1. Introduction
Waveform design has received considerable attention in recent years [
1] and been employed in many applications, including polarimetric radar [
2], multiple-input multiple-output (MIMO) radar [
3,
4], stealth communications [
5] and the code-division multiple-access (CDMA) system. As a category of the general waveform design research [
1], the design of waveform sets (i.e., multidimensional waveforms) is an important research content of MIMO radar. Generally, waveform sets are desired to have a good correlation property, which can effectively improve radar resolution, detection performance, imaging quality, the ability to obtain information and the accuracy of MIMO channel estimation [
6,
7,
8]. In recent years, a large number of scholars has been devoted to designing waveform sets with a good correlation property. The main research covers two aspects: one is the waveform sets with good auto- and cross-correlation properties [
9,
10,
11,
12,
13,
14,
15,
16], and the other is the complementary sets of sequences (CSS) [
17,
18,
19,
20,
21,
22,
23,
24,
25].
Waveform sets with good auto- and cross-correlation properties, also known as the orthogonal waveform set (OWS), have low autocorrelation sidelobes and low cross-correlation levels. In the early stage, the simulated annealing- [
9] and cross entropy-based [
10] methods were proposed for OWS design. However, due to the high computational complexity, these methods are not suited to design long waveforms. To improve the computational efficiency, the CAN (cyclic algorithm-new) algorithm [
11] based on fast Fourier transform (FFT) is proposed to minimize the autocorrelation sidelobes and the cross-correlation. This algorithm is computationally efficient and can be used for the design of long waveforms. As it is impossible to design completely orthogonal (i.e., both the autocorrelation sidelobes and the cross-correlation are zeroes) waveform sets [
11,
16], [
11] proposes to design the waveform sets that are orthogonal only at the specified intervals and extends the classical WeCAN (weighted cyclic algorithm-new) [
26] algorithm to MIMO radar. To solve the same problem, [
12] develops the LBFGS (limited-memory Broyden, Fletcher, Goldfarb and Shanno) iterative algorithm, which is more efficient than the WeCAN algorithm. However, because of the complicated linear search rule for determining the step size, the LBFGS iterative algorithm is still time consuming. Recently, the majorization-minimization (MM)-based algorithms (i.e., MM-Corr (MM-correlation) and MM-WeCorr (MM-weighted correlation)) are proposed in [
14]. These two algorithms are also based on FFT operations and much faster than the CAN and WeCAN algorithms [
11].
Another waveform set with a good correlation property is the complementary sets of sequences (CSS). A waveform set is called CSS if and only if the autocorrelation sum of the waveforms is a delta function [
17]. CSS design is proposed to overcome the difficulty of generating a single unimodular waveform with ideal (impulse-like) autocorrelation. A common application of the CSS is the pulse compression [
18,
19,
20]. In pulse compression radar, the complementary sequences are used to modulate consecutive pulses in a coherent pulse train. Then, the autocorrelation sidelobes can be reduced via the coherent [
19] or noncoherent [
18] accumulation, which can be regarded as the process of obtaining the autocorrelation sum of the complementary sequences. Moreover, due to the good correlation property, CSS has been widely applied to the CDMA system [
21], ISI (intersymbol interference) channel estimation [
22], orthogonal frequency division multiplexing (OFDM) [
23], and many other areas. The main methods of designing CSS are the analytical construction methods, which have great limitation in generating long waveforms. To overcome this problem, [
24] introduces a computational framework based on an iterative twisted approximation (ITROX) for periodically complementary sets of sequences design. Subsequently, [
25] extends the CAN algorithm [
26] and develops a fast algorithm named CANARY (CAN complementary). Additionally, [
14] applies the MM method to the design of CSS.
In addition to the good correlation property, waveform sets are expected to have a good stopband property when the radar systems work in a crowded electromagnetic environment. Waveforms with the stopband property, also known as the sparse frequency waveforms (SFW) in many literature works, are a kind of waveforms with several frequency stopbands. The applications of SFW include ultra-wide bandwidth (UWB) systems [
27], high frequency surface wave radar (HFSWR) [
28,
29] and cognitive radar [
30]. By designing waveforms with the stopband property, it can effectively overcome the narrowband interference in the congested frequency bands. At present, there are many research works on the design of a single waveform with the stopband property [
31,
32,
33,
34,
35,
36], but they cannot be used in MIMO systems. Therefore, [
36] proposes an iterative algorithm combined with the steepest descent (SD) method for MIMO waveform design. By searching along the gradient direction, the convergence speed of this algorithm is improved. However, the computation of the step size along the gradient direction is complicated, which makes the algorithm still costly. In order to improve the computational efficiency, [
37] proposes an algorithm named MDISAA-SFW (multi-dimensional iterative spectral approximation algorithm-SFW) based on alternating projection and phase retrieval.
In this paper, we consider the problem of designing unimodular waveform sets with good correlation and stopband properties and propose a gradient-based algorithm, i.e., Gra-WeCorr-SFW (gradient-weighted correlation-SFW). By using the relationship between the correlation function and the power spectrum density (PSD), the design problem is formulated as an unconstrained minimization problem in the frequency domain. Then, the phase gradient is deduced, and its matrix form is given. In order to avoid searching the step size, we use the Taylor series expansion to derive the step size, which is more efficient than the traditional searching methods. Since both the gradient and the step size can be implemented via the FFT operations and the Hadamard product, the proposed algorithm has high computational efficiency. We also deduce the simplified algorithm named Gra-Corr-SFW (gradient-correlation-SFW), which is faster than the Gra-WeCorr-SFW, for the design problem without considering the correlation weights.
The rest of the paper is organized as follows. In
Section 2, the design problem is formulated. In
Section 3, we develop a gradient-based algorithm by deducing the phase gradient and the step size and then summarize the algorithm. In
Section 4, the simplified algorithm for the design problem without considering the correlation weights is derived.
Section 5 provides several numerical experiments to verify the effectiveness of the proposed algorithms. Finally,
Section 6 gives the conclusions.
Notation: Boldface upper case letters denote matrices, while boldface lower case letters denote column vectors. , and denote the complex conjugate, transpose and conjugate transpose, respectively. and denote the Euclidean norm and the Frobenius norm. and denote the real and imaginary part, respectively. denotes a diagonal matrix formed with the column vector . ∘ denotes the Hadamard product. denotes the m-th element of the vector . is the l-th iteration of . is the all-one vectors of length N. denotes the identity matrix. and denote the -point FFT and IFFT (inverse FFT) operations of , respectively. and represent the FFT and IFFT of each column of the matrix , respectively. In the (I)FFT operations, if the length of is less than , is padded with trailing zeros to length . is the exponent arithmetic applied to the scalar, vector or matrix.