###### Abstract  
This paper presents a simplified joint synchronization scheme for the integer carrier frequency offset and physical cell identity using a primary synchronization signal (PSS) in 5G new radio (NR) communication systems. We demonstrate the efficiency of our proposed NR-PSS synchronization scheme by deriving its simplified implementation, which exploits the near-zero autocorrelation feature between cyclically shifted NR-PSS symbols. As a figure of merit, we compute the probability of detection failure of the proposed NR-PSS synchronization scheme and validate its accuracy via simulations. To illustrate the benefits and limitations of the proposed NR-PSS synchronization scheme, we compare it with the conventional NR-PSS synchronization scheme, considering factors such as detection performance and computation complexity. Numerical results indicate that, regardless of the channel environment, the proposed NR-PSS synchronization scheme achieves a significant reduction in arithmetic complexity while maintaining the same detection capability as existing NR-PSS synchronization schemes. 5G new radio system; primary synchronization signal; physical cell identity; integer carrier frequency offset 2023114326[[https://doi.org/10.3390/math11204326](https://doi.org/10.3390/math11204326)]([https://doi.org/10.3390/math11204326](https://doi.org/10.3390/math11204326)) Academic Editor: [PERSON] Received: 27 September 202312 October 202316 October 202316 October 202316 October 202317 October 202317 October 202317 October 202317 October 202317 October 202317 October 202317 December 202317 December 202317 December  
extend network coverage, 5G NR incorporates innovative technologies such as massive multiple-input multiple-output (MIMO) and beamforming [5; 6]. An essential distinctive feature of NR is its ability to facilitate beam-based operations. This technology proves highly advantageous for mmWave and massive MIMO applications, both of which are widely acknowledged as key enabling technologies for 5G NR mobile networks [7; 8]. Despite significantly increasing available bandwidth and spectral efficiency, these advancements pose challenges to the cell search for NR initial access between the user terminal and next-generation NodeB (gNodeB) [9; 10; 11]. More precisely, in mmWave communication systems, the use of high-frequency carriers may lead to a significant increase in sensitivity to time and frequency offsets, which can become a potential drawback [9; 10]. Furthermore, beam management entails complex algorithms and extensive computational processing at user equipment (UE) to achieve initial access and beam tracking [11]. When a terminal is powered on or loses its connection, it will typically attempt to connect or reconnect to the cellular network. During this process, the UE may have limited information about potential serving cells. To obtain essential synchronization information like symbol timing offset (STO), carrier frequency offset (CFO), and physical cell identity (PCI) of potential serving cells, the UE performs the cell search procedure [12; 13; 14; 15; 16]. For the UE to synchronize with the network, it relies on various specific synchronization signals (SSs) periodically broadcasted from a serving gNodeB. These SSs include an NR primary synchronization signal (NR-PSS), an NR secondary synchronization signal (NR-SSS), and a physical broadcast channel (PBCH), which together carry information about the PCI of the transmitting gNodeB and provide synchronization information. In the initial phase of communication, these SSs are used for the STO and CFO detection [13]. By detecting the SS broadcasted by a best-serving gNodeB, the UE can estimate the STO and CFO [14]. After removing coarse STO and CFO in the time domain, the UE tries to identify a specific PCI by retrieving the synchronization identity (SID) message broadcasted by the SS, which contains a cell ID sector (CIDS) and a cell ID group (CIDG) [15; 16].