# An Under-Sampling Array Signal Processing Method Based on Improved Hadamard Matrix
## 1 Introduction  
Array signal processing [1] plays an important role in many aspects of applications such as radar, sonar, communication, and so on. Direction of arrival (DOA) estimation, as an important branch of array signal processing, has become a hot research topic. There are three main types of DOA estimation algorithms [2]: beam scanning algorithm; high-resolution methods; and sparse signal-processing algorithm. Conventional beamforming (CBF) is the most widely used beam scanning algorithm, it has high robustness against signal mismatch [3], but the resolution of the algorithm is limited by the Rayleigh limit [4], which means that it has wide beam width. Typical high-resolution methods include minimum variance distortionless response (MVDR) [5], multiple signal classification (MUSIC) [6], DOA estimation based on eigenspace (ES-DOA) [7], and estimation signal parameter via rotational invariance techniques (ESPRIT), etc. These algorithms mainly rely on the statistical characteristics of the signal received by the array to complete the estimation of the direction of arrival of the signal. Therefore, to obtain higher estimation accuracy, more hardware circuits are often required to obtain a large amount of measurement data, and a large amount of calculation time is required. Compressive sensing (CS) [8], a sparse signal-processing algorithm, can obtain all the information of the signal using less signal observation data. Its appearance improves the performance of high-resolution DOA estimation under low sampling conditions. Compressive beamforming methods have higher angular resolution than traditional high-resolution beamforming methods [9].Furthermore, the case where the number of sources is larger than the number of physical sensors has been shown to be much more difficult in the DOA estimation problems studied [10, 11]. For this uncertain DOA estimation task, various sparse array structures have been proposed aspossible solutions [12; 13], such as the nested array [14; 15] and coprime array [16] and their various extensions, for both second-order and fourth-order problems. In recent years, CS has received widespread attention, and the measurement matrix is an important technology to implement an under-sampling array signal processing. Reference [17] proposed a compressive sampling array (CSA) using a random measurement matrix, such as the Gaussian, or Bernoulli matrix, to compress large-size array elements into small-size array elements and estimated the angle spectrum of uncorrelated sources by the CSA-MVDR method. Reference [18] proposed a novel compressive sensing beamforming (CSB) algorithm based on CSA for DOA estimation. Reference [19] investigated DOA estimation by compressive sampling array and compressive beamformer algorithm and analyzed their advantages and limitations. DOA estimation of coherent sources was addressed in reference [20] by compressing the array in the spatial domain, but the effect of estimation could be improved. Reference [21] proposed the approach of CS array DOA estimation based on eigenspace (CSA-ES-DOA), and it used a very small number of measurements to resolve the DOA estimation of the coherent signals and two closely adjacent signals. The measurement matrix plays a vital role in CSA, and the random matrix is difficult in hardware implementation, so the deterministic matrix based on m-sequence [22], the sequential partial Hadamard matrix [23], the deterministic binary block diagonal matrix [24], the Polynomial deterministic matrix [25] and Toeplitz and circulant matrices [26] were researched. The above deterministic measurement matrix can be realized by hardware, but they have certain conditions and limitations, and the reconstruction resolution after compressing is not ideal enough, especially at a small compression ratio.