## 1 Introduction  
Global Positioning System (GPS) signal deteriorates greatly in urban environments, and its signal power often degrades down to \(-\)144 dBm, resulting in difficult acquisition. With the advent of wearable applications and the Internet of Things (IoT), the energy consumption of GPS receivers is becoming increasingly important. Currently, the power consumption of commercialized GPS products is high--up to 20-30 mW in continually position mode [1]. Thus, they are not suitable for energy-sensitive applications, such as smart watches and sharing bikes. In a real GPS receiver, the acquisition engine consumes over 70% energy due to its complex computation and repeated acquisition in urban areas. Until now, researchers have studied several methods of GPS signal acquisition. These acquisition methods mainly focus on the reduction of acquisition time, thus reducing The Time to First Fix (TTFF), or increase the sensitivity while ignoring the energy consumption [2]. Research on GPS acquisition mainly addresses three aspects, which are correlation, integration, and searching strategy. For correlation, the fast Fourier transform (FFT)-based parallel code acquisition method has been the most popular algorithm in recent years due to its fast searching compared with serial code acquisition [3; 4; 5]. In a weak signal environment, 1 ms correlation results are not reliable because of noise interference. In this case, methods such as coherent integration, non-coherent integration, and differentially coherent integration should be applied to further improve the signal-to-noise ratio (SNR) [6]. Coherent integration is an efficient method to improve sensitivity, but the time is limited because of the bit transition. Non-coherent integration is less sensitive, but infers extra squaring loss. A multiple of one data bit interval is chosen as predetection integration time to estimate the most likely data bit combination and reverse the data signs subsequently for very weak GPS signals [7; 8]. A combined coherent and differentially coherent acquisition scheme avoids complex FFT post-processing of correlation results and acquires \(-147\) dBm GPS signal in 200 ms without any assistance data [9]. Rigorous mathematical analysis and Monte-Carlo simulation shows that differentially combined integration can decrease approximately 3 dB loss of non-coherent integration for weak GPS signals [10]. Since received weak signals may suffer self-interference from other strong satellite signals, a partitioned subspace projection method is proposed to mitigate the self-interference [11]. A new space-borne GPS receiver called Navigator employs special signal processing algorithms in radiation-hardened hardware that enable fast acquisition and high sensitivity for high-altitude satellites [12]. The maximum/threshold crossing (MAX/TC) method is the most commonly used acquisition searching strategy [13]. It is a hybrid criterion where the uncertainty region is divided into N\({}_{\rm s}\) sectors with M cells each. The cell is selected according to the MAX criterion inside a sector. In detail, in a sector, select cell C(i) if u(i) \(\geq\) u(i), for \(i\) = 1,..., M; choose cell C(i) if u(i) \(\geq\) k, else go to next sector. Note that for M = N\({}_{\rm c}\), N\({}_{\rm s}\) = 1, \(\xi\) = \(-\infty\), the MAX/TC criterion reduces to the MAX criterion, while for M = N\({}_{\rm c}\), N\({}_{\rm s}\) = N\({}_{\rm c}\), it reduces to the TC criterion. The usefulness of dealing with the MAX/TC criterion lies in the fact that the sector size M is a significant parameter to be optimized in order to achieve minimum acquisition time for a serial architecture, or reasonable complexity for a parallel architecture [14]. A proper selection of the probability of a false alarm using constant false alarm rate (CFAR) criteria can obtain a lesser mean acquisition time than fixed threshold method or MAX method without threshold [15]. A threshold function approach was proposed to simplify the CFAR criteria by a modeling procedure and an approximation procedure [16].