## 1 Introduction  
As the most mature Global Navigation Satellite System (GNSS), Global Positioning System (GPS) has occupied the dominated position in many aspects, such as civil application, scientific research, and military defense. Further, the GPS modernization plan has drawn remarkable improvement that adds new civil signals transmitted by parts of GPS satellites for high-accuracy navigating and positioning in complex environments. These new signals possess better performance of self-/cross-correlation, as well as the ability to suppress narrowband interference and correct transmission delay of the ionosphere [1]. The longer code period of new signals, especially GPS L2C signals, indicates that computational complexity and time consumption will become greater for conventional acquisition algorithms [2]. Besides, multimode multi-frequency GNSS receivers are applied in many fields nowadays. Broadband receiving of multi-frequency signals is the popular and feasible technique to minimize the energy consumption and physical size of those receivers [3, 4]. However, broadband receiving requires higher sampling frequency, which dramatically increases computation and time consumption of the subsequent signal processing. It brings an inevitable dilemma for signal acquisition. In principle, signal acquisition is the chief operation to identify the visibility of satellites and provide estimated values of carrier frequency and code phase of received signals. Based on these estimated values, signal tracking can be successfully activated for following baseband operations. Actually, signal acquisition is a two-dimensional search process over carrier frequency and code phase uncertainty [5], by correlating with local replicated ones. Successful acquisitions are affirmed if acquisition decision statistics exceed the acquisition threshold. Conceivably, signal acquisition is quite time-consuming and requires tremendous computation [6]. Meanwhile, computation acts as the bottleneck of GNSS receiver applications, as it determines the lowest sensitivity of the baseband processing. Therefore, requirements for faster, less computationally demanding, and more efficient acquisition approaches are put forward to GNSS receivers in an extensive research field [1; 5; 7]. Attributed to the primary convolution and correlation operations in the acquisition process, various sophisticated acquisition algorithms have been investigated to enhance the efficiency of Fast Fourier Transform (FFT) to confront the challenge of computation and time cost. For instance, ideas of sparse FFT [8], fast FFT [9], reduced-size FFT [10], and averaging correlation [11] are attempted to reduce computational complexity when acquiring satellite signals. Specific to \(n\)-point FFT operations of parallel search acquisition algorithms, various divisions have presented their respective effects on relieving the burden of computation and time cost, such as radix-2 and radix-4 FFT [12] or by replacing with \(m\) sets \(2k\)-point FFT [13]. Besides, multi-step acquisition methods [14; 15; 16; 17; 18] also are studied to seek for better performance of signal acquisition on computation and time consumption. Taking advantage of the double-channel structure of modernized GPS signals, the joint data-pilot channel strategy [19] outperforms other individual channel strategies on signal acquisition. In fact, computational complexity and time consumption of signal acquisition are highly dependent on the data size of sampled signals used for convolution and correlation operations. Although extensive study has been conducted, most of them concentrate on reducing instant processing points of FFT. Only a few investigations [20; 21; 22] have carried out attempts to directly shorten or down-sample the received original signals. The bandpass sampling theory [23], which is a direct expansion of the traditional Shannon sampling theorem, has been widely applied in the fields of communication and radar signal processing [24; 25; 26].