# GNSS Signal Compression Acquisition Algorithm Based on Sensing Matrix Optimization
## 1 Introduction  
Global navigation satellite system (GNSS) [1] is now widely used in civil, commercial, and military applications [2; 3; 4] that allows users to compute their position, velocity, and time (PVT) based on signals transmitted by satellites worldwide. The obvious example is global positioning system (GPS), but this term includes other systems such as GLONASS, Galileo, and BeiDou. The GNSS receiver performs many complex operations, out of which acquisition has great importance. It provides a rough estimate of the code delay and Doppler frequency values of the received signal, which plays a crucial role in synchronizing the local signal with the received one.  
Serial searches are conventional search techniques that use a correlator to test every hypothesis in the two-dimensional (2D) search space. The techniques based on a parallel process method (fast Fourier transform (FFT), etc.) can improve the acquisition effect. However, the improvement of mean acquisition time (MAT) owes to the usage of dedicated hardware resources. To further reduce the computational burden of signal acquisition, the folding techniques [5; 6; 7] are adopted to fold the code replica, and the folded code is correlated with the received signal. However, they are all based on Nyquist sampling and cannot fundamentally solve the high sampling rate and sizeable computational complexity of GNSS signal acquisition.  
The data size of the compressed sensing (CS) [8] algorithm is independent of the signal bandwidth. Meanwhile, it points out that sparse signals can be sampled at a lower rate than Nyquist sampling [9]. Thus, the data storage and computational complexity can be significantly reduced by using the CS theory. It has been applied in many fields including image processing [10], audio signal processing [11], wireless communication [12], etc. In recent researches, several CS methods have been applied to GNSS signal acquisition. By means of the Walsh-Hadamard matrix, [PERSON] et al. [13] proposed a two-stage compression of GNSSsignal, which achieved fast acquisition with fewer correlations than traditional acquisition methods. [PERSON] et al. [14] estimated the Doppler frequency using the strapdown inertial navigation system (SINS) speed information to reduce the frequency search range. After that, a CS model is constructed to lower the sampling rate and computation complexity. He et al. [15] directly selected the maximum inner product value as the sparse value. The GPS signal can be significantly compressed, and the transmission energy consumption is reduced accordingly. [PERSON] et al. [16] designed the sparse representation dictionary according to the same pseudo-random noise (PRN) code and the different satellite carrier frequencies. Then, a new multiple measurement vector acquisition algorithm is adopted to acquire the signal. However, the above algorithms construct the sensing matrix based on a random measurement matrix, which leads to the detection probability is low under low SNR condition. In this paper, a GNSS signal compression acquisition algorithm based on sensing matrix optimization is proposed.  
The rest of this paper is organized as follows. Section 2 describes the sparsification of GNSS signal, the compression acquisition theory, and the existing compression acquisition algorithm. Section 3 proposes the optimization method of sensing matrix and theoretically analyzes the advantages of the proposed method, including the noise power and computational complexity. Section 4 compares the performances of the proposed algorithm with other acquisition algorithms and discusses the performances with different compression parameters. Section 5 summarizes the simulation results and the different advantages of the proposed algorithm.