Complexity-Effective Joint Detection of Physical Cell Identity and Integer Frequency Offset in 5G New Radio Communication Systems

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.

1. Introduction

The 3rd generation partnership project (3GPP) has standardized fifth-generation new radio (5G NR) with the primary focus of enhancing the energy consumption and transmission efficiency of long-term evolution-advanced (LTE-A) while supporting ultra-wide bandwidths in millimeter-wave (mmWave) frequency bands [1]. As the demand for ubiquitous connectivity continues to grow, it is expected that 5G NR technology will extend its support to non-terrestrial networks (NTN), improve terrestrial coverage, and offer upgraded services [2,3]. To address the limitations of the 5G NR standard and introduce the NTN, the 3GPP aims to establish an NTN standard based on the 5G NR standard, thereby facilitating the integration of terrestrial network (TN) and NTN specifications. In the terrestrial downlink, NR uses orthogonal frequency division multiplexing (OFDM) with a cyclic prefix (CP), similar to LTE-A. In the uplink (UL), NR can also employ OFDM, as well as OFDM with discrete Fourier transform (DFT) precoding, which improves UL coverage but results in a lower peak-to-average power ratio [4]. To enhance spectral efficiency and 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]. With this information at hand, the terminal can further exploit other important system information and complete its initial access to a serving cell.

During the NR initial access procedure, the NR-PSS detection can be challenging due to the presence of substantial initial time and frequency offsets, which may result in performance degradation of the UE receiver [17,18,19]. To solve such a problem, a differential correlation strategy was devised to jointly detect the integer CFO (ICFO) and CIDS in the frequency domain [20,21,22,23]. One approach presented in [20] is based on maximum likelihood (ML) detection, utilizing minimum mean-square-error (MMSE) reduced-rank approximation of the channel. Despite the ML approach offering optimal detection performance, its computational intensity makes it less suitable for practical implementations. Conversely, the joint synchronization of ICFO and NR-PSS presented in [21,22] offers sub-optimal synchronization performance with moderate computational complexity as an alternative. In one paper [23], a reduced-complexity joint detection method for ICFO and NR-PSS was proposed using the cyclic-shifted property of NR-PSS. While this approach significantly reduces computational load, it does compromise performance to some extent. In 5G NR systems, the utilization of more affordable crystal oscillators in the UE may lead to considerable STO and CFO between the UE and the gNodeB. This misalignment introduces complexities in the process of performing the initial synchronization [13]. Moreover, utilizing beam-formed SS burst transmission with beam sweeping is essential for mitigating the increased channel propagation loss in the mmWave frequency band. However, due to beam sweeping, it may not be feasible to average NR-PSS synchronization over several time slots [9]. Since NR-SSS detection can only be carried out once the NR-PSS is completely recognized, the detection of NR-PSS becomes one of the utmost important and demanding tasks in the entire cell search procedure. As a result, developing a complexity-effective and still high-performance NR-PSS synchronization scheme becomes a crucial and difficult task in 5G NR communication systems.

In this paper, we present a simplified formulation for NR-PSS synchronization in the 5G NR communication system. Using a cyclic-shifted feature of NR-PSS generated from m-sequence, we devise an effective ICFO and CIDS synchronization scheme without compromising detection performance. We theoretically derive the probability of detection failure of the proposed ICFO and CIDS synchronization scheme to assess its performance. Numerical examples indicate that the proposed ICFO and CIDS detection approach achieves comparable performance while significantly reducing complexity by over 90% compared to sub-optimal NR-PSS synchronization methods, regardless of the channel model. This makes it a promising candidate for the initial cell search in 5G NR communication systems.

The remaining part of this paper is outlined as follows: In the next section, we introduce the signal model, SS, and cell search procedure in the 5G NR system. Section 3 presents our proposed joint ICFO and CIDS detection method, along with the theoretical performance analysis. Section 4 presents the experimental examples to verify the merits of the proposed NR-PSS synchronization scheme. Finally, in Section 5, we conclude with discussion.

2. System Description

2.1. Signal Model

We consider an OFDM system consisting of M equi-spaced subcarriers [15,16,17,18]. To create a time-domain OFDM symbol with a duration of T, we apply an M-point inverse DFT (IDFT) to the complex-valued data symbol $X_q\left(m\right)$ with symbol energy $E_X=\mathbb{E}\left\{\right|X_q\left(m\right){|}^2\}$, where $m=0,1,\cdots ,M-1$ is the subcarrier index and $\mathbb{E}\{·\}$ denotes the expectation operator. In order to ensure interference-free transmission, we append a CP of a duration $T_g$ to the effective part of the OFDM signal, producing one OFDM symbol of a duration $T_u=T+T_g$. With a duration of $T_u=(M+M_g)/f_s$, the transmitted signal in the time domain for the q-th symbol period can be written as

$$\begin{array}{cc}\hfill x_q\left(n\right)& =\sum _{m=0}^{M-1}{X}_q\left(m\right)e^{j2\pi mn/M},n=-M_g,-M_g+1,\cdots ,M-1,\hfill \end{array}$$

(1)

where $f_s$ represents the sampling frequency, $j=\sqrt{-1}$ is the square root of minus one, and $M_g$ is the CP length.

The transmitted signal undergoes multipath fading and is distorted by additive white Gaussian noise (AWGN). Moreover, the received signal frequently experiences the CFO arising from the discrepancy between the local oscillators of the transmitter and receiver. This paper considers a scenario where uncompensated ICFO, residual CFO (RCFO), and residual sampling time offset (RSTO) persist after the initial synchronization process has been performed in the pre-DFT phase [24,25]. As studied in [23], the impact of RSTO on the synchronization performance during the post-DFT detection phase is minimal. Assuming the receiver perfectly knows the RSTO, the transmitted signal $x_q\left(n\right)$ undergoes linear convolution with the multi-path fading channel. Hence, the received signal in the time domain for the q-th symbol period can be given by

$$\begin{array}{cc}\hfill y_q\left(n\right)& =e^{j2\pi q(\upsilon +\eta )M_u/M}{e}^{j2\pi (\upsilon +\eta )n/M}{x}_q\left(n\right)\otimes h_q\left(n\right)\hfill \\ & +z_q\left(n\right),n=-M_g,-M_g+1,\cdots ,M-1,\hfill \end{array}$$

(2)

where $\upsilon$ represents the ICFO normalized by the subcarrier spacing denoted by ${\Delta }_f$, $\eta$ is the RCFO normalized by ${\Delta }_f$, $M_u=M+M_g$, ⊗ denotes the linear convolution operator, $h_q\left(n\right)$ is the discrete channel impulse response with L resolvable multipaths, and $z_q\left(n\right)$ is the zero-mean AWGN with variance ${\sigma }_z^2$. If we assume that the CP duration is substantially longer to mitigate the impact of the multipath channel, it can be ideally removed from the received time-domain signal. By conducting the DFT process on the CP-discarded signal, the frequency domain received signal can be expressed as [26]

$$\begin{array}{cc}\hfill Y_q\left(m\right)& \approx G_q(m-\upsilon )X_q(m-\upsilon )e^{j2\pi ϵ(q{M}_u+M_g)/M}+I_q\left(m\right)+Z_q\left(m\right),\left|\upsilon \right|\le U,\hfill \end{array}$$

(3)

where $Y_q\left(m\right)$ denotes the demodulated OFDM symbol for the symbol index q, $ϵ=\upsilon +\eta$, $G_q\left(m\right)$ represents the channel state information (CSI) following a zero-mean Gaussian distribution with variance ${\sigma }_G^2$, $I_q\left(m\right)$ represents the inter-carrier interference (ICI) term with variance ${\sigma }_I^2\approx E_X{\sigma }_G^2{\eta }^2{\pi }^2/3$ for relatively small values of $\eta$, $Z_q\left(m\right)$ represents the zero-mean AWGN with variance ${\sigma }_Z^2$, and U corresponds to the largest value of ICFO.

2.2. Synchronization Signal

Upon the initial power-up of the UE, it retrieves two synchronization parameters, CIDS w and CIDG g, to find and identify the network. The NR system specifies 1008 PCIs, each uniquely derived by concatenating one NR-PSS and one NR-SSS from the available candidates. The NR-PSS is characterized by the CIDS value, represented as $w\in \{0,1,2\}$, while the NR-SSS is distinguished by the CIDG value, denoted as $g\in \{0,1,2,\cdots ,335\}$. The assignment of PCIs to UEs is carried out using the formula $N_{PCI}=g+336w$.

The frequency domain NR-PSS can be generated using a maximum length sequence (m-sequence) of length 127 points. This m-sequence occupies 127 subcarriers of the synchronization signal block (SSB) bandwidth. By introducing a cyclic shift to a base m-sequence denoted as $c\left(m^{\prime }\right)$, three unique NR-PSS symbols are obtained. The generation of these symbols can be described as

$$\begin{array}{c}\hfill p_w\left(m^{\prime }\right)=1-2c\left([m^{\prime }+43w]\mathrm{mod}N\right),0\le m^{\prime }<M_s,\end{array}$$

(4)

where $w\in \{0,1,2\}$ is an identifier of the NR-PSS among the potential candidates, $M_s=127$ is the length of the NR-PSS symbol. We denote $P_w\left(m\right)$ as the NR-PSS broadcasted at subcarrier m, where it spans across 240 subcarriers in the frequency domain. With this notation in mind, the NR-PSS sequence equipped with CIDS w can be expressed as

$$\begin{array}{cc}\hfill P_w\left(m\right)& =\left\{\begin{array}{cc}{p}_w(m-56),\hfill & 56\le m\le 182\hfill \\ 0,\hfill & \mathrm{otherwise},\hfill \end{array}\right.\hfill \end{array}$$

(5)

which says that out of the total 240 available subcarriers, the NR-PSS binary modulates only 127 subcarriers, with the remaining subcarriers being zero-inserted. Importantly, three NR-PSSs, each corresponding to a different CIDS value, are cyclically shifted versions of one another. This property ensures that the cross-correlation between any two NR-PSSs approaches zero when $M_s\gg 1$.

The NR-SSS is derived by combining two m-sequences to address the issue of poor cross-correlation. The resulting sequence produces a gold sequence with a length of 127 points and thus the NR-SSS is formulated as

$$\begin{array}{cc}\hfill s_g\left(m^{\prime }\right)& =[1-2c_0\left([m^{\prime }+15\lfloor g/112\rfloor +5w]\mathrm{mod}{M}_s\right)]\hfill \\ & \times [1-2c_1\left([m^{\prime }+\left(g\mathrm{mod}112\right)]\mathrm{mod}{N}_s\right)],0\le m^{\prime }<M_s,\hfill \end{array}$$

(6)

where $g\in \{0,1,2,\cdots ,335\}$ represents an identifier of the NR-SSS among the potential candidates, the operator $\lfloor x\rfloor$ returns the largest integer that does not exceed x, and the operator $\left(x\mathrm{mod}y\right)$ returns the remainder after dividing x by y. The generation of two sequences $c_0\left(m^{\prime }\right)$ and $c_1\left(m^{\prime }\right)$ is thoroughly formulated in [1]. The sequence $s_g\left(m^{\prime }\right)$ is then allocated onto $M_s$ NR-SSS subcarriers, where $m\in S=\{56\le m\le 182\}$, resulting in $S_g\left(m\right)$ as follows

$$S_g\left(m\right)=\left\{\begin{array}{cc}{s}_g(m-56),\hfill & 56\le m\le 182\hfill \\ 0,\hfill & 48\le m\le 55,183\le m\le 191,\hfill \end{array}\right.$$

(7)

where the NR-SSS occupies 127 subcarriers within the frequency domain and the other subcarriers out of the 240 available subcarriers are allocated exclusively for the PBCH signal.

2.3. Cell Search Procedure

In general, the initial cell search procedure consists of three steps [17]. The first step in the UE-side synchronization procedure is to obtain coarse STO, CFO, and frame timing based on the time-domain NR-PSS sequence. Once the pre-DFT synchronization processing is successfully completed, NR-PSS synchronization can be performed to obtain the CIDS in the frequency domain. Since the CSI is available only after the UE detects the ICFO, a differential detection technique is usually employed for non-coherent NR-PSS detection, thus eliminating the need for CSI estimation. This approach effectively removes the effects of channel fading and STO from the NR-SSS without requiring the CSI, consequently enhancing the synchronization process. Assuming $G_q\left(m\right)\approx G_q(m-1)$, the differential cost function is given by [21]

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_{pss}(\tilde{\upsilon },\tilde{w})& =\sum _{m\in \mathcal{D}}{\overline{Y}}_q(m+\tilde{\upsilon })D_{\tilde{w}}^{*}\left(m\right),\hfill \end{array}$$

(8)

where the notation of the type $\tilde{x}$ stands for trial value of x, $\tilde{\upsilon }\in \{-U,-U+1,\cdots ,U\}$, $\tilde{w}\in \{0,1,2\}$, $\mathcal{D}=\left\{m\right|57\le m\le 182\}$ whose cardinality is $M_s-1$, $D_{\tilde{w}}\left(m\right)=P_{\tilde{w}}\left(m\right)P_{\tilde{w}}(m-1)$ is the differential NR-PSS, and ${\overline{Y}}_q\left(m\right)=Y_q\left(m\right)Y_q^{*}(m-1)$. Based on the mentioned observations, the estimated values $(\widehat{\upsilon },\widehat{w})$ can be found by identifying the maximum of the cost function ${\mathrm{\Omega }}_{pss}(\tilde{\upsilon },\tilde{w})$ over $(\tilde{\upsilon },\tilde{w})$ as follows

$$\begin{array}{cc}\hfill (\widehat{\upsilon },\widehat{w})=& \underset{(\tilde{\upsilon },\tilde{w})}{\arg \max }\Re \left\{{\mathrm{\Omega }}_{pss}(\tilde{\upsilon },\tilde{w})\right\},\hfill \end{array}$$

(9)

where $\widehat{x}$ is the estimated value of parameter x and $\Re \{·\}$ denotes the real component of a complex number. After the completion of NR-PSS detection, the UE is prepared to estimate the CSI for the purpose of coherent detection of the subsequent NR-SSS signal. This can be achieved by performing the following task

$$\begin{array}{cc}\hfill (\widehat{g},\widehat{\eta })=& \underset{(\tilde{g},\tilde{\eta })}{\arg \max }\Re \left\{{\mathrm{\Omega }}_{sss}\left(\tilde{g}\right)e^{-j4\pi \tilde{\eta }(M+M_g)/M}\right\},\hfill \end{array}$$

(10)

where

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_{sss}\left(\tilde{g}\right)& =\sum _{m\in S}{Y}_q^{*}(m+\widehat{\upsilon })Y_{q+2}(m+\widehat{\upsilon })P_{\widehat{w}}\left(m\right)S_{\tilde{g}}^{*}\left(m\right)e^{-j4\pi \widehat{\upsilon }(M+M_g)/M}.\hfill \end{array}$$

(11)

For each trial value of $\tilde{g}$, (10) carries out an exhaustive search of $\eta$ by quantizing the potential RCFO values appropriately, resulting in a significant computational burden. To address the complexity issue, a sub-optimal solution has been proposed in [16].

As can be seen in (6), since NR-SSS is generated through a combination of CIDS w and CIDG g in the frequency domain, NR-SSS detection is possible only after complete recognition of ICFO and CIDS. Therefore, the accurate detection of the NR-PSS is essential in the initial synchronization procedure, and this paper concentrates on the detection of ICFO and CIDS in the frequency domain.

3. Reduced-Complexity ICFO and CIDS Detection Method

In this section, we present a simplified formulation for joint ICFO and CIDS synchronization in 5G NR communication systems, which exploits a good autocorrelation feature of differential NR-PSS sequence. In addition, we validate the viability of the proposed NR-PSS detection method by evaluating the probability of detection failure and computational complexity.

3.1. Detection Algorithm

For a simple presentation, denote $q_l$ as the transmitted symbol index of the l-th NR-PSS with CIDS w located in OFDM blocks. Therefore, we have $X_{q_l}\left(m\right)=P_w\left(m\right)$ and $X_{q_l+2}\left(m\right)=S_g\left(m\right)$ within each SSB. As shown in (9), we perform hypothesis testing for potential ICFO and CIDS candidates, which involves significant computational demands. To address such complexity issues, a computationally efficient ICFO and CIDS synchronization scheme is proposed, without sacrificing the detection performance of (9). Figure 1 depicts the receiver structure outlining the initial synchronization process for the 5G NR communication system.

Utilizing the property that the differential relation of an m-sequence does not alter the characteristics of the m-sequence, we observe that three m-sequences $D_{\tilde{w}}\left(m\right)$’s ($\tilde{w}=0,1,2$) exhibit cyclically shifted replicas of one another. Consequently, when the number of subcarriers is sufficiently large, i.e., $M_s\gg 1$, the cross-correlation between any two of three m-sequences becomes essentially zero. More importantly, it can be observed that three m-sequences are interrelated as $D_{\tilde{w}}\left(m\right)=D_0\left(m\right)B_{\tilde{w}}\left(m\right)$, where $B_{\tilde{w}}\left(m\right)=D_{\tilde{w}}\left(m\right)/D_0\left(m\right)\in \{-1,1\}$ for $\tilde{w}=0,1,2$. The phase rotation indicator $B_{\tilde{w}}\left(m\right)$ captures the cyclic shift between $D_0\left(m\right)$ and its cyclic-shifted versions $D_{\tilde{w}}\left(m\right)$ for $\tilde{w}=1,2$. In order to mitigate the impact of interference on the estimated parameter, adopting the average estimate yields the following cost function given by

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_{red}(\tilde{\upsilon },\tilde{w})& =\sum _{l=1}^{M_v}\sum _{m\in \mathcal{D}}{\overline{Y}}_{q_l}(m+\tilde{\upsilon })D_0^{*}\left(m\right)B_{\tilde{w}}^{*}\left(m\right)\hfill \end{array}$$

(12)

with

$${\overline{Y}}_{q_l}\left(m\right)\approx |G_{q_l}{(m-\upsilon )|}^2{D}_w(m-\upsilon )+{\overline{I}}_{q_l}\left(m\right)+{\overline{Z}}_{q_l}\left(m\right),$$

(13)

where $M_v$ is the number of average estimates,

$$\begin{array}{cc}\hfill {\overline{I}}_{q_l}\left(m\right)& =G_{q_l}(m-\upsilon )X_{q_l}(m-\upsilon )I_{q_l}^{*}(m-1)e^{j2\pi ϵ(q_l{M}_u+M_g)/M}\hfill \\ & +G_{q_l}^{*}(m-\upsilon -1)X_{q_l}^{*}(m-\upsilon -1)I_{q_l}\left(m\right)e^{-j2\pi ϵ(q_l{M}_u+M_g)/M}+I_{q_l}\left(m\right)Z_{q_l}^{*}(m-1)\hfill \\ & +I_{q_l}^{*}(m-1)Z_{q_l}\left(m\right)+I_{q_l}\left(m\right)I_{q_l}^{*}(m-1),\hfill \end{array}$$

(14)

and

$$\begin{array}{cc}\hfill {\overline{Z}}_{q_l}\left(m\right)& =G_{q_l}(m-\upsilon )X_{q_l}(m-\upsilon )Z_{q_l}^{*}(m-1)e^{j2\pi ϵ(q_l{M}_u+M_g)/M}\hfill \\ & +G_{q_l}^{*}(m-\upsilon -1)X_{q_l}^{*}(m-\upsilon -1)Z_{q_l}\left(m\right)e^{-j2\pi ϵ(q_l{M}_u+M_g)/M}+Z_{q_l}\left(m\right)Z_{q_l}^{*}(m-1).\hfill \end{array}$$

(15)

If we replace ${\mathrm{\Omega }}_{pss}(\tilde{\upsilon },\tilde{w})$ with ${\mathrm{\Omega }}_{red}(\tilde{\upsilon },\tilde{w})$ in (9), the estimates $(\widehat{\upsilon },\widehat{w})$ are similarly obtained. Utilizing the cyclic-shifted feature of m-sequences, we can implement autocorrelation with the local template $D_{\tilde{w}}\left(m\right)$ for the remaining hypothetical candidates $\tilde{w}\in \{1,2\}$ and $\tilde{\upsilon }\in \{-U,-U+1,\cdots ,U\}$ by compensating the polarity of the pre-calculated autocorrelation for $\tilde{w}=0$ on a subcarrier-wise basis. This compensation is performed based on the phase rotation indicator $B_{\tilde{w}}\left(m\right)$, eliminating the need for hardware multipliers, and all operations involve only shift accumulation. As a result, the completion of (12) necessitates the computation of only one autocorrelation for each trial value of $\tilde{w}$. This formulation results in a decrease in the arithmetic complexity, as it involves performing fewer multiplication operations.

Upon substituting (3) into (12), one obtains

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_{red}(\tilde{\upsilon },\tilde{w})& =\sum _{l=1}^{M_v}\sum _{m\in \mathcal{D}}{\left|G_{q_l}(m+\tilde{\upsilon }-\upsilon )\right|}^2{D}_w(m+\tilde{\upsilon }-\upsilon )D_0^{*}\left(m\right)B_{\tilde{w}}^{*}\left(m\right)\hfill \\ & +\sum _{l=1}^{M_v}\sum _{m\in \mathcal{D}}{\overline{I}}_{q_l}\left(m\right)D_0^{*}\left(m\right)B_{\tilde{w}}^{*}\left(m\right)+\sum _{l=1}^{M_v}\sum _{m\in \mathcal{D}}{\overline{Z}}_{q_l}\left(m\right)D_0^{*}\left(m\right)B_{\tilde{w}}^{*}\left(m\right).\hfill \end{array}$$

(16)

Under the hypothesis that $D_{\tilde{w}}\left(m\right)$ is a perfect match with the $\tilde{\upsilon }$-th shifted sequence of the received NR-PSS, equivalently, $(\tilde{\upsilon },\tilde{w})=(\upsilon ,w)$, (16) can be rewritten as

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_{red}(\tilde{\upsilon },\tilde{w})& =\sum _{l=1}^{M_v}\sum _{m\in \mathcal{D}}{\left|G_{q_l}\left(m\right)\right|}^2{E}_X^2+\sum _{l=1}^{M_v}\sum _{m\in \mathcal{D}}{\overline{I}}_{q_l}\left(m\right)D_0^{*}\left(m\right)B_{\tilde{w}}^{*}\left(m\right)\hfill \\ & +\sum _{l=1}^{M_v}\sum _{m\in \mathcal{D}}{\overline{Z}}_{q_l}\left(m\right)D_0^{*}\left(m\right)B_{\tilde{w}}^{*}\left(m\right),\hfill \end{array}$$

(17)

where $E_X=\mathbb{E}\left\{\right|X_{q_l}\left(m\right){|}^2\}$ represents the symbol energy, and ${\overline{I}}_{q_l}\left(m\right)$ and ${\overline{Z}}_{q_l}\left(m\right)$ are considered as zero-mean complex Gaussian random variables (RVs) with variances ${\sigma }_{\overline{I}}^2$ and ${\sigma }_{\overline{Z}}^2$, respectively. As the amount of frequency selective fading increases, the first component on the right-hand side (RHS) of (17) contributes to multipath diversity. When $(\tilde{\upsilon },\tilde{w})\ne (\upsilon ,w)$, assuming a flat fading channel, we have

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_{red}(\tilde{\upsilon },\tilde{w})& =\sum _{l=1}^{M_v}{\left|G_{q_l}(m+\tilde{\upsilon }-\upsilon )\right|}^2\sum _{m\in \mathcal{D}}{D}_w(m+\tilde{\upsilon }-\upsilon )D_0^{*}\left(m\right)B_{\tilde{w}}^{*}\left(m\right)\hfill \\ & +\sum _{l=1}^{M_v}\sum _{m\in \mathcal{D}}{\overline{I}}_{q_l}\left(m\right)D_0^{*}\left(m\right)B_{\tilde{w}}^{*}\left(m\right)+\sum _{l=1}^{M_v}\sum _{m\in \mathcal{D}}{\overline{Z}}_{q_l}\left(m\right)D_0^{*}\left(m\right)B_{\tilde{w}}^{*}\left(m\right).\hfill \end{array}$$

(18)

In this case, as the first component of the RHS in (18) becomes negligible due to the nearly orthogonal property of differential m-sequence at all other lags, ${\mathrm{\Omega }}_{red}(\tilde{\upsilon },\tilde{w})$ can be approximated as

$$\begin{array}{cc}\hfill {\mathrm{\Omega }}_{red}(\tilde{\upsilon },\tilde{w})& \approx \sum _{l=1}^{M_v}\sum _{m\in \mathcal{D}}{\overline{I}}_{q_l}\left(m\right)D_0^{*}\left(m\right)B_{\tilde{w}}^{*}\left(m\right)+\sum _{l=1}^{M_v}\sum _{m\in \mathcal{D}}{\overline{Z}}_{q_l}\left(m\right)D_0^{*}\left(m\right)B_{\tilde{w}}^{*}\left(m\right).\hfill \end{array}$$

(19)

Please note that (19) can be treated as a zero-mean complex Gaussian RV with variance ${\sigma }_1^2\approx M_v(M_s-1)E_X^2({\sigma }_{\overline{I}}^2+{\sigma }_{\overline{Z}}^2)/2$.

3.2. Performance Analysis

This subsection presents a closed-form solution for the probability of detection failure of the proposed ICFO and CIDS synchronization scheme. For ease of derivation, we assume a slowly time-variant flat fading channel, i.e., $G_{q_l}\left(m\right)\approx G_{q_l+1}\left(m\right)$. Let $a\sim \mathcal{N}(b,c)$ represent that a is a Gaussian RV with mean b and variance c. The probability of detection failure denoted by $P_f=Prob\{(\widehat{\upsilon },\widehat{w})\ne (\upsilon ,w)\}$ refers to the probability of erroneously deciding both the ICFO and CIDS. Conditioned on $\mathcal{G}=|G_{q_l}{\left(m\right)|}^2$, if we assume that $(\widehat{\upsilon },\widehat{w})=(\upsilon ,w)$ (hypothesis $H_1$) in (17), then $\Re \left\{{\mathrm{\Omega }}_{red}(\tilde{\upsilon },\tilde{w})\right\}\sim \mathcal{N}(\mu ,{\sigma }_1^2)$ because of near-zero autocorrelation feature of m-sequence and $M_s\gg 1$, where $\mu =M_v(M_s-1)E_X^2\mathcal{G}$ and ${\sigma }_1^2\approx M_v(M_s-1)E_X^2({\sigma }_{\overline{I}}^2+{\sigma }_{\overline{Z}}^2)/2$. Here, ${\sigma }_{\overline{I}}^2=2E_X{\sigma }_I^2\mathcal{G}+2{\sigma }_I^2{\sigma }_Z^2+{\sigma }_I^4$ is the variance of (14) and ${\sigma }_{\overline{Z}}^2=2E_X{\sigma }_Z^2\mathcal{G}+{\sigma }_Z^4$ is the variance of (15). Under hypothesis $H_0$ that $(\widehat{\upsilon },\widehat{w})\ne (\upsilon ,w)$ in (19), $\Re \left\{{\mathrm{\Omega }}_{red}(\tilde{\upsilon },\tilde{w})\right\}\sim \mathcal{N}(0,{\sigma }_0^2)$ with ${\sigma }_0^2\approx {\sigma }_1^2$ when $M_s\gg 1$.

Let $y=\Re \left\{{\mathrm{\Omega }}_{red}(\tilde{\upsilon },\tilde{w})\right\}$, and assuming that ICFOs have an equal probability of occurrence, the probability of detection failure of the proposed NR-PSS synchronization method is the probability that the RV y under hypothesis $H_0$ exceeds all other $3(2U+1)-1$ RVs under hypothesis $H_0$ and hypothesis $H_1$. Conditioned on $\mathcal{G}$, the probability of incorrectly detecting both the ICFO and CIDS can be expressed as

$$\begin{array}{cc}\hfill P_f\left(\mathcal{G}\right)& =1-{\int }_{-\infty }^{\infty }\frac{1}{\sqrt{2\pi {\sigma }_1^2}}{e}^{-\frac{{(y-\mu )}^2}{2{\sigma }_1^2}}{\left[{\int }_{-\infty }^y\frac{1}{\sqrt{2\pi {\sigma }_0^2}}{e}^{-\frac{x^2}{2{\sigma }_0^2}}dx\right]}^{6U+2}dy.\hfill \end{array}$$

(20)

Thus, $P_f\left(\mathcal{G}\right)$ can be obtained as

$$\begin{array}{cc}\hfill P_f\left(\mathcal{G}\right)& =1-{\int }_{-\infty }^{\infty }\frac{1}{\sqrt{2\pi }}{e}^{-\frac{y^2}{2}}{\left[1-Q\left(y+\lambda \right)\right]}^{6U+2}dy,\hfill \end{array}$$

(21)

where $Q(·)$ is the Q-function and $\lambda =\mu /{\sigma }_1$. In (21), we have

$$\begin{array}{c}\hfill \lambda =\sqrt{\frac{2M_v(M_s-1)}{2(1/{\gamma }_z+/{\gamma }_i)+{(1/{\gamma }_z+/{\gamma }_i)}^2}},\end{array}$$

(22)

where ${\gamma }_z=E_X\mathcal{G}/{\sigma }_Z^2$ is the received signal-to-noise ratio (SNR) and ${\gamma }_i=E_X\mathcal{G}/{\sigma }_I^2$ is the received signal-to-ICI ratio (SIR). To simplify the derivation of (21), the following upper-bound presented in [27,28] is adopted

$$\begin{array}{c}\hfill Q\left(y+\lambda \right)\le \sum _{i=1}^{M_e}{a}_i{e}^{-\frac{b_i{(y+\lambda )}^2}{2}},\end{array}$$

(23)

where $a_i$ and $b_i$ represent suitable coefficients for deciding the accuracy of the approximation. It is important to note that the bound approaches the precise value as $M_e$ increases. By substituting (23) into (21) and applying the multinomial theorem, we obtain the lower bound for $P_f\left(\mathcal{G}\right)$

$$\begin{array}{cc}\hfill P_f\left(\mathcal{G}\right)& \ge 1-\sum _{u_1+\cdots +u_{M_e+1}=6U+2}\prod _{i=1}^{M_e}{(-a_i)}^{u_i}\left(\genfrac{}{}{0pt}{}{6U+2}{u_1,u_2,\cdots ,u_{M_e+1}}\right)\hfill \\ & \times \frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{\infty }{e}^{-\frac{y^2}{2}}{e}^{-\frac{{\left(y+\lambda \right)}^2}{2}{\sum }_{i=1}^{M_e}{b}_i{u}_i}dy,\hfill \end{array}$$

(24)

which can be further derived as

$$\begin{array}{cc}\hfill P_f\left(\mathcal{G}\right)& \ge 1-\sum _{u_1+\cdots +u_{M_e+1}=6U+2}\prod _{i=1}^{M_e}{(-a_i)}^{u_i}\left(\genfrac{}{}{0pt}{}{6U+2}{u_1,u_2,\cdots ,u_{M_e+1}}\right)\hfill \\ & \times {\left(1+\sum _{i=1}^{M_e}{b}_i{u}_i\right)}^{-1/2}{e}^{-\frac{{\lambda }^2}{2}\left\{1-{\left(1+{\sum }_{i=1}^{M_e}{b}_i{u}_i\right)}^{-1}\right\}},\hfill \end{array}$$

(25)

where

$$\begin{array}{cc}\hfill \left(\genfrac{}{}{0pt}{}{6U+2}{u_1,u_2,\cdots ,u_{M_e+1}}\right)=& \left(\genfrac{}{}{0pt}{}{u_1}{u_1}\right)\left(\genfrac{}{}{0pt}{}{u_1+u_2}{u_2}\right)\left(\genfrac{}{}{0pt}{}{u_1+u_2+\cdots +u_{M_e+1}}{u_{M_e+1}}\right).\hfill \end{array}$$

(26)

By averaging $P_f\left(\mathcal{G}\right)$ over $\mathcal{G}$, where the probability density function of $\mathcal{G}$ is represented as $f\left(\mathcal{G}\right)=(1/{\sigma }_G^2)e^{-\mathcal{G}/{\sigma }_G^2}$, one arrives at

$$\begin{array}{cc}\hfill P_f& \ge 1-\sum _{u_1+\cdots +u_{M_e+1}=6U+2}\prod _{i=1}^{M_e}{(-a_i)}^{u_i}\left(\genfrac{}{}{0pt}{}{6U+2}{u_1,u_2,\cdots ,u_{M_e+1}}\right)\hfill \\ & \times {\left(1+\sum _{i=1}^{M_e}{b}_i{u}_i\right)}^{-1/2}F(u_1,u_2,\cdots ,u_{M_e}),\hfill \end{array}$$

(27)

where

$$\begin{array}{c}\hfill F(u_1,u_2,\cdots ,u_{M_e})={\int }_0^{\infty }\frac{1}{{\sigma }_G^2}{e}^{-\frac{{\lambda }^2}{2}\left\{1-{\left(1+{\sum }_{i=1}^{M_e}{b}_i{u}_i\right)}^{-1}\right\}}{e}^{-\frac{\mathcal{G}}{{\sigma }_G^2}}d\mathcal{G}.\end{array}$$

(28)

When the SNR is sufficiently large, the second squared term in the denominator on the RHS of (22) can be considered insignificant. Consequently, by neglecting this term, we can approximate $\lambda$ as

$$\begin{array}{c}\hfill \lambda \approx \sqrt{\frac{M_v(M_s-1){\gamma }_z{\gamma }_i}{{\gamma }_z+{\gamma }_i}}.\end{array}$$

(29)

By substituting (29) into (28), and performing some mathematical simplifications, we can get

$$\begin{array}{c}\hfill F(u_1,u_2,\cdots ,u_{M_e})\approx \frac{2({\overline{\gamma }}_z+{\overline{\gamma }}_i)}{(M_s-1){\overline{\gamma }}_z{\overline{\gamma }}_i\left\{1-{\left(1+{\sum }_{i=1}^{M_e}{b}_i{u}_i\right)}^{-1}\right\}+2({\overline{\gamma }}_z+{\overline{\gamma }}_i)},\end{array}$$

(30)

where ${\overline{\gamma }}_z=\mathbb{E}\left\{\mathcal{G}\right\}{E}_X/{\sigma }_Z^2$ denotes the average SNR and ${\overline{\gamma }}_i=\mathbb{E}\left\{\mathcal{G}\right\}{E}_X/{\sigma }_I^2$ denotes the average SIR. Therefore, the unconditional probability of detection failure is computed as

$$\begin{array}{cc}\hfill P_f& \approx 1-\sum _{u_1+\cdots +u_{M_e+1}=6U+2}\prod _{i=1}^{M_e}{(-a_i)}^{u_i}\left(\genfrac{}{}{0pt}{}{6U+2}{u_1,u_2,\cdots ,u_{M_e+1}}\right){\left(1+\sum _{i=1}^{M_e}{b}_i{u}_i\right)}^{-1/2}\hfill \\ & \times \frac{2({\overline{\gamma }}_z+{\overline{\gamma }}_i)}{(M_s-1){\overline{\gamma }}_z{\overline{\gamma }}_i\left\{1-{\left(1+{\sum }_{i=1}^{M_e}{b}_i{u}_i\right)}^{-1}\right\}+2({\overline{\gamma }}_z+{\overline{\gamma }}_i)}.\hfill \end{array}$$

(31)

Please note that the detection probability denoted by $P_d=\mathrm{Prob}\{(\widehat{\upsilon },\widehat{w})=(\upsilon ,w)\}$ is the probability that the RV y under hypothesis $H_1$ exceeds all other $3(2U+1)-1$ RVs under hypothesis $H_0$, which is the complementary probability of $P_f$ and $P_d$ is expressed in a upper-bound form due to the use of (23). As a global performance metric, we will assess the performance of the presented NR-PSS detection methods equivalently, using $P_d=1-P_f$, in the simulation section.

3.3. Complexity Analysis

To show the benefits of the proposed approach, we compare it with three representative NR-PSS synchronization schemes as baselines [20,21,22,23]. These benchmark methods can be easily extended to enable $M_v$-average estimation. The first benchmark is the approximated MMSE (AMMSE) detection formulated as [20]

$$\begin{array}{cc}\hfill (\widehat{\upsilon },\widehat{w})& =\underset{(\tilde{\upsilon },\tilde{w})}{\arg \max }\sum _{l=1}^{M_v}{∥{\mathbf{U}}^H{\overline{\mathbf{Y}}}_{q_l}(\tilde{\upsilon },\tilde{w})∥}^2,\hfill \end{array}$$

(32)

where ${\parallel ·\parallel }^2$ denotes the two-norm of vector, the columns of $\mathbf{U}$ consists of the $M_r$ normalized eigenvectors of $\mathbf{F}{\mathbf{F}}^H$ involved to the $M_r$ largest eigenvalues, $M_r$ stands for the rank order of the reduced channel matrix, F denotes the $M_s\times L$ normalized DFT matrix whose $(m,n)$-th component is $e^{-j2\pi mn/M}$, and ${\overline{\mathbf{Y}}}_{q_l}(\tilde{\upsilon },\tilde{w})$ is the $M_s$-dimensional vector with components $\{{\overline{Y}}_{q_l}(m+\tilde{\upsilon })D_{\tilde{w}}^{*}\left(m\right);m\in \mathcal{D}\}$. The second reference is the joint detection of ICFO and CIDS (JDICID), which can be formulated as [21,22]

$$\begin{array}{cc}\hfill (\widehat{\upsilon },\widehat{w})=& \underset{(\tilde{\upsilon },\tilde{w})}{\arg \max }\Re \left\{\sum _{l=1}^{M_v}\sum _{m\in \mathcal{D}}{\overline{Y}}_{q_l}(m+\tilde{\upsilon })D_{\tilde{w}}^{*}\left(m\right)\right\}.\hfill \end{array}$$

(33)

Finally, the third benchmark is the sequential detection of ICFO and CIDS (SDICID) [23] and applying $M_v$-times average estimation, the ICFO can initially be estimated as

$$\begin{array}{cc}\hfill \widehat{\upsilon }=& \underset{\tilde{\upsilon }}{\arg \max }\Re \left\{\sum _{l=1}^{M_v}\sum _{m\in \mathcal{D}}{\overline{Y}}_{q_l}(m+\tilde{\upsilon }){\overline{D}}^{*}\left(m\right)\right\},\hfill \end{array}$$

(34)

where $\overline{D}\left(m\right)={\sum }_{\tilde{w}=0}^2{D}_{\tilde{w}}\left(m\right)$. Using the estimate $\widehat{\upsilon }$, the CIDS is sequentially detected as

$$\begin{array}{cc}\hfill \widehat{w}=& \underset{\tilde{w}}{argmax}\Re \left\{\sum _{l=1}^{M_v}\sum _{m\in \mathcal{D}}{\overline{Y}}_{q_l}(m+\widehat{\upsilon })D_{\tilde{w}}^{*}\left(m\right)\right\}.\hfill \end{array}$$

(35)

To ensure fair comparisons, we consider the number of floating point operations (flops) required for every arithmetic operation. Specifically, we assume six flops for every complex multiplication, two flops for every complex addition, and three flops for every complex magnitude, as specified in [29]. Regarding the JDICID scheme, it requires $(5M_s-4)M_v-2$ flops to compute the quantity $\Re \left\{{\sum }_{l=1}^{M_v}{\sum }_{m\in \mathcal{D}}{\overline{Y}}_q(m+\tilde{\upsilon })D_{\tilde{w}}^{*}\left(m\right)\right\}$ for every hypothesis. Adopting $M_v$-times average estimation, therefore, the total count of flops associated with (33) is obtained as $(6U+3)(5M_s-6)M_v$. For the AMMSE scheme, the overall processing complexity is given by $(6U+3)(10M_s+2)M_v{M}_r$. Identically, the SDICID method involves computing $\Re \left\{{\sum }_{l=1}^{M_v}{\sum }_{m\in \mathcal{D}}{\overline{Y}}_{q_l}(m+\tilde{\upsilon }){\overline{D}}^{*}\left(m\right)\right\}$ and $\Re \left\{{\sum }_{l=1}^{M_v}{\sum }_{m\in \mathcal{D}}{\overline{Y}}_{q_l}(m+\widehat{\upsilon })D_{\tilde{w}}^{*}\left(m\right)\right\}$ with $(5M_s-4)M_v-2$ and $(2M_s-1)M_v-2$ flops for each trial, respectively. As a consequence, (34) requires $(2U+1)(5M_s-6)M_v$ flops to estimate the ICFO, whereas (35) requires $3(2M_s-3)M_v$ flops to identify the CIDS. To summarize, the SDICID method needs $(2U+1)[(5M_s-4)M_v-2]+3[(2M_s-1)M_v-2]$ flops to sequentially obtain the ICFO and CIDS. In the end, we examine the arithmetic operations associated with the proposed detection method. When the trial value $\tilde{w}=0$, the proposed method uses $(2U+1)[(4M_s-4)M_v-2]$ flops to calculate the cost function, while it requires additional $2(2U+1)[(M_s-1)M_v-1]$ real multiplications to realize the cost function in the case when $\tilde{w}=1,2$. Consequently, the overall count of flops amounts to $(2U+1)[(6M_s-6)M_v-3]$.

Table 1. The number of flops for the NR-PSS synchronization schemes.

Algorithm	Number of Flops
AMMSE scheme	$(6U+3)(10M_s+2)M_v{M}_r$
JDICID scheme	$(6U+3)(5M_s-6)M_v$
SDICID scheme	$(2U+1)[(5M_s-4)M_v-2]+3[(2M_s-1)M_v-2]$
Proposed scheme	$(2U+1)[(6M_s-6)M_v-3]$

presents the total count of flops used in the NR-PSS synchronization schemes.

4. Simulation Results

In this section, a comprehensive simulation is conducted to compare the effectiveness of the proposed NR-PSS synchronization scheme with that of existing synchronization methods. The proposed NR-PSS detector is validated using the Matlab software R2018a (MathWorks Inc., Natick, MA, USA) as the simulation platform.

4.1. Simulation Environment

We employ an OFDM system with a DFT size of $M=2048$ and a CP length of $M_g=144$. It operates in the 6 GHz frequency band, utilizing a transmission bandwidth of 40 MHz and a subcarrier spacing of ${\Delta }_f=30$ kHz. As a consequence, the sampling frequency is equal to $f_s=61.44$ MHz. In our simulations, we employ a 5G tapped delay line-B model for non-line-of-sight scenarios, as described in [30]. We consider two representative channel profiles: Very short delay (Vsd) spread and Very long delay (Vld) spread, each characterized by a different delay spread.

Table 2. Simulation parameters.

Parameter	Value
Carrier frequency	$f_c=6$ GHz
DFT size	$M=2048$
CP size	$M_g=144$
NR-PSS length	$M_s=128$
Channel bandwidth	$B_w=40$ MHz
Subcarrier spacing	${\Delta }_f=30$ kHz
Sampling frequency	$f_s=61.44$ MHz
Doppler frequency	$D_f=300$ Hz
Channel model	Flat and Rayleigh fading

shows the values of the system and channel parameters used in the simulation. To account for the stability of commercial oscillators used in mobile applications, which is approximately $\pm 20$ ppm, we choose the value of U to 4. For each realization, the ICFO is uniformly generated from the range $[-U,U]$. In the AMMSE scheme, we set the rank order of the reduced channel to be $M_r=5$. To compute (31), we consider the scenario where $M_e=2$ [27]. Unless otherwise stated, the Doppler frequency is equal to $D_f=300$ Hz, which is equivalent to the UE speed of 54 km/h [31], $\eta =0.1$, and $U=4$.

As an overall performance metric, we assess the probability of detection failure denoted by $P_f=\mathrm{Prob}\{(\widehat{\upsilon },\widehat{w})\ne (\upsilon ,w)\}$ of the presented NR-PSS synchronization schemes, or equivalently, the detection probability denoted by $P_d=\mathrm{Prob}\{(\widehat{\upsilon },\widehat{w})=(\upsilon ,w)\}$. Also, we evaluate the probability of incorrectly detecting the ICFO and CIDS, each denoted by $P_{f1}=\mathrm{Prob}\{\widehat{\upsilon }\ne \upsilon \}$ and $P_{f2}=\mathrm{Prob}\{\widehat{w}\ne w\}$, respectively.

4.2. Performance Evaluation

In Figure 2, the probability of detection failure defined as $\mathrm{Prob}\{(\widehat{\upsilon },\widehat{w})\ne (\upsilon ,w)\}$ is depicted for both the conventional SDICID and proposed detection algorithms versus SNR in the flat fading channel. To focus on verifying our theoretical analysis, we consider scenarios in which $D_f=0$ Hz and RSTO are completely estimated. Despite not being shown in Figure 2, the error probabilities of both the JDICID and the proposed methods show a significant similarity because of the use of the identical cost function. It is noteworthy that the proposed NR-PSS detector shows a significant performance enhancement compared to the SDICID method with respect to the probability of detection failure. Moreover, the theoretical results exhibit a good correspondence with the results obtained through simulation, even when confronted with frequency offset values. These observations offer ample evidence to support the effectiveness of the proposed NR-PSS detection method and the exactness of the numerical analysis presented in Section 3.2. As expected, using average estimation improves the detection ability of the NR-PSS detection methods, and the trade-off for achieving this enhanced performance is the additional processing time and arithmetic load.

Figure 3 presents $P_f$ of the various NR-PSS synchronization schemes versus SNR in the multipath fading channel model. By employing the ML strategy, the AMMSE method shows significant improvement over the proposed NR-PSS detection method. However, despite the adoption of approximated detection to decrease complexity, the arithmetic burden of the AMMSE method remains considerably high, thereby affecting its overall efficiency. As we can see in Figure 3, it is evident that the proposed NR-SSS detection method provides comparable or even superior detection performance compared to other baseline methods like JDICID and SDICID. Furthermore, the performance difference between the SDICID and proposed methods widens with an increase in the frequency selectivity of the channel, as observed in the Vld model. This enhancement can be attributed to the presence of multipath diversity, while the widening gap is a result of the degradation of the autocorrelation property of NR-PSS as the maximum channel delay spread increases. Based on these observations, it is clear that the proposed NR-PSS synchronization scheme constitutes an efficient method to strike a balance between estimation accuracy and arithmetic load. Regardless of the detection method, an irreducible error floor is observed because of the presence of the ICI term introduced by uncompensated RCFO, which informs us that RCFO-induced ICI is ultimately the limiting cause for performance in both multipath channel environments. Therefore, it is essential to periodically estimate the RCFO during the subsequent cell search procedure [16].

Figure 4 shows $P_{f1}$ and $P_{f2}$ of the various NR-PSS synchronization schemes versus SNR in the Vsd channel model. The simulation environment is the same as that used in Figure 3. It is evident from Figure 4 that the shape of the probability curves exhibits qualitative similarity to that illustrated in Figure 3. Examining the results in Figure 3 and Figure 4, one can observe that the ICFO detector plays a predominant role in determining the overall capability of the NR-PSS synchronization receiver than the CIDS detector, with the latter exhibiting better detection performance. This observation can be explained by the fact that more hypotheses are required for ICFO detection than for CIDS detection, consequently resulting in a higher probability of detection failure. By employing a 127-length m-sequence with its near-zero autocorrelation feature, the probability of accurately detecting ICFO and CIDS under a non-zero CFO hypothesis increases in comparison to LTE-A systems [20,21].

Figure 5 offers a comparative analysis of the complexity and performance associated with the NR-PSS synchronization schemes versus $M_v$ and U, respectively, when $SNR=-5$ dB. When $M_v$ is fixed, it becomes apparent from Figure 5a that the proposed method significantly reduces computational complexity by 60.0% and 96.1% over the JDICID and AMMSE methods, respectively, demonstrating the tradeoff between detection capability and arithmetic operation. On the other hand, the SDICID method can be implemented with approximately 0.9 times the complexity on average compared to the proposed NR-PSS detector, at the cost of detection performance. In Figure 5b, the detection probability denoted by $\mathrm{Prob}\{(\widehat{\upsilon },\widehat{w})=(\upsilon ,w)\}$ is evaluated in the Vsd channel. As predicted, it is shown from Figure 5b that the performance of the NR-PSS detection methods undergoes degradation with an increase in U. However, this decline in performance is relatively minor thanks to the adoption of the m-sequence. This phenomenon becomes apparent as $M_v$ increases. For detection methods other than the AMMSE approach, achieving a target detection probability of 90% necessitates averaging over more NR-PSS symbols. Hence, employing the average estimate across multiple SSBs serves as a potential solution to enhance poor coverage areas, which will be further investigated in the following example.

Figure 6 depicts $P_f$ for the various NR-PSS synchronization schemes as a function of $M_v$ at SNR values of $-10$ and $-5$ dB. To achieve the target detection probability of $90\%$, we denote the required number of averages as $M_a$. When $\mathrm{SNR}=-5$ dB, it is evident from Figure 6a that the AMMSE and SDICID approaches use $M_a=6$ and 17, respectively, while both JDICID and the proposed method require an average estimation of 11 NR-PSS symbols to achieve the target probability under the Vds channel condition. With this observation at hand, the arithmetic burden of the proposed method required to achieve the same target probability is reduced by 23.9%, 60.0%, and 93.0% compared with that of the SDICID, JDICID, and AMMSE approaches, respectively. More interestingly, the number of averaging symbols is substantially decreased in the case of the Vld channel, thanks to the time diversity effect across multiple SSBs. Compared to the results in Figure 6a, a larger $M_a$ is required to obtain the target probability when $\mathrm{SNR}=-10$ dB, thereby leading to an increase in processing latency. On the other hand, the number of average estimations needed in the AMMSE method increases from $M_a=1$ to 8 in the Vld channel, while the SDICID approach experiences a similar increase from $M_a=5$ to 30. For both the JDICID and proposed methods, $M_a$ increases from 3 to 18. In this case, the proposed method exhibits a 28.7%, 60.0%, and 91.4% reduction in complexity compared to the SDICID, JDICID, and AMMSE methods, respectively. Regardless of the channel and SNR values, it can be concluded that the proposed synchronization method has a lower computational burden than the benchmark methods.

5. Conclusions

This paper presents a computationally effective NR-PSS synchronization scheme for 5G NR communication systems. By exploiting the autocorrelation feature of the differential NR-PSS sequence, we formulate a simplified variation of the proposed joint ICFO and CIDS synchronization method. In order to confirm the viability of the proposed NR-PSS synchronization scheme, we analytically compute the probability of detection failure, and further evaluate its accuracy by comparing it with simulation results. Numerical examples show that the proposed ICFO and CIDS detection method considerably reduces arithmetic operations while achieving the same level of detection performance as benchmark detection methods. Interestingly, the number of NR-PSS symbols used for the average estimate decreases significantly with the increase in the frequency selectivity of the channel, while still maintaining the complexity–performance trade-off. Upon implementing our proposed synchronization receiver in 5G NR systems, the arithmetic operation can be decreased by more than 90% irrespective of the channel model and SNR condition, which in turn leads to reduced power consumption and eventually contributes to improving the battery lifetime of UE devices.