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Analysis of Gaussian phase noise effects in DFTsOFDM systems for subTHz transmissions
EURASIP Journal on Wireless Communications and Networking volume 2024, Article number: 60 (2024)
Abstract
The future generations of communication technologies envision the transmission of signals across the millimeter wave and subTHz spectrums. However, the characteristics of the propagation channel at such high frequencies differ from what is observed in the conventional lowfrequency spectrum with for instance, the apparition of stronger phase noise (PN) induced by the Radio Frequency (RF) transceivers and more especially by the oscillators. That is why there is growing interest in evaluating and adapting the 5G new radio (5GNR) physical layer to the presence of PN. This article is dedicated to the study of discrete Fourier transformspreadOrthogonal Frequency Division Multiplexing (DFTsOFDM) under uncorrelated Gaussian PN (GPN) impairments. We show that the presence of GPN induces two distortions: (i) a frequencydependent random rotation of data, namely the subcarrier phase error (SPE) and (ii) a frequencydependent intercarrier interference (ICI) that are analytically expressed. Then, we investigate the design of the adapted and optimal detection criterion according to the baseband model we derived in this paper. We demonstrate that (i) the proposed polar detector outperforms the conventional Euclidean detector and (ii) contrary to legacy OFDM, DFTsOFDM is a promising solution when strong GPN is involved.
1 Introduction
The deployment of cellular technologies in wider bandwidths at higher frequencies is an appealing solution to increase the capacity of wireless transmissions. The “subTHz” spectrum ([100–300] GHz) is identified as a potential candidate for 6G [1]. Indeed, total but fragmented bandwidth of 58.6 GHz has been identified [2]. The 5G New Radio (\(5\)GNR) standard today covers transmissions up to 71 GHz [3]. However, the 3\(^{\text {rd}}\) generation partnership project (3GPP) is looking forward to the use of new spectrum for future releases. On the one hand, the IEEE 802.15.3d specification standardizes a SingleCarrier (SC) waveform with carrier aggregation [1]. On the other hand, research is being conducted to study how 5GNR MultiCarrier (MC) waveforms such as Orthogonal Frequency Division Multiplexing (OFDM) or discrete Fourier transform (DFT)spreadOFDM (DFTsOFDM) can be adapted to the specificities of the subTHz bands [4].
OFDM is a MC waveform standardized in 4G and 5GNR for wireless transmissions. It consists in sending the signal simultaneously on many orthogonal subcarriers by using fast Fourier transform (FFT) [5]. OFDM reduces the design complexity of equalizers by allowing a “1tap” channel equalization per subcarrier in the frequencydomain^{Footnote 1} [6]. However, the main drawback of OFDM is its high PeaktoAverage Power Ratio (PAPR) [7] which puts a lot of constraints on the Power Amplifier (PA). If the advantages of OFDM have to be maintained, the use of the DFTsOFDM seems to be a good compromise. The waveform is also standardized in 4G and 5GNR but rather for uplink transmissions and also a candidate for subTHz [8]. It can be seen as a DFTprecoded OFDM and emulates a SC system. It thus offers a lower PAPR compared to OFDM [9] and allows to operate PAs with reduced power back off, resulting in a higher energy efficiency [10].
Transmitting in the subTHz bands presents some specific impairments such as the Phase Noise (PN) which is mainly generated by the high frequency oscillators suffering from random and shortterm frequency instabilities. This phenomenon may lead to synchronization issues in clocked and sampleddata digital systems, as documented in [11]. The severity of PN amplifies as the carrier frequency increases, posing a significant threat to system performance transmitting in subTHz bands. The induced effects of PN on OFDM transmission have been massively studied [12,13,14,15]. It has been proved that PN causes a common symbol rotation to all subcarriers, namely common phase error (CPE), and additive InterCarrier Interference (ICI). To correct the CPE, Phase Tracking Reference Signals (PTRS) have been introduced in 5GNR [15, 16]. However, dealing with the ICI remains tricky and requires sophisticated cancellation techniques [17]. That is the reason why ICI mitigation techniques have been proposed in recent literature [17,18,19]. However, those techniques cannot be directly extended to the DFTsOFDM systems because they rely on the presence of the CPE which is no longer the case with DFTsOFDM. In [8], the authors show that increasing the ICS up to 3840 kHz for DFTsOFDM systems, provides enhanced robustness against PN. However, increasing the InterCarrier Spacing (ICS) implies shortening the CP duration which results in lowered robustness against delay spread in the case of multipath channel [20] or reduce the system’s ability to handle time synchronization errors. It should also be mentioned that \(5\)GNR standard imposes a minimum number of subcarriers to be allocated. If this constraint is maintained, increasing the ICS necessarily implies increasing the bandwidth and the sampling frequency. PN mitigation in DFTsOFDM is discussed in [15, 21, 22]. Additionally, works were carried to enhance the channel estimation in the context of OFDM by proposing new PTRS scheme [18].
1.1 Related stateoftheart and motivations
PN can be modeled in various ways. Indeed, the PN can be defined as the sum of multiple noises [23]: some with a correlated nature, such as Wiener noise, and others without, like the white Gaussian noise. Many studies consider models with correlated nature [4, 12, 14, 17, 18] such as the Wiener PN [22] and the 3GPP PN models [24]. The correlated nature of the PN allows to track and thus to compensate its correlated part to limit its impact on the radio link performance [25]. However, when the signal bandwidth is very large as planned for subTHz and THz communications, the impact of the Gaussian nature of the PN becomes dominant [23]. It occurs: i) when the oscillator corner frequency remains small compared to the system bandwidth and ii) when the signal duration is short enough to fulfill the criterion presented in [23]. Those observations have been made at 200 GHz with a highfrequency oscillator presented in [26] and the simplified PN model detailed in [27].
As the system bandwidths are expected to grow for 6G transmissions in upper mmWave and subTHz transmissions [28], the question of the design of physical layer robust to white Gaussian PN (GPN) rises. In this context, many studies focus on SC systems with constellation optimizations [29, 30], optimal detection criterion [31] and performance evaluation and comparison [32]. MC systems are also considered which is more in line with the 3GPP vision. Link and system performance of the OFDM and DFTsOFDM have been evaluated under GPN channels [33]. However, the GPN effects on MC systems are not yet comprehensively understood especially for DFTsOFDM systems. That is the reason why, we propose in this work to better study the effects of GPN on DFTsOFDM systems. The correlated nature of the PN will be thus considered fully and ideally compensated. Needless to mention that, in practice, perfect compensation is not feasible [34]. But in this work, it will be assumed that stateoftheart algorithms are sufficiently efficient to make the correlated PN negligible compared to GPN. Table 1 provides a brief summary of PN compensation techniques presented in the literature for different waveforms, and highlights the interest of the work presented in this paper.
Overall, the main contributions of this paper can be described as follows:

1.
We model the distortions induced by the GPN on the received DFTsOFDM baseband signal.

2.
We develop a novel detection criterion adapted to DFTsOFDM systems in the presence of GPN by assuming the high SignaltoInterferenceplusNoise Ratio (SINR) approximation.

3.
We propose a suboptimal implementation of the proposed detector more suitable for realistic systems [29].

4.
We evaluate the link performance of a DFTsOFDM systems and compare them with legacy OFDM (in terms of packet error rates with 5GNR channel coding techniques). To provide a fair comparison between the MC techniques, ICI mitigation techniques proposed for the OFDM [17] will be considered.
The remainder of the paper is as follows. In Sect. 2.1, we present the system model. Section 2.2 provides the analysis of the baseband received signal impaired by GPN. Detection criteria are derived in Sect. 3.1. Simulation results and comparison with stateoftheart techniques are carried out in Sect. 3.2. A discussion is opened in Sect. 4 and finally, we end up with the conclusion in Sect. 5.
1.2 Notations
In what follows, underlined lower boldface letters, \(\underline{\textbf{a}}\), indicate column vectors, with \(a_k\) denoting the \(k\)th element of the column vector. The terms \({\textbf{diag}}\{ {\textbf{A}} \}\) and \(\textbf{diag}_k\lbrace \textbf{A} \rbrace\) respectively denote the diagonal and the \(k\)th element of the diagonal of the matrix \(\textbf{A}\). The term \(\mathbb {E}[\cdot ]\) denotes the expectation operator of the argument. The term \(\textbf{A}^{1}\) represents the inverse of the matrix \(\textbf{A}\). The symbols \({\mathfrak {R}}{\mathfrak {e}}\{\cdot \}\), \({\mathfrak {I}}{\mathfrak {m}}\{\cdot \}\), \((\cdot )^*\) and \((\cdot )^H\) respectively denote the real part, the imaginary part, the conjugate and the transposeconjugate of the argument. The operator \(\odot\) and the symbol \(\parallel \cdot \parallel\) respectively represent the Hadamard product and the norm operator. \(\underline{\textbf{a}}_\rho\) (resp. \(\underline{\textbf{a}}_\theta\)) represents the column vector which contains all the magnitude values (resp. the phase values) of the vector \(\underline{\textbf{a}}\). The term \(a_{k_\rho }\) (resp. \(a_{k_\theta }\)) denotes the magnitude value (resp. phase value) of the symbol \(a_k\). The symbols det(\(\cdot\)) and \(\cdot\) respectively denote the determinant value of the argument and the product operator. \(\textbf{N}_X\) indicates matrices of size \(X \times X\) that fully contain the number \(\textbf{N}\). The term \(\textbf{F}_{X}\) is the DFT matrix of size X defined as:
2 Methods
2.1 System model
2.1.1 Channel model
A coherent communication receiver with perfect time/frequency synchronization will be assumed in this work. The radio propagation channels in subTHz frequencies are expected to be sparse [42]. It has been demonstrated that the LineofSight (LoS) provides most of the energy contribution in subTHz scenarios [42, 43]. It can be achieved by the use of high gain directional antennas which spatially filter the channel. Hence, we will assume a LoSonly propagation channel in this paper.
2.1.2 PN model
Regarding the PN model, we consider an uncorrelated GPN model. The phase error induced by the GPN can be modeled as follows
where \(\sigma _{\phi }^2\) denotes the sum variance of the Tx and Rx GPNs, i.e.,
The term \(K_0\) represents the power spectral density (PSD) of the generated GPN and \(B_W\) the bandwidth. For GPN, the phase error is white which means that its PSD is independent of the frequency. Regarding the bandwidth, it can be expressed as function of MC parameters \(B_W = N_s \Delta _f\) where \(N_s\) denotes the DFT size block and \(\Delta _f\) the ICS. The model (3) is based on the channel model considered in Sect. 2.1.1, i.e., absence of multipath components. Three levels of GPN regimes are considered in this work as detailed in Table 2.
2.1.3 DFTsOFDM waveform with the presence of PN
The expression of the baseband discretetime received signal impaired by the GPN at the \(k\)th subcarrier is expressed as follows
where \(s_k\) is the transmitted signal at the \(k\)th subcarrier and \(N_f\) is the Fast Fourier Transform (FFT) block size. The terms \(\beta (k,N_s,N_f,\phi )\), \(\gamma (k,N_s,N_f,\phi )\) and \(\eta (k,N_s,N_f)\) are, respectively, the channel coefficient, the additive ICI and the thermal noise after Fourier transformations at the receiver and are defined in (5) by
The term \(n_p \backsim \mathbb {C} {\mathcal {N}}\left( 0,\sigma _n^2\right)\) represents the discretetime thermal noise in timedomain which follows an independent and identically distributed complex Gaussian random process of zeromean and variance \(\sigma _n^2\). The term \(\eta (k,N_s,N_f)\) is the thermal noise after Fourier transformations at the receiver. To illustrate the baseband model (4) and to compare it with the equivalent SC and OFDM models, we represent in Fig. 1 the probability of the Error Vector Magnitude obtained for the different waveforms under strong GPN regime and without thermal noise. Yellow areas denote the region of highly probable occurrences and blue zones represent the regions of almost null probability of appearance.
One can observe that, with Fig. 1a, GPN only causes the rotation of the transmitted symbols for SC systems (without pulse shaping filter). Regarding OFDM systems, in Fig. 1b, the received constellation looks really noisy. The distortion is isotropic and results from the presence of ICI. Concerning the DFTsOFDM systems, the behavior is frequencydependent (or subcarrierdependent) and that is why we decided to illustrate with two distinct subcarriers \(k=1\) and \(k=10\). For the subcarrier \(k=1\), depicted in Fig. 1c, one can observe that the symbol rotation effect dominates the ICI effect. Whereas for the subcarrier \(k=10\), depicted in Fig. 1d, the ICI is stronger.
The OFDM model can thus not be directly extended to DFTsOFDM. That is the reason why we believe that studying the particularities of the DFTsOFDM model is worthy. We propose in the next section a probabilistic study of the distortion terms \(\beta (k,N_s,N_f,\phi )\) and \(\gamma (k,N_s,N_f,\phi )\).
2.2 Model analysis
In this section, we study the stochastic properties of distortion terms \(\beta (k,N_s,N_f,\phi )\) and \(\gamma (k,N_s,N_f,\phi )\). Thereafter, for sake of clarity, we denote \(\beta _k = \beta (k,N_s,N_f,\phi )\), \(\gamma _k =\gamma (k,N_s,N_f,\phi )\) and \(\eta _k=\eta (k,N_s,N_f)\). The terms \(\beta _k\) and \(\gamma _k\) can be expressed as function of the one order Dirichlet kernel K and G as follows
where
According to these expressions, one can notice the dependence on the subcarrier index k, DFT spreading size \(N_s\) and FFT size \(N_f\). One should not forget that the variables \(\beta _k\) and \(\gamma _k\) are functions of time as well (the time index is omitted in (6) for the sake of conciseness). The objective of the section is twofold: (i) determining their probability density functions and (ii) studying their dependence on the frequency (the subcarrier). For OFDMlike systems, the central limit theorem states that the random noises converge to the standard normal distribution when observed in the frequency domain (postFFT at the receiver side). To verify this assumption, we estimate the mean value \(m_X = \mathbb {E}[X[n]]\) and the variance \(\sigma _X^2 = \mathbb {E}[X[n]  m_X^2]\) of the random complex sequences X[n] (where n is the time index) and we apply a normality test. When it comes to the frequency dependence, the variations of the means and variances obtained for each subcarrier are analyzed. We propose to compare the maximum, minimum and arithmetic mean values. As long as those three values are almost equal, there is no dependence on frequency. On the contrary, if those values differ, it implies that the subcarriers do not experiences the same stochastic distortion i.e., a subcarrierdependent behavior. Because the variables \(\beta _k\) and \(\gamma _k\) are complex, we propose to simplify the problem by analyzing their real coordinates. The polar representation has been preferred for \(\beta _k\) and the Cartesian^{Footnote 2} for \(\gamma _k\). Two scenarios, namely first and second configurations, are defined in Tables 3 and 4.
2.2.1 Two evaluation scenarios
This first scenario aims at evaluating the impact of the system bandwidth \(B_W\). To do so, we consider different DFT precoding sizes \(N_s\) for the DFTsOFDM with a constant ICS (i.e., a constant numerology^{Footnote 3}. index \(\mu =9\)). The parameter list is given in Table 3.
The second configuration, detailed in Table 4, focuses on the impact of the ICS and on the ratio between the DFT size and the FFT size^{Footnote 4} denoted by \(\zeta =\frac{N_s}{N_f}\). All configurations share the same GPN PSD level with \(K_0 = 108.67\) dBc/Hz.
2.2.2 Stochastic properties of the subcarrier phase error
Let us express the term \(\beta _k\) in the following form
where \(\rho _k\) and \(\Phi _k\) are, respectively, the magnitude and the phase of \(\beta _k\). Therefore, one obtains the following mean and variance vectors
First configuration: Fig. 2a, b presents the dependence of stochastic properties of \(\rho _k\) on the DFT spreading value \(N_s\) by depicting the variations of the maximum, the minimum and the arithmetic mean of \(\underline{\varvec{m}}_\rho\) and \(\underline{\varvec{\sigma }}_\rho ^2\) as function of \(N_s\). One can observe that the three curves are superimposed around 1 for the mean and around 0 (or rather a negligible value) for the variance. We can therefore conclude that \(\rho _k\) is a deterministic value equal to 1 for each subcarrier and whatever the DFT spreading value \(N_s\) is.
Figure 2c, d shows the similar study for \(\underline{\varvec{m}}_\Phi\) and \(\underline{\varvec{\sigma }}_\Phi ^2\). In Fig. 2c, the three curves are also superimposed around 0. It implies that the rotation angle \(\Phi _k\) is zeromean for all subcarriers k and whatever the DFT spreading length \(N_s\) is. However, when one looks at the curves for the variances depicted on Fig. 2d, they can observe that the curves start standing out from \(N_s=240\) and the difference between the maximum and minimum values increases with \(N_s\). It means that:

for \(\zeta = \frac{N_s}{N_f} \le \frac{240}{512}\), the variances \(\underline{\varvec{\sigma }}_\Phi ^2\) are frequencyindependent and all the subcarriers experience the same random rotation. The behavior tends to what is observed for a classic OFDM system.

otherwise, the variances \(\underline{\varvec{\sigma }}_\Phi ^2\) are frequencydependent, i.e., appearance of a subcarrierdependent behavior. The subcarriers experience different random rotations, hence the name “subcarrier phase error” (SPE). This behavior is characteristic of DFTsOFDM and is not observed in classic OFDM systems.
Second configuration: Fig. 3 presents the stochastic analysis of \(\rho _k\) and \(\Phi _k\) by considering the second configuration. By paying attention to the scales, one can notice that the stochastic parameters of \(\underline{\varvec{m}}_\rho\) remain frequencyindependent and, thus \(\rho _k\) is a deterministic value equal to 1. Regarding \(\underline{\varvec{\sigma }}_\Phi ^2\), the subcarrierdependent behavior is this time observed for all the values of \(\zeta\). It means that \(\Phi _k\) are zeromean random rotations with a frequencydependent variance. In addition to that, if one assesses the dependence on frequency by evaluating the difference between the maximum and minimum values, they can observe that it remains constant for all values of the ratio \(\zeta\). It implies that the variances of the rotation \(\underline{\varvec{\sigma }}_\Phi ^2\) are actually functions of \(\zeta\). This observation was not expected from the analysis of the analytical expressions (6).
Gaussianity test: \(\beta _k\) is therefore reduced to a random rotation of angle \(\Phi _k\). In the previous paragraphs, we numerically estimated the mean and variance of the angles \(\Phi _k\) as function of the ratio \(\zeta\). The aim of this paragraph is to validate the Gaussian distribution hypothesis. To do so, we consider a specific ratio \(\zeta = 0.9375\) (with (\(N_s,N_f\)) = (480, 512)) and we compare the cumulative distribution function (CDF) of \(\Phi _k\) with the CDF of a Gaussian random process (zeromean and variance \(\sigma _{\Phi _k}^2\)).
The results are illustrated in Fig. 4 for two distinct subcarriers \(k = \{1,8\}\), chosen arbitrarily. One can notice the good matching between the CDFs of \(\Phi _k\) and the equivalent Gaussian random distribution. To further validate the hypothesis, we perform a Kolmogorov–Smirnov test [45], with significance factor^{Footnote 5}\(\alpha = 0.1\), to evaluate the Gaussianity property of simulated \(\Phi _k\). As expected, the test is validated and therefore we can assume that the SPE \(\Phi _k\) follows an uncorrelated zeromean Gaussian random process with a variance \(\sigma _{\Phi _k}^2\)
2.2.3 Stochastic properties of the intercarrier interference
In this paragraph, we study the stochastic properties and the distribution of the ICI term \(\gamma _k\). To do so, we define the mean and variance of the real and imaginary parts
Because the results for the real and imaginary parts are the same, only the analysis for the real part is presented hereafter.
First configuration: Fig. 5a, b presents the dependence of stochastic properties of \({\mathfrak {R}}{\mathfrak {e}}\{\gamma _k\}\) on the DFT spreading value \(N_s\) assuming the first configuration. One can observe that the real part of the ICI is a zeromean random variable. Additionally:

for \(\zeta = \frac{N_s}{N_f} \le \frac{240}{512}\), one can notice a frequencyindependent variance, meaning that all the subcarriers experience the same random noise. The behavior tends to what is observed for a classic OFDM system.

otherwise, the variances \(\underline{\varvec{\sigma }}_{{\mathfrak {R}}{\mathfrak {e}}\{\gamma \}}^2\) are frequencydependent. The subcarriers experience a different random additive noise. This behavior is characteristic of DFTsOFDM and was not observed in classic OFDM systems.
As for the SPE (configuration 1), one can observe the subcarrierdependent behavior of ICI increases with \(\zeta\).
Second configuration: With Fig. 6a, b, we analyze the stochastic property of the real part of the ICI for a constant ratio \(\zeta\). One can observe that the real part of the ICI is a zeromean random variable with frequencydependent variance. The difference between the maximum and the minimum values is independent of the ratio \(\zeta\). It thus means that the variances of the real part of the ICI is function of the subcarrier index and the ratio \(\zeta\). This observation was not expected and is not trivial from the analytical expression (6).
Gaussianity test: \(\gamma _k\) is therefore a complex zeromean random noise. In the previous paragraphs, we numerically estimated the mean and variance of its real part \({\mathfrak {R}}{\mathfrak {e}}\{\gamma _k\}\) as function of the ratio \(\zeta\). The objective of this paragraph is to validate the hypothesis of a Gaussian distribution. To do so, we consider a specific ratio \(\zeta = 0.9375\) (with (\(N_s,N_f\)) = (480, 512)) and we compare the cumulative distribution function (CDF) of \({\mathfrak {R}}{\mathfrak {e}}\{\gamma _k\}\) with the CDF of a Gaussian random process (zeromean and variance \(\sigma _{{\mathfrak {R}}{\mathfrak {e}}\{\gamma _k\}}^2\)). The results are illustrated in Fig. 7 for two distinct subcarriers \(k = \{1,8\}\), chosen arbitrarily. One can notice the curves do not perfectly match. Indeed, applying the KolmogorovSmirnov test (with the significance factor \(\alpha = 0.1\)) to determine if the variable \(\gamma _k\) follows an uncorrelated Gaussian distribution would result in the hypothesis being rejected. Thereby, the variable \(\gamma _k\) is correlated which is different compared to the previously studied SPE term \(\Phi _k\). The ICI correlated nature is expected and corresponds to the first sum of (6).
2.2.4 Lessons learnt
In this section, we characterized the term \(\beta _k\) and the ICI term \(\gamma _k\). We showed that the term \(\beta _k\) corresponds to a random phase shift where the SPE (i.e., the phase of \(\beta _k\)) follows a zeromean Gaussian random process. The variance of the SPE distribution depends on the ratio \(\zeta = \frac{N_s}{N_f}\), the power of the PN \(\sigma _\phi ^2\), and on the subcarrier index k (from a given ratio \(\zeta\)). Moreover, the ICI is a zeromean random variable with a correlated distribution. All these observations cannot be easily analytically studied. The expression (4) can be simplified as follows
The baseband model (12) is interesting because it reflects the compromise DFTsOFDM operates between SC and CPOFDM. Indeed, one can observe i) the random rotation of the data, \(e^{j\Phi _k}\), specific of SC systems [29, 30] and ii) the additive ICI, \(\gamma _k\), characteristic of MC systems, such as OFDM systems [13]. For the CPOFDM, a detector based on Euclidean distances is most suited because of the isotropic distribution of the additive ICI and thermal noise [14]. When it comes to the SC case, specific detection criteria have been proposed, such as a detection in the polar domain [29, 30]. The question of the optimal detection criterion for DFTsOFDM systems with GPN impairments is therefore still open.
3 Results
3.1 Optimum demodulation in the presence of GPN
In this section, we derive an accurate detection criterion adapted to DFTsOFDM baseband model (12). For that, we can rewrite the expression (12) in a vector form as follows
with \(\underline{\textbf{r}} = [r_0, \cdots , r_{N_s1}]^T, \; \underline{\textbf{s}} = [s_0, \cdots , s_{N_s1}]^T \; \text {and} \; \underline{\varvec{\Phi }} = [\Phi _0, \cdots , \Phi _{N_s1}]^T\). The term \(\underline{\textbf{w}} = \underline{\varvec{\eta }} \; + \; \underline{\varvec{\gamma }}\) such that \(\underline{\varvec{\eta }} = [\eta _0, \cdots , \eta _{N_s1}]^T\) and \(\underline{\varvec{\gamma }} = [\gamma _0, \cdots , \gamma _{N_s1}]^T\), denotes the additive interferenceplusnoise after DFTsOFDM demodulator including the thermal noise and the ICI term. The model of the baseband received signal (13) is similar to the model of a SC system in the presence of the PN \(\underline{\varvec{\Phi }}\) and an additive noise \(\underline{\textbf{w}}\). However, it seems important to remind that in the case of SC, the noise follows an independent complex Gaussian distribution while for the DFTsOFDM the ICI, and by extension the term \(\underline{\textbf{w}}\), exhibits a correlated nature. The authors in [29, 30] propose an optimal detector for a SC system impaired by GPN. We therefore propose to extend their results to DFTsOFDM systems. To do so, we need to explicit the probability distribution of the random rotation \(\underline{\varvec{\Phi }}\) and the additive interferenceplusnoise term \(\underline{\textbf{w}}\).
3.1.1 Probability distribution
According to the results presented in Sect. 2.2, we define the SPE vector \(\underline{\varvec{\Phi }}\) as a real Gaussian vector with a mean \(\underline{\textbf{m}}_\Phi\) and a correlation matrix \(\varvec{\Gamma }_\Phi\) as follows
where \(\sigma _{\Phi _k}^2\),^{Footnote 6} is defined in (10) and \(\textbf{I}_{N_s}\) is the identity matrix with size \(N_s\times N_s\). The noise vector \(\underline{\textbf{w}}\) is the sum of the thermal noise vector \(\underline{\varvec{\eta }}\) and the ICI vector \(\underline{\varvec{\gamma }}\). According to the linear property and power conservation of the Fourier transform, the thermal noise vector \(\underline{\varvec{\eta }}\) is defined as a complex Gaussian vector (independent and identically distributed) given by
where \(\underline{\textbf{m}}_{\varvec{\eta }}\) denotes the mean vector, \(\varvec{\Gamma }_{\varvec{\eta }}\) the correlation matrix and \(\sigma ^2_n\) the variance of the thermal noise. If one considers that the ICI follows a correlated complex Gaussian distribution, it leads to
where \(\underline{\textbf{m}}_\gamma\) is the mean vector and \(\varvec{\Gamma }_\gamma\) the correlation matrix of the ICI vector \(\underline{\varvec{\gamma }}\). Considering (15) and (16), the noise vector \(\underline{\textbf{w}}\) can be defined as a Gaussian vector with a mean vector \(\underline{\textbf{m}}_w\) and a correlation matrix \(\varvec{\Gamma }_w\) given by
3.1.2 Optimal detection criterion
The Symbol Error Probability is minimized by the Maximum Likelihood (ML) decision criterion for equiprobable and independent symbols. The channel likelihood function can be written as follows [29, 30]
Thus, we have to express more clearly the ML decision criterion. Before studying the amplitude and phase of the received signal vector \(\underline{\textbf{r}}\), we denote \(\underline{\textbf{w}} \odot e^{j\left( \underline{\varvec{\Phi }} + \underline{\textbf{s}}_\theta \right) }\) by \(\underline{\textbf{w}}'\). The term \(\underline{\textbf{w}}'\) has the same stochastic properties as \(\underline{\textbf{w}}\).^{Footnote 7} The amplitude of the received DFTsOFDM symbol is given by
and the phase by
According to the demonstration of the high SINR approximation given in “Appendix 1”, the expressions (19) and (20) become
The symbol \(\underline{\textbf{s}}^{1}_\rho\) is the vector containing the inverse of elements of \(\underline{\textbf{s}}_\rho\). The following terms \({\mathfrak {R}}{\mathfrak {e}}\{\underline{\textbf{w}}'\}\) and \(\underline{\varvec{\Phi }} + {\mathfrak {I}}{\mathfrak {m}}\{\underline{\textbf{w}}'\} \odot \underline{\textbf{s}}^{1}_\rho\) are, respectively, the additive noise in the amplitude channel and the additive Gaussian noise in the phase channel. In order to express the ML decision criterion in (18), we have to derive the Probability Density Function (PDF) of the expressions in (21). The PDF of the first expression can be expressed as follows
where \(z = (\underline{\textbf{r}}_\rho  \underline{\textbf{s}}_\rho )^T {\mathfrak {R}}{\mathfrak {e}}\lbrace \varvec{\Gamma }_w\rbrace ^{1}(\underline{\textbf{r}}_\rho  \underline{\textbf{s}}_\rho )\). When it comes to the phase domain, the term \(\underline{\varvec{\Phi }} + {\mathfrak {I}}{\mathfrak {m}}\{\underline{\textbf{w}}'\} \odot \underline{\textbf{s}}^{1}_\rho\) is expressed as follows
Therefore, we can express the PDF of the second expression in (21) by
where \(l = (\underline{\textbf{r}}_\theta  \underline{\textbf{s}}_\theta )^T \varvec{\Gamma }^{1}(\underline{\textbf{r}}_\theta  \underline{\textbf{s}}_\theta )\). From (22) and (24), the conditional PDF is written as
where z and l are expressed above. The correlation matrices \(\varvec{\Gamma }_w\) and \(\varvec{\Gamma }\) can be considered as apriori known by the detector (at least in estimation). Based on the joint observation of amplitude and phase of the received signal vector, the ML decision rule can be expressed as
where the estimated vector \(\underline{\mathbf {\hat{\textbf{s}}}}\) is supposed to be the transmitted symbols sequence. It means that for DFTsOFDM systems in presence of a GPN, the ML decision rule involves the symbols transmitted over all the \(N_s\) subcarriers. The set \({\mathcal {C}}^{'}\) thus contains all the possible transmitted sequences with a size of \((N_s)^{M_s}\), where \(M_s\) is the modulation order. For practical values, such as a FFT with 128 subcarriers and 16QAM, i.e., \(N_s=128\) and \(M_s=16\), one realizes that the number of possibilities is far too large to perform the ML expression in (26) and thus, the decision rule cannot be implemented in practice. This severe complexity comes from the ICI correlated nature. That is why we propose to investigate a suboptimal decision rule assuming that the ICI follows an uncorrelated Gaussian random process.
3.1.3 Suboptimal detection criterion
For the suboptimal detection, let us suppose that the ICI follows an uncorrelated complex Gaussian random process. This means that its correlation matrix is a diagonal matrix and we can defined it as follows
where \(m_{\gamma _k}\) and \(\sigma _{\gamma _k}^2\) are, respectively, the mean and the variance of \(\varvec{\gamma }_k\). The expression (12) becomes
where \(\Phi _k\backsim {\mathcal {N}}(0,\sigma _{\Phi _k}^2)\) and \(w_k = \eta _k + \gamma _k\) is defined as follows
By considering the high SINR approximation, i.e., \({\mathfrak {R}}{\mathfrak {e}}\{w_k'\}\) and \({\mathfrak {I}}{\mathfrak {m}}\{w_k'\}\) almost 0; and \(\text {arctan}\left( x\right) =x\) if x small (see “Appendix 1”), the expressions (19) and (20) respectively become
where \(w_k'= w_k\cdot e^{j(\Phi _k + s_{k_\theta })} \backsim \mathbb {C}{\mathcal {N}}(0,\sigma _{w_k}^2)\). So, (21) can be simplified as follows
The joint PDF of expressions in (32) is expressed as follows
Hence, the ML decision rule using the channel likelihood expression (33) can be expressed as
Contrary to what was presented earlier, the ML decision rule independently involves each subcarrier k. The number of possibilities is reduced to the size of the constellation just like for the conventional Euclidean decision rule. The proposed polar detector based on the ML metric (34) will be denoted by SUBD from now.
3.1.4 Estimation of the required apriori knowledge
The challenge in applying this detector relies in estimating \(\sigma _{\Phi _k}^2\) and \(\sigma _{\gamma _k}^2\) in realistic transmission scenarios. Let us first focus on the estimation of \(\sigma _{\Phi _k}^2\). The main idea is to express the relation between the phase error at the signal level \(e^{j\underline{\varvec{\Phi }}}\) and the phase error at the oscillator level \(e^{j\underline{\varvec{\phi }}}\)
for \(i\in \llbracket 0,\cdots ,N_s1 \rrbracket\) and \(k\in \llbracket 0,\cdots ,N_f1 \rrbracket\). The matrices \(\textbf{A}_{TX}\) and \(\textbf{A}_{RX}\) are defined as follows
where \(\textbf{M}_{TX}\) and \(\textbf{D}_{RX}\) respectively represent the \(N_s\)to\(N_f\) mapping matrix and the \(N_f\)to\(N_s\) demapping matrix. By using the small angle approximation, the expression (35) gives
because \(\textbf{B} \in \mathbb {R}^{N_s\times N_f}\). Then, the correlation matrix of \(\underline{\varvec{\Phi }}\), \(\textbf{R}_{\underline{\varvec{\Phi }}} = \mathbb {E}\left[ \underline{\varvec{\Phi }}\cdot \underline{\varvec{\Phi }}^H \right]\), can be expressed as function of the correlation matrix \(\textbf{R}_{\underline{\varvec{\phi }}}\) of the PN \(\underline{\varvec{\phi }}\) as bellow
Consequently, we compute the expression (14) upon (38) to obtain the different \(\sigma _{\Phi _k}^2\) for each DFT subcarrier k as follows
The expression (39) states that it is possible to evaluate the variance of the SPE, \(\sigma _{\Phi _k}^2\), directly from the stochastic model of the local oscillators \(\textbf{R}_{\underline{\varvec{\phi }}}\). Therefore, if one samples the response of a intermediatefrequency oscillator and measures the variance of the phase jitter, they can determine the variance of the SPE [41].
Regarding the demonstration presented in “Appendix 2”, the ICI power is defined for \(k\in \llbracket 0,\cdots ,N_s1 \rrbracket\) by the following relation
We already know that the Local Oscillator (LO)induced PN results in a random rotation and the generation of ICI after Fourier transformations. However, (40) goes further and states that the variance of the LO Phase error \(\sigma _{\phi }^2\) corresponds to the sum of the variance of the SPE \(\sigma _{\Phi _k}^2\) and the power of the ICI \(\sigma _{\gamma _k}^2\). As a consequence, when a subcarrier k suffers from strong ICI, the SPE is limited and conversely. In addition, because we know that the SPE is frequencydependent, it proves that for a DFTsOFDM waveform, some subcarriers are dominated by the rotation effects while other by the presence of ICI.
3.2 Performance evaluation
In this section, we evaluate the performance of following systems under GPN channels:

OFDM with Euclidean detector (EUCD)

DFTsOFDM with Euclidean detector (EUCD)

DFTsOFDM with the proposed suboptimal polar detector (SUBD)
The Euclidean detector EUCD correspond to the ML detection defined by
where symbols \(s_i \in {\mathcal {C}}\), with \({\mathcal {C}}\) the set of the selected modulation scheme. The suboptimal polar detector (SUBD) applies the decision rule expressed in (34). The latter detection criterion is only evaluated with DFTsOFDM signals as it is directly derived from DFTsOFDM baseband model. The performance of DFTsOFDM with SUBD will be assessed first in Genieaided conditions and then in realistic conditions (i.e., with estimation of unknown variances as detailed in Sect. 3.1.4). The performance indicator is the Transport Block Error Rate (TBLER) as a function of the SignaltoNoise Ratio (SNR) while considering a 16QAM modulation and a low density parity check (LDPC) following the 5GNR specifications [44] with a code rate (CR) equal to 0.7.
3.2.1 Impact of DFT parameters
As we mentioned in this paper, DFTsOFDM operates as a compromise between SC and OFDM systems depending on the value of the ratio \(\zeta\). Therefore, we propose in a first time to observe the achieved performance of DFTsOFDM systems with configurations defined in Table 3 (i.e., by varying \(\zeta\)) and in Table 4 (i.e., with constant \(\zeta\)) for the conventional EUCD detector. The results are, respectively, depicted in Figs. 8 and 9.
From Fig. 8, one can notice that reducing the DFT spreading size \(N_s\) allows to reach the targeted TBLER of \(10^{2}\) with a low SNR. Indeed, enlarging the useful system bandwidth \(B_W\) (by increasing \(N_s\)) integrates more GPN power and thus degrades the signal quality. That is why for higher values of \(N_s\), higher SNR are required to successfully decode the signal or not (if the GPN power is very high). Then for a fixed system bandwidth, there is no performance penalty for numerologies from 6 to 9 as one can observe by looking at Fig. 9. Therefore, enlarging the ICS does not help and DFTsOFDM with the EUCD decision rule does not work under strong GPN regime.
3.2.2 Comparison of detection techniques
The results presented above demonstrate the inefficiency of the Euclidean detector when the PN power is high (larger \(\zeta\)). In this section, we compare our proposed detector SUBD with the Euclidean detector EUCD. We compare both DFTsOFDM and OFDM assuming the same bandwidth, i.e., by considering \(N_s\) active subcarriers among the \(N_f\) for the OFDM modulation.
Figure 10 shows the system performance for the different GPN regimes and genieaided SUBD detector. In strong GPN regime, one can first observe that the two EUCD receivers do not perform well as they cannot achieve a TBLER \(<10^{1}\) with SNR \(=30\) dB. Fortunately, the proposed DFTsOFDM with SUBD receiver succeeds in decoding packets and can even reach a TBLER of \(10^{2}\) for SNR \(\simeq 21\) dB. In medium GPN regime, our proposed detector still outperforms the EUCD but with a small SNR gain compared to the strong GPN case. For low GPN, the EUCD detector has almost the same performance as the SUBD, but it is slightly better than the SUBD.
We have demonstrated the gain of the SUBD detector over the EUCD detector in a strong GPN regime and genieaided conditions (perfect knowledge of SPE and ICI stochastic properties). Indeed, the polar detection criterion makes sense when phase distortion is dominant. As a consequence, DFTsOFDM with the proposed SUBD detector is an enabler for communications under strong GPN regime. As a consequence, for the considered scenarios, the assumption of independent ICI is sufficiently valid.
Figure 11 presents and compares the performance achieved by the genieaided and the realistic DFTsOFDM/SUBD receiver, i.e., by considering the expressions (39) and (40). One can notice the estimated SUBD yields the same results as the genieaided one. These results validate the proposed estimation approach.
4 Discussions
ICI cancellation techniques have also been investigated, especially for OFDM systems [17, 19]. In those studies, PN is generally assumed partially correlated [24]. We decide to evaluate the performance of the commonly used De_ICI algorithm [17] under GPN channel. The results are depicted in Fig. 12. The De_ICI algorithm means filtering the OFDM received signal in the frequency domain (postFFT domain) with a filter of length \(L_f\). In [17], the authors show that the increase of \(L_f\) enhances the mitigation of the ICI at the price of increased complexity. In what follows, De_ICI\(L_f\) denotes the application of the De_ICI technique with a filter of length \(L_f\). For comparison, the case without PN (ideal scenario), with GPN but without any compensation techniques and with GPN and CPE Estimation (CPEE) algorithm [15] are presented. The CPEE algorithm compensates the common phase error (i.e., the constellation rotation) of the received signal (no ICI mitigation). The De_ICI1 is equivalent to the CPE compensation [17].
In Fig. 12, the uncoded bit error rate (BER) is evaluated as function of the SNR under GPN channel. First, one can notice that the CPEE algorithm does not provide any performance gain with respect to uncompensated case. This result is expected because GPN is a zeromean phase error noise, therefore the CPE tends to zero with probability 1 (meaning that with GPN there is only a negligible symbol rotation for OFDM constellations). CPEE algorithm is only beneficial when correlated PN are involved. When it comes to the De_ICI, the algorithm fails at mitigating the ICI power even for filter lengths up to \(N_f=17\). Indeed, with GPN the correlated nature of the ICI is limited, as we observed with DFTsOFDM in this article and therefore, the De_ICI does not successfully remove the ICI from the received signal. It implies that, for OFDM systems, GPN leads to the generation of ICI (and no CPE), whereas with DFTsOFDM, it leads to both a random rotation and generation of ICI. The power of the resulting ICI in DFTsOFDM is thus lower than for OFDM. It allows to adapt the decision rule as presented in this article which results in significant performance gain especially when strong GPN regime is involved. That is the reason why DFTsOFDM is more suited than OFDM for transmissions under strong GPN channels.
5 Conclusion
In this paper, we investigated the effects of GPN on OFDM and DFTsOFDM waveforms. In DFTsOFDM systems, the presence of GPN results in two distortions i) a frequencydependent random rotation of the received constellation named SPE and ii) the generation of additive ICI. Their random distributions have been characterized. It appears that, in DFTsOFDM, the random rotation dominates against the generation of ICI for some subcarriers and conversely. That is why we studied and derived the optimal decision rule to be applied on each subcarrier. This optimal decision rule is too complex to be implemented in practical devices, thus, we derived a suboptimal detector and compared its achievable performance with the conventional Euclideandistancebased criterion. We can draw two conclusions from the obtained results: i) known PN mitigation techniques designed when the PN exhibits a partially correlated nature (especially for MC systems) prove to be inefficient under GPN channels and, ii) DFTsOFDM with the proposed polar detector significantly outperforms its OFDM contender when strong GPN regimes are involved. That is the reason why we believe that DFTsOFDM is a promising enabler of subTHz transmissions under strong GPN channels.
Nonetheless, all the work presented above is based on the assumption that the channel is dominated by its LoS contribution. Even if this assumption has been validated by recent studies [46,47,48,49], some scenarios with a second strong resolvable path can exist such as indoor environment, vehicular communications with the ground rebound. In such cases, our work cannot be trivially extended because the propagation channels become frequency selective. Furthermore, when the PN has an uncorrelated nature, it could interfere with tracking and channel estimation in real time. Joint channel estimation and GPNinduced impairments must be considered which is one interesting study perspective of the presented work.
Availability of data and materials
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
Notes
By assuming the Cyclic Prefix (CP) size is higher than the channel delay spread and assuming a frequencyselective channel.
The choice of the system coordinate is quite clear from the study and for sake of simplicity we do not provide all the results but to give a hint as \(\beta _k\) represents a rotation and \(\gamma _k\) is an additive noise the respective choice for polar and Cartesian systems makes sense.
Numerology is a feature introduced in \(5\)GNR and corresponds to the ICS \(\Delta _f\) is the following \(\Delta _f = 15 \times 2 ^\mu\) kHz [44]
\(N_s\) also denotes the size of the inverse DFT (IDFT) despreading block size and \(N_f\) the inverse FFT (IFFT) block size.
It represents the validity factor of the Gaussianity test.
The SPE power \(\sigma _{\Phi _k}^2\) can be obtained by taking the diagonal of the correlation matrix \(\varvec{\Gamma }_\Phi\), i.e. \(\sigma _{\Phi _k}^2 = \textbf{diag}_k\lbrace \varvec{\Gamma }_\Phi \rbrace\).
by assuming the complex Gaussian noise vector is circularly symmetric.
Abbreviations
 \(N_s\) :

Size of the DFT spreading block
 \(N_f\) :

Size of the FFT block
 \(\mu\) :

Numerology
 \(B_W\) :

System bandwidth
 \(F_s\) :

Sampling frequency
 \(K_0\) :

Power spectral density of the phase noise
 5GNR:

5G new radio
 BER:

Bit error rate
 CDF:

Cumulative distribution function
 CP:

Cyclic Prefix
 CPE:

Common phase error
 CR:

Coding rate
 DFT:

Discrete Fourier transform
 DFTsOFDM:

DFTspreadOFDM
 FFT:

Fast Fourier transform
 GPN:

Gaussian PN
 ICI:

Intercarrier interference
 ICS:

Intercarrier spacing
 LoS:

Lineofsight
 MC:

Multicarrier
 ML:

Maximum likelihood
 OFDM:

Orthogonal frequency division multiplexing
 PAPR:

Peaktoaverage power ratio
 PDF:

Probability density function
 PN:

Phase noise
 PSD:

Power spectral density
 PTRS:

Phase tracking reference signals
 RF:

Radio frequency
 SC:

Singlecarrier
 SINR:

Signaltointerferenceplusnoise ratio
 SNR:

Signaltonoise ratio
 SPE:

Subcarrier phase error
 TBLER:

Transport block error rate
References
V. Petrov, T. Kurner, I. Hosako, IEEE 802.15.3d: first standardization efforts for subterahertz band communications toward 6G. IEEE Commun. Mag. 58(11), 28–33 (2020). https://doi.org/10.1109/MCOM.001.2000273
J.B. Doré, Y. Corre, S. Bicais, J. Palicot, E. Faussurier, D. Kténas, F. Bader, Above90GHz spectrum and singlecarrier waveform as enablers for efficient Tbit/s wireless communications, in Proceedings of International Conference on Telecommunications (ICT), SaintMalo, France (2018)
3GPP: NR; User Equipement (UE) radio transmission and reception; Part 1: Range 1 Standalone. Technical Specification (TS) 38.1011, 3rd Generation Partnership Project (3GPP) (December 2023). Version 17.12.0
O. Tervo, T. Levanen, K. Pajukoski, J. Hulkkonen, P. Wainio, M. Valkama, 5G New Radio Evolution Towards SubTHz Communications, in Proceedings 2nd 6G wireless summit (6G SUMMIT), pp. 1–6 (2020). https://doi.org/10.1109/6GSUMMIT49458.2020.9083807
T. Pollet, M. Moeneclaey, Synchronizability of OFDM signals, in Proceedings IEEE Global Communications Conference (GLOBECOM), vol. 3, pp. 2054–20583 (1995). https://doi.org/10.1109/GLOCOM.1995.502767
M.V. Clark, Adaptive frequencydomain equalization and diversity combining for broadband wireless communications. IEEE J. Sel. Areas Commun. 16(8), 1385–1395 (1998). https://doi.org/10.1109/49.730448
N. Dinur, D. Wulich, Peak to average power ratio in amplitude clipped high order OFDM, in Proceedings of IEEE Military Communications Conference (MILCOM) (Cat. No.98CH36201), vol. 2, pp. 684–6872 (1998). https://doi.org/10.1109/MILCOM.1998.722213
T. Oskari, N. Ilmari, N.I. Peruga, T. Esa, H. Jari, On the potential of using subTHz frequencies for beyond 5G, in Joint European Conference on Networks and Communications & 6G Summit (EuCNC/6G Summit), pp. 37–42 (2022). https://doi.org/10.1109/EuCNC/6GSummit54941.2022.9815761
G. Berardinelli, K.I. Pedersen, T.B. Sorensen, P. Mogensen, Generalized DFTspreadOFDM as 5G waveform. IEEE Commun. Mag. 54(11), 99–105 (2016). https://doi.org/10.1109/MCOM.2016.1600313CM
S. Merchan, A.G. Armada, J.L. Garcia, OFDM performance in amplifier nonlinearity. IEEE Trans. Broadcast. 44(1), 106–114 (1998). https://doi.org/10.1109/11.713060
A. Demir, Computing timing jitter from phase noise spectra for oscillators and phaselocked loops with white and 1/f noise. IEEE Trans. Circuits Syst. I Regular Papers 53(9), 1869–1884 (2006). https://doi.org/10.1109/TCSI.2006.881184
T. Pollet, M. Van Bladel, M. Moeneclaey, BER sensitivity of OFDM systems to carrier frequency offset and Wiener phase noise. IEEE Trans. Commun. 43(2/3/4), 191–193 (1995). https://doi.org/10.1109/26.380034
A. Garcia Armada, Understanding the effects of phase noise in orthogonal frequency division multiplexing (OFDM). IEEE Trans. Broadcast. 47(2), 153–159 (2001). https://doi.org/10.1109/11.948268
D. Petrovic, W. Rave, G. Fettweis, Intercarrier interference due to phase noise in OFDM  estimation and suppression, in Proceedings of IEEE 60th Vehicular Technology Conference (VTCFall), vol. 3, pp. 2191–2195 (2004). https://doi.org/10.1109/VETECF.2004.1400429
J.C. Sibel, Pilotbased phase noise tracking for uplink DFTsOFDM in 5G, in Proceeding International Conference Telecommunications (ICT), pp. 52–56 (2018). https://doi.org/10.1109/ICT.2018.8464891
Y. Qi, M. Hunukumbure, H. Nam, H. Yoo, S. Amuru, On the phase tracking reference signal (PTRS) design for 5G new radio (NR), in Proceedings of IEEE Vehicular Technology Conference (VTCFall) (2018). https://doi.org/10.1109/VTCFall.2018.8690852
M. Afshang, D. Hui, J.F.T. Cheng, S. Grant, On phase noise compensation for OFDM operation in 5G and beyond, in Proceedings IEEE Wireless Communications and Networking Conference (WCNC), pp. 2166–2171 (2022). https://doi.org/10.1109/WCNC51071.2022.9771982
J.C. Sibel, Tracking the phase noise in subTHz bands, in Proceedings IEEE Wireless Communications and Networking Conference (WCNC), pp. 932–937 (2022). https://doi.org/10.1109/WCNC51071.2022.9771863
J.L. Hernando, A.S. Tan, A. Shojaeifard, Mitigation of Phase Noiseinduced ICI at THz bands using CPOFDM PTRS signals, in Proceedings IEEE Globecom Workshops (GC Wkshps), pp. 1766–1771 (2022). https://doi.org/10.1109/GCWkshps56602.2022.10008741
L.H. Nguyen, V. Braun, H. Halbauer, T. Wild, Waveform comparison under hardware limitations for 6G SubTHz communications, in Proceedings IEEE Annual Consumer Communications & Networking Conference (CCNC), pp. 1–6 (2022). https://doi.org/10.1109/CCNC49033.2022.9700588
Y. Li, H.G. Ryu, L. Zhou, D. Sun, H. Liu, J. Li, Frequency offset and phase noise influence and its compensation in the DFTspread OFDM system, in Proceedings of International Conference on Communication, Circuits, and Systems (ICCCAS), pp. 52–56 (2008). https://doi.org/10.1109/ICCCAS.2008.4657726
V. Syrjälä, T. Levanen, M. Valkama, Methods for phase noise mitigation for DFTSOFDM waveforms, in Proceedings IEEE Global Conference on Signal and Information Processing (GlobalSIP), pp. 660–664 (2016). https://doi.org/10.1109/GlobalSIP.2016.7905924
S. Bicais, J.B. Doré, Phase noise model selection for subTHz communications, in Proceedings IEEE Global Communications Conference (GLOBECOM), pp. 1–6 (2019). https://doi.org/10.1109/GLOBECOM38437.2019.9013189
3GPP: Study on new radio access technology: Radio Frequency (RF) and coexistence aspects. Technical Specification (TS) 38.803, 3rd Generation Partnership Project (3GPP) (March 2017). Release 14 version 2.0.0
Y. Bello, J.B. Doré, D. Demmer, Wiener interpolation filter for phase noise estimation in subTHz transmission, in Proceedings IEEE 97th Vehicular Technology Conference (VTCSpring), pp. 1–5 (2023). https://doi.org/10.1109/VTC2023Spring57618.2023.10200443
S. Li, D. Fritsche, C. Carta, F. Ellinger, A 200GHz subharmonic injectionlocked oscillator with 0dBm output power and 3.5% DCtoRFefficiency, in Proceedings of IEEE Radio Frequency Integrated Circuits symposium (RFIC), pp. 212–215 (2018). https://doi.org/10.1109/RFIC.2018.8428978
M.R. Khanzadi, D. Kuylenstierna, A. Panahi, T. Eriksson, H. Zirath, Calculation of the performance of communication systems from measured oscillator phase noise. IEEE Trans. Circuits Syst. I Reg. Pap. 61(5), 1553–1565 (2014). https://doi.org/10.1109/TCSI.2013.2285698
M. Sarajlic, N. Tervo, A. Pärssinen, L.H. Nguyen, H. Halbauer, K. Roth, V. Kumar, T. Svensson, A. Nimr, S. Zeitz, M. Dörpinghaus, G. Fettweis, Waveforms for subTHz 6G: Design Guidelines. In: Proceedings Joint European Conference on Networks and Communcation & 6G Summit (EuCNC/6G Summit), pp. 168–173 (2023). https://doi.org/10.1109/EuCNC/6GSummit58263.2023.10188254
R. Krishnan, M.R. Khanzadi, T. Eriksson, T. Svensson, Soft metrics and their performance analysis for optimal data detection in the presence of strong oscillator phase noise. IEEE Trans. Commun. 61, 2385–2395 (2013). https://doi.org/10.1109/TCOMM.2013.042313.120670
S. Bicaïs, J.B. Doré, Design of digital communications for strong phase noise channels. IEEE Open J. Veh. Technol. 1, 227–243 (2020). https://doi.org/10.1109/OJVT.2020.2994626
S. Bicaïs, J.B. Doré, J.L. Gonzalez Jimenez, On the optimum demodulation in the presence of Gaussian phase noise, in Proceedings 25th International Conference on Telecommunication (ICT), pp. 269–273 (2018). https://doi.org/10.1109/ICT.2018.8464897
P. Desombre, H. Farès, Y. Louët, Performance comparison of digital modulations in the presence of Gaussian phase noise in the subTHz context, in Proceedings of International Workshop on Mobile Terahertz Systems (IWMTS), pp. 1–5 (2021). https://doi.org/10.1109/IWMTS51331.2021.9486786
S. Tarboush, H. Sarieddeen, M.S. Alouini, T.Y. AlNaffouri, Singleversus multicarrier terahertzband communications: a comparative study. IEEE Open J. Commun. Soc. 3, 1466–1486 (2022). https://doi.org/10.1109/OJCOMS.2022.3201038
B. Picinbono, J.M. Kerilis, Some properties of prediction and interpolation errors. IEEE Trans. Acoust. Speech Signal Process. 36(4), 525–531 (1988). https://doi.org/10.1109/29.1556
B.J. Kwak, N.O. Song, B. Park, D.S. Kwon, Spiral QAM: a novel modulation scheme robust in the presence of phase noise, in Proceedings of IEEE 68th Vehicular Technology Conference (VTCFall), pp. 1–5 (2008). https://doi.org/10.1109/VETECF.2008.221
A.J. Viterbi, A.M. Viterbi, Nonlinear estimation of PSKmodulated carrier phase with application to burst digital transmission. IEEE Trans. Inf. Theory 29(4), 543–551 (1983). https://doi.org/10.1109/TIT.1983.1056713
U. Mengali, A.N. D’Andrea, Synchronization Techniques for Digital Receivers, vol. 1 (Springer, Berlin, 1997). https://doi.org/10.1007/9781489918079
J. Bhatti, M. Moeneclaey, Pilotaided carrier synchronization using an approximate DCTbased phase noise model, in IEEE international symposium on signal processes and information technology (ISSPIT), pp. 1143–1148 (2007). https://doi.org/10.1109/ISSPIT.2007.4458068
A. Ugolini, A. Piemontese, T. Eriksson, Spiral constellations for phase noise channels. IEEE Trans. Commun. 67(11), 7799–7810 (2019). https://doi.org/10.1109/TCOMM.2019.2937293
Y. Bello, D. Demmer, J.B. Doré, Time domain phase noise mitigation in OFDM systems for subTHz bands, in Proceedings of IEEE Global Communications Conference (GLOBECOM) (2023)
Y. Bello, D. Demmer, A. Hamani, A. Siligaris, C. Dehos, N. Cassiau, J.B. Doré, J.L. GonzálezJiménez, Performance assessment of a 5GNR Dband CMOS transceiver with phase noise impairments, in Proceedings of Joint European conference on networks and communications & 6G summit (EuCNC/6G Summit), pp. 246–251 (2023). https://doi.org/10.1109/EuCNC/6GSummit58263.2023.10188317
L. Pometcu, R. D’Errico, An indoor channel model for high datarate communications in Dband. IEEE Access 8, 9420–9433 (2020). https://doi.org/10.1109/ACCESS.2019.2960614
Y. Xing, T.S. Rappaport, Propagation measurements and path loss models for subTHz in urban microcells, in Proceedings of IEEE International Conference on Communications (ICC), pp. 1–6 (2021). https://doi.org/10.1109/ICC42927.2021.9500385
3GPP: NR; NR and NGRAN overall description; Stage 2. Technical Specification (TS) 38.300, 3rd Generation Partnership Project (3GPP) (September 2023). Version 17.6.0
F.J. Massey, The Kolmogorov–Smirnov test for goodness of fit. J. Am. Stat. Assoc. 46(253), 68–78 (1951). https://doi.org/10.2307/2280095
L. Pometcu, R. D’Errico, Characterization of subTHz and mmwave propagation channel for indoor scenarios, in Proceedings European Conference on Antennas and Propagation (EuCAP), pp. 1–4 (2018). https://doi.org/10.1049/cp.2018.0991
L. Pometcu, R. D’Errico, Channel model characteristics in Dband for NLOS indoor scenarios, in Proceedings of European Conference on Antennas and Propagation (EuCAP), pp. 1–4 (2019)
Y. Xing, T.S. Rappaport, Propagation measurement system and approach at 140 GHzmoving to 6G and above 100 GHz, in Proceedings IEEE Global Communications Conference (GLOBECOM), pp. 1–6 (2018). https://doi.org/10.1109/GLOCOM.2018.8647921
J.M. Eckhardt, T. Doeker, S. Rey, T. Kürner, Measurements in a real data centre at 300 GHz and recent results, in Proceedings of European conference on antennas and propagation (EuCAP), pp. 1–5 (2019)
Acknowledgements
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This work was funded by the French National Research Agency as part of France 2030 and the NFSYSTERA project (ANR22PEFT0006 NFSYSTERA).
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YB, JBD and DD contributed to the formulation of the research problem, development of algorithms, numerical simulations and writing of the paper. All authors read and approved the manuscript.
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Appendices
Appendix 1: High SINR approximation
The Symbol Error Probability is minimized by the Maximum Likelihood (ML) decision criterion for equiprobable and independent symbols. The channel likelihood function can be written as follows [29, 30]:
Thus, we have to express more clearly the ML decision criterion. Before studying the amplitude and phase of the received signal vector \(\underline{\textbf{r}}\), we denote \(\underline{\textbf{w}} \odot e^{j\left( \underline{\varvec{\Phi }} + \underline{\textbf{s}}_\theta \right) }\) by \(\underline{\textbf{w}}'\). The postFFT noise \(\underline{\textbf{w}}'\) has the same stochastic properties as the additive channel noise \(\underline{\textbf{w}}\) (i.e., zeromean circular symmetric complex Gaussian noise). The amplitude and phase of the received DFTsOFDM symbol is, respectively, given by (43) and (44).
One can express \(\underline{\textbf{w}}'[k]\) in its polar form by setting \(\underline{\textbf{w}}'[k] = \rho _w[k] e^{j \Phi _w[k]}\). The high SNR approximation means that the noise power becomes negligible with respect to the signal power, i.e., \(\rho _w[k]/\underline{\textbf{s}}_\rho [k] \rightarrow 0\). By applying this approximation on (43) and (44), one obtains
and
It eventually leads to:
and
Thus, we obtain
Appendix 2: Relation between ICI power and variance of the SPE
Let us consider \(\mathbf {\underline{s}}\) as the data symbol vector before the DFT transformation at the transmitter. The expression of the received signal \(\mathbf {\underline{r}}\) after Fourier transformations can be expressed by
where \(\textbf{C} = \textbf{A}_{RX}\textbf{Q}\textbf{A}_{TX}\). The terms \(\textbf{F}_{N_s}\) and \(\textbf{F}_{N_f}\) respectively are the Fourier transform matrix of size \(N_s\) and \(N_f\). The matrices \(\textbf{M}_{TX}\) and \(\textbf{D}_{RX}\) respectively represent the \(N_s\)to\(N_f\) mapping and \(N_f\)to\(N_s\) demapping. The matrix \(\textbf{Q}\) is a diagonal matrix such that \(\textbf{diag}\lbrace \textbf{Q} \rbrace = e^{j\underline{\varvec{\phi }}}\), where \(\underline{\varvec{\phi }}\) represents the GPN generated by the oscillators at transmitter and receiver side. We can develop the expression (51) as follows
where \(\varvec{\underline{\beta }}\) is the SPE vector and \(\varvec{\underline{\gamma }}\) the ICI vector. The term \(\textbf{G}\) represents a diagonal matrix such that
and \(\textbf{K} = \textbf{C}  \textbf{G}\), where \(\textbf{K}\) is the ICI matrix. The expression of the ICI correlation matrix is given by
The second term of (54) in bracket can be simplified as follows
by realizing that \(\textbf{GK}^H=\textbf{0}_{N_s}\) is the product of a diagonal matrix \(\textbf{G}\) with a zero diagonal elements matrix \(\textbf{K}\) (similarly \(\textbf{KG}^H=\textbf{0}_{N_s}\)). By combining (55) and (54), one obtains
Let’s simplify the expressions of \(\mathbb {E}\left[ \textbf{Q}\textbf{Q}^H\right]\) and \(\mathbb {E}\left[ \textbf{G}\textbf{G}^H\right]\). We have
The small angle approximation [41] leads to
because \(\textbf{R}_{\underline{\varvec{\phi }}} = \sigma _{\varvec{\phi }}^2 \cdot \textbf{I}_{N_f}\). Now, we will derive the second expectation term \(\mathbb {E}\left[ \textbf{G}\textbf{G}^H\right]\) as bellow
where \(\textbf{Z}\) is a diagonal matrix such that \(\textbf{diag} \left\{ \textbf{Z} \right\} = \textbf{diag} \left\{ \textbf{R}_{\underline{\varvec{\Phi }}} \right\}\). Therefore, the expression (56) becomes
Since we use normalized QAM schemes for data modulation, i.e., \(\mathbb {E}\left[ \mathbf {\underline{s}}\;\mathbf {\underline{s}}^H \right] = \textbf{I}_{N_s}\), the correlation matrix of the ICI vector is given by
Finally, the expression of the ICI power at a subcarrier k according to the expressions (27) and (60) is given by
Finally, since \(\textbf{R}_{\underline{\varvec{\Phi }}}\) is a diagonal matrix such that \(\sigma _{\Phi _k}^2 = \textbf{diag}_k \left\{ \textbf{R}_{\underline{\varvec{\Phi }}} \right\}\), the ICI power at subcarrier k is
where \(\sigma _{\phi }^2\) represents the GPN power generated by oscillators in (3) and \(\sigma _{\Phi _k}^2\) the SPE power at subcarrier k.
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Bello, Y., Doré, JB. & Demmer, D. Analysis of Gaussian phase noise effects in DFTsOFDM systems for subTHz transmissions. J Wireless Com Network 2024, 60 (2024). https://doi.org/10.1186/s1363802402350y
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DOI: https://doi.org/10.1186/s1363802402350y