# Energy Efficient GNSS Signal Acquisition Using Singular Value Decomposition (SVD)
## 1 Introduction
[PERSON] and [PERSON] identified GNSS signal characteristics and addressed them by a universal acquisition and tracking channel, proposing an architecture that allows sequential acquisition and tracking of any chipping rate, carrier frequency, FDMA channel, modulation--i.e., BPSK(q) or QPSK(q), sin/cos BOC(p, q), CBOC(r, p, Pr\(\pm\)), and TMBOC(r, p, \(w_{r}\))--or constellation, where a mobile device could integrate fewer universal channels, securing signal availability and minimizing power consumption and chip size, the results showing a 66% increase in power consumption compared with the established reference [18]. The design principles align very well with this research in the sense that they identify the need to design new receivers to accommodate the increasing demands of new GNSS signals. In recent years, the CS approach has been proven to effectively reduce the number of measurement samples required for digital signal acquisition systems. Compressed sensing, also known as compressive sensing, is a signal processing technique for efficiently acquiring and reconstructing a signal by finding solutions to underdetermined linear systems. This is based on the principle that, through optimization, the sparsity of a signal can be exploited to recover it from far fewer samples than required by the Shannon-Nyquist sampling theorem [19]. This research recommends an efficient method to acquire a GNSS signal using compressed sensing. Fortunately, the GPS signal, as any wireless RF signal, is relatively sparse [20]. The topic proposed in this paper is a novel CS method that requires low computation and regular hardware size, completes the acquisition process faster, and acquires weak signals until about \(-160\) dBm. An extensive description of CS theory is described in Sections 2 and 3. The central problem of compressed sensing is the reconstruction of the high-dimensional sparse signal representation of \(x\) from a low-dimensional linear observation \(y\). A study from [PERSON] and [PERSON] performed a preliminary exploration of CS theory applied in GPS systems in 2012 [21]. They utilized the classic random binary matrix to observe the GPS signal and then adopt the reduced multiple measurement vector boost algorithm to reconstruct the signal. However, the signal reconstruction algorithm is very complex as the scheme is based on the multiple measurement CS theory. [PERSON] proposed a two-stage compressed sensing algorithm taking a specifically structured matrix as the measurement matrix and employing multiple Walsh-Hadamard transforms as the signal reconstruction algorithm in 2012 [22], though the two-stage compressed sensing will lead to much higher algorithmic complexity. Additionally, the algorithm can be used only to acquire strong GPS signals, which is not always the case. [PERSON] et al. developed a novel technique scheme based on CS achieves the transform sparsity of GNSS signals by utilizing the Gaussian random matrix and recovers the signal by using the single measurement OMP (orthogonal matching pursuit) algorithm [23]. This scheme has an extra carrier to noise(CNR) loss problem, and the extra CNR caused by the CS algorithm is inversely proportional to the compressed ratio. The research is useful in the sense that it implies how to select a better anti-noise performance measurement matrix and how to choose the best performance of a signal reconstruction algorithm based on different compression ratios, increasing the coherent integration and the number of non-coherent integration. To solve the problem mentioned previously, a novel GNSS signal acquisition scheme based on compressed sensing is proposed in this research. The main focus is on \(\ell_{1}\) minimization decoding models because \(\ell_{1}\) minimization has the following two advantages: (a) the flexibility to incorporate prior information into decoding models; and (b) uniform recoverability [24]. A critical aspect regarding uniform recoverability is that recoverability is essentially invariant concerning different types of random matrices.