Analysis of Gaussian phase noise effects in DFT-s-OFDM systems for sub-THz transmissions

Abstract

The future generations of communication technologies envision the transmission of signals across the millimeter wave and sub-THz spectrums. However, the characteristics of the propagation channel at such high frequencies differ from what is observed in the conventional low-frequency 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 (5G-NR) physical layer to the presence of PN. This article is dedicated to the study of discrete Fourier transform-spread-Orthogonal Frequency Division Multiplexing (DFT-s-OFDM) under uncorrelated Gaussian PN (GPN) impairments. We show that the presence of GPN induces two distortions: (i) a frequency-dependent random rotation of data, namely the subcarrier phase error (SPE) and (ii) a frequency-dependent 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, DFT-s-OFDM 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 “sub-THz” 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 ($5$G-NR) 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 Single-Carrier (SC) waveform with carrier aggregation [1]. On the other hand, research is being conducted to study how 5G-NR Multi-Carrier (MC) waveforms such as Orthogonal Frequency Division Multiplexing (OFDM) or discrete Fourier transform (DFT)-spread-OFDM (DFT-s-OFDM) can be adapted to the specificities of the sub-THz bands [4].

OFDM is a MC waveform standardized in 4G and 5G-NR 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 “1-tap” channel equalization per subcarrier in the frequency-domain¹ [6]. However, the main drawback of OFDM is its high Peak-to-Average 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 DFT-s-OFDM seems to be a good compromise. The waveform is also standardized in 4G and 5G-NR but rather for uplink transmissions and also a candidate for sub-THz [8]. It can be seen as a DFT-precoded 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 sub-THz bands presents some specific impairments such as the Phase Noise (PN) which is mainly generated by the high frequency oscillators suffering from random and short-term frequency instabilities. This phenomenon may lead to synchronization issues in clocked and sampled-data 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 sub-THz 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 (PT-RS) have been introduced in 5G-NR [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 DFT-s-OFDM systems because they rely on the presence of the CPE which is no longer the case with DFT-s-OFDM. In [8], the authors show that increasing the ICS up to 3840 kHz for DFT-s-OFDM 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 multi-path channel [20] or reduce the system’s ability to handle time synchronization errors. It should also be mentioned that $5$G-NR standard imposes a minimum number of sub-carriers to be allocated. If this constraint is maintained, increasing the ICS necessarily implies increasing the bandwidth and the sampling frequency. PN mitigation in DFT-s-OFDM is discussed in [15, 21, 22]. Additionally, works were carried to enhance the channel estimation in the context of OFDM by proposing new PT-RS scheme [18].

1.1 Related state-of-the-art 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 sub-THz 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 high-frequency 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 sub-THz 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 DFT-s-OFDM have been evaluated under GPN channels [33]. However, the GPN effects on MC systems are not yet comprehensively understood especially for DFT-s-OFDM systems. That is the reason why, we propose in this work to better study the effects of GPN on DFT-s-OFDM 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 state-of-the-art 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.

Table 1 Summary of the state-of-the-art phase noise mitigation techniques

Overall, the main contributions of this paper can be described as follows:

• 1. We model the distortions induced by the GPN on the received DFT-s-OFDM baseband signal.

• 2. We develop a novel detection criterion adapted to DFT-s-OFDM systems in the presence of GPN by assuming the high Signal-to-Interference-plus-Noise Ratio (SINR) approximation.

• 3. We propose a sub-optimal implementation of the proposed detector more suitable for realistic systems [29].

• 4. We evaluate the link performance of a DFT-s-OFDM systems and compare them with legacy OFDM (in terms of packet error rates with 5G-NR 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 state-of-the-art 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 transpose-conjugate 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:

$$\begin{aligned} \textbf{F}^{(i,k)}_{X}=\frac{1}{\sqrt{X}}e^{-j2\pi \frac{\left( i-1\right) \left( k-1\right) }{X}} \cdot \end{aligned}$$

(1)

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 sub-THz frequencies are expected to be sparse [42]. It has been demonstrated that the Line-of-Sight (LoS) provides most of the energy contribution in sub-THz 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 LoS-only 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

$$\begin{aligned} \phi [\tau ] \backsim {\mathcal {N}}(0,\sigma _{\phi }^2), \end{aligned}$$

(2)

where $\sigma _{\phi }^2$ denotes the sum variance of the Tx and Rx GPNs, i.e.,

$$\begin{aligned} \sigma _{\phi }^2&= \sigma _{\phi _{TX}}^2 + \sigma _{\phi _{RX}}^2 \nonumber \\&= 2K_0 B_W \cdot \end{aligned}$$

(3)

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 multi-path components. Three levels of GPN regimes are considered in this work as detailed in Table 2.

Table 2 Evaluation of $K_0$ depending on signal bandwidth and GPN regime

2.1.3 DFT-s-OFDM waveform with the presence of PN

The expression of the baseband discrete-time received signal impaired by the GPN at the $k$th subcarrier is expressed as follows

$$\begin{aligned} r_k = s_k\beta (k,N_s,N_f,\phi ) \; + \; \gamma (k,N_s,N_f,\phi ) + \;\eta (k,N_s,N_f), \end{aligned}$$

(4)

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

$$\begin{aligned} \begin{aligned} \beta (k,N_s,N_f,\phi )&=\frac{1}{N_s}\frac{1}{N_f}\sum \limits _{f=0}^{N_s-1}\sum \limits _{m=0}^{N_s-1} \sum \limits _{p=0}^{N_f-1}e^{j2\pi \frac{(m-f)}{N_f}p}e^{-j2\pi \frac{(m-f)}{N_s}k}e^{j\phi _p} \\ \gamma (k,N_s,N_f,\phi )&= \frac{1}{N_s}\frac{1}{N_f}\sum \limits _{n=0,n\ne k}^{N_s-1}s_n\sum \limits _{f=0}^{N_s-1}\sum \limits _{m=0}^{N_s-1}\sum \limits _{p=0}^{N_f-1}e^{j2\pi \frac{(m-f)}{N_f}p}e^{j2\pi \frac{kf-nm}{N_s}}e^{j\phi _p} \\ \eta (k,N_s,N_f)&= \frac{1}{\sqrt{N_s}}\frac{1}{\sqrt{N_f}}\sum \limits _{f=0}^{N_s-1}\sum \limits _{p=0}^{N_f-1} n_pe^{-j2\pi \frac{fp}{N_f}}e^{j2\pi \frac{kf}{N_s}}\cdot \end{aligned} \end{aligned}$$

(5)

The term $n_p \backsim \mathbb {C} {\mathcal {N}}\left( 0,\sigma _n^2\right)$ represents the discrete-time thermal noise in time-domain which follows an independent and identically distributed complex Gaussian random process of zero-mean 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 DFT-s-OFDM systems, the behavior is frequency-dependent (or subcarrier-dependent) 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.

Fig. 1

GPN distortion on 16-QAM modulated signal with SNR = 50 dB

The OFDM model can thus not be directly extended to DFT-s-OFDM. That is the reason why we believe that studying the particularities of the DFT-s-OFDM 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

$$\begin{aligned} \begin{aligned} \beta (k,N_s,N_f,\phi )&= \sum \limits _{p=0}^{N_f-1}e^{j\phi _p}K^{N_s}_{N_s}\left( \frac{p}{N_f}-\frac{k}{N_s}\right) G^{N_s}_{N_f}\left( \frac{k}{N_s}-\frac{p}{N_f}\right) \\ \gamma (k,N_s,N_f,\phi )&= \sum \limits _{n=0,n\ne k}^{N_s-1}s_n\sum \limits _{p=0}^{N_f-1}e^{j\phi _p}K^{N_s}_{N_s}\left( \frac{p}{N_f}-\frac{n}{N_s}\right) G^{N_s}_{N_f}\left( \frac{k}{N_s}-\frac{p}{N_f}\right) , \end{aligned} \end{aligned}$$

(6)

where

$$\begin{aligned} K^{P}_{X}\left( y\right) = \frac{1}{X}\sum \limits _{m=0}^{P-1}e^{j2\pi my} \quad \text {and} \quad G^{P}_{X}\left( z\right) = \frac{1}{X}\sum \limits _{f=0}^{P-1}e^{j2\pi fz} \cdot \end{aligned}$$

(7)

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 OFDM-like systems, the central limit theorem states that the random noises converge to the standard normal distribution when observed in the frequency domain (post-FFT 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 subcarrier-dependent 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 Cartesian² for $\gamma _k$. Two scenarios, namely first and second configurations, are defined in Tables 3 and 4.

Table 3 Size of FFT and DFT blocks

Table 4 Size of FFT and DFT blocks

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 DFT-s-OFDM with a constant ICS (i.e., a constant numerology³. 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 size⁴ 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

$$\begin{aligned} \beta _k =\rho _k e^{j\Phi _k}, \end{aligned}$$

(8)

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

$$\begin{aligned} \begin{aligned} \underline{\varvec{m}}_\rho&=\left[ m_{\rho _0},\cdots ,m_{\rho _{N_s-1}}\right] ^T, \quad \underline{\varvec{\sigma }}_\rho ^2 =\left[ \sigma _{\rho _0}^2,\cdots ,\sigma _{\rho _{N_s-1}}^2 \right] ^T \\ \underline{\varvec{m}}_\Phi&=\left[ m_{\Phi _0},\cdots ,m_{\Phi _{N_s-1}}\right] ^T, \quad \underline{\varvec{\sigma }}_\Phi ^2 =\left[ \sigma _{\Phi _0}^2,\cdots ,\sigma _{\Phi _{N_s-1}}^2 \right] ^T \cdot \end{aligned} \end{aligned}$$

(9)

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.

Fig. 2

Statistical analysis of $\rho _k$ and $\Phi _k$ using the First configuration

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 zero-mean 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 frequency-independent 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 frequency-dependent, i.e., appearance of a subcarrier-dependent behavior. The subcarriers experience different random rotations, hence the name “subcarrier phase error” (SPE). This behavior is characteristic of DFT-s-OFDM 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 frequency-independent and, thus $\rho _k$ is a deterministic value equal to 1. Regarding $\underline{\varvec{\sigma }}_\Phi ^2$, the subcarrier-dependent behavior is this time observed for all the values of $\zeta$. It means that $\Phi _k$ are zero-mean random rotations with a frequency-dependent 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).

Fig. 3

Statistical analysis of $\rho _k$ and $\Phi _k$ using the Second configuration

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 (zero-mean 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 factor⁵$\alpha = 0.1$, 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 zero-mean Gaussian random process with a variance $\sigma _{\Phi _k}^2$

$$\begin{aligned} \Phi _k \backsim {\mathcal {N}}(0,\sigma _{\Phi _k}^2), \forall k, \forall (N_s,N_f). \end{aligned}$$

(10)

Fig. 4

CDF of $\Phi _k$

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

$$\begin{aligned} \begin{aligned} \underline{\varvec{m}}_{{\mathfrak {R}}{\mathfrak {e}}\{\gamma \}}&=\left[ m_{{\mathfrak {R}}{\mathfrak {e}}\{\gamma _0\}},\cdots ,m_{{{\mathfrak {R}}{\mathfrak {e}}\{\gamma _{N_s-1}\}}}\right] ^T, \quad \underline{\varvec{\sigma }}_{{\mathfrak {R}}{\mathfrak {e}}\{\gamma \}}^2 =\left[ \sigma _{{\mathfrak {R}}{\mathfrak {e}}\{\gamma _0\}}^2,\cdots ,\sigma _{{{\mathfrak {R}}{\mathfrak {e}}\{\gamma _{N_s-1}\}}}^2 \right] ^T \\ \underline{\varvec{m}}_{{\mathfrak {I}}{\mathfrak {m}}\{\gamma \}}&=\left[ m_{{\mathfrak {I}}{\mathfrak {m}}\{\gamma _0\}},\cdots ,m_{{{\mathfrak {I}}{\mathfrak {m}}\{\gamma _{N_s-1}\}}}\right] ^T, \quad \underline{\varvec{\sigma }}_{{\mathfrak {I}}{\mathfrak {m}}\{\gamma \}}^2 =\left[ \sigma _{{\mathfrak {I}}{\mathfrak {m}}\{\gamma _0\}}^2,\cdots ,\sigma _{{{\mathfrak {I}}{\mathfrak {m}}\{\gamma _{N_s-1}\}}}^2 \right] ^T \cdot \end{aligned} \end{aligned}$$

(11)

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 zero-mean random variable. Additionally:

• for $\zeta = \frac{N_s}{N_f} \le \frac{240}{512}$, one can notice a frequency-independent 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 frequency-dependent. The subcarriers experience a different random additive noise. This behavior is characteristic of DFT-s-OFDM and was not observed in classic OFDM systems.

As for the SPE (configuration 1), one can observe the subcarrier-dependent behavior of ICI increases with $\zeta$.

Fig. 5

Statistical analysis of ${\mathfrak {R}}{\mathfrak {e}}\{\gamma _k\}$ using the First configuration

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 zero-mean random variable with frequency-dependent 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 zero-mean 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 (zero-mean 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 Kolmogorov-Smirnov 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 zero-mean 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 zero-mean random variable with a correlated distribution. All these observations cannot be easily analytically studied. The expression (4) can be simplified as follows

$$\begin{aligned} r_k \backsimeq \; s_ke^{j\Phi _k} \; + \; \gamma _k + \;\eta _k \cdot \end{aligned}$$

(12)

The baseband model (12) is interesting because it reflects the compromise DFT-s-OFDM operates between SC and CP-OFDM. 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 CP-OFDM, 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 DFT-s-OFDM 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 DFT-s-OFDM baseband model (12). For that, we can rewrite the expression (12) in a vector form as follows

$$\begin{aligned} \underline{\textbf{r}} \simeq \; \underline{\textbf{s}} \odot e^{j\underline{\varvec{\Phi }}} \; + \; \underline{\textbf{w}}, \end{aligned}$$

(13)

with $\underline{\textbf{r}} = [r_0, \cdots , r_{N_s-1}]^T, \; \underline{\textbf{s}} = [s_0, \cdots , s_{N_s-1}]^T \; \text {and} \; \underline{\varvec{\Phi }} = [\Phi _0, \cdots , \Phi _{N_s-1}]^T$. The term $\underline{\textbf{w}} = \underline{\varvec{\eta }} \; + \; \underline{\varvec{\gamma }}$ such that $\underline{\varvec{\eta }} = [\eta _0, \cdots , \eta _{N_s-1}]^T$ and $\underline{\varvec{\gamma }} = [\gamma _0, \cdots , \gamma _{N_s-1}]^T$, denotes the additive interference-plus-noise after DFT-s-OFDM 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 DFT-s-OFDM 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 DFT-s-OFDM systems. To do so, we need to explicit the probability distribution of the random rotation $\underline{\varvec{\Phi }}$ and the additive interference-plus-noise 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

$$\begin{aligned} \begin{aligned} \underline{\varvec{\Phi }} \backsim {\mathcal {N}}(\underline{\textbf{m}}_\Phi ,\varvec{\Gamma }_\Phi ) = \; \left\{ \begin{array}{ll} \underline{\textbf{m}}_\Phi = \underline{\textbf{0}}_{N_s} \\ \\ \varvec{\Gamma }_\Phi = \sigma _{\Phi _k}^2\textbf{I}_{N_s}, \end{array} \right. \end{aligned} \end{aligned}$$

(14)

where $\sigma _{\Phi _k}^2$,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

$$\begin{aligned} \begin{aligned} \underline{\varvec{\eta }} \backsim \mathbb {C}{\mathcal {N}}(\underline{\textbf{m}}_\eta ,\varvec{\Gamma }_\eta ) = \; \left\{ \begin{array}{ll} \underline{\textbf{m}}_\eta = \underline{\textbf{0}}_{N_s} \\ \\ \varvec{\Gamma }_\eta = \sigma _{n}^2\textbf{I}_{N_s}, \end{array} \right. \end{aligned} \end{aligned}$$

(15)

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

$$\begin{aligned} \underline{\varvec{\gamma }} \backsim {{\mathbb {C}}}{\mathcal{N}}(\underline{\textbf{m}}_\gamma ,\varvec{\Gamma }_\gamma ) = \; \left\{ \begin{array}{ll} \underline{\textbf{m}}_\gamma = \underline{\textbf{0}}_{N_s} \\ \\ \varvec{\Gamma }_\gamma = \varvec{\Gamma }_\gamma , \end{array} \right. \end{aligned}$$

(16)

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

$$\begin{aligned} \begin{aligned} \underline{\textbf{w}} \backsim \mathbb {C}{\mathcal {N}}(\underline{\textbf{m}}_w,\varvec{\Gamma }_w) = \; \left\{ \begin{array}{ll} \underline{\textbf{m}}_w = \underline{\textbf{0}}_{N_s} \\ \\ \varvec{\Gamma }_w = \sigma _{n}^2\textbf{I}_{N_s} \; + \; \varvec{\Gamma }_\gamma \cdot \end{array} \right. \end{aligned} \end{aligned}$$

(17)

3.1.2 Optimal detection criterion

The Symbol Error Probability is minimized by the Maximum Likelihood (ML) decision criterion for equi-probable and independent symbols. The channel likelihood function can be written as follows [29, 30]

$$\begin{aligned} \begin{aligned} p(\underline{\textbf{r}}|\underline{\textbf{s}}) = p(\underline{\textbf{r}}_\rho ,\underline{\textbf{r}}_\theta \;|\; \underline{\textbf{s}}_\rho ,\underline{\textbf{s}}_\theta )\cdot \end{aligned} \end{aligned}$$

(18)

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 $\underline{\textbf{w}}$.⁷ The amplitude of the received DFT-s-OFDM symbol is given by

$$\begin{aligned} \begin{aligned} \underline{\textbf{r}}_\rho&= \left|\; (\underline{\textbf{s}}_\rho + \underline{\textbf{w}}') \odot e^{j\left( \underline{\varvec{\Phi }} + \underline{\textbf{s}}_\theta \right) } \; \right|\\&= \sqrt{(\underline{\textbf{s}}_\rho + {\mathfrak {R}}{\mathfrak {e}}\{\underline{\textbf{w}}'\})^2 + {\mathfrak {I}}{\mathfrak {m}}\{\underline{\textbf{w}}'\}^2}, \end{aligned} \end{aligned}$$

(19)

and the phase by

$$\begin{aligned} \begin{aligned} \underline{\textbf{r}}_\theta&= \text {arg}\left( \; (\underline{\textbf{s}}_\rho + \underline{\textbf{w}}') \odot e^{j\left( \underline{\varvec{\Phi }} + \underline{\textbf{s}}_\theta \right) } \; \right) \\&= \underline{\textbf{s}}_\theta + \underline{\varvec{\Phi }} + \text {arctan}\left( {\mathfrak {I}}{\mathfrak {m}}\{\underline{\textbf{w}}'\} \odot (\underline{\textbf{s}}_\rho + {\mathfrak {R}}{\mathfrak {e}}\{\underline{\textbf{w}}'\})^{-1} \right) \cdot \end{aligned} \end{aligned}$$

(20)

According to the demonstration of the high SINR approximation given in “Appendix 1”, the expressions (19) and (20) become

$$\begin{aligned} \left\{ \begin{array}{ll} \underline{\textbf{r}}_\rho - \underline{\textbf{s}}_\rho \backsimeq {\mathfrak {R}}{\mathfrak {e}}\{\underline{\textbf{w}}'\} \\ \\ \underline{\textbf{r}}_\theta - \underline{\textbf{s}}_\theta \backsimeq \underline{\varvec{\Phi }} + {\mathfrak {I}}{\mathfrak {m}}\{\underline{\textbf{w}}'\} \odot \underline{\textbf{s}}^{-1}_\rho \cdot \\ \end{array} \right. \end{aligned}$$

(21)

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

$$\begin{aligned} \begin{aligned} p(\underline{\textbf{r}}_\rho \; |\; \underline{\textbf{s}}_\rho ) = \frac{1}{\sqrt{(2\pi )^{N_s}|\text {det}\left( {\mathfrak {R}}{\mathfrak {e}}\lbrace \varvec{\Gamma }_w\rbrace \right) }|}\cdot e^{-\frac{1}{2}z}, \end{aligned} \end{aligned}$$

(22)

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

$$\begin{aligned} \begin{aligned} \underline{\varvec{\Phi }} + {\mathfrak {I}}{\mathfrak {m}}\{\underline{\textbf{w}}'\} \odot \underline{\textbf{s}}^{-1}_\rho \backsim {\mathcal {N}}(\underline{\textbf{m}},\varvec{\Gamma }) = \; \left\{ \begin{array}{ll} \underline{\textbf{m}} = \underline{\textbf{0}}_{N_s} \\ \\ \varvec{\Gamma } = \varvec{\Gamma }_\Phi + {\mathfrak {I}}{\mathfrak {m}}\lbrace \varvec{\Gamma }_w\rbrace \odot \underline{\textbf{s}}^{-2}_\rho . \end{array} \right. \end{aligned} \end{aligned}$$

(23)

Therefore, we can express the PDF of the second expression in (21) by

$$\begin{aligned} \begin{aligned} p(\underline{\textbf{r}}_\theta \;|\; \underline{\textbf{s}}_\theta ) = \frac{1}{\sqrt{(2\pi )^{N_s}|\text {det}(\varvec{\Gamma })|}}\cdot e^{-\frac{1}{2}l}, \end{aligned} \end{aligned}$$

(24)

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

$$\begin{aligned} \begin{aligned} p(\underline{\textbf{r}} \;|\; \underline{\textbf{s}}) = \frac{e^{\left( -\frac{1}{2}\left( z+l\right) \right) }}{(2\pi )^{N_s}\sqrt{|\text {det}(\varvec{\Gamma }_w)|\cdot |\text {det}(\varvec{\Gamma })|}}, \end{aligned} \end{aligned}$$

(25)

where z and l are expressed above. The correlation matrices $\varvec{\Gamma }_w$ and $\varvec{\Gamma }$ can be considered as a-priori 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

$$\begin{aligned} \begin{aligned} \underline{\hat{\textbf{s}}}&= \underset{\mathbf {\underline{s}} \in {\mathcal {C}}^{'}}{\text {argmax}} \quad p\left( \underline{\textbf{r}} \;|\; \underline{\textbf{s}} \right) \\&= \underset{\mathbf {\underline{s}} \in {\mathcal {C}}^{'}}{\text {argmax}} \quad \text {ln} \left( p\left( \underline{\textbf{r}} \;|\; \underline{\textbf{s}} \right) \right) \\&= \underset{\mathbf {\underline{s}} \in {\mathcal {C}}^{'}}{\text {argmin}}\; (\underline{\textbf{r}}_\rho - \underline{\textbf{s}}_\rho )^T {{\mathfrak {R}}{\mathfrak {e}}\lbrace \varvec{\Gamma }_w\rbrace }^{-1}(\underline{\textbf{r}}_\rho - \underline{\textbf{s}}_\rho ) + (\underline{\textbf{r}}_\theta - \underline{\textbf{s}}_\theta )^T \varvec{\Gamma }^{-1}(\underline{\textbf{r}}_\theta - \underline{\textbf{s}}_\theta ) + \log \left( |\text {det}(\varvec{\Gamma })|\right) , \end{aligned} \end{aligned}$$

(26)

where the estimated vector $\underline{\mathbf {\hat{\textbf{s}}}}$ is supposed to be the transmitted symbols sequence. It means that for DFT-s-OFDM 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 16-QAM, 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 sub-optimal decision rule assuming that the ICI follows an uncorrelated Gaussian random process.

3.1.3 Sub-optimal detection criterion

For the sub-optimal 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

$$\begin{aligned} \gamma _k \backsim \mathbb {C}{\mathcal {N}}(m_{\gamma _k},\sigma _{\gamma _k}^2) = \; \left\{ \begin{array}{ll} m_{\gamma _k} = 0 \\ \\ \sigma _{\gamma _k}^2 = \textbf{diag}_k\lbrace {\varvec{\Gamma }_{\varvec{\gamma }}}\rbrace , \end{array} \right. \end{aligned}$$

(27)

where $m_{\gamma _k}$ and $\sigma _{\gamma _k}^2$ are, respectively, the mean and the variance of $\varvec{\gamma }_k$. The expression (12) becomes

$$\begin{aligned} \begin{aligned} r_k \simeq&\; s_k\cdot e^{j\Phi _k} \; + \; w_k, \end{aligned} \end{aligned}$$

(28)

where $\Phi _k\backsim {\mathcal {N}}(0,\sigma _{\Phi _k}^2)$ and $w_k = \eta _k + \gamma _k$ is defined as follows

$$\begin{aligned} \begin{aligned} w_k \backsim \mathbb {C}{\mathcal {N}}(m_{w_k},\sigma _{w_k}^2) = \; \left\{ \begin{array}{ll} m_{w_k} = m_{\eta _k} + m_{\gamma _k} \; \backsimeq 0 \\ \\ \sigma _{w_k}^2 = \sigma _{n}^2 + \sigma _{\gamma _k}^2 \cdot \end{array} \right. \end{aligned} \end{aligned}$$

(29)

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

$$\begin{aligned} \begin{aligned} r_{k_\rho }&= \left|\;(s_{k_\rho } + w_k')\cdot e^{j(\Phi _k + s_{k_\theta })} \; \right|\\&\backsimeq s_{k_\rho } + {\mathfrak {R}}{\mathfrak {e}}\{w_k'\} \end{aligned} \end{aligned}$$

(30)

$$\begin{aligned} \begin{aligned} r_{k_\theta }&= \text {arg}\left( \; (s_{k_\rho } + w_k')\cdot e^{j(\Phi _k + s_{k_\rho })} \; \right) \\&\backsimeq s_{k_\theta } + \Phi _k + \frac{{\mathfrak {I}}{\mathfrak {m}}\{w_k'\}}{s_{k_\rho }}, \end{aligned} \end{aligned}$$

(31)

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

$$\begin{aligned} \left\{ \begin{array}{ll} r_{k_\rho } - s_{k_\rho } \backsim {\mathcal {N}}(0,\sigma _{w_k}^2/2) \\ \\ r_{k_\theta } - s_{k_\theta } \backsim {\mathcal {N}}(0, \sigma _{\Phi _k}^2 + \frac{\sigma _{w_k}^2}{2s^2_{k_\rho }}).\\ \end{array} \right. \end{aligned}$$

(32)

The joint PDF of expressions in (32) is expressed as follows

$$\begin{aligned} \begin{aligned} p(r_k \;|\; s_k) = \frac{\text {exp} \left( -\frac{1}{2}\left( \frac{(r_{k_\rho } - s_{k_\rho })^2}{\sigma _{w_k}^2/2} + \frac{(r_{k_\theta } - s_{k_\theta })^2}{\sigma _{\Phi _k}^2 + \sigma _{w_k}^2/ 2s_{k_\rho }^2}\right) \right) }{2\pi \sqrt{\frac{\sigma _{w_k}^2}{2}(\sigma _{\Phi _k}^2 + \sigma _{w_k}^2/2 s_{k_\rho }^2)}} \cdot \end{aligned} \end{aligned}$$

(33)

Hence, the ML decision rule using the channel likelihood expression (33) can be expressed as

$$\begin{aligned} \begin{aligned} {\hat{s}}_k&= \underset{s_i \in {\mathcal {C}}}{\text {argmax}}\; p(r \;|\; s_i) \\&= \underset{s_i \in {\mathcal {C}}}{\text {argmax}}\; \text {ln} \left( p(r \;|\; s_i)\right) \\&= \underset{s_i \in {\mathcal {C}}}{\text {argmin}}\; \frac{(r_{k_\rho } - s_{i_\rho } )^2}{\sigma _{w_k}^2/2} + \frac{(r_{k_\theta } - s_{i_\theta })^2}{\sigma _{\Phi _k}^2 + \frac{\sigma _{w_k}^2}{2 s_{i_\rho }^2}} \;+ \;\log \left( \sigma _{\Phi _k}^2 + \frac{\sigma _{w_k}^2}{2 s_{i_\rho }^2}\right) \cdot \end{aligned} \end{aligned}$$

(34)

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 SUB-D from now.

Fig. 6

Statistical analysis of ${\mathfrak {R}}{\mathfrak {e}}\{\gamma _k\}$ using the Second configuration

3.1.4 Estimation of the required a-priori 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 $e^{j\underline{\varvec{\phi }}}$

$$\begin{aligned} e^{j\underline{\varvec{\Phi }}} = \textbf{B}e^{j\underline{\varvec{\phi }}} \quad \text {with} \quad \textbf{B}^{(i,k)} = \textbf{A}^{(i,k)}_{RX}\textbf{A}^{(k,i)}_{TX}, \end{aligned}$$

(35)

for $i\in \llbracket 0,\cdots ,N_s-1 \rrbracket$ and $k\in \llbracket 0,\cdots ,N_f-1 \rrbracket$. The matrices $\textbf{A}_{TX}$ and $\textbf{A}_{RX}$ are defined as follows

$$\begin{aligned} \textbf{A}_{TX}=\textbf{F}^H_{N_f}\textbf{M}_{TX}\textbf{F}_{N_s} \quad \text {and} \quad \textbf{A}_{RX}=\textbf{F}^H_{N_s}\textbf{D}_{RX}\textbf{F}_{N_f}, \end{aligned}$$

(36)

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

$$\begin{aligned} \begin{aligned} \underline{\textbf{1}}_{N_s} + j\underline{\varvec{\Phi }} = \textbf{B}\left( \underline{\textbf{1}}_{N_f} + j\underline{\varvec{\phi }}\right) \Longleftrightarrow \underline{\varvec{\Phi }} = \textbf{B}\underline{\varvec{\phi }}, \end{aligned} \end{aligned}$$

(37)

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

$$\begin{aligned} \textbf{R}_{\underline{\varvec{\Phi }}} = \textbf{B}\textbf{R}_{\underline{\varvec{\phi }}}\textbf{B}^H \; \cdot \end{aligned}$$

(38)

Consequently, we compute the expression (14) upon (38) to obtain the different $\sigma _{\Phi _k}^2$ for each DFT subcarrier k as follows

$$\begin{aligned} \sigma _{\Phi _k}^2 = \textbf{diag}_k\lbrace \textbf{B}\textbf{R}_{\underline{\varvec{\phi }}}\textbf{B}^H\rbrace \cdot \end{aligned}$$

(39)

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 intermediate-frequency 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 $k\in \llbracket 0,\cdots ,N_s-1 \rrbracket$ by the following relation

$$\begin{aligned} \sigma _{\gamma _k}^2 \simeq \sigma _{\phi }^2 - \sigma _{\Phi _k}^2\cdot \end{aligned}$$

(40)

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 frequency-dependent, it proves that for a DFT-s-OFDM waveform, some subcarriers are dominated by the rotation effects while other by the presence of ICI.

Fig. 7

CDF of ${\mathfrak {R}}{\mathfrak {e}}\{\gamma _k\}$

3.2 Performance evaluation

In this section, we evaluate the performance of following systems under GPN channels:

• OFDM with Euclidean detector (EUC-D)

• DFT-s-OFDM with Euclidean detector (EUC-D)

• DFT-s-OFDM with the proposed sub-optimal polar detector (SUB-D)

The Euclidean detector EUC-D correspond to the ML detection defined by

$$\begin{aligned} {\hat{s}}_k = \underset{s_i \in {\mathcal {C}}}{\text {argmin}}\; \parallel r_k - s_i\parallel _2^2, \end{aligned}$$

(41)

where symbols $s_i \in {\mathcal {C}}$, with ${\mathcal {C}}$ the set of the selected modulation scheme. The sub-optimal polar detector (SUB-D) applies the decision rule expressed in (34). The latter detection criterion is only evaluated with DFT-s-OFDM signals as it is directly derived from DFT-s-OFDM baseband model. The performance of DFT-s-OFDM with SUB-D will be assessed first in Genie-aided 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 Signal-to-Noise Ratio (SNR) while considering a 16-QAM modulation and a low density parity check (LDPC) following the 5G-NR specifications [44] with a code rate (CR) equal to 0.7.

3.2.1 Impact of DFT parameters

As we mentioned in this paper, DFT-s-OFDM 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 DFT-s-OFDM systems with configurations defined in Table 3 (i.e., by varying $\zeta$) and in Table 4 (i.e., with constant $\zeta$) for the conventional EUC-D 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 DFT-s-OFDM with the EUC-D decision rule does not work under strong GPN regime.

Fig. 8

DFT-s-OFDM with EUC-D, 16-QAM, LDPC CR$=0.7$ for configurations defined in Table 3

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 SUB-D with the Euclidean detector EUC-D. We compare both DFT-s-OFDM and OFDM assuming the same bandwidth, i.e., by considering $N_s$ active subcarriers among the $N_f$ for the OFDM modulation.

Fig. 9

DFT-s-OFDM with EUC-D, 16-QAM, LDPC CR$=0.7$ for configurations defined in Table 4

Figure 10 shows the system performance for the different GPN regimes and genie-aided SUB-D detector. In strong GPN regime, one can first observe that the two EUC-D receivers do not perform well as they cannot achieve a TBLER $<10^{-1}$ with SNR $=30$ dB. Fortunately, the proposed DFT-s-OFDM with SUB-D 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 EUC-D but with a small SNR gain compared to the strong GPN case. For low GPN, the EUC-D detector has almost the same performance as the SUB-D, but it is slightly better than the SUB-D.

We have demonstrated the gain of the SUB-D detector over the EUC-D detector in a strong GPN regime and genie-aided conditions (perfect knowledge of SPE and ICI stochastic properties). Indeed, the polar detection criterion makes sense when phase distortion is dominant. As a consequence, DFT-s-OFDM with the proposed SUB-D 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.

Fig. 10

Comparison of achieved performance for the three GPN regimes, 16-QAM, LDPC with CR$=$ $0.7$ and $N_s=480$, $N_f=512$ ($\zeta =0.9375$ for DFT-s-OFDM)

Figure 11 presents and compares the performance achieved by the genie-aided and the realistic DFT-s-OFDM/SUB-D receiver, i.e., by considering the expressions (39) and (40). One can notice the estimated SUB-D yields the same results as the genie-aided one. These results validate the proposed estimation approach.

Fig. 11

Comparison of achieved performance with genie-aided and realistic DFT-s-OFDM SUB-D system for the three GPN regimes, 16-QAM, LDPC with CR$=$ $0.7$ and $\zeta =0.9375$ ($N_s=480$, $N_f=512$)

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 (post-FFT 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_ICI-1 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 zero-mean 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 DFT-s-OFDM 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 DFT-s-OFDM, it leads to both a random rotation and generation of ICI. The power of the resulting ICI in DFT-s-OFDM 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 DFT-s-OFDM is more suited than OFDM for transmissions under strong GPN channels.

Fig. 12

OFDM performance: raw BER in function of the SNR, 16-QAM, FFT size of 512 with 480 active subcarriers and 3.686-GHz bandwidth

5 Conclusion

In this paper, we investigated the effects of GPN on OFDM and DFT-s-OFDM waveforms. In DFT-s-OFDM systems, the presence of GPN results in two distortions i) a frequency-dependent 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 DFT-s-OFDM, 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 sub-optimal detector and compared its achievable performance with the conventional Euclidean-distance-based 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) DFT-s-OFDM with the proposed polar detector significantly outperforms its OFDM contender when strong GPN regimes are involved. That is the reason why we believe that DFT-s-OFDM is a promising enabler of sub-THz 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 GPN-induced impairments must be considered which is one interesting study perspective of the presented work.

Appendices

Appendix 1: High SINR approximation

The Symbol Error Probability is minimized by the Maximum Likelihood (ML) decision criterion for equi-probable and independent symbols. The channel likelihood function can be written as follows [29, 30]:

$$\begin{aligned} \begin{aligned} p(\underline{\textbf{r}}|\underline{\textbf{s}}) = p(\underline{\textbf{r}}_\rho ,\underline{\textbf{r}}_\theta \;|\; \underline{\textbf{s}}_\rho ,\underline{\textbf{s}}_\theta )\cdot \end{aligned} \end{aligned}$$

(42)

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 post-FFT noise $\underline{\textbf{w}}'$ has the same stochastic properties as the additive channel noise $\underline{\textbf{w}}$ (i.e., zero-mean circular symmetric complex Gaussian noise). The amplitude and phase of the received DFT-s-OFDM symbol is, respectively, given by (43) and (44).

$$\begin{aligned} \underline{\textbf{r}}_\rho [k]&= \sqrt{(\underline{\textbf{s}}_\rho [k] + {\mathfrak {R}}{\mathfrak {e}}\{\underline{\textbf{w}}'[k]\})^2 + {\mathfrak {I}}{\mathfrak {m}}\{\underline{\textbf{w}}'[k]\}^2}\;, \nonumber \\&= (\underline{\textbf{s}}_\rho [k] + {\mathfrak {R}}{\mathfrak {e}}\{\underline{\textbf{w}}'[k]\}) \sqrt{1+ \left( \frac{{\mathfrak {I}}{\mathfrak {m}}\{\underline{\textbf{w}}'[k]\}}{(\underline{\textbf{s}}_\rho [k] + {\mathfrak {R}}{\mathfrak {e}}\{\underline{\textbf{w}}'[k]\}}\right) ^2} \end{aligned}$$

(43)

$$\begin{aligned} \underline{\textbf{r}}_\theta [k]&= \text {arg}\left( \; (\underline{\textbf{s}}_\rho [k] + \underline{\textbf{w}}'[k]) \odot e^{j\left( \underline{\varvec{\Phi }}[k] + \underline{\textbf{s}}_\theta [k] \right) } \; \right) \nonumber \\&= \underline{\textbf{s}}_\theta [k] + \underline{\varvec{\Phi }}[k] + \text {arctan}\left( \frac{{\mathfrak {I}}{\mathfrak {m}}\{\underline{\textbf{w}}'[k]\}}{\underline{\textbf{s}}_\rho [k] + {\mathfrak {R}}{\mathfrak {e}}\{\underline{\textbf{w}}'[k]\}} \right) \end{aligned}$$

(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

$$\begin{aligned} \lim _{\rho _w[k]/\underline{\textbf{s}}_\rho [k] \rightarrow 0} \; \frac{{\mathfrak {I}}{\mathfrak {m}}\{\underline{\textbf{w}}'[k]\}}{\underline{\textbf{s}}_\rho [k] + {\mathfrak {R}}{\mathfrak {e}}\{\underline{\textbf{w}}'[k]\}}&= \lim _{\rho _w[k]/\underline{\textbf{s}}_\rho [k] \rightarrow 0} \; \frac{\rho _w[k] \sin (\Phi _w[k])}{\underline{\textbf{s}}_\rho [k] + \rho _w[k] \cos (\Phi _w[k])} \nonumber \\&=0 \end{aligned}$$

(45)

and

$$\begin{aligned} \lim _{\rho _w[k]/\underline{\textbf{s}}_\rho [k] \rightarrow 0} \; \left( {\underline{\textbf{s}}_\rho [k] + {\mathfrak {R}}{\mathfrak {e}}\{\underline{\textbf{w}}'[k]\}} \right)&= \lim _{\rho _w[k]/\underline{\textbf{s}}_\rho [k] \rightarrow 0} \; \left( {\underline{\textbf{s}}_\rho [k] + \rho _w[k] \cos (\Phi _w[k])} \right) \nonumber \\&=\underline{\textbf{s}}_\rho [k] \; . \end{aligned}$$

(46)

It eventually leads to:

$$\begin{aligned} \underline{\textbf{r}}_\rho [k]&= \sqrt{(\underline{\textbf{s}}_\rho [k] + {\mathfrak {R}}{\mathfrak {e}}\{\underline{\textbf{w}}'[k]\})^2 + {\mathfrak {I}}{\mathfrak {m}}\{\underline{\textbf{w}}'[k]\}^2} \nonumber \\&\approx \underline{\textbf{s}}_\rho [k] + {\mathfrak {R}}{\mathfrak {e}}\{\underline{\textbf{w}}'[k]\} \end{aligned}$$

(47)

and

$$\begin{aligned} \underline{\textbf{r}}_\theta [k]&= \underline{\textbf{s}}_\theta [k] + \underline{\varvec{\Phi }}[k] + \text {arctan}\left( \frac{{\mathfrak {I}}{\mathfrak {m}}\{\underline{\textbf{w}}'[k]\}}{\underline{\textbf{s}}_\rho [k] + {\mathfrak {R}}{\mathfrak {e}}\{\underline{\textbf{w}}'[k]\}} \right) \nonumber \\&\approx \underline{\textbf{s}}_\theta [k] + \underline{\varvec{\Phi }}[k] + \frac{{\mathfrak {I}}{\mathfrak {m}}\{\underline{\textbf{w}}'[k]\}}{\underline{\textbf{s}}_\rho [k] + {\mathfrak {R}}{\mathfrak {e}}\{\underline{\textbf{w}}'[k]\}} \end{aligned}$$

(48)

$$\begin{aligned}&\approx \underline{\textbf{s}}_\theta [k] + \underline{\varvec{\Phi }}[k] + \frac{{\mathfrak {I}}{\mathfrak {m}}\{\underline{\textbf{w}}'[k]\}}{\underline{\textbf{s}}_\rho [k] } \cdot \end{aligned}$$

(49)

Thus, we obtain

$$\begin{aligned} \left\{ \begin{array}{ll} \underline{\textbf{r}}_\rho - \underline{\textbf{s}}_\rho \backsimeq {\mathfrak {R}}{\mathfrak {e}}\{\underline{\textbf{w}}'\} \\ \\ \underline{\textbf{r}}_\theta - \underline{\textbf{s}}_\theta \backsimeq \underline{\varvec{\Phi }} + {\mathfrak {I}}{\mathfrak {m}}\{\underline{\textbf{w}}'\} \odot \underline{\textbf{s}}^{-1}_\rho \cdot \\ \end{array} \right. \end{aligned}$$

(50)

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

$$\begin{aligned} \mathbf {\underline{r}} = \underbrace{\textbf{F}^H_{N_s}\textbf{D}_{RX}\textbf{F}_{N_f}}_{\textbf{A}_{RX}}\textbf{Q}\underbrace{\textbf{F}^H_{N_f}\textbf{M}_{TX}\textbf{F}_{N_s}}_{\textbf{A}_{TX}}\mathbf {\underline{s}} = \textbf{C}\mathbf {\underline{s}}, \end{aligned}$$

(51)

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

$$\begin{aligned} \begin{aligned} \mathbf {\underline{r}}&= \textbf{G}\mathbf {\underline{s}} + \textbf{K}\mathbf {\underline{s}} \\&= \mathbf {\underline{s}} \odot \varvec{\underline{\beta }} + \varvec{\underline{\gamma }} \end{aligned} \end{aligned}$$

(52)

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

$$\begin{aligned} \textbf{diag}\lbrace \textbf{G} \rbrace = \textbf{diag}\lbrace \textbf{C} \rbrace = \varvec{\underline{\beta }}, \end{aligned}$$

(53)

and $\textbf{K} = \textbf{C} - \textbf{G}$, where $\textbf{K}$ is the ICI matrix. The expression of the ICI correlation matrix is given by

$$\begin{aligned} \begin{aligned} \textbf{R}_{\textbf{K}} = \mathbb {E}\left[ \textbf{K} \textbf{K}^H \right]&= \mathbb {E}\left[ \left( \textbf{C} - \textbf{G}\right) \left( \textbf{C} - \textbf{G}\right) ^H \right] \\&=\mathbb {E}\left[ \textbf{C}\textbf{C}^H - \textbf{C}\textbf{G}^H - \textbf{G}\textbf{C}^H + \textbf{G}\textbf{G}^H\right] \\&=\mathbb {E}\left[ \textbf{C}\textbf{C}^H - \left( \textbf{C}\textbf{G}^H + \textbf{G}\textbf{C}^H\right) + \textbf{G}\textbf{G}^H\right] \cdot \end{aligned} \end{aligned}$$

(54)

The second term of (54) in bracket can be simplified as follows

$$\begin{aligned} \begin{aligned} \textbf{C}\textbf{G}^H + \textbf{G}\textbf{C}^H&= ( \textbf{GG}^H + \textbf{KG}^H) + ( \textbf{GG}^H + \textbf{GK}^H) \\&= 2\textbf{G}\textbf{G}^H +\textbf{0}_{N_s}, \end{aligned} \end{aligned}$$

(55)

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

$$\begin{aligned} \begin{aligned} \textbf{R}_{\textbf{K}}&= \mathbb {E}\left[ \textbf{C}\textbf{C}^H - \textbf{G}\textbf{G}^H\right] \\&= \mathbb {E}\left[ \textbf{C}\textbf{C}^H\right] - \mathbb {E}\left[ \textbf{G}\textbf{G}^H\right] \\&= \textbf{A}_{RX}\mathbb {E}\left[ \textbf{Q}\textbf{Q}^H\right] \textbf{A}^H_{RX} - \mathbb {E}\left[ \textbf{G}\textbf{G}^H\right] \cdot \end{aligned} \end{aligned}$$

(56)

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

$$\begin{aligned} \begin{aligned} \mathbb {E}\left[ \textbf{Q}\textbf{Q}^H\right]&= \mathbb {E}\left[ \textbf{diag} \left\{ e^{j\underline{\varvec{\phi }}} \left( e^{j\underline{\varvec{\phi }}}\right) ^H \right\} \right] \; \text {because} \;\textbf{diag}\lbrace \textbf{Q} \rbrace = e^{j\underline{\varvec{\phi }}} \\&= \textbf{diag} \left\{ \mathbb {E}\left[ e^{j\underline{\varvec{\phi }}} \left( e^{j\underline{\varvec{\phi }}}\right) ^H \right] \right\} \\&= \textbf{diag} \left\{ \textbf{R}_{e^{\underline{\varvec{\phi }}}} \right\} \cdot \textbf{I}_{N_f} \cdot \end{aligned} \end{aligned}$$

(57)

The small angle approximation [41] leads to

$$\begin{aligned} \begin{aligned} \mathbb {E}\left[ \textbf{Q}\textbf{Q}^H\right]&\simeq \textbf{diag} \left\{ \textbf{1}_{N_f}+\textbf{R}_{\underline{\varvec{\phi }}} \right\} \cdot \textbf{I}_{N_f} \\&\simeq \left( 1+\sigma _{\phi }^2 \right) \cdot \textbf{I}_{N_f}, \end{aligned} \end{aligned}$$

(58)

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

$$\begin{aligned} \begin{aligned} \mathbb {E} \left[ \textbf{G}\textbf{G}^H\right]&= \mathbb {E}\left[ \textbf{diag} \left\{ \varvec{\underline{\beta }} \;\varvec{\underline{\beta }}^H \right\} \right] , \; \text {because} \;\textbf{diag}\lbrace \textbf{G} \rbrace = \varvec{\underline{\beta }} \\&= \textbf{diag} \left\{ \mathbb {E}\left[ \varvec{\underline{\beta }} \;\varvec{\underline{\beta }}^H \right] \right\} \\&\simeq \textbf{diag} \left\{ \textbf{R}_{e^{\underline{\varvec{\Phi }}}} \right\} , \; \text {because} \; \varvec{\underline{\beta }} \simeq e^{\underline{\varvec{\Phi }}} \;\text {based on results obtained in} 2.2 \\&\simeq \textbf{diag} \left\{ \textbf{1}_{N_s}+\textbf{R}_{\underline{\varvec{\Phi }}} \right\} , \; \text {based on the small angle approximation} \\&\simeq \textbf{I}_{N_s} + \textbf{diag} \left\{ \textbf{R}_{\underline{\varvec{\Phi }}} \right\} \\&\simeq \textbf{I}_{N_s} + \textbf{Z}, \end{aligned} \end{aligned}$$

(59)

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

$$\begin{aligned} \begin{aligned} \textbf{R}_{\textbf{K}}&\simeq \left( 1+\sigma _{\phi }^2 \right) \cdot \textbf{A}_{RX} \textbf{A}^H_{RX} - \left( \textbf{I}_{N_s} + \textbf{Z} \right) \\&\simeq \left( 1+\sigma _{\phi }^2 \right) \cdot \textbf{I}_{N_s} - \left( \textbf{I}_{N_s} + \textbf{Z} \right) \cdot \end{aligned} \end{aligned}$$

(60)

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

$$\begin{aligned} \begin{aligned} \textbf{R}_{\varvec{\underline{\gamma }}} = \mathbb {E}\left[ \varvec{\underline{\gamma }}\;\varvec{\underline{\gamma }}^H \right]&= \mathbb {E}\left[ \textbf{K}\mathbf {\underline{s}}\; \mathbf {\underline{s}}^H \textbf{K}^H \right] \\&= \mathbb {E}\left[ \textbf{K} \textbf{K}^H \right] \\&= \textbf{R}_{\textbf{K}} \cdot \end{aligned} \end{aligned}$$

(61)

Finally, the expression of the ICI power at a subcarrier k according to the expressions (27) and (60) is given by

$$\begin{aligned} \begin{aligned} \sigma _{\gamma _k}^2&= \textbf{diag}_k\lbrace \textbf{R}_{\textbf{K}}\rbrace \\&\simeq \textbf{diag}_k \left\{ \left( 1+\sigma _{\phi }^2 \right) \cdot \textbf{I}_{N_s} \right\} - \textbf{diag}_k \left\{ \left( \textbf{I}_{N_s} + \textbf{Z} \right) \right\} \\&\simeq 1 + \sigma _{\phi }^2 - 1 - \textbf{diag}_k \left\{ \textbf{Z} \right\} \\&\simeq \sigma _{\phi }^2 - \textbf{diag}_k \left\{ \textbf{R}_{\underline{\varvec{\Phi }}} \right\} \cdot \end{aligned} \end{aligned}$$

(62)

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

$$\begin{aligned} \begin{aligned} \sigma _{\gamma _k}^2&\simeq \sigma _{\phi }^2 - \sigma _{\Phi _k}^2, \end{aligned} \end{aligned}$$

(63)

where $\sigma _{\phi }^2$ represents the GPN power generated by oscillators in (3) and $\sigma _{\Phi _k}^2$ the SPE power at subcarrier k.