In [

In Section

We name the subspace defined by (

In addition to the elimination of the modular constraint, another advantage of PONLP is that it transforms the complex vector optimization problem into a real vector optimization problem, and many optimization software packages only support the optimization of real vectors. Utilizing the chain rule, the phase-only gradient of (

In littoral zones, solving (

Notably, for all

To yield a waveform that can suppress range sidelobes and active interference simultaneously, we combine

As mentioned in Section

(a) Autocorrelation of a designed unimodular waveform. (b) A range profile of a highly spiky clutter. (c) A range profile of a pervasive clutter.

In Figure

In order to construct the descriptor set

After the set

Unlike version II of CSSA, the number of the suppressed range cells is determined by multiple factors, including the threshold

From Figures

According to (

ISAA is based on alternating projections (AP), and thus the basic modules are the sets. For ISAA, there are two kinds of sets: objective sets and constraint sets. The objective sets are comprised of all the vectors with some good property, and the constraint sets are comprised of all the vectors with the must-have property. To derive the objective set for filter design, we derive the objective function first. According to (

To constraint the SNR loss, we introduce the Inner Product Constraint (IPC) of which the constraint set can be expressed by the following set-valued function:

A design example is shown in Figure

A design example. (a) The autocorrelation magnitude of the waveform. (b) The spectrum of the waveform.

The suppressed band is [0.1, 0.3] in normalized frequency. From Figure

To quantify the clutter suppression performance of the waveform, we define the performance index as follows:

Average PI of the designed waveform.

The numerical simulation has shown that this method based on waveform design can mitigate the clutter spikes sidelobe interferences. However, these results are obtained under the assumption that the positions of the spikes are fixed during the processing interval. In fact, sea clutter is highly dynamic; the spikes can only survive for seconds or even shorter, and sometimes they move in range at the group velocity of the sea waves. Thus, compared with mitigating the sidelobes from targets with large RCS, it seems very difficult to apply the method for sea spikes suppression. Here we discuss the scheme and strategy on spikes suppression using specifically designed waveforms.

As mentioned in Section

Unfortunately, the scheme discussed above will have no clutter suppression effect if the subdwells cannot be accomplished during the lifetime of the spikes. It may perform even worse than traditional radar for the sidelobes outside the suppressed intervals are sometimes higher than traditional waveforms. In other words, the related algorithms must be effective enough to reduce time consumption. It is obvious that the waveform design algorithm is time-consuming, and thus it is a major concern to improve its computational efficiency. Compared with air-borne and space-borne systems, the limitations of the size and power consumption of the signal processing systems mounted on ship-borne or coastal radars are not so rigorous, which allows us to introduce powerful computers for waveform design. For example, we can build a hybrid system with CPUs and GPUs; these kinds of systems can handle floating point data stream in massive parallel computing. With the floating point operation capability, the computer can reduce the quantization error of the gradients and will converge faster than a fixed point computer. With the parallel computing capability, the time consumption of the gradients calculation can be reduced.

We provide a detection example to demonstrate the usage of the methods provided in this paper. We setup a simplified radar scene described as follows:

the signal and interference are normalized with respect to the noise level;

a large still target with

according to intelligence, a small target may exist near this large target, and its position is unknown to us;

the echoes of the targets are modeled as Gaussian processes.

Assume that the actual position of the small target is the 84th range cell, and the autocorrelation level of the LFM waveform in this range cell is −34 dB; thus, the sidelobe interference is

The detection performance.

Secondly, we utilize a two stage method. In the first stage, considering that the position of the small target is unknown, we cannot utilize the method provided in Section

The correlation of the probe waveform and the filter. The suppressed intervals are (a) [

For the filter shown in Figure

After the suspicious range of the small target is determined, we start the second stage and run the waveform design procedure. Considering that the small target may have moved, we broaden the suppressed interval. The autocorrelation of the designed waveform is shown in Figure ^{−6}, which means that the result is much reliable. Compared with the traditional method (dash line in Figure

The autocorrelation of the designed waveform.

Notably, the waveform designed here is phase modulated, which means it is Doppler sensitive. This property of the waveform may lead to SNR loss when the radar encounters moving targets. By constructing a receive filter of moving target [

Assume that the small target is staying still as before, while the large target starts to move with its normalized Doppler frequency that equals −0.016. In the first stage of detection, we design different filters for testing different range Doppler intervals using ISAA-COIPC. One of the filter’s discrete cross-ambiguity functions is shown in Figure

The discrete cross-ambiguity function of a designed receive filter.

The discrete ambiguity function of the designed waveform.

From Figures

The average null depths of the quantized filters and waveforms.

Quantization (bits) | 8 | 12 | 16 |

Average null depth of designed filter (dB) | −45 | −61 | −93 |

Average null depth of designed waveform (dB) | −59 | −88 | −112 |

From Table

In this paper, we provide a method for designing waveforms and filters with the capability to suppress range sidelobe interference in the CUT from nearby large scatterers and active interference from other wireless devices. By utilizing information from secondary database, the method can suppress the correlation sidelobes in specified zones and can have a stop band in the desired frequency intervals. In scenarios that the secondary database is not adequate to use, the proposed filter design method can be used to determine the zone in which the sidelobe should be suppressed. The numerical examples have shown the usage of these methods, and the result shows that the detection performance is improved about 7 dB compared with the traditional method under the specified radar scene.

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

This work is supported by the National Natural Science Foundation of China (61371181), Shandong Provincial Natural Science Foundation, China (ZR2012FQ007), and the Natural Scientific Research Innovation Foundation in Harbin Institute of Technology (HIT.NSRIF.2011118).