Extending Fibre Nonlinear Interference Power Modelling to Account for General Dual-Polarisation 4D Modulation Formats

Abstract

In optical communications, four-dimensional (4D) modulation formats encode information onto the quadrature components of two arbitrary orthogonal states of polarisation of the optical field. Many analytical models available in the optical communication literature allow, within a first-order perturbation framework, the computation of the average power of the nonlinear interference (NLI) accumulated in coherent fibre-optic transmission systems. However, all such models only operate under the assumption of transmitted polarisation-multiplexed two-dimensional (PM-2D) modulation formats, which only represent a limited subset of the possible dual-polarisation 4D (DP-4D) formats. Namely, only those where data transmitted on each polarisation channel are mutually independent and identically distributed. This paper presents a step-by-step mathematical derivation of the extension of existing NLI models to the class of arbitrary DP-4D modulation formats. In particular, the methodology adopted follows the one of the popular enhanced Gaussian noise model, albeit dropping most assumptions on the geometry and statistic of the transmitted 4D modulation format. The resulting expressions show that, whilst in the PM-2D case the NLI power depends only on different statistical high-order moments of each polarisation component, for a general DP-4D constellation, several other cross-polarisation correlations also need to be taken into account.

1. Introduction

With the resurgence of polarisation-diverse, optical coherent detection, transmission of information over an optical fibre is typically performed exploiting four degrees of freedom of the optical field: two quadrature components over two orthogonal states of polarisation. The standard approach consists in encoding data independently over the two polarisation channels using the same two-dimensional (2D) modulation format. The resulting four-dimensional (4D) constellation is often referred to as a polarisation-multiplexed 2D (PM-2D) modulation format. The strong point of PM-2D formats is their simplicity of generation and performance analysis: as the two polarisation channels are independent and under the assumption of data-independent cross-polarisation interference in the fibre channel, transmission performance can be evaluated using the 2D component format.

Despite the popularity of PM-2D formats, a substantial amount of research work in the literature has been devoted to more general 4D formats, i.e., 4D constellations which are not necessarily generated as Cartesian products of a component 2D constellation [1,2]. These formats have recently regained attention due to their potential power efficiency, nonlinearity tolerance, and ultimately their still unexplored shaping gains. The reason for that relies on the fact that, by exploiting the full 4D space, sensitivity and other relevant performance metrics such as mutual information or generalised mutual information can be improved compared to traditional PM-2D formats [3,4,5,6,7].

Previous works on optimised 4D modulation formats have either operated under an additive white Gaussian noise channel hypothesis [1,2,3] or exploited some heuristic approaches to derive nonlinearly tolerant formats in the fibre-optic channel [5,6,7]. However, accurately predicting the amount of nonlinear interference generated by transmission of a given constellation in an optical fibre is key to optimising its shape in N dimensions.

Modelling of nonlinear interference (NLI) in optical fibre transmission is quite a mature field of research where an impressive amount of progress was made in the first half of the 2010s, e.g., in [8,9,10,11]. In particular, [10,11] introduced for the first time the possibility of predicting the dependency of nonlinear interference power as a function of the modulation format features, i.e., geometrical shape and statistical properties. Among other assumptions, one underlying key point of all previous models is the transmission of PM-2D modulation formats, where data on the two polarisation channels are assumed to be independent and identically distributed. Under this constraint, one can predict the NLI power using the statistical properties of the 2D component modulation format. It is clear, however, that this approach ceases to be applicable to general DP-4D formats, where a single 2D component format might not even exist.

In this work, we extend the existing analytical expressions for NLI power to account for DP-4D constellations where the two 2D polarisation components are not identically distributed or when there is statistical dependency between them. The undertaken approach is the same as in [9], i.e., a frequency-domain, first-order perturbation study. Unlike [9], no assumptions are made on either the marginal or joint statistics of the two polarisation components of the transmitted 4D constellation (besides being zero-mean). The final expressions reveal the impact of several different (nontrivial) cross-polarisation statistics on the NLI power.

The formulas presented in this work enable an accurate computation of the NLI power for all possible dual-polarisation formats in optical fibre transmission. As a result, a reliable optimisation of both geometry and symbol probability of occurrence of such 4D formats is also enabled for the optical fibre channel.

2. Organisation of the Manuscript and Notation

The manuscript is organised as follows: (i) in Section 3, the investigated system model is described and the model assumptions are presented; (ii) Section 4, Section 5, Section 6, Section 7 and Section 8 are devoted to a step-by-step analytical derivation of the model; and (iii) ultimately, the main model expression is presented in Section 8. In particular: in Section 4, the regular perturbation (RP) solution to the frequency domain Manakov equation is derived for a multi-span fibre system and its power spectral density (PSD) is evaluated in the case of a transmitted periodic signal; in Section 5, the contributions of the different high-order moments and cross-polarisation correlations of the transmitted 4D modulation format are highlighted; finally, Section 8 derives, via Theorem 2, an expression for the PSD as the signal period is extended into infinity. A flowchart of the main derivation steps performed in this work, with their corresponding references in the manuscript, is shown in Figure 1.

Throughout this manuscript, we denote 2D (column) vectors with boldface letters (e.g., $\mathit{a}$), whereas 2D column vector functions are indicated with boldface capital letters (e.g., $\mathit{E}(f,z),\tilde{\mathit{E}}(t,z)$, etc.). For indicating the optical field, the first variable of represents either the time or frequency variable whereas the second one represents the fibre propagation section. An exception is made for the multi-span system case, where second and third variables are assigned to the number of spans and span length, respectively. This highlights the joint dependence of the output optical field on these two variables, as shown later in the paper. $\mathcal{F}\{·\}$, $\mathbb{E}\{·\}$, and $Re\{·\}$ indicate the Fourier transform, the statistical expectation, and the real part operators, respectively. The delta distribution is indicated by $\delta (·)$, whereas ${\delta }_k$ denotes the Kronecker delta defined as

$${\delta }_k\triangleq \left\{\begin{array}{c}1\mathrm{for}k=0,\\ 0\mathrm{elsewhere}.\end{array}\right.$$

Finally, $\mathbb{Z}$, $\mathbb{R}$, and $\mathbb{C}$ denote the integer, real, and complex fields, respectively, and j is the imaginary unit.

Figure 1. Flowchart of the analytical derivation in this work.

Figure 1. Flowchart of the analytical derivation in this work.

3. Model Assumptions

3.1. System Model

The baseband equivalent model of the optical fibre system under investigation in this work is shown in Figure 2. The fibre channel is a multi-span fibre system using Erbium-doped fibre amplification (EDFA). In this manuscript, it is assumed that a single-channel signal is transmitted. The transmitter is assumed to generate for each symbol period n the 4D symbol ${\mathit{a}}_n={[a_{x,n},a_{y,n}]}^T$, where $a_{x,n},a_{y,n}\in \mathbb{C}$ are complex symbols modulated on two arbitrary orthogonal polarisation states x and y, respectively. The sequence of symbols ${\mathit{a}}_n$ for $n\in \mathbb{Z}$ is assumed to be a cyclostationary process of period W. The set of random variables (RVs) within each period of such process are also assumed to be statistically independent. Linear modulation with a single, real pulse $p\left(t\right)$ on x and y polarisation is adopted. The pulse $p\left(t\right)$ with spectrum $P\left(f\right)$ is assumed to be strictly band-limited within the range of frequencies $[-R_s/2,R_s/2]$. As discussed in Section 3.3, the transmitted signal $\tilde{\mathit{E}}(t,0)$ is assumed to be periodic with period T, such that

$$\tilde{\mathit{E}}(t,0)=\sum _{n=-(W-1)/2}^{(W-1)/2}{\mathit{a}}_{n}p(t-n{T}_s),\mathrm{for}0\le t\le T,$$

(1)

$T_s=1/R_s=T/W$ represents the symbol period, and $R_s$ is the symbol rate. A schematic representation of the transmitted signal is shown in Figure 3.

Figure 2. System model under investigation in this work which consists of an optical fibre system model and a nonlinear interference (NLI) variance estimation block: the two branches in the NLI variance estimator block indicate alternative ways of estimating the NLI variance ${\Sigma }_{\mathrm{NLI}}$.

Figure 2. System model under investigation in this work which consists of an optical fibre system model and a nonlinear interference (NLI) variance estimation block: the two branches in the NLI variance estimator block indicate alternative ways of estimating the NLI variance ${\Sigma }_{\mathrm{NLI}}$.

The signal $\tilde{\mathit{E}}(t,0)$ is transmitted over $N_s$ (homogeneous) fibre spans, each of length $L_s$ and each followed by an ideal lumped optical amplifier for which the gain exactly recovers from the span losses. Since in this work we are only concerned about the prediction of NLI arising from the signal–signal nonlinear interactions along the fibre propagation, the optical noise added by the amplifier plays no role in the model and will be entirely neglected. The signal at the channel output $\tilde{\mathit{E}}(t,N_s,L_s)$ is ideally compensated for accumulated chromatic dispersion in the link (see Section 4). In the frequency-domain output of the chromatic dispersion compensation (CDC) block $\mathring{\mathit{E}}(f,N_s,L_s)$ (Figure 2), we ideally isolate the first-order RP term ${\mathit{E}}_1(f,N_s,L_s)$ (see Section 4) and we compute its PSD $\mathit{S}(f,N_s,L_s)$. The vector of the NLI powers ${\Sigma }_{\mathrm{NLI}}\triangleq {[{\sigma }_{\mathrm{NLI},x}^2,{\sigma }_{\mathrm{NLI},y}^2]}^T$ for both x and y polarisations is obtained by integrating over the frequency interval $[-R_s/2,R_s/2]$ the NLI PSD weighted by the function ${|P\left(f\right)|}^2$, where $P^{\ast }\left(f\right)$ is the frequency response of a matched filter (MF) for the system under consideration. As shown in Figure 2, this quantity is equivalent to the variance of the output of the MF followed by symbol-rate sampling, which more naturally arises when assessing the transmission performance of systems employing an MF at the receiver. The model in this manuscript provides an analytical relationship between the statistical features of the transmitted symbols ${\mathit{a}}_n$ and ${\Sigma }_{\mathrm{NLI}}$.

3.2. DP-4D vs. PM-2D Formats

The model presented in this paper allows for prediction of the NLI for generic 4D real modulation formats. A 4D format is defined as a set

$$\mathcal{A}\triangleq \{{\mathit{a}}^{\left(i\right)}={[a_x^{\left(i\right)},a_y^{\left(i\right)}]}^T\in {\mathbb{C}}^2,i=1,2,\dots ,M\},$$

(2)

where $a_x$ and $a_y$ are the symbols modulated on two orthogonal polarisation states x and y, respectively, and M is the modulation cardinality. It can be seen that the elements in $\mathcal{A}$ are 2D vectors in $\mathbb{C}$ as opposed to 4D. This is only due the to baseband-equivalent representation of signals used throughout this paper, while it is common to refer to a modulation format dimensionality based on the real signal dimensions, which justifies the 4D format label.

Two important particular cases of the formats in (2) are (i) the so-called polarisation-multiplexed 2D (PM-2D) modulation formats, which are characterised by $\mathcal{A}={\mathcal{X}}^2$, $\mathcal{X}\in \mathbb{C}$, where $\mathcal{X}$ represents the 2D component constellation, and (ii) polarisation-hybrid 2D modulation formats characterized by $\mathcal{A}=\mathcal{X}\times \mathcal{Y}$, with $\mathcal{X},\mathcal{Y}\in \mathbb{C}$, $\mathcal{X}\ne \mathcal{Y}$, where $\mathcal{X}$ and $\mathcal{Y}$ are two distinct component 2D formats in x and y polarisation, respectively. PM-2D formats are the most common ones in optical communications due to their generation’s simplicity. Both PM-2D and polarisation-hybrid 2D formats are often analysed in terms of their 2D polarisation components. This is because $\mathcal{A}$ can be factorised in two component formats which are independently encoded. Hence, if the generic transmitted constellation point is regarded as a random variable, in a conventional PM-2D format, the two polarisation components are statistically independent. In the remainder of this paper, no specific assumption on either the geometry or the statistic of the transmitted 4D symbols will be made, except the zero-mean feature $\mathbb{E}\left\{{\mathit{a}}^{\left(i\right)}\right\}=\mathbf{0}.$

Figure 3. Schematic representation of the periodic signal assumption, where W symbols are transmitted every T [s], each symbol with a duration of $T_s$ [s]: the periodicity assumption will be lifted in Section 8 by letting ${\Delta }_f\to 0$.

Figure 3. Schematic representation of the periodic signal assumption, where W symbols are transmitted every T [s], each symbol with a duration of $T_s$ [s]: the periodicity assumption will be lifted in Section 8 by letting ${\Delta }_f\to 0$.

3.3. Transmitted Signal Form

Let $\tilde{\mathit{E}}(t,z)={\tilde{E}}_x(t,z){\mathit{i}}_x+{\tilde{E}}_y(t,z){\mathit{i}}_y$ be the complex envelope of the optical field vector at time t and fibre section z, and let ${\mathit{i}}_x$, ${\mathit{i}}_y$ denote 2 orthonormal polarisations of the transversal plane of propagation. Let also $\mathit{E}(f,z)=E_x(f,z){\mathit{i}}_x+E_y(f,z){\mathit{i}}_y$ be the (vector) Fourier transform of $\tilde{\mathit{E}}(t,z)$ defined as

$$\mathit{E}(f,z)=\mathcal{F}\left\{\tilde{\mathit{E}}(t,z)\right\}\triangleq {\int }_{-\infty }^{\infty }\tilde{\mathit{E}}(t,z)e^{-j2\pi ft}dt.$$

Because of the periodicity assumption made in (1) (see Figure 3), we can write $\tilde{\mathit{E}}(t,0)$ as

$$\tilde{\mathit{E}}(t,0)=\sum _{k=-\infty }^{\infty }{\mathit{C}}_k{e}^{j2\pi k{\Delta }_{f}t},$$

(3)

where ${\mathit{C}}_k={[C_{x,k},C_{y,k}]}^T$, $C_{x/y,k}$ are the Fourier series coefficients of $\tilde{\mathit{E}}(t,0)$ and ${\Delta }_f=1/T$ is the frequency spacing of the spectral lines in $E_{x/y}(f,z)$. Hence, $\mathit{E}(f,0)$ can be then written as

$$\begin{array}{c}\hfill \mathit{E}(f,0)=\sum _{k=-\infty }^{\infty }{\mathit{C}}_k\delta (f-k{\Delta }_f).\end{array}$$

(4)

Since each component of $\tilde{\mathit{E}}(t,0)$ is periodic with period T, we can write

$$\tilde{\mathit{E}}(t,0)=\sum _{n=-\infty }^{\infty }\widehat{\mathit{E}}(t-nT,0),$$

where, as per assumption in (1), we have

$$\widehat{\mathit{E}}(t,0)\triangleq \left\{\begin{array}{cc}{\displaystyle \sum _{n=-(W-1)/2}^{(W-1)/2}}{\mathit{a}}_{n}p(t-n{T}_s),\hfill & \mathrm{for}-\frac{T}{2}\le t\le \frac{T}{2}\\ 0,\hfill & \mathrm{otherwise}\end{array}\right.,$$

and, W is assumed to be odd without loss of generality. Under the above assumptions, the Fourier coefficients in (3), for $k\in \mathbb{Z}$, are given by

$$\begin{array}{cc}\hfill {\mathit{C}}_k& ={\Delta }_f{\int }_{-\frac{T}{2}}^{\frac{T}{2}}\widehat{\mathit{E}}(t,0)e^{-j2\pi k{\Delta }_{f}t}dt\hfill \end{array}$$

(5a)

$$\begin{array}{cc}\hfill & ={\Delta }_f{\int }_{-\frac{T}{2}}^{\frac{T}{2}}\sum _{n=-(W-1)/2}^{(W-1)/2}{\mathit{a}}_{n}p(t-n{T}_s)e^{-j2\pi k{\Delta }_{f}t}dt\hfill \end{array}$$

(5b)

$$\begin{array}{cc}\hfill & ={\Delta }_f\sum _{n=-(W-1)/2}^{(W-1)/2}{\mathit{a}}_n{\int }_{-\frac{T}{2}}^{\frac{T}{2}}p(t-n{T}_s)e^{-j2\pi k{\Delta }_{f}t}dt\hfill \end{array}$$

(5c)

$$\begin{array}{cc}\hfill & \approx {\Delta }_f\sum _{n=-(W-1)/2}^{(W-1)/2}{\mathit{a}}_{n}P\left(k{\Delta }_f\right)e^{-j2\pi k{\Delta }_{f}n{T}_s}\hfill \end{array}$$

(5d)

$$\begin{array}{cc}\hfill & ={\Delta }_{f}P\left(k{\Delta }_f\right)\sum _{n=-(W-1)/2}^{(W-1)/2}{\mathit{a}}_n{e}^{-j2\pi \frac{kn}{W}}\hfill \end{array}$$

(5e)

$$\begin{array}{cc}\hfill & =\sqrt{{\Delta }_f}P\left(k{\Delta }_f\right){\nu }_{\mathit{k}},\hfill \end{array}$$

(5f)

where $P\left(f\right)\triangleq \mathcal{F}\left\{p\right(t\left)\right\}$ and

$${\nu }_k={[{\nu }_{x,k},{\nu }_{y,k}]}^T=\sqrt{{\Delta }_f}\sum _{n=-(W-1)/2}^{(W-1)/2}{\mathit{a}}_n{e}^{-j2\pi \frac{kn}{W}},\forall k\in \mathbb{Z},$$

(6)

are the discrete Fourier transforms of the sequence ${\mathit{a}}_n$, $n=0,1,\dots ,W-1$. Note that the approximation in (5c)–(5d) is justified only for large enough values of T as

$$\underset{T\to \infty }{lim}{\int }_{-\frac{T}{2}}^{\frac{T}{2}}p(t-n{T}_s)e^{-j2\pi k{\Delta }_{f}t}dt=\mathcal{F}\left\{p(t-n{T}_s)\right\}{|}_{f=k{\Delta }_f},\mathrm{for}n,k\in \mathbb{Z},$$

and letting $T\to \infty$ will be the approach taken at a later stage in this derivation.

Finally, combining (4) and (5f), we obtain

$$\mathit{E}(f,0)=\sqrt{{\Delta }_f}\sum _{k=-\infty }^{\infty }P\left(k{\Delta }_f\right){\nu }_k\delta (f-k{\Delta }_f)\approx \sum _{k=-(W-1)/2}^{(W-1)/2}P\left(k{\Delta }_f\right){\nu }_k\delta (f-k{\Delta }_f),$$

(7)

where the approximate equality on the right-hand side of (7) stems from the fact that $p\left(t\right)$ is assumed to be strictly or quasi-strictly band-limited (see Section 3.1). Hence, $P\left(k{\Delta }_f\right)$ is effectively equal to zero for $k=-W/2,-W/2+1,\dots ,W/2$.

4. PSD of the First-Order NLI for Periodic Transmitted Signals

To find an analytical expression for NLI power, first, a solution as explicit as possible to the Manakov equation [12]

$$\begin{array}{cc}\hfill \frac{\partial \tilde{\mathit{E}}(t,z)}{\partial z}& =-\frac{\alpha }{2}\tilde{\mathit{E}}(t,z)-j\frac{{\beta }_2}{2}\frac{{\partial }^2\tilde{\mathit{E}}(t,z)}{\partial t^2}+j\frac{8}{9}\gamma {|\tilde{\mathit{E}}(t,z)|}^2\tilde{\mathit{E}}(t,z),\hfill \end{array}$$

(8)

must be found. Equation (8) describes the propagation of the optical field $\tilde{\mathit{E}}(t,z)$ in a single strand of fibre (e.g., a fibre span with no amplifier in the system in Figure 2). In this case, $\alpha$, ${\beta }_2$, and $\gamma$ representing the attenuation, group velocity dispersion, and nonlinearity coefficients, respectively, can be assumed to be spatially constant. As it is well-known, general closed-form solutions are not available for (8). Like most of the existing NLI power models in the literature, the model derived here operates within a first-order perturbative framework. In particular, a frequency-domain first-order regular perturbation (RP) approach in the $\gamma$ coefficient is performed [13,14], i.e., the Fourier transform of the solution in (8) is expressed as

$$\mathit{E}(f,z)=\sum _{n=0}^{\infty }{\gamma }^n{\mathit{A}}_n(f,z)\approx {\mathit{A}}_0(f,z)+\gamma {\mathit{A}}_1(f,z),$$

(9)

where

$${\mathit{E}}_n(f,z)={\gamma }^n{\mathit{A}}_n(f,z)\mathrm{for}n=0,1,\dots ,$$

(10)

represents the so-called nth order term of the expansion.

In the following theorem, we present the expressions for ${\mathit{E}}_0(f,z),$ and ${\mathit{E}}_1(f,z),$ when a multiple fibre span system like the one in Figure 2 is considered. These expressions are well-known in the literature (see, e.g., [13]). Nevertheless, we present the proof in Appendix A for completeness.

Theorem 1

(First-order frequency-domain RP solution for a multi-span fibre system). Let $\mathit{E}(f,z)$ be the solution in the frequency domain of the Manakov equation for the system in Figure 2 with initial condition at distance $z=0$ given by the transmitted signal $\mathit{E}(f,0)$. Then, the first-order RP solution after $N_s$ spans $\mathit{E}(f,N_s,L_s)$ is given by

$$\mathit{E}(f,N_s,L_s)\approx {\mathit{E}}_0(f,N_s,L_s)+{\mathit{E}}_1(f,N_s,L_s),$$

where the zeroth-order term is given by

$${\mathit{E}}_0(f,N_s,L_s)={\mathit{E}}_0(f,0)e^{j2{\pi }^2{f}^2{\beta }_2{N}_s{L}_s},$$

and the first-order term is

$${\mathit{E}}_1(f,N_s,L_s)=-j\frac{8}{9}\gamma e^{j2{\pi }^2{f}^2{\beta }_2{N}_s{L}_s}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{\mathit{E}}^T(f_1,0){\mathit{E}}^{\ast }(f_2,0)\mathit{E}(f-f_1+f_2,0)\eta (f_1,f_2,f,N_s,L_s)d{f}_{1}d{f}_2,$$

(11)

with

$$\eta (f_1,f_2,f,N_s,L_s)\triangleq \frac{1-e^{-\alpha L_s}{e}^{j4{\pi }^2{\beta }_2(f-f_1)(f_2-f_1)L_s}}{\alpha -j4{\pi }^2{\beta }_2(f-f_1)(f_2-f_1)}\sum _{l=1}^{N_s}{e}^{-j4{\pi }^2{\beta }_2(l-1)(f-f_1)(f_2-f_1)L_s},$$

(12)

where $N_s$ and $L_s$ are the number of spans and the span length of each span, respectively.

Proof. 

See Appendix A. □

While Theorem 1 gives an approximation for the field at the output of the fibre, we are interested in the field after ideal CDC (see Figure 2). Ideal CDC ideally removes the exponential $e^{j2{\beta }_2{\pi }^2{f}^2{N}_s{L}_s}$ from (11), leading to a first-order term in the RP solution for the system in Figure 2 given by

$$\begin{array}{c}{\mathring{\mathit{E}}}_1(f,N_s,L_s)={[{\mathring{E}}_{1,x},{\mathring{E}}_{1,y}]}^T=-j\frac{8}{9}\gamma {\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{\mathit{E}}^T(f_1,0){\mathit{E}}^{\ast }(f_2,0)\mathit{E}(f-f_1+f_2,0)\\ ·\eta (f_1,f_2,f,N_s,L_s)d{f}_{1}d{f}_2.\end{array}$$

(13)

Substituting the spectrum of the transmitted periodic signal (7) in (13), we obtain, for instance, for the x component in (13),

$$\begin{array}{cc}\hfill {\mathring{E}}_{1,x}(f,N_s,L_s)& =-j\frac{8}{9}\gamma {\Delta }_f^{3/2}\sum _{k=-\infty }^{\infty }\sum _{m=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }P\left(k{\Delta }_f\right)P^{\ast }\left(m{\Delta }_f\right)P\left(n{\Delta }_f\right)\left({\nu }_{x,k}{\nu }_{x,m}^{\ast }{\nu }_{x,n}+{\nu }_{y,k}{\nu }_{y,m}^{\ast }{\nu }_{x,n}\right)\hfill \\ \hfill & ·{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\delta (f_1-k{\Delta }_f)\delta (f_2-m{\Delta }_f)\delta (f-f_1+f_2-n{\Delta }_f)\eta (f_1,f_2,f,N_s,L_s)d{f}_{1}d{f}_2.\hfill \end{array}$$

(14)

Although the product of Dirac’s deltas in (14) is not well-defined in the standard distribution theory framework, in this case, such product can be dealt with in the same way as products between distributions and smooth functions. This approach was formalised by Colombeau in his theory of product between distributions [15]. Thus, integrating in $f_1$ and $f_2$, we obtain

$$\begin{array}{cc}\hfill {\mathring{E}}_{1,x}& (f,N_s,L_s)=-j\frac{8}{9}\gamma {\Delta }_f^{3/2}\sum _{k=-\infty }^{\infty }\sum _{m=-\infty }^{\infty }\sum _{n=-\infty }^{\infty }P\left(k{\Delta }_f\right)P^{\ast }\left(m{\Delta }_f\right)P\left(n{\Delta }_f\right)\hfill \\ \hfill & ·\left({\nu }_{x,k}{\nu }_{x,m}^{\ast }{\nu }_{x,n}+{\nu }_{y,k}{\nu }_{y,m}^{\ast }{\nu }_{x,n}\right)\eta (k{\Delta }_f,m{\Delta }_f,(k-m+n){\Delta }_f,N_s,L_s)\delta (f-(k-m+n){\Delta }_f).\hfill \end{array}$$

(15)

Setting $i=k-m+n$ and defining

$$\begin{array}{cc}\hfill {\eta }_{k,m,n}& \triangleq \eta (k{\Delta }_f,m{\Delta }_f,(k-m+n){\Delta }_f,N_s,L_s)\hfill \\ \hfill & =\frac{1-e^{-\alpha L_s}{e}^{j4{\pi }^2{\Delta }_f^2{\beta }_2(n-m)(m-k)L_s}}{\alpha -j4{\pi }^2{\Delta }_f^2{\beta }_2(n-m)(m-k)}\sum _{l=1}^{N_s}{e}^{-j4{\pi }^2{\Delta }_f^2{\beta }_2(l-1)(n-m)(m-k)L_s}\hfill \end{array}$$

(16)

(15) can be rewritten as

$${\mathring{E}}_{1,x}(f,N_s,L_s)=\sum _{i=-\infty }^{\infty }{c}_i\delta (f-i{\Delta }_f),$$

(17)

where

$$c_i\triangleq -j\frac{8}{9}\gamma {\Delta }_f^{3/2}\sum _{\begin{array}{c}(k,m,n)\in {\mathcal{S}}_i\end{array}}P\left(k{\Delta }_f\right)P^{\ast }\left(m{\Delta }_f\right)P\left(n{\Delta }_f\right)\left({\nu }_{x,k}{\nu }_{x,m}^{\ast }{\nu }_{x,n}+{\nu }_{y,k}{\nu }_{y,m}^{\ast }{\nu }_{x,n}\right){\eta }_{k,m,n},$$

(18)

and

$${\mathcal{S}}_i\triangleq \{(k,m,n)\in {\mathbb{Z}}^3:k-m+n=i\}.$$

(19)

The PSD of the received nonlinear interference (to the 1st-order) is defined as

$$\mathit{S}(f,N_s,L_s)={[S_x(f,N_s,L_s),S_y(f,N_s,L_s)]}^T\triangleq {\left[\mathbb{E}\left\{|{\mathring{E}}_{1,x}(f,N_s,L_s){|}^2\right\},\mathbb{E}\left\{|{\mathring{E}}_{1,y}(f,N_s,L_s){|}^2\right\}\right]}^T.$$

For periodic signals, which in the frequency domain can be expressed as in (17), the PSD can be expressed as [16] (Section 4.1.2)

$$S_x(f,N_s,L_s)=\sum _{i=-\infty }^{\infty }\mathbb{E}\{|c_i{|}^2\}\delta (f-i{\Delta }_f).$$

(20)

Substituting the expression (18) for $c_i$ in (20), we obtain

$$\begin{array}{cc}\hfill S_x(f,N_s,L_s)& ={\left(\frac{8}{9}\right)}^2{\gamma }^2{\Delta }_f^3\sum _{i=-\infty }^{\infty }\delta (f-i{\Delta }_f)\mathbb{E}\{\sum _{(k,m,n)\in {\mathcal{S}}_i}P\left(k{\Delta }_f\right)P^{\ast }\left(m{\Delta }_f\right)P\left(n{\Delta }_f\right)\left({\nu }_{x,k}{\nu }_{x,m}^{\ast }{\nu }_{x,n}\right.\hfill \end{array}$$

$$\begin{array}{cc}\hfill \left.+{\nu }_{y,k}{\nu }_{y,m}^{\ast }{\nu }_{x,n}\right)& {\eta }_{k,m,n}\sum _{(k^{\prime },m^{\prime },n^{\prime })\in {\mathcal{S}}_i}{P}^{\ast }\left(k^{\prime }{\Delta }_f\right)P\left(m^{\prime }{\Delta }_f\right)P^{\ast }\left(n^{\prime }{\Delta }_f\right)\left({\nu }_{x,k^{\prime }}^{\ast }{\nu }_{x,m^{\prime }}{\nu }_{x,n^{\prime }}^{\ast }+{\nu }_{y,k^{\prime }}^{\ast }{\nu }_{y,m^{\prime }}{\nu }_{x,n^{\prime }}^{\ast }\right){\eta }_{k^{\prime },m^{\prime },n^{\prime }}^{\ast }\}\hfill \\ \hfill & ={\left(\frac{8}{9}\right)}^2{\gamma }^2{\Delta }_f^3\sum _{i=-\infty }^{\infty }\delta (f-i{\Delta }_f)\mathbb{E}\{\sum _{\begin{array}{c}(k,m,n)\in {\mathcal{S}}_i\\ (k^{\prime },m^{\prime },n^{\prime })\in {\mathcal{S}}_i\end{array}}{\mathcal{P}}_{k,m,n,k^{\prime },m^{\prime },n^{\prime }}\left({\nu }_{x,k}{\nu }_{x,m}^{\ast }{\nu }_{x,n}{\nu }_{x,k^{\prime }}^{\ast }{\nu }_{x,m^{\prime }}{\nu }_{x,n^{\prime }}^{\ast }\right.\hfill \end{array}$$

(21a)

$$+{\nu }_{x,k}{\nu }_{x,m}^{\ast }{\nu }_{x,n}{\nu }_{y,k^{\prime }}^{\ast }{\nu }_{y,m^{\prime }}{\nu }_{x,n^{\prime }}^{\ast }+{\nu }_{y,k}{\nu }_{y,m}^{\ast }{\nu }_{x,n}{\nu }_{x,k^{\prime }}^{\ast }{\nu }_{x,m^{\prime }}{\nu }_{x,n^{\prime }}^{\ast }+\left.{\nu }_{y,k}{\nu }_{y,m}^{\ast }{\nu }_{x,n}{\nu }_{y,k^{\prime }}^{\ast }{\nu }_{y,m^{\prime }}{\nu }_{x,n^{\prime }}^{\ast }\right){\eta }_{k,m,n}{\eta }_{k^{\prime },m^{\prime },n^{\prime }}^{\ast }\},$$

(21b)

where we have defined

$${\mathcal{P}}_{k,m,n,k^{\prime },m^{\prime },n^{\prime }}\triangleq P\left(k{\Delta }_f\right)P^{\ast }\left(m{\Delta }_f\right)P\left(n{\Delta }_f\right)P^{\ast }\left(k^{\prime }{\Delta }_f\right)P\left(m^{\prime }{\Delta }_f\right)P^{\ast }\left(n^{\prime }{\Delta }_f\right).$$

(22)

The following proposition can be used to make (21b) more compact, and in particular, it will be used to group the two inner correlation terms in (21b) (${\nu }_{x,k}{\nu }_{x,m}^{\ast }{\nu }_{x,n}{\nu }_{y,k^{\prime }}^{\ast }{\nu }_{y,m^{\prime }}{\nu }_{x,n^{\prime }}^{\ast }$ and ${\nu }_{y,k}{\nu }_{y,m}^{\ast }{\nu }_{x,n}{\nu }_{x,k^{\prime }}^{\ast }{\nu }_{x,m^{\prime }}{\nu }_{x,n^{\prime }}^{\ast }$).

Proposition 1.

For ${\mathcal{P}}_{k,m,n,k^{\prime },m^{\prime },n^{\prime }}$ in (22), we have

$$\begin{array}{cc}\hfill & \sum _{\begin{array}{c}(k,m,n)\in {\mathcal{S}}_i\\ (k^{\prime },m^{\prime },n^{\prime })\in {\mathcal{S}}_i\end{array}}{\mathcal{P}}_{k,m,n,k^{\prime },m^{\prime },n^{\prime }}{\nu }_{x,k}{\nu }_{x,m}^{\ast }{\nu }_{x,n}{\nu }_{y,k^{\prime }}^{\ast }{\nu }_{y,m^{\prime }}{\nu }_{x,n^{\prime }}^{\ast }{\eta }_{k,m,n}{\eta }_{k^{\prime },m^{\prime },n^{\prime }}^{\ast }\hfill \\ \hfill & =(\sum _{\begin{array}{c}(k,m,n)\in {\mathcal{S}}_i\\ (k^{\prime },m^{\prime },n^{\prime })\in {\mathcal{S}}_i\end{array}}{\mathcal{P}}_{k,m,n,k^{\prime },m^{\prime },n^{\prime }}{\nu }_{y,k}{\nu }_{y,m}^{\ast }{\nu }_{x,n}{\nu }_{x,k^{\prime }}^{\ast }{\nu }_{x,m^{\prime }}{\nu }_{x,n^{\prime }}^{\ast }{\eta }_{k,m,n}{\eta }_{k^{\prime },m^{\prime },n^{\prime }}^{\ast }{)}^{\ast }.\hfill \end{array}$$

(23)

Proof. 

See Appendix B. □

Using (23) and (21b) can be written as

$$\begin{array}{cc}\hfill S_x(f,N_s,L_s)& ={\left(\frac{8}{9}\right)}^2{\gamma }^2{\Delta }_f^3\sum _{i=-\infty }^{\infty }\delta (f-i{\Delta }_f)\mathbb{E}\{\sum _{\begin{array}{c}(k,m,n)\in {\mathcal{S}}_i\\ (k^{\prime },m^{\prime },n^{\prime })\in {\mathcal{S}}_i\end{array}}{\mathcal{P}}_{k,m,n,k^{\prime },m^{\prime },n^{\prime }}\hfill \\ \hfill & ·\left({\nu }_{x,k}{\nu }_{x,m}^{\ast }{\nu }_{x,n}{\nu }_{x,k^{\prime }}^{\ast }{\nu }_{x,m^{\prime }}{\nu }_{x,n^{\prime }}^{\ast }+{\nu }_{y,k}{\nu }_{y,m}^{\ast }{\nu }_{x,n}{\nu }_{y,k^{\prime }}^{\ast }{\nu }_{y,m^{\prime }}{\nu }_{x,n^{\prime }}^{\ast }\right){\eta }_{k,m,n}{\eta }_{k^{\prime },m^{\prime },n^{\prime }}^{\ast }\hfill \\ \hfill & +2Re\left\{{\mathcal{P}}_{k,m,n,k^{\prime },m^{\prime },n^{\prime }}{\nu }_{x,k}{\nu }_{x,m}^{\ast }{\nu }_{x,n}{\nu }_{y,k^{\prime }}^{\ast }{\nu }_{y,m^{\prime }}{\nu }_{x,n^{\prime }}^{\ast }{\eta }_{k,n,m}{\eta }_{k^{\prime },n^{\prime },m^{\prime }}^{\ast }\right\}\}\hfill \\ \hfill & ={\left(\frac{8}{9}\right)}^2{\gamma }^2{\Delta }_f^3\sum _{i=-\infty }^{\infty }\delta (f-i{\Delta }_f)\sum _{\begin{array}{c}(k,m,n)\in {\mathcal{S}}_i\\ (k^{\prime },m^{\prime },n^{\prime })\in {\mathcal{S}}_i\end{array}}{\mathcal{P}}_{k,m,n,k^{\prime },m^{\prime },n^{\prime }}\left(\mathbb{E}\left\{{\nu }_{x,k}{\nu }_{x,m}^{\ast }{\nu }_{x,n}{\nu }_{x,k^{\prime }}^{\ast }{\nu }_{x,m^{\prime }}{\nu }_{x,n^{\prime }}^{\ast }\right\}\right.\hfill \\ \hfill & \left.+\mathbb{E}\left\{{\nu }_{y,k}{\nu }_{y,m}^{\ast }{\nu }_{x,n}{\nu }_{y,k^{\prime }}^{\ast }{\nu }_{y,m^{\prime }}{\nu }_{x,n^{\prime }}^{\ast }\right\}\right){\eta }_{k,m,n}{\eta }_{k^{\prime },m^{\prime },n^{\prime }}^{\ast }\hfill \\ \hfill & +2Re\left\{{\mathcal{P}}_{k,m,n,k^{\prime },m^{\prime },n^{\prime }}\mathbb{E}\left\{{\nu }_{x,k}{\nu }_{x,m}^{\ast }{\nu }_{x,n}{\nu }_{y,k^{\prime }}^{\ast }{\nu }_{y,m^{\prime }}{\nu }_{x,n^{\prime }}^{\ast }\right\}{\eta }_{k,n,m}{\eta }_{k^{\prime },n^{\prime },m^{\prime }}^{\ast }\right\},\hfill \end{array}$$

(24)

where the real part operator arises from the sum of the complex conjugate terms discussed in the Proposition section (Section 1).

According to (24), calculation of the PSD of the NLI reduces to the computation of a four-dimensional summation (per frequency component $i{\Delta }_f$) of three sixth-order correlations of the sequence of random variables ${\nu }_{x/y,n},n=0,1,\dots ,W-1$. The y-component $S_y(f,N_s,L_s)$ of the PSD can be calculated once $S_x(f,N_s,L_s)$ is obtained by simply swapping the polarisation labels $x\to y$ and $y\to x$. This is due to the invariance of the Manakov equation in (8) to such a transformation.

5. Classification of the Modulation-Dependent Contributions in the 6th-Order Frequency-Domain Correlation

In this section, we will break down the frequency-domain sixth-order correlation terms in (24) to highlight different contributions in terms of 4D modulation-dependent cross-polarisation correlations.

5.1. Expansion in Terms of the Stochastic Moments of the Transmitted Modulation Format

To relate the PSD in (24) to the statistical properties of the transmitted modulation format, we replace (6) into (24), obtaining

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill S_x(f,N_s,L_s)& ={\left(\frac{8}{9}\right)}^2{\gamma }^2{\Delta }_f^3\sum _{i=-\infty }^{\infty }\delta (f-i{\Delta }_f)\sum _{\begin{array}{c}(k,m,n)\in {\mathcal{S}}_i\\ (k^{\prime },m^{\prime },n^{\prime })\in {\mathcal{S}}_i\end{array}}[{\mathcal{P}}_{k,m,n,k^{\prime },m^{\prime },n^{\prime }}{\eta }_{k,m,n}{\eta }_{k^{\prime },m^{\prime },n^{\prime }}^{\ast }\hfill \\ \hfill & ·\sum _{\mathit{i}\in {\{0,1,\dots ,W-1\}}^6}{\mathsf{S}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })+2Re\{{\mathcal{P}}_{k,m,n,k^{\prime },m^{\prime },n^{\prime }}{\eta }_{k,m,n}{\eta }_{k^{\prime },m^{\prime },n^{\prime }}^{\ast }\hfill \\ \hfill & ·\sum _{\mathit{i}\in {\{0,1,\dots ,W-1\}}^6}{\mathsf{T}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })\left\}\right],\hfill \end{array}\end{array}$$

(25)

where $\mathit{i}\triangleq (i_1,i_2,\dots ,i_6)$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathsf{S}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })& \triangleq {\Delta }_f^3\left[\mathbb{E}\left\{a_{x,i_1}{a}_{x,i_2}^{\ast }{a}_{x,i_3}{a}_{x,i_4}^{\ast }{a}_{x,i_5}{a}_{x,i_6}^{\ast }\right\}+\mathbb{E}\left\{a_{y,i_1}{a}_{y,i_2}^{\ast }{a}_{x,i_3}{a}_{y,i_4}^{\ast }{a}_{y,i_5}{a}_{x,i_6}^{\ast }\right\}\right]\hfill \\ \hfill & ·e^{-j\frac{2\pi }{W}(k{i}_1-m{i}_2+n{i}_3-k^{\prime }{i}_4+m^{\prime }{i}_5-n^{\prime }{i}_6)},\hfill \end{array}\end{array}$$

(26)

and

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & {\mathsf{T}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })\triangleq {\Delta }_f^3\mathbb{E}\left\{a_{x,i_1}{a}_{x,i_2}^{\ast }{a}_{x,i_3}{a}_{y,i_4}^{\ast }{a}_{y,i_5}{a}_{x,i_6}^{\ast }\right\}{e}^{-j\frac{2\pi }{W}(k{i}_1-m{i}_2+n{i}_3-k^{\prime }{i}_4+m^{\prime }{i}_5-n^{\prime }{i}_6)}.\hfill \end{array}\end{array}$$

(27)

The terms ${\mathsf{S}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })$ and ${\mathsf{T}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })$ give rise to several correlations among the transmitted symbols $a_{x,i}$ and $a_{y,j}$ at different time-slots $i,j$, each weighted by a complex exponential. As discussed in Section 3, in this work, we operate under the assumption that the sequence of vector RVs ${\mathit{a}}_i$ for $i=0,1,\dots ,W-1$ are independent, identically distributed (i.i.d.), and with $\mathbb{E}\left\{{\mathit{a}}_i\right\}=\mathbb{E}\left\{\mathit{a}\right\}=\mathbf{0}$. As shown in the following example, this assumption allows us to discard the ${\mathsf{S}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })$ and ${\mathsf{T}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })$ terms which are identically zero for some values of $\mathit{i}$. Moreover, as it will be shown in Example 2 for all other values of $\mathit{i}$, ${\mathsf{S}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })$, and ${\mathsf{T}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })$ can be expressed as a product of high-order statistical moments of the RVs $a_x$ and $a_y$, which enables a more compact expression for (25).

Example 1.

Under the i.i.d. assumption for the sequence of vector RVs ${\mathit{a}}_i$, $i=0,1,\dots W-1$ made in this work, in any of the cases where

$$i_{{\kappa }_1}\ne i_{{\kappa }_2}=i_{{\kappa }_3}=\dots =i_{{\kappa }_6}\mathrm{for}{\kappa }_1,{\kappa }_2,\dots ,{\kappa }_6=1,2,\dots ,6;{\kappa }_1\ne {\kappa }_2\ne \dots \ne {\kappa }_6,$$

(28)

any of the sixth-order correlations in (26) and (27) degenerate into a product between a first-order moment and a fifth-order correlation. Such a product is equal to zero under our assumption $\mathbb{E}\left\{a_{x,i}\right\}=\mathbb{E}\left\{a_x\right\}=0$. For example, for $i_1\ne i_2=i_3=\dots =i_6$, we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \mathbb{E}\left\{a_{x,i_1}{a}_{x,i_2}^{\ast }{a}_{x,i_3}{a}_{x,i_4}^{\ast }{a}_{x,i_5}{a}_{x,i_6}^{\ast }\right\}=\mathbb{E}\left\{a_{x,i_1}\right\}\mathbb{E}\{|a_{x,i_2}{|}^4{a}_{x,i_2}^{\ast }\}=\mathbb{E}\left\{a_x\right\}\mathbb{E}\{|a_x{|}^4{a}_x^{\ast }\}=0.\hfill \end{array}\end{array}$$

From this follows that, for all elements in the set defined in (28), ${\mathsf{S}}_i(k,m,n,k^{\prime },m^{\prime },n^{\prime })=0$ and ${\mathsf{T}}_i(k,m,n,k^{\prime },m^{\prime },n^{\prime })=0$. This example highlights a zero-contribution region in the 6D space ${\{0,1,\dots ,W-1\}}^6$, as illustrated in Figure 4.

The ${\mathsf{S}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })$ and ${\mathsf{T}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })$ contributions for the set in (28) are identically zero regardless of the values taken by $k,m,n,k^{\prime },m^{\prime }$, and $n^{\prime }$. However, as it will be shown in Section 6, for a specific subset of values $k,m,n,k^{\prime },m^{\prime }$, and $n^{\prime }$, such contributions cancel each other in the inner sums in (25) due to the complex exponential weights.

Example 2.

Under the i.i.d. assumption for the sequence of vector RVs ${\mathit{a}}_i$, $i=0,1,\dots W-1$ made in this work, we have that, for all elements in the set $\{\mathit{i}\in {\{0,1,\dots ,W-1\}}^6,i_1=i_2,i_3=i_4=i_5=i_6,i_1\ne i_3\}$

$$\begin{array}{cc}\hfill {\mathsf{S}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })& ={\Delta }_f^3\left[\mathbb{E}\right\{|a_{x,i_1}{|}^2\left\}\mathbb{E}\right\{|a_{x,i_3}{|}^4\}+\mathbb{E}\{|a_{y,i_1}{|}^2\left\}\mathbb{E}\right\{|a_{x,i_3}{|}^2|a_{y,i_3}{|}^2\left\}\right]\hfill \\ \hfill & ·e^{-j\frac{2\pi }{W}((k-m)i_1+(n-k^{\prime }+m^{\prime }-n^{\prime })i_3)}\hfill \\ \hfill & ={\Delta }_f^3\left[\mathbb{E}\right\{|a_x{|}^2\left\}\mathbb{E}\right\{|a_x{|}^4\}+\mathbb{E}\{|a_y{|}^2\left\}\mathbb{E}\right\{|a_x{|}^2|a_y{|}^2\left\}\right]e^{-j\frac{2\pi }{W}((k-m)i_1+(n-k^{\prime }+m^{\prime }-n^{\prime })i_3)},\hfill \\ \hfill {\mathsf{T}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })& ={\Delta }_f^3\mathbb{E}\{|a_{x,i_1}{|}^2\left\}\mathbb{E}\right\{|a_{x,i_3}{|}^2|a_{y,i_3}{|}^2\}{e}^{-j\frac{2\pi }{W}(k{i}_1-m{i}_2+n{i}_3-k^{\prime }{i}_4+m^{\prime }{i}_5-n^{\prime }{i}_6)}\hfill \\ \hfill & ={\Delta }_f^3\mathbb{E}\{|a_x{|}^2\left\}\mathbb{E}\right\{|a_x{|}^2|a_y{|}^2\}{e}^{-j\frac{2\pi }{W}(k{i}_1-m{i}_2+n{i}_3-k^{\prime }{i}_4+m^{\prime }{i}_5-n^{\prime }{i}_6)}.\hfill \end{array}$$

It can be noted that (i) the sixth-order correlation degenerates into a product of marginal high-order moments of $a_x$ and $a_y$ and into the cross-polarisation correlation $\mathbb{E}\{|a_x{|}^2|a_y{|}^2\}$ and (ii) all elements within the set in this example contribute to the inner summation in (25) with the same set of moments, cross-polarisation correlations, and products thereof (i.e., $\mathbb{E}\{|a_x{|}^2\},\mathbb{E}\{|a_x{|}^4\},\mathbb{E}\{|a_y{|}^2\},and\mathbb{E}\{|a_x{|}^2|a_y{|}^2\}$). This example illustrates how to break down each instance of the contributions ${\mathsf{S}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })$ and ${\mathsf{T}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })$, which will be then added up in Section 6.

In the remainder of this section, we first partition the six-dimensional space $\mathit{i}\in {\{0,1,\dots ,W-1\}}^6$ and list all sets corresponding to nonzero elements of ${\mathsf{S}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })$ and ${\mathsf{T}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })$. As shown in Example 2, this will help highlight the contribution of a specific set in terms of high-order moments of the transmitted symbols $\mathit{a}$ in (25). Then, we proceed to list all such contributions.

5.2. Set Partitioning

The six-dimensional space $\mathit{i}\in {\{0,1,\dots ,W-1\}}^6$ can be partitioned in different subsets, each one uniquely defined by a partition on the set of indices ($i_1$, $i_2$, $i_3$, $i_4$, $i_5$, and $i_6$). Each partition defines its corresponding subset in ${\{0,1,\dots ,W-1\}}^6$ as follows: for each index partition, the indices belonging to the same subset all take the same value, whilst the indices belonging to different subsets have distinct values. This is schematically illustrated in Figure 4. For example, the subset of ${\{0,1,\dots ,W-1\}}^6$ labelled by the index partition $\{(i_1,i_2),(i_3,i_4),(i_5,i_6)\}$ is defined as $\{\mathit{i}\in {\{0,1,\dots ,W-1\}}^6:i_1=i_2,i_3=i_4,i_5=i_6,i_1\ne i_3\ne i_5\}$. This subset is shown in Figure 4 as part of ${\mathcal{L}}_1$.

In Figure 4, the families of subsets of ${\{0,1,\dots ,W-1\}}^6$ labelled ${\mathcal{L}}_i$, $i=1,2,\dots ,4$, are also highlighted. These families are characterised by subsets sharing the same cardinality of elements associated to their corresponding index partition. For example, in ${\mathcal{L}}_1$, all index partitions are characterised by 3 subsets, each one containing 2 indices. As shown in Example 2, this way of partitioning the set ${\{0,1,\dots ,W-1\}}^6$ is useful as it separates out the different contributions of (25) based on the high-order moments of $\mathit{a}$, as it is highlighted in region ${\mathcal{L}}_3$ of Figure 4.

Since we have 6 different indices, the number of subsets in a partition can vary from 1 to 6. Each of these subsets can contain a number of elements also ranging from 1 to 6. However, the subsets of ${\{0,1,\dots ,W-1\}}^6$, where the corresponding index partition has one or more index subsets with only one element, bring no contribution to (25) and thus can be discarded. This is illustrated in Example 1. The above class of index partitions then forms a zero contribution region, as shown in Figure 4. Such a region also includes all subsets where the corresponding index partitions contain 4 or more index subsets, as at least one of these subsets will have to contain only one element.

As shown in Figure 4, by removing the zero contribution region from ${\{0,1,\dots ,W-1\}}^6$, only 4 different families of subsets are left:

(i)

${\mathcal{L}}_1=\{\mathit{i}\in {\{0,1,\dots ,W-1\}}^6:i_{{\kappa }_1}=i_{{\kappa }_2},i_{{\kappa }_3}=i_{{\kappa }_4},i_{{\kappa }_5}=i_{{\kappa }_6};{\kappa }_1,{\kappa }_2,\dots {\kappa }_6=1,2,\dots ,6;{\kappa }_1\ne {\kappa }_2\ne {\kappa }_3\ne {\kappa }_4\ne {\kappa }_5\ne {\kappa }_6\}$. This set contains all sets of elements where the indices $i_1,i_2,\dots ,i_6$, can be grouped in 3 pairs. The indices take up the same value within each pair but different values across different pairs. It can be found that this set can be partitioned in 15 different subsets ${\mathcal{C}}_1^{\left(i\right)},i=1,2,\dots ,15$, representing all possible distinct ways of pairing the $i_k$ indices for $k=1,2,\dots 6$. These sets are listed in Table A1 in Appendix C, where each column shows a subgroup of indices taking the same value;

(ii)

${\mathcal{L}}_2:\{\mathit{i}\in {\{0,1,\dots ,W-1\}}^6,i_{{\kappa }_1}=i_{{\kappa }_2}=i_{{\kappa }_3},i_{{\kappa }_4}=i_{{\kappa }_5}=i_{{\kappa }_6};{\kappa }_1,{\kappa }_2,\dots ,{\kappa }_6=1,2,\dots ,6;{\kappa }_1\ne {\kappa }_2\ne {\kappa }_3\ne {\kappa }_4\ne {\kappa }_5\ne {\kappa }_6\}$ which can be broken down in 10 subsets ${\mathcal{C}}_2^{\left(i\right)},i=1,2,\dots ,10$, listed in Table A2 in Appendix C. Each index subgroup identifies a triplet of indices assuming the same value;

(iii)

${\mathcal{L}}_3=\{\mathit{i}\in {\{0,1,\dots ,W-1\}}^6:i_{{\kappa }_1}=i_{{\kappa }_2},i_{{\kappa }_3}=i_{{\kappa }_4}=i_{{\kappa }_5}=i_{{\kappa }_6};{\kappa }_1,{\kappa }_2,\dots ,{\kappa }_6=1,2,\dots ,6,{\kappa }_1\ne {\kappa }_2\ne {\kappa }_3\ne {\kappa }_4\ne {\kappa }_5\ne {\kappa }_6\}$ which can be partitioned in 15 subsets ${\mathcal{C}}_3^{\left(i\right)},i=1,2,\dots ,15$, listed in Table A3 in Appendix C. Each of the two index subgroups identifies the pair and the quadruple of indices assuming the same value;

(iv)

${\mathcal{L}}_4:\{\mathit{i}\in {\{0,1,\dots ,W-1\}}^6:i_1=i_2=i_3=i_4=i_5=i_6\}$.

6. Evaluation of the $\mathcal{L}$-Based Contributions

In this section, we provide three examples for the computation of the contributions of a generic element in ${\mathcal{L}}_1$, ${\mathcal{L}}_2$, and ${\mathcal{L}}_3$. The full list of contributions in these three sets and the ones in ${\mathcal{L}}_4$ are given in Section 6.1, Section 6.2, Section 6.3, Section 6.4.

We label each contribution as ${\mathsf{M}}_g^{\left(h\right)}(k,m,n,k^{\prime },m^{\prime },n^{\prime })$ and ${\mathsf{N}}_g^{\left(h\right)}(k,m,n,k^{\prime },m^{\prime },n^{\prime })$, where

$$\begin{array}{cc}\hfill {\mathsf{M}}_g^{\left(h\right)}(k,m,n,k^{\prime },m^{\prime },n^{\prime })& \triangleq \sum _{\mathit{i}\in {\mathcal{C}}_g^{\left(h\right)}}{\mathsf{S}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime }),\hfill \\ \hfill {\mathsf{N}}_g^{\left(h\right)}(k,m,n,k^{\prime },m^{\prime },n^{\prime })& \triangleq \sum _{\mathit{i}\in {\mathcal{C}}_g^{\left(h\right)}}{\mathsf{T}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime }),\hfill \end{array}$$

(29)

and the subsets ${\mathcal{C}}_g^{\left(h\right)}$ are taken from Table A1, Table A2, Table A3 in Appendix C.

Example 3

(Contributions in ${\mathcal{L}}_1$). ${\mathsf{M}}_1^{\left(1\right)}$, i.e., one of the 2 contributions for the set ${\mathcal{C}}_1^{\left(1\right)}=\{\mathit{i}\in {\{0,1,\dots ,W-1\}}^6:i_1=i_2,i_3=i_4,i_5=i_6,i_1\ne i_3,i_1\ne i_5,i_3\ne i_5\}$ is given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathsf{M}}_1^{\left(1\right)}& \triangleq \sum _{\mathit{i}\in {\mathcal{C}}_1^{\left(1\right)}}{\mathsf{S}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })\hfill \\ \hfill & ={\Delta }_f^3\left[{\mathbb{E}}^3\left\{|a_x{|}^2\right\}+\mathbb{E}\left\{|a_y{|}^2\right\}{|\mathbb{E}\left\{a_x{a}_y^{\ast }\right\}|}^2\right]\sum _{i_1=0}^{W-1}{e}^{-j\frac{2\pi }{W}(k-m)i_1}\sum _{i_3\ne i_1}{e}^{-j\frac{2\pi }{W}(n-k^{\prime })i_3}\hfill \\ \hfill & ·\sum _{\begin{array}{c}{i}_5\ne i_1,\\ i_5\ne i_3\end{array}}{e}^{-j\frac{2\pi }{W}(m^{\prime }-n^{\prime })i_5}.\hfill \end{array}\end{array}$$

(30)

Since

$$\sum _{k=0}^{W-1}{e}^{jnk\frac{2\pi }{W}}=\left\{\begin{array}{cc}W,\hfill & \mathrm{for}n=pW,p\in \mathbb{Z}\hfill \\ 0,\hfill & \mathrm{elsewhere}\hfill \end{array}\right.,$$

(31)

we can compute (30) using the following approach:

• we add up the terms for all $i_1,i_3,i_5$ values including all cases when $i_1$, $i_3$, and $i_5$ are equal among each other. Because of (31), these terms sum up to $W^3$ only when $k=m+pW,n=k^{\prime }+pW,and{m}^{\prime }=n^{\prime }+pW$, $p\in \mathbb{Z}$; otherwise, they sum to 0;
• we subtract the terms corresponding to the cases: $i_1=i_3,i_1\ne i_5$; $i_1=i_5,i_1\ne i_3$; and $i_3=i_5,i_1\ne i_3$. As an example, the number of terms defined by $i_1=i_3,i_1\ne i_5$ is given by the difference between the number of all pairs $i_1,i_5\in \{0,1,2,\dots ,W-1\}$ and the number of terms for $i_1=i_5$. According to (31), the former terms sum to $W^2$ only for $k-m+n-k^{\prime }=pW,m^{\prime }-n^{\prime }=pW$, whereas the latter sum to W only for $k-m+n-k^{\prime }+m^{\prime }-n^{\prime }=pW$, with $p\in \mathbb{Z}$. In all other cases, they all bring zero contribution. Similar results are obtained for $i_1=i_5,i_1\ne i_3$ and $i_3=i_5,i_1\ne i_3$;
• we finally subtract the terms $i_1=i_3=i_5$, which sum to W only for $k-m+n-k^{\prime }+m^{\prime }-n^{\prime }=pW$, $p\in \mathbb{Z}$; otherwise, they sum to 0 (see (31)).

Hence, we obtain

$$\begin{array}{cc}\hfill {\mathsf{M}}_1^{\left(1\right)}& ={\Delta }_f^3[{\mathbb{E}}^3\{|a_x{|}^2\}+2{\mathbb{E}}^2\{|a_x{|}^2\}|\mathbb{E}\left\{a_x{a}_y^{\ast }\right\}{|}^2+\mathbb{E}\{|a_y{|}^2\}|\mathbb{E}\left\{a_x{a}_y^{\ast }\right\}{|}^2\left]\right[W^3{\delta }_{k-m-pW}{\delta }_{n-k^{\prime }-pW}{\delta }_{m^{\prime }-n^{\prime }-pW}\hfill \\ & -[W^2({\delta }_{k-m+n-k^{\prime }-pW}{\delta }_{m^{\prime }-n^{\prime }-pW}+{\delta }_{k-m+m^{\prime }-n^{\prime }-pW}{\delta }_{n-k^{\prime }-pW}+{\delta }_{m^{\prime }-n^{\prime }+n-k^{\prime }-pW}{\delta }_{k-m-pW})\hfill \\ \hfill & -3W{\delta }_{k-m+m^{\prime }-n^{\prime }+n-k^{\prime }-pW}]-W{\delta }_{k-m+m^{\prime }-n^{\prime }+n-k^{\prime }-pW}]\hfill \\ \hfill & =[{\mathbb{E}}^3\{|a_x{|}^2\}+2{\mathbb{E}}^2\{|a_x{|}^2\}|\mathbb{E}\left\{a_x{a}_y^{\ast }\right\}{|}^2+\mathbb{E}\{|a_y{|}^2\}|\mathbb{E}\left\{a_x{a}_y^{\ast }\right\}{|}^2\left]\right[R_s^3{\delta }_{k-m+pW}{\delta }_{n-k^{\prime }-pW}{\delta }_{m^{\prime }-n^{\prime }-pW}\hfill \\ \hfill & -R_s^2{\Delta }_f({\delta }_{k-m+n-k^{\prime }-pW}{\delta }_{m^{\prime }-n^{\prime }-pW}+{\delta }_{k-m+m^{\prime }-n^{\prime }-pW}{\delta }_{n-k^{\prime }-pW}+{\delta }_{m^{\prime }-n^{\prime }+n-k^{\prime }-pW}{\delta }_{k-m-pW})\hfill \\ \hfill & +2R_s{\Delta }_f^2{\delta }_{k-m+m^{\prime }-n^{\prime }+n-k^{\prime }-pW}],\hfill \end{array}$$

where we have used $R_s=W{\Delta }_f$.

The same approach can be followed to compute ${\mathsf{N}}_1^{\left(1\right)}$, which is, thus, given by

$$\begin{array}{cc}\hfill {\mathsf{N}}_1^{\left(1\right)}& \triangleq \sum _{\mathit{i}\in {\mathcal{C}}_1^{\left(1\right)}}{\mathsf{T}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })={\mathbb{E}}^2\{|a_x{|}^2\}|\mathbb{E}\left\{a_x{a}_y^{\ast }\right\}{|}^2[R_s^3{\delta }_{k-m-pW}{\delta }_{n-k^{\prime }-pW}{\delta }_{m^{\prime }-n^{\prime }-pW}\hfill \\ & -R_s^2{\Delta }_f({\delta }_{k-m+n-k^{\prime }-pW}{\delta }_{m^{\prime }-n^{\prime }-pW}+{\delta }_{k-m+m^{\prime }-n^{\prime }-pW}{\delta }_{n-k^{\prime }-pW}+{\delta }_{m^{\prime }-n^{\prime }+n-k^{\prime }-pW}{\delta }_{k-m-pW})\hfill \\ \hfill & +2R_s{\Delta }_f^2{\delta }_{k-m+m^{\prime }-n^{\prime }+n-k^{\prime }-pW}].\hfill \end{array}$$

All other contributions in ${\mathcal{L}}_1$ can be computed using the approach used in this example.

Example 4

(Contributions in ${\mathcal{L}}_2$). ${\mathsf{M}}_2^{\left(1\right)}$, i.e., the contribution for the set ${\mathcal{C}}_2^{\left(1\right)}=\{\mathit{i}\in {\{0,1,\dots ,W-1\}}^6:i_1=i_2=i_3,i_4=i_5=i_6,i_1\ne i_4\}$ is given by

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\mathsf{M}}_2^{\left(1\right)}& =\sum _{\mathit{i}\in {\mathcal{C}}_2^{\left(1\right)}}{\mathsf{S}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })\hfill \\ \hfill & ={\Delta }_f^3\left[\mathbb{E}\right\{a_{x,i_1}|a_{x,i_1}{|}^2\}{\mathbb{E}}^{\ast }\{a_{x,i_4}|a_{x,i_4}{|}^2\}+\mathbb{E}\{a_{x,i_1}|a_{y,i_1}{|}^2\}{\mathbb{E}}^{\ast }\{a_{x,i_4}|a_{y,i_4}{|}^2\left\}\right]\hfill \\ \hfill & ·\sum _{i_1=0}^{W-1}{e}^{-j\frac{2\pi }{W}(k-m+n)i_1}\sum _{i_4\ne i_1}{e}^{-j\frac{2\pi }{W}(-k^{\prime }+m^{\prime }-n^{\prime })i_4}\hfill \\ \hfill & ={\Delta }_f^3[|\mathbb{E}\{a_x|a_x{|}^2{\}|}^2+|\mathbb{E}\{a_x|a_y{|}^2{\}|}^2]\sum _{i_1=0}^{W-1}{e}^{-j\frac{2\pi }{W}(k-m+n)i_1}\sum _{i_4\ne i_1}{e}^{-j\frac{2\pi }{W}(-k^{\prime }+m^{\prime }-n^{\prime })i_4}.\hfill \end{array}\end{array}$$

(32)

Following a similar approach as in Example 3, we compute (32) by

• adding up the terms for all $i_1$ and $i_4$ values including all cases when $i_1$ and $i_4$ are equal to each other. These terms sum up to $W^2$ only when $k-m+n=pW$ and $-k^{\prime }+m^{\prime }-n^{\prime }=pW$, with $p\in \mathbb{Z}$; otherwise, they sum to 0;
• subtracting the terms corresponding to the cases $i_1=i_4$. These terms sum to W only for $k-m+n-k^{\prime }+m^{\prime }-n^{\prime }=pW$, $p\in \mathbb{Z}$; otherwise, they sum to zero.

We, thus, obtain

$${\mathsf{M}}_2^{\left(1\right)}=[|\mathbb{E}\{a_x|a_x{|}^2{\}|}^2+|\mathbb{E}\{a_x|a_y{|}^2{\}|}^2][R_s^2{\Delta }_f{\delta }_{k-m+n-pW}{\delta }_{k^{\prime }-m^{\prime }+n^{\prime }-pW}-R_s{\Delta }_f^2{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }-pW}],$$

Following the same approach for ${\mathsf{N}}_2^{\left(1\right)}$, we have

$$\begin{array}{cc}\hfill {\mathsf{N}}_2^{\left(1\right)}& \triangleq \sum _{\mathit{i}\in {\mathcal{C}}_1^{\left(1\right)}}{\mathsf{T}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })\hfill \\ \hfill & =\mathbb{E}\{a_x|a_x{|}^2\left\}\mathbb{E}\right\{a_x|a_y{|}^2\}[R_s^2{\Delta }_f{\delta }_{k-m+n}{\delta }_{k^{\prime }-m^{\prime }+n^{\prime }-pW}-R_s{\Delta }_f^2{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }-pW}].\hfill \end{array}$$

All other contributions in ${\mathcal{L}}_2$ can be computed using the approach used in this example.

Example 5

(Contributions in ${\mathcal{L}}_3$). ${\mathsf{M}}_3^{\left(3\right)}$, i.e., the contribution for the values in the set ${\mathcal{C}}_3^{\left(3\right)}=\{\mathit{i}\in {\{0,1,\dots ,W-1\}}^6:i_1=i_4,i_2=i_3=i_5=i_6,i_1\ne i_4,i_2\ne i_3,i_5\ne i_6\}$ is given by

$$\begin{array}{cc}\hfill {\mathsf{M}}_3^{\left(3\right)}& =\sum _{\mathit{i}\in {\mathcal{C}}_3^{\left(3\right)}}{\mathsf{S}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })\hfill \\ \hfill =& {\Delta }_f^3\left[\mathbb{E}\right\{|a_{x,i_1}{|}^2\left\}\mathbb{E}\right\{|a_{x,i_2}{|}^4\}+\mathbb{E}\{|a_{y,i_1}{|}^2\left\}\mathbb{E}\right\{|a_{x,i_2}{|}^2|a_{y,i_2}{|}^2\left\}\right]\sum _{i_1=1}^{W-1}{e}^{-j\frac{2\pi }{W}(k-k^{\prime })i_1}\sum _{i_2\ne i_1}{e}^{-j\frac{2\pi }{W}(-m+n+m^{\prime }-n^{\prime })i_2}\hfill \\ \hfill =& {\Delta }_f^3\left[\mathbb{E}\right\{|a_x{|}^2\left\}\mathbb{E}\right\{|a_x{|}^4\}+\mathbb{E}\{|a_y{|}^2\left\}\mathbb{E}\right\{|a_x{|}^2|a_y{|}^2\left\}\right]\sum _{i_1=1}^{W-1}{e}^{-j\frac{2\pi }{W}(k-k^{\prime })i_1}\sum _{i_2\ne i_1}{e}^{-j\frac{2\pi }{W}(-m+n+m^{\prime }-n^{\prime })i_2}.\hfill \end{array}$$

(33)

As in the ${\mathcal{L}}_2$ case described in Example 4, in ${\mathcal{L}}_3$, each subset is characterized by 2 subgroups of indices. Hence, the approach followed to compute (33) is identical to (32) and gives

$$\begin{array}{cc}\hfill {\mathsf{M}}_3^{\left(3\right)}& =\left[\mathbb{E}\right\{|a_x{|}^4\left\}\mathbb{E}\right\{|a_x{|}^2\}+\mathbb{E}\{|a_x{|}^2|a_y{|}^2\left\}\mathbb{E}\right\{|a_y{|}^2\left\}\right][R_s^2{\Delta }_f{\delta }_{k-k^{\prime }-pW}{\delta }_{m-n-m^{\prime }+n^{\prime }-pW}\hfill \\ \hfill & -R_s{\Delta }_f^2{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }-pW}].\hfill \end{array}$$

Similarly,

$$\begin{array}{cc}\hfill {\mathsf{N}}_3^{\left(3\right)}& =\mathbb{E}\left\{a_x{a}_y^{\ast }\right\}\mathbb{E}\{a_x^{\ast }{a}_y|a_x{|}^2\}[R_s^2{\Delta }_f{\delta }_{k-k^{\prime }-pW}{\delta }_{m-n-m^{\prime }+n^{\prime }-pW}-R_s{\Delta }_f^2{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }-pW}].\hfill \end{array}$$

All other contributions in ${\mathcal{L}}_3$ can be computed using the approach used in this example.

As shown in the above examples, each contribution ${\mathsf{M}}_g^{\left(h\right)}$ and ${\mathsf{N}}_g^{\left(h\right)}$ is nonzero only for a specific set of $(k,m,n,k^{\prime },m^{\prime },n^{\prime })$ values which is spanned by $p\in \mathbb{Z}$. However, the terms $(k,m,n,k^{\prime },m^{\prime },n^{\prime })$ arising for all $p\ne 0$ bring a total contribution to (25) that can be considered negligible. This is due to our assumption on $P\left(f\right)$ being strictly band-limited (see Section 3.1) and to the magnitude of the functions product ${\eta }_{k,m,n}{\eta }_{k^{\prime },m^{\prime },n^{\prime }}^{\ast }$ (see definitions (16) and (22)). Thus, in the computations performed in the following subsections, we will restrict ourselves to the case $p=0$.

6.1. Contributions in ${\mathcal{L}}_1$

In this section, the contributions ${\mathsf{M}}_1^{\left(i\right)}$, ${\mathsf{N}}_1^{\left(i\right)}$ for $i=1,2,\dots ,15$ are computed following Example 3. These contributions are listed in Table 1.

6.2. Contributions in ${\mathcal{L}}_2$

Following Example 4, the contributions ${\mathsf{M}}_2^{\left(h\right)},{\mathsf{N}}_2^{\left(h\right)},h=1,2,\dots ,10$, are computed and listed in Table 2.

6.3. Contributions in ${\mathcal{L}}_3$

Following Example 5, the contributions ${\mathsf{M}}_3^{\left(h\right)},{\mathsf{N}}_3^{\left(h\right)},h=1,2,\dots ,15$, are computed and listed in Table 3.

6.4. Contributions in ${\mathcal{L}}_4$

Since ${\mathcal{L}}_4$ comprises a single subset characterised by the single subgroup of all 6 indices (see Section 5.2), only one pair of contributions ${\mathsf{M}}_4^{\left(1\right)}$, ${\mathsf{N}}_4^{\left(1\right)}$ exists, and it is given by

$$\begin{array}{cc}\hfill {\mathsf{M}}_4^{\left(1\right)}& \triangleq \sum _{\mathit{i}\in {\mathcal{C}}_4^{\left(1\right)}}{\mathsf{S}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })\hfill \\ \hfill & =\sum _{i_1=0}^{W-1}\left[\mathbb{E}\right\{|a_x{|}^6\}+\mathbb{E}\{|a_x{|}^2|a_y{|}^4\left\}\right]e^{-j\frac{2\pi }{W}(k-m)i_1}\hfill \\ \hfill & =\left[\mathbb{E}\right\{|a_x{|}^6\}+\mathbb{E}\{|a_x{|}^2|a_y{|}^4\left\}\right]R_s{\Delta }_f^2{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }},\hfill \\ \hfill {\mathsf{N}}_4^{\left(1\right)}& \triangleq \sum _{\mathit{i}\in {\mathcal{C}}_4^{\left(1\right)}}{\mathsf{S}}_{\mathit{i}}(k,m,n,k^{\prime },m^{\prime },n^{\prime })=\mathbb{E}\{|a_x{|}^4|a_y{|}^2\}{R}_s{\Delta }_f^2{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}.\hfill \end{array}$$

7. Sum of All Contributions

In Section 5, we evaluated all contributions ${\mathsf{M}}_g^{\left(h\right)}$ and ${\mathsf{N}}_g^{\left(h\right)}$ to the PSD in (25). In particular, from (25)–(27), and (29), we have

$$\begin{array}{cc}\hfill & S_x(f,N_s,L_s)={\left(\frac{8}{9}\right)}^2{\gamma }^2{\Delta }_f^3\sum _{i=-\infty }^{\infty }\delta (f-i{\Delta }_f)\sum _{\begin{array}{c}(k,m,n)\in {\mathcal{S}}_i\\ (k^{\prime },m^{\prime },n^{\prime })\in {\mathcal{S}}_i\end{array}}\left[\mathsf{P}\sum _{g=1}^4\sum _{h=1}^{H\left(g\right)}{\mathsf{M}}_g^{\left(h\right)}+2Re\left\{\mathsf{P}\sum _{g=1}^4\sum _{h=1}^{H\left(g\right)}{\mathsf{N}}_g^{\left(h\right)}\right\}\right],\hfill \end{array}$$

(34)

where $H\left(g\right)$ is the number of subsets in the partitions of ${\mathcal{L}}_g$, $g=1,2,3,4$, described in Section 5.2 ($H\left(g\right)=15,10,15,1$ for $g=1,2,3,4$, resp.) and

$$\mathsf{P}\triangleq {\mathcal{P}}_{k,m,n,k^{\prime },m^{\prime },n^{\prime }}{\eta }_{k,m,n}{\eta }_{k^{\prime },m^{\prime },n^{\prime }}^{\ast }.$$

(35)

In this section, we evaluate ${\sum }_{g=1}^4{\sum }_{h=1}^{H\left(g\right)}{\mathsf{M}}_g^{\left(h\right)}$ and ${\sum }_{g=1}^4{\sum }_{h=1}^{H\left(g\right)}{\mathsf{N}}_g^{\left(h\right)}$ as well as compact the resulting expression as much as possible.

Before we proceed with computing the abovementioned summation, we remove the Kronecker deltas in ${\mathsf{M}}_g^{\left(h\right)}$ and ${\mathsf{N}}_g^{\left(h\right)}$ corresponding to contributions in the following subspaces: (i) $k=m$; (ii) $n=m$; (iii) $k^{\prime }=m^{\prime }$; and (iv) $n^{\prime }=m^{\prime }$. These contributions correspond to so-called bias terms, i.e., they arise from a component of the field ${\mathit{E}}_1(f,z)$ which is fully correlated with the transmitted field ${\mathit{E}}_0(t,0)$. This component, after CDC and MF, only results in a deterministic and static complex scaling of the received constellation, which is typically compensated at the receiver even in the presence of other noise sources in the system. Thus, it does not contribute to the power of the additive zero-mean interference component that we observe at the output of the MF + sampling stage once the received constellation is synchronised (in phase and amplitude) with the transmitted one. A more detailed discussion on these bias terms can be found in [8] (Appendix A), [14] (Appendix C). Moreover, the component ${\delta }_{k-m+n}{\delta }_{k^{\prime }-m^{\prime }+n^{\prime }}$ in ${\mathsf{M}}_g^{\left(h\right)}$ and ${\mathsf{N}}_g^{\left(h\right)}$ is also removed as it only gives nonzero contribution to the PSD for frequency $f=i{\Delta }_f=0$; hence, its effect on the total NLI variance vanishes as we let ${\Delta }_f\to 0$ (see Section 8). A total of 23 terms from the last columns of Table 1, Table 2 and Table 3 are thus removed. The remaining contributions are given in Table A4 in Appendix D.

We now compact the contributions in Table A4 by grouping the Kronecker delta products based on each correlation term they multiply. We use three pairs of curly brackets $\{·\}$ to denote the terms multiplying $R_s^3$, $R_s^2$, and $R_s$. The list of all Kronecker delta products multiplying each correlation term is shown in Table 4. The correlation terms are divided into intra-polarisation (expectations containing only $a_x$) and cross-polarisation terms (expectations containing both $a_x$ and $a_y$). Moreover, the correlations are categorised based on the specific contribution (either $\mathsf{M}$ or $\mathsf{N}$) in (34) to which they belong.

As it can be observed in Table 4, each correlation term is associated with different delta functions. To compact these terms, we exploit a property introduced in the following proposition.

Proposition 2.

Let $D_1(k,m,n,k^{\prime },m^{\prime },n^{\prime })$ and $D_2(k,m,n,k^{\prime },m^{\prime },n^{\prime })$ be two Kronecker delta products of the kind shown in Table 4. If

$$D_1(k,m,n,k^{\prime },m^{\prime },n^{\prime })=D_2(n,m,k,n^{\prime },m^{\prime },k^{\prime }),$$

(36)

then

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \sum _{\begin{array}{c}(k,m,n)\in {\mathcal{S}}_i\\ (k^{\prime },m^{\prime },n^{\prime })\in {\mathcal{S}}_i\end{array}}{\mathcal{P}}_{k,m,n,k^{\prime },m^{\prime },n^{\prime }}{\eta }_{k,m,n}{\eta }_{k^{\prime },m^{\prime },n^{\prime }}^{\ast }{D}_1(k,m,n,k^{\prime },m^{\prime },n^{\prime })\hfill \\ \hfill & =\sum _{\begin{array}{c}(k,m,n)\in {\mathcal{S}}_i\\ (k^{\prime },m^{\prime },n^{\prime })\in {\mathcal{S}}_i\end{array}}{\mathcal{P}}_{k,m,n,k^{\prime },m^{\prime },n^{\prime }}{\eta }_{k,m,n}{\eta }_{k^{\prime },m^{\prime },n^{\prime }}^{\ast }{D}_2(k,m,n,k^{\prime },m^{\prime },n^{\prime }).\hfill \end{array}\end{array}$$

(37)

This property also holds when applying the transformations $k=n$, $n=k$, $k^{\prime }=n^{\prime }$, and $n^{\prime }=k^{\prime }$, individually.

Proof. 

See Appendix E. □

The property in (37) allows us to group many of the Kronecker function products in Table 4 under a single term. Namely, the Kronecker delta products in Table 4 can be grouped in subsets that are closed to property (36), since they all result in the same value of summations in (37). In particular, 14 distinct subsets can be identified for the list of Kronecker delta products in Table 4. We label these subsets, which are shown in Table 5, as ${\mathcal{D}}_l$ for $l=1,2,\dots ,14$.

Summing all the contributions in Table 4, using Proposition 2 for the elements in the subsets listed in Table 4, and finally ordering by Kronecker delta product, we obtain from (34)

$$\begin{array}{cc}\hfill S_x(f,N_s& ,L_s)={\left(\frac{8}{9}\right)}^2{\gamma }^2{\Delta }_f\sum _{i=-\infty }^{\infty }\delta (f-i{\Delta }_f)\hfill \\ \hfill & ·\sum _{\begin{array}{c}(k,m,n)\in {\mathcal{S}}_i\\ (k^{\prime },m^{\prime },n^{\prime })\in {\mathcal{S}}_i\end{array}}[R_s^3{\Delta }_f^2[({\mathsf{a}}_1\mathsf{P}+2Re\left\{{\mathsf{a}}_1^{\prime }\mathsf{P}\right\}){\delta }_{k-k^{\prime }}{\delta }_{m-m^{\prime }}{\delta }_{n-n^{\prime }}+({\mathsf{a}}_2\mathsf{P}+2Re\left\{{\mathsf{a}}_2^{\prime }\mathsf{P}\right\}){\delta }_{k-k^{\prime }}{\delta }_{m+n^{\prime }}{\delta }_{n+m^{\prime }}\hfill \\ \hfill & +({\mathsf{a}}_3\mathsf{P}+2Re\left\{{\mathsf{a}}_3^{\prime }\mathsf{P}\right\}){\delta }_{k+n}{\delta }_{m-m^{\prime }}{\delta }_{k^{\prime }+n^{\prime }}]+R_s^2{\Delta }_f^3[({\mathsf{b}}_1\mathsf{P}+2Re\left\{{\mathsf{b}}_1^{\prime }\mathsf{P}\right\}){\delta }_{k-m-k^{\prime }}{\delta }_{n+m^{\prime }-n^{\prime }}\hfill \\ \hfill & +({\mathsf{b}}_2\mathsf{P}+2Re\left\{{\mathsf{b}}_2^{\prime }\mathsf{P}\right\}){\delta }_{k-m+m^{\prime }}{\delta }_{n-k^{\prime }-n^{\prime }}+({\mathsf{b}}_2^{\ast }\mathsf{P}+2Re\left\{{\mathsf{b}}_3^{\prime }\mathsf{P}\right\}){\delta }_{k+n-k^{\prime }}{\delta }_{m-m^{\prime }+n^{\prime }}\hfill \\ \hfill & +({\mathsf{b}}_4\mathsf{P}+2Re\left\{{\mathsf{b}}_4^{\prime }\mathsf{P}\right\}){\delta }_{k+n+m^{\prime }}{\delta }_{m+k^{\prime }+n^{\prime }}+({\mathsf{c}}_1\mathsf{P}+2Re\left\{{\mathsf{c}}_1^{\prime }\mathsf{P}\right\}){\delta }_{k+n}{\delta }_{m+k^{\prime }-m^{\prime }+n^{\prime }}\hfill \\ \hfill & +({\mathsf{c}}_2\mathsf{P}+2Re\left\{{\mathsf{c}}_2^{\prime }\mathsf{P}\right\}){\delta }_{k-k^{\prime }}{\delta }_{m-n-m^{\prime }+n^{\prime }}+({\mathsf{c}}_3\mathsf{P}+2Re\left\{{\mathsf{c}}_3^{\prime }\mathsf{P}\right\}){\delta }_{k+m^{\prime }}{\delta }_{m-n+k^{\prime }+n^{\prime }}\hfill \\ \hfill & +({\mathsf{c}}_4\mathsf{P}+2Re\left\{{\mathsf{c}}_4^{\prime }\mathsf{P}\right\}){\delta }_{m-m^{\prime }}{\delta }_{k+n-k^{\prime }-n^{\prime }}+({\mathsf{c}}_3^{\ast }\mathsf{P}+2Re\left\{{\mathsf{c}}_5^{\prime }\mathsf{P}\right\}){\delta }_{m+k^{\prime }}{\delta }_{k+n+m^{\prime }-n^{\prime }}\hfill \\ \hfill & +({\mathsf{c}}_1^{\ast }\mathsf{P}+2Re\left\{{\mathsf{c}}_6^{\prime }\mathsf{P}\right\}){\delta }_{k^{\prime }+n^{\prime }}{\delta }_{k-m+n+m^{\prime }}]+R_s{\Delta }_f^4({\mathsf{d}}_1\mathsf{P}+2Re\left\{{\mathsf{d}}_1^{\prime }\mathsf{P}\right\}){\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}],\hfill \end{array}$$

(38)

where the coefficients multiplying $\mathsf{P}$ are listed in Table A5 in Appendix F and where coset leaders in Table 5 have been used.

Table 4. List of Kronecker delta contributions ordered by the corresponding high-order moment or correlation of the transmitted modulation format.

Correlation Terms	Kronecker Delta Products
Intra-Polarisation Terms
In ${\mathsf{M}}_{\mathit{g}}^{\left(\mathit{h}\right)}$
${\mathbb{E}}^3\{|a_x{|}^2\}$	{${\delta }_{k-k^{\prime }}{\delta }_{m-m^{\prime }}{\delta }_{n-n^{\prime }},{\delta }_{k-n^{\prime }}{\delta }_{m-m^{\prime }}{\delta }_{n-k^{\prime }}$},{$-2{\delta }_{n-k^{\prime }}{\delta }_{k-m+m^{\prime }-n^{\prime }},-2{\delta }_{n-n^{\prime }}{\delta }_{k-m-k^{\prime }+m^{\prime }},$
	$-2{\delta }_{k-k^{\prime }}{\delta }_{m-n-m^{\prime }+n^{\prime }},-2{\delta }_{m-m^{\prime }}{\delta }_{k+n-k^{\prime }-n^{\prime }},-2{\delta }_{k-n^{\prime }}{\delta }_{m-n+k^{\prime }-m^{\prime }}$},{$12{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}$}
$\mathbb{E}\{|a_x{|}^2\}|\mathbb{E}\left\{a_x^2\right\}{|}^2$	{${\delta }_{k+n}{\delta }_{m-m^{\prime }}{\delta }_{k^{\prime }+n^{\prime }},{\delta }_{k-k^{\prime }}{\delta }_{m+n^{\prime }}{\delta }_{n+m^{\prime }},{\delta }_{k+m^{\prime }}{\delta }_{m+k^{\prime }}{\delta }_{n-n^{\prime }},{\delta }_{k+m^{\prime }}{\delta }_{m+n^{\prime }}{\delta }_{n-k^{\prime }},$
	${\delta }_{k-n^{\prime }}{\delta }_{m+k^{\prime }}{\delta }_{n+m^{\prime }}$},{$-{\delta }_{n-k^{\prime }}{\delta }_{k-m+m^{\prime }-n^{\prime }},-3{\delta }_{m+k^{\prime }}{\delta }_{k+n+m^{\prime }-n^{\prime }},-3{\delta }_{k+n}{\delta }_{m+k^{\prime }-m^{\prime }+n^{\prime }}$
	$-3{\delta }_{k^{\prime }+n^{\prime }}{\delta }_{k+n-m+m^{\prime }},-{\delta }_{m-m^{\prime }}{\delta }_{k+n-k^{\prime }-n^{\prime }},-3{\delta }_{m+n^{\prime }}{\delta }_{k+n-k^{\prime }+m^{\prime }},-3{\delta }_{n+m^{\prime }}{\delta }_{k-m-k^{\prime }-n^{\prime }},$
	$-{\delta }_{k-k^{\prime }}{\delta }_{m-n-m^{\prime }+n^{\prime }},-3{\delta }_{k+m^{\prime }}{\delta }_{m-n+k^{\prime }+n^{\prime }},-{\delta }_{n-n^{\prime }}{\delta }_{k-m-k^{\prime }+m^{\prime }},-{\delta }_{k-n^{\prime }}{\delta }_{m-n+k^{\prime }-m^{\prime }},$}
	{$18{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}$}
$|\mathbb{E}\{a_x|a_x{|}^2{\}|}^2$	{}, {${\delta }_{k-m-k^{\prime }}{\delta }_{n+m^{\prime }-n^{\prime }},{\delta }_{k-m+m^{\prime }}{\delta }_{n-k^{\prime }-n^{\prime }},{\delta }_{k-m-n^{\prime }}{\delta }_{n-k^{\prime }+m^{\prime }},{\delta }_{k+n-k^{\prime }}{\delta }_{m-m^{\prime }+n^{\prime }},$
	${\delta }_{k+n-n^{\prime }}{\delta }_{m+k^{\prime }-m^{\prime }},{\delta }_{k-k^{\prime }+m^{\prime }}{\delta }_{m-n+n^{\prime }},{\delta }_{k-k^{\prime }-n^{\prime }}{\delta }_{m-n+m^{\prime }},{\delta }_{k+m^{\prime }-n^{\prime }}{\delta }_{m-n+k^{\prime }}$},
	{$-9{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}$}
$|\mathbb{E}\left\{a_x^3\right\}{|}^2$	{}, $\left\{{\delta }_{k+n+m^{\prime }}{\delta }_{m+k^{\prime }+n^{\prime }}\right\},\{-{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
$\mathbb{E}\{|a_x{|}^4\left\}\mathbb{E}\right\{|a_x{|}^2\}$	{}, $\{{\delta }_{k-k^{\prime }}{\delta }_{m-n-m^{\prime }+n^{\prime }},{\delta }_{k-n^{\prime }}{\delta }_{m-n+k^{\prime }-m^{\prime }},{\delta }_{m-m^{\prime }}{\delta }_{k+n-k^{\prime }-n^{\prime }},{\delta }_{n-k^{\prime }}{\delta }_{k-m+m^{\prime }-n^{\prime }},$
	${\delta }_{n-n^{\prime }}{\delta }_{k-m-k^{\prime }+m^{\prime }}\},\{-9{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
${\mathbb{E}}^{\ast }\{a_x^2|a_x{|}^2\}\mathbb{E}\left\{a_x^2\right\}$	{}, $\{{\delta }_{k+n}{\delta }_{m+k^{\prime }-m^{\prime }+n^{\prime }},{\delta }_{k+m^{\prime }}{\delta }_{m-n+k^{\prime }+n^{\prime }},{\delta }_{n+m^{\prime }}{\delta }_{k-m-k^{\prime }-n^{\prime }}\}$,
	$\{-3{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
$\mathbb{E}\{a_x^2|a_x{|}^2\}{\mathbb{E}}^{\ast }\left\{a_x^2\right\}$	{}, $\{{\delta }_{m+k^{\prime }}{\delta }_{k-n+m^{\prime }-n^{\prime }},{\delta }_{m+n^{\prime }}{\delta }_{k+n-k^{\prime }+m^{\prime }},{\delta }_{k^{\prime }+n^{\prime }}{\delta }_{k-m+n+m^{\prime }}\},\{-3{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
$\mathbb{E}\{|a_x{|}^6\}$	$\left\{\right\}$, $\left\{\right\}$, $\left\{{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\right\}$
Cross-polarisation terms
In ${\mathsf{M}}_{\mathit{g}}^{\left(\mathit{h}\right)}$
$\mathbb{E}\{|a_x{|}^2\}{\mathbb{E}}^2\{|a_y{|}^2\}$	$\left\{{\delta }_{k-k^{\prime }}{\delta }_{m-m^{\prime }}{\delta }_{n-n^{\prime }}\right\},\{-2{\delta }_{n-n^{\prime }}{\delta }_{k-m-k^{\prime }+m^{\prime }},-{\delta }_{m-m^{\prime }}{\delta }_{k+n-k^{\prime }-n^{\prime }},-{\delta }_{k-k^{\prime }}{\delta }_{m-n-m^{\prime }+n^{\prime }}\},$
	$\left\{4{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\right\}$
$\mathbb{E}\{|a_x{|}^2\}|\mathbb{E}\left\{a_y^2\right\}{|}^2$	$\left\{{\delta }_{k+m^{\prime }}{\delta }_{m+k^{\prime }}{\delta }_{n-n^{\prime }}\right\},\{-{\delta }_{n-n^{\prime }}{\delta }_{k-m-k^{\prime }+m^{\prime }},-{\delta }_{m+k^{\prime }}{\delta }_{k+n+m^{\prime }-n^{\prime }},-{\delta }_{k+m^{\prime }}{\delta }_{m-n+k^{\prime }+n^{\prime }}\},$
	$\left\{2{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\right\}$
$\mathbb{E}\{|a_x{|}^2\left\}\mathbb{E}\right\{|a_y{|}^4\}$	{}, {${\delta }_{n-n^{\prime }}{\delta }_{k-m-k^{\prime }+m^{\prime }}\},\{-{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
$\mathbb{E}\{|a_x{|}^2{a}_y^{\ast }\left\}\mathbb{E}\right\{a_y|a_y{|}^2\}$	{}, $\{{\delta }_{k-m+m^{\prime }}{\delta }_{n-k^{\prime }-n^{\prime }},{\delta }_{k-k^{\prime }+m^{\prime }}{\delta }_{m-n+n^{\prime }}\},\{-2{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
$\mathbb{E}\{|a_x{|}^2|a_y{|}^2\left\}\mathbb{E}\right\{|a_y{|}^2\}$	{}, $\{{\delta }_{k-k^{\prime }}{\delta }_{m-n-m^{\prime }+n^{\prime }},{\delta }_{m-m^{\prime }}{\delta }_{k+n-k^{\prime }-n^{\prime }}\},\{-4{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
$\mathbb{E}\left\{a_x{a}_y\right\}{\mathbb{E}}^{\ast }\{a_x{a}_y|a_x{|}^2\}$	{}, $\{{\delta }_{k+m^{\prime }}{\delta }_{m-n+k^{\prime }+n^{\prime }},{\delta }_{m+k^{\prime }}{\delta }_{k-n+m^{\prime }-n^{\prime }},{\delta }_{n+m^{\prime }}{\delta }_{k-m-k^{\prime }-n^{\prime }},{\delta }_{k^{\prime }+n^{\prime }}{\delta }_{k-m+n+m^{\prime }}\},$
	$\{-4{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
${\mathbb{E}}^{\ast }\{|a_x{|}^2{a}_y^2\}\mathbb{E}\left\{a_y^2\right\}$	{}, $\left\{{\delta }_{k+m^{\prime }}{\delta }_{m-n+k^{\prime }+n^{\prime }}\right\},\{-{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
$\mathbb{E}\{|a_x{|}^2{a}_y^2\}{\mathbb{E}}^{\ast }\left\{a_y^2\right\}$	{}, $\left\{{\delta }_{m+k^{\prime }}{\delta }_{k-n+m^{\prime }-n^{\prime }}\right\},\{-{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
$\mathbb{E}\{|a_x{|}^2|a_y{|}^4\}$	{}, {}, $\left\{{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\right\}$
$|\mathbb{E}\left\{a_x{a}_y^{\ast }\right\}{|}^2\mathbb{E}\{|a_y{|}^2\}$	$\left\{{\delta }_{k-n^{\prime }}{\delta }_{m-m^{\prime }}{\delta }_{n-k^{\prime }}\right\},\{-2{\delta }_{n-k^{\prime }}{\delta }_{k-m+m^{\prime }-n^{\prime }},-2{\delta }_{k-n^{\prime }}{\delta }_{m-n+k^{\prime }-m^{\prime }},-{\delta }_{k-k^{\prime }}{\delta }_{m-n-m^{\prime }+n^{\prime }},$
	$-{\delta }_{m-m^{\prime }}{\delta }_{k+n-k^{\prime }-n^{\prime }}\},\left\{8{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\right\}$
$\mathbb{E}\left\{a_x{a}_y\right\}\mathbb{E}\left\{a_x^{\ast }{a}_y\right\}{\mathbb{E}}^{\ast }\left\{a_y^2\right\}$	$\left\{{\delta }_{k-n^{\prime }}{\delta }_{m+k^{\prime }}{\delta }_{n+m^{\prime }}\right\},\{-2{\delta }_{m+k^{\prime }}{\delta }_{k+n+m^{\prime }-n^{\prime }},-{\delta }_{k+n}{\delta }_{m+k^{\prime }-m^{\prime }+n^{\prime }},-{\delta }_{n+m^{\prime }}{\delta }_{k-m-k^{\prime }-n^{\prime }},$
	$-{\delta }_{k-n^{\prime }}{\delta }_{m-n+k^{\prime }-m^{\prime }}\},\left\{4{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\right\}$
${\mathbb{E}}^{\ast }\left\{a_x{a}_y\right\}\mathbb{E}\left\{a_x{a}_y^{\ast }\right\}\mathbb{E}\left\{a_y^2\right\}$	$\left\{{\delta }_{k+m^{\prime }}{\delta }_{m+n^{\prime }}{\delta }_{n-k^{\prime }}\right\},\{-2{\delta }_{k+m^{\prime }}{\delta }_{m-n+k^{\prime }+n^{\prime }},-{\delta }_{k^{\prime }+n^{\prime }}{\delta }_{k-m+n+m^{\prime }},-{\delta }_{n-k^{\prime }}{\delta }_{k-m+m^{\prime }-n^{\prime }},$
	$-{\delta }_{m+n^{\prime }}{\delta }_{k+n-k^{\prime }+m^{\prime }}\},\left\{4{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\right\}$
$|\mathbb{E}\left\{a_x{a}_y\right\}{|}^2\mathbb{E}\{|a_y{|}^2\}$	{${\delta }_{k+n}{\delta }_{m-m^{\prime }}{\delta }_{k^{\prime }+n^{\prime }},{\delta }_{k-k^{\prime }}{\delta }_{m+n^{\prime }}{\delta }_{n+m^{\prime }}\},\{-2{\delta }_{k^{\prime }+n^{\prime }}{\delta }_{k-m+n+m^{\prime }},-2{\delta }_{k+n}{\delta }_{m-m^{\prime }+k^{\prime }+n^{\prime }},$
	$-2{\delta }_{n+m^{\prime }}{\delta }_{k-m-k^{\prime }-n^{\prime }},-2{\delta }_{m+n^{\prime }}{\delta }_{k+n-k^{\prime }+m^{\prime }},-{\delta }_{m-m^{\prime }}{\delta }_{k+n-k^{\prime }-n^{\prime }},-{\delta }_{k-k^{\prime }}{\delta }_{m-n-m^{\prime }+n^{\prime }}\}$,
	$\left\{8{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\right\}$
$|\mathbb{E}\{a_x|a_y{|}^2{\}|}^2$	{}, $\{{\delta }_{k-m-n^{\prime }}{\delta }_{n-k^{\prime }+m^{\prime }},{\delta }_{k+n-k^{\prime }}{\delta }_{m-m^{\prime }+n^{\prime }},{\delta }_{k-k^{\prime }-n^{\prime }}{\delta }_{m-n+m^{\prime }}\},\{-4{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
$|\mathbb{E}\left\{a_x{a}_y^2\right\}{|}^2$	{}, $\left\{{\delta }_{k+n+m^{\prime }}{\delta }_{m+k^{\prime }+n^{\prime }}\right\},\{-{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
$|\mathbb{E}\left\{a_x^{\ast }{a}_y^2\right\}{|}^2$	{}, $\left\{{\delta }_{k+m^{\prime }-n^{\prime }}{\delta }_{m-n+k^{\prime }}\right\},\{-{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
In ${\mathsf{N}}_{\mathit{g}}^{\left(\mathit{h}\right)}$
$\mathbb{E}\{|a_x{|}^2\}|\mathbb{E}\left\{a_x{a}_y^{\ast }\right\}{|}^2$	{${\delta }_{k-k^{\prime }}{\delta }_{m-m^{\prime }}{\delta }_{n-n^{\prime }},{\delta }_{k-n^{\prime }}{\delta }_{m-m^{\prime }}{\delta }_{n-k^{\prime }}\},\{-2{\delta }_{n-k^{\prime }}{\delta }_{k-m+m^{\prime }-n^{\prime }},-2{\delta }_{k-k^{\prime }}{\delta }_{m-n-m^{\prime }+n^{\prime }},$
	$-2{\delta }_{m-m^{\prime }}{\delta }_{k+n-k^{\prime }-n^{\prime }},-{\delta }_{n-n^{\prime }}{\delta }_{k-m-k^{\prime }+m^{\prime }},-{\delta }_{k-n^{\prime }}{\delta }_{m-n+k^{\prime }-m^{\prime }}\},\left\{8{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\right\}$
$\mathbb{E}\{|a_x{|}^2\}|\mathbb{E}\left\{a_x{a}_y\right\}{|}^2$	$\{{\delta }_{k+m^{\prime }}{\delta }_{m+k^{\prime }}{\delta }_{n-n^{\prime }},{\delta }_{k-n^{\prime }}{\delta }_{m+k^{\prime }}{\delta }_{n+m^{\prime }}$

Table 4. List of Kronecker delta contributions ordered by the corresponding high-order moment or correlation of the transmitted modulation format.

Correlation Terms	Kronecker Delta Products
Intra-Polarisation Terms
In ${\mathsf{M}}_{\mathit{g}}^{\left(\mathit{h}\right)}$
${\mathbb{E}}^3\{|a_x{|}^2\}$	{${\delta }_{k-k^{\prime }}{\delta }_{m-m^{\prime }}{\delta }_{n-n^{\prime }},{\delta }_{k-n^{\prime }}{\delta }_{m-m^{\prime }}{\delta }_{n-k^{\prime }}$},{$-2{\delta }_{n-k^{\prime }}{\delta }_{k-m+m^{\prime }-n^{\prime }},-2{\delta }_{n-n^{\prime }}{\delta }_{k-m-k^{\prime }+m^{\prime }},$
	$-2{\delta }_{k-k^{\prime }}{\delta }_{m-n-m^{\prime }+n^{\prime }},-2{\delta }_{m-m^{\prime }}{\delta }_{k+n-k^{\prime }-n^{\prime }},-2{\delta }_{k-n^{\prime }}{\delta }_{m-n+k^{\prime }-m^{\prime }}$},{$12{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}$}
$\mathbb{E}\{|a_x{|}^2\}|\mathbb{E}\left\{a_x^2\right\}{|}^2$	{${\delta }_{k+n}{\delta }_{m-m^{\prime }}{\delta }_{k^{\prime }+n^{\prime }},{\delta }_{k-k^{\prime }}{\delta }_{m+n^{\prime }}{\delta }_{n+m^{\prime }},{\delta }_{k+m^{\prime }}{\delta }_{m+k^{\prime }}{\delta }_{n-n^{\prime }},{\delta }_{k+m^{\prime }}{\delta }_{m+n^{\prime }}{\delta }_{n-k^{\prime }},$
	${\delta }_{k-n^{\prime }}{\delta }_{m+k^{\prime }}{\delta }_{n+m^{\prime }}$},{$-{\delta }_{n-k^{\prime }}{\delta }_{k-m+m^{\prime }-n^{\prime }},-3{\delta }_{m+k^{\prime }}{\delta }_{k+n+m^{\prime }-n^{\prime }},-3{\delta }_{k+n}{\delta }_{m+k^{\prime }-m^{\prime }+n^{\prime }}$
	$-3{\delta }_{k^{\prime }+n^{\prime }}{\delta }_{k+n-m+m^{\prime }},-{\delta }_{m-m^{\prime }}{\delta }_{k+n-k^{\prime }-n^{\prime }},-3{\delta }_{m+n^{\prime }}{\delta }_{k+n-k^{\prime }+m^{\prime }},-3{\delta }_{n+m^{\prime }}{\delta }_{k-m-k^{\prime }-n^{\prime }},$
	$-{\delta }_{k-k^{\prime }}{\delta }_{m-n-m^{\prime }+n^{\prime }},-3{\delta }_{k+m^{\prime }}{\delta }_{m-n+k^{\prime }+n^{\prime }},-{\delta }_{n-n^{\prime }}{\delta }_{k-m-k^{\prime }+m^{\prime }},-{\delta }_{k-n^{\prime }}{\delta }_{m-n+k^{\prime }-m^{\prime }},$}
	{$18{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}$}
$|\mathbb{E}\{a_x|a_x{|}^2{\}|}^2$	{}, {${\delta }_{k-m-k^{\prime }}{\delta }_{n+m^{\prime }-n^{\prime }},{\delta }_{k-m+m^{\prime }}{\delta }_{n-k^{\prime }-n^{\prime }},{\delta }_{k-m-n^{\prime }}{\delta }_{n-k^{\prime }+m^{\prime }},{\delta }_{k+n-k^{\prime }}{\delta }_{m-m^{\prime }+n^{\prime }},$
	${\delta }_{k+n-n^{\prime }}{\delta }_{m+k^{\prime }-m^{\prime }},{\delta }_{k-k^{\prime }+m^{\prime }}{\delta }_{m-n+n^{\prime }},{\delta }_{k-k^{\prime }-n^{\prime }}{\delta }_{m-n+m^{\prime }},{\delta }_{k+m^{\prime }-n^{\prime }}{\delta }_{m-n+k^{\prime }}$},
	{$-9{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}$}
$|\mathbb{E}\left\{a_x^3\right\}{|}^2$	{}, $\left\{{\delta }_{k+n+m^{\prime }}{\delta }_{m+k^{\prime }+n^{\prime }}\right\},\{-{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
$\mathbb{E}\{|a_x{|}^4\left\}\mathbb{E}\right\{|a_x{|}^2\}$	{}, $\{{\delta }_{k-k^{\prime }}{\delta }_{m-n-m^{\prime }+n^{\prime }},{\delta }_{k-n^{\prime }}{\delta }_{m-n+k^{\prime }-m^{\prime }},{\delta }_{m-m^{\prime }}{\delta }_{k+n-k^{\prime }-n^{\prime }},{\delta }_{n-k^{\prime }}{\delta }_{k-m+m^{\prime }-n^{\prime }},$
	${\delta }_{n-n^{\prime }}{\delta }_{k-m-k^{\prime }+m^{\prime }}\},\{-9{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
${\mathbb{E}}^{\ast }\{a_x^2|a_x{|}^2\}\mathbb{E}\left\{a_x^2\right\}$	{}, $\{{\delta }_{k+n}{\delta }_{m+k^{\prime }-m^{\prime }+n^{\prime }},{\delta }_{k+m^{\prime }}{\delta }_{m-n+k^{\prime }+n^{\prime }},{\delta }_{n+m^{\prime }}{\delta }_{k-m-k^{\prime }-n^{\prime }}\}$,
	$\{-3{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
$\mathbb{E}\{a_x^2|a_x{|}^2\}{\mathbb{E}}^{\ast }\left\{a_x^2\right\}$	{}, $\{{\delta }_{m+k^{\prime }}{\delta }_{k-n+m^{\prime }-n^{\prime }},{\delta }_{m+n^{\prime }}{\delta }_{k+n-k^{\prime }+m^{\prime }},{\delta }_{k^{\prime }+n^{\prime }}{\delta }_{k-m+n+m^{\prime }}\},\{-3{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
$\mathbb{E}\{|a_x{|}^6\}$	$\left\{\right\}$, $\left\{\right\}$, $\left\{{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\right\}$
Cross-polarisation terms
In ${\mathsf{M}}_{\mathit{g}}^{\left(\mathit{h}\right)}$
$\mathbb{E}\{|a_x{|}^2\}{\mathbb{E}}^2\{|a_y{|}^2\}$	$\left\{{\delta }_{k-k^{\prime }}{\delta }_{m-m^{\prime }}{\delta }_{n-n^{\prime }}\right\},\{-2{\delta }_{n-n^{\prime }}{\delta }_{k-m-k^{\prime }+m^{\prime }},-{\delta }_{m-m^{\prime }}{\delta }_{k+n-k^{\prime }-n^{\prime }},-{\delta }_{k-k^{\prime }}{\delta }_{m-n-m^{\prime }+n^{\prime }}\},$
	$\left\{4{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\right\}$
$\mathbb{E}\{|a_x{|}^2\}|\mathbb{E}\left\{a_y^2\right\}{|}^2$	$\left\{{\delta }_{k+m^{\prime }}{\delta }_{m+k^{\prime }}{\delta }_{n-n^{\prime }}\right\},\{-{\delta }_{n-n^{\prime }}{\delta }_{k-m-k^{\prime }+m^{\prime }},-{\delta }_{m+k^{\prime }}{\delta }_{k+n+m^{\prime }-n^{\prime }},-{\delta }_{k+m^{\prime }}{\delta }_{m-n+k^{\prime }+n^{\prime }}\},$
	$\left\{2{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\right\}$
$\mathbb{E}\{|a_x{|}^2\left\}\mathbb{E}\right\{|a_y{|}^4\}$	{}, {${\delta }_{n-n^{\prime }}{\delta }_{k-m-k^{\prime }+m^{\prime }}\},\{-{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
$\mathbb{E}\{|a_x{|}^2{a}_y^{\ast }\left\}\mathbb{E}\right\{a_y|a_y{|}^2\}$	{}, $\{{\delta }_{k-m+m^{\prime }}{\delta }_{n-k^{\prime }-n^{\prime }},{\delta }_{k-k^{\prime }+m^{\prime }}{\delta }_{m-n+n^{\prime }}\},\{-2{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
$\mathbb{E}\{|a_x{|}^2|a_y{|}^2\left\}\mathbb{E}\right\{|a_y{|}^2\}$	{}, $\{{\delta }_{k-k^{\prime }}{\delta }_{m-n-m^{\prime }+n^{\prime }},{\delta }_{m-m^{\prime }}{\delta }_{k+n-k^{\prime }-n^{\prime }}\},\{-4{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
$\mathbb{E}\left\{a_x{a}_y\right\}{\mathbb{E}}^{\ast }\{a_x{a}_y|a_x{|}^2\}$	{}, $\{{\delta }_{k+m^{\prime }}{\delta }_{m-n+k^{\prime }+n^{\prime }},{\delta }_{m+k^{\prime }}{\delta }_{k-n+m^{\prime }-n^{\prime }},{\delta }_{n+m^{\prime }}{\delta }_{k-m-k^{\prime }-n^{\prime }},{\delta }_{k^{\prime }+n^{\prime }}{\delta }_{k-m+n+m^{\prime }}\},$
	$\{-4{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
${\mathbb{E}}^{\ast }\{|a_x{|}^2{a}_y^2\}\mathbb{E}\left\{a_y^2\right\}$	{}, $\left\{{\delta }_{k+m^{\prime }}{\delta }_{m-n+k^{\prime }+n^{\prime }}\right\},\{-{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
$\mathbb{E}\{|a_x{|}^2{a}_y^2\}{\mathbb{E}}^{\ast }\left\{a_y^2\right\}$	{}, $\left\{{\delta }_{m+k^{\prime }}{\delta }_{k-n+m^{\prime }-n^{\prime }}\right\},\{-{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
$\mathbb{E}\{|a_x{|}^2|a_y{|}^4\}$	{}, {}, $\left\{{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\right\}$
$|\mathbb{E}\left\{a_x{a}_y^{\ast }\right\}{|}^2\mathbb{E}\{|a_y{|}^2\}$	$\left\{{\delta }_{k-n^{\prime }}{\delta }_{m-m^{\prime }}{\delta }_{n-k^{\prime }}\right\},\{-2{\delta }_{n-k^{\prime }}{\delta }_{k-m+m^{\prime }-n^{\prime }},-2{\delta }_{k-n^{\prime }}{\delta }_{m-n+k^{\prime }-m^{\prime }},-{\delta }_{k-k^{\prime }}{\delta }_{m-n-m^{\prime }+n^{\prime }},$
	$-{\delta }_{m-m^{\prime }}{\delta }_{k+n-k^{\prime }-n^{\prime }}\},\left\{8{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\right\}$
$\mathbb{E}\left\{a_x{a}_y\right\}\mathbb{E}\left\{a_x^{\ast }{a}_y\right\}{\mathbb{E}}^{\ast }\left\{a_y^2\right\}$	$\left\{{\delta }_{k-n^{\prime }}{\delta }_{m+k^{\prime }}{\delta }_{n+m^{\prime }}\right\},\{-2{\delta }_{m+k^{\prime }}{\delta }_{k+n+m^{\prime }-n^{\prime }},-{\delta }_{k+n}{\delta }_{m+k^{\prime }-m^{\prime }+n^{\prime }},-{\delta }_{n+m^{\prime }}{\delta }_{k-m-k^{\prime }-n^{\prime }},$
	$-{\delta }_{k-n^{\prime }}{\delta }_{m-n+k^{\prime }-m^{\prime }}\},\left\{4{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\right\}$
${\mathbb{E}}^{\ast }\left\{a_x{a}_y\right\}\mathbb{E}\left\{a_x{a}_y^{\ast }\right\}\mathbb{E}\left\{a_y^2\right\}$	$\left\{{\delta }_{k+m^{\prime }}{\delta }_{m+n^{\prime }}{\delta }_{n-k^{\prime }}\right\},\{-2{\delta }_{k+m^{\prime }}{\delta }_{m-n+k^{\prime }+n^{\prime }},-{\delta }_{k^{\prime }+n^{\prime }}{\delta }_{k-m+n+m^{\prime }},-{\delta }_{n-k^{\prime }}{\delta }_{k-m+m^{\prime }-n^{\prime }},$
	$-{\delta }_{m+n^{\prime }}{\delta }_{k+n-k^{\prime }+m^{\prime }}\},\left\{4{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\right\}$
$|\mathbb{E}\left\{a_x{a}_y\right\}{|}^2\mathbb{E}\{|a_y{|}^2\}$	{${\delta }_{k+n}{\delta }_{m-m^{\prime }}{\delta }_{k^{\prime }+n^{\prime }},{\delta }_{k-k^{\prime }}{\delta }_{m+n^{\prime }}{\delta }_{n+m^{\prime }}\},\{-2{\delta }_{k^{\prime }+n^{\prime }}{\delta }_{k-m+n+m^{\prime }},-2{\delta }_{k+n}{\delta }_{m-m^{\prime }+k^{\prime }+n^{\prime }},$
	$-2{\delta }_{n+m^{\prime }}{\delta }_{k-m-k^{\prime }-n^{\prime }},-2{\delta }_{m+n^{\prime }}{\delta }_{k+n-k^{\prime }+m^{\prime }},-{\delta }_{m-m^{\prime }}{\delta }_{k+n-k^{\prime }-n^{\prime }},-{\delta }_{k-k^{\prime }}{\delta }_{m-n-m^{\prime }+n^{\prime }}\}$,
	$\left\{8{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\right\}$
$|\mathbb{E}\{a_x|a_y{|}^2{\}|}^2$	{}, $\{{\delta }_{k-m-n^{\prime }}{\delta }_{n-k^{\prime }+m^{\prime }},{\delta }_{k+n-k^{\prime }}{\delta }_{m-m^{\prime }+n^{\prime }},{\delta }_{k-k^{\prime }-n^{\prime }}{\delta }_{m-n+m^{\prime }}\},\{-4{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
$|\mathbb{E}\left\{a_x{a}_y^2\right\}{|}^2$	{}, $\left\{{\delta }_{k+n+m^{\prime }}{\delta }_{m+k^{\prime }+n^{\prime }}\right\},\{-{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
$|\mathbb{E}\left\{a_x^{\ast }{a}_y^2\right\}{|}^2$	{}, $\left\{{\delta }_{k+m^{\prime }-n^{\prime }}{\delta }_{m-n+k^{\prime }}\right\},\{-{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\}$
In ${\mathsf{N}}_{\mathit{g}}^{\left(\mathit{h}\right)}$
$\mathbb{E}\{|a_x{|}^2\}|\mathbb{E}\left\{a_x{a}_y^{\ast }\right\}{|}^2$	{${\delta }_{k-k^{\prime }}{\delta }_{m-m^{\prime }}{\delta }_{n-n^{\prime }},{\delta }_{k-n^{\prime }}{\delta }_{m-m^{\prime }}{\delta }_{n-k^{\prime }}\},\{-2{\delta }_{n-k^{\prime }}{\delta }_{k-m+m^{\prime }-n^{\prime }},-2{\delta }_{k-k^{\prime }}{\delta }_{m-n-m^{\prime }+n^{\prime }},$
	$-2{\delta }_{m-m^{\prime }}{\delta }_{k+n-k^{\prime }-n^{\prime }},-{\delta }_{n-n^{\prime }}{\delta }_{k-m-k^{\prime }+m^{\prime }},-{\delta }_{k-n^{\prime }}{\delta }_{m-n+k^{\prime }-m^{\prime }}\},\left\{8{\delta }_{k-m+n-k^{\prime }+m^{\prime }-n^{\prime }}\right\}$
$\mathbb{E}\{|a_x{|}^2\}|\mathbb{E}\left\{a_x{a}_y\right\}{|}^2$	$\{{\delta }_{k+m^{\prime }}{\delta }_{m+k^{\prime }}{\delta }_{n-n^{\prime }},{\delta }_{k-n^{\prime }}{\delta }_{m+k^{\prime }}{\delta }_{n+m^{\prime }}$