A Gamma-Mixture Solution for Definite Noncentral Complex Gaussian Quadratic Forms

Introduction

This work was funded by the Brasil 6G Project with support from RNP/M-CTI (Grant 01245.010604/2020-14), the xGMobile Project (XGM-AFCCT2025-8-1-1 and XGM-AFCCT-2024-9-1-1) with resources from EMBRAPII/MCТI (Grant 052/2023 PPI IoT/Manufatura 4.0), CNPq (302085/2025-4, 306199/2025-4), FAPEMIG (APQ-03162-24, PPE-00124-23, RED-00194-23), and FINEP (nº 1060/2024, contract 25.0883.00). We also acknowledge the support of ABRINT, MODIRUM|GESPI, NIC.br, Ericsson Brasil, and Volkswagen Truck & Bus Ltda.

F. D. A. García, F. A. P. de Figueiredo, and R. A. A. de Souza are with the Wireless and Artificial Intelligence (WAI) laboratory, National Institute of Telecommunications (INATEL), Santa Rita do Sapucaí, MG, 37536-001, Brazil (e-mail: [email protected]; [email protected]; [email protected]). C. R. Nogueira da Silva is with the Department of Electrical Engineering, Federal University of Triângulo Mineiro, Uberaba, 38064-200, Brazil (e-mail: [email protected]). Michel Daoud Yacoub and José Cândido Silveira Santos Filho are with the Department of Communications, School of Electrical and Computer Engineering, State University of Campinas, Campinas, SP, 13083-852, Brazil (e-mail: [email protected]; [email protected]).

Gaussian quadratic forms (GQFs) arise in a broad range of wireless communications and signal-processing applications, including optimum combining [2], noncoherent modulation [3, 4, 5], diversity techniques [6, 7, 8, 9], beamforming [10, 11], multibeam transmission strategies [12], orthogonal space–time block coding [13], relay-assisted communications [14, 15, 16, 17], analysis of variance [18], power-spectrum estimation [19], differential detection [20], matched-field processing [21], cooperative communications [22], and adaptive filtering [23, 24, 25]. GQFs can be classified according to the nature of the underlying Gaussian random variables, the definiteness of the associated quadratic-form matrix, and the presence or absence of noncentrality. Accordingly, they may be real or complex; positive definite, negative definite, or indefinite; and central or noncentral [26].¹ As the model incorporates a broader combination of these characteristics, deriving its fundamental distributional quantities—particularly the probability density function (PDF) and cumulative distribution function (CDF)—becomes more challenging. In this paper, we focus on positive- and negative-definite noncentral complex Gaussian quadratic forms (CGQFs).²

Related Work on Gaussian Quadratic Forms

Exact Approaches

The distribution of GQFs has been studied extensively since the mid-twentieth century. Early contributions focused primarily on quadratic forms (QFs) in real Gaussian variables. Robbins [27] and Gurland [28, 29, 30] developed distributional representations and characteristic-function inversion methods for definite and indefinite quadratic forms, whereas Shah [31] extended these analyses to noncentral normal variables. The study of CGQFs can be traced to Turin [32], who derived the characteristic function of Hermitian quadratic forms in complex Gaussian variables. Kotz et al. [33, 34] later derived series expansions for central and noncentral positive-definite real GQFs using Maclaurin-series, Laguerre-polynomial, and chi-squared-series expansions. Tziritas [35] derived series representations for the PDF and CDF of central and noncentral positive-definite GQFs involving real and complex Gaussian variables. ables, whereas Reifler et al. [36] derived a multidimensional Hermite-polynomial series representation for the CDF of noncentral positive-definite real GQFs. For indefinite CGQFs, Biyari and Lindsey [37] derived exact series expressions for a particular class of noncentral Hermitian GQFs. Raphaeli [1] subsequently derived series representations for the PDF and CDF of noncentral indefinite CGQFs using contour integration and residue calculus, providing a computationally efficient and fast-converging exact approach.

Approximate Approaches

Approximate solutions were later developed to reduce the computational burden associated with exact series and numerical inversion methods. Saddlepoint approximations were investigated in [38], [39]. Al-Naffouri et al. subsequently proposed a general framework encompassing definite and indefinite, as well as central and noncentral GQFs [40]. Their central-case results are available in closed form, whereas the noncentral case is represented by a one-dimensional integral that can be efficiently evaluated using a saddle-point approximation. Ramírez-Espinosa et al. proposed an alternative approximation for noncentral positive-definite real GQFs based on a linear combination of derivatives of the moment-generating function [41]. Ramírez-Espinosa et al. later introduced an auxiliary-variable framework applicable to both definite and indefinite noncentral CGQFs [42]. The resulting PDF and CDF expressions define valid approximate distributions for every value of the accuracy-controlling parameter and converge to the target distribution as this parameter increases.

Numerical Methods

Imhof introduced a numerical characteristic-function inversion method for evaluating the distributions of real GQFs [43]. Davies later developed a related inversion procedure with numerical error control [44], whereas Rice proposed numerical-integration techniques for computing GQF distributions [45]. These methods avoid the evaluation of complicated series coefficients and can provide accurate pointwise estimates of the PDF or CDF. More recently, Issaid et al. developed an importance-sampling method for the efficient estimation of rare-event and tail probabilities associated with positive GQFs [46].

Drawbacks and Limitations

Despite substantial progress, existing exact and approximate approaches involve important tradeoffs among scope, computational efficiency, convergence behavior, and analytical tractability. Their principal limitations can be summarized as follows:

• Restricted applicability: Several formulations are tailored to specific subclasses of GQFs and therefore do not provide a general treatment of real and complex, central and noncentral, and definite and indefinite cases. For example, the results in [33], [34], [36] are restricted to central or noncentral positive-definite real GQFs, while the approximation in [41] is likewise limited to noncentral positive-definite real GQFs. Tziritas [35], in turn, considered central and noncentral positive-definite GQFs involving both real and complex Gaussian variables. The formulation of Biyari and Lindsey [37] applies only to a particular class of noncentral Hermitian GQFs. Raphaeli’s exact formulation [1] and the approximate frameworks of Ramírez-Espinosa et al. [42] and Al-Naffouri et al. [40] represent the broadest class of available approaches, encompassing noncentral indefinite CGQFs.

• High computational complexity: The cost of exact nested-series methods generally increases with the quadratic-form dimension, eigenvalue multiplicities, noncentrality parameters, and required truncation order [1], [26], [31], [33], [34], [37]. In Raphaeli’s formulation [1], for example, clusters of closely spaced eigenvalues may further slow convergence, thereby increasing the required truncation order and the overall computational burden. Approximate methods are subject to similar scaling limitations. In [41], the computational cost increases with the quadratic-form dimension and the auxiliary shape parameter, as larger values require higher-order derivatives of the moment-generating function. Likewise, in the formulation of Ramírez-Espinosa et al. [42], the cost of evaluating the series coefficients grows combinatorially with the quadratic-form dimension and the accuracy-controlling parameters.

• Validity of the PDF and CDF: For practical implementation, the exact PDF and CDF series must be truncated. However, the resulting truncated versions are not automatically guaranteed to preserve all the defining properties of a probability distribution. In particular, the truncated PDF may become negative or fail to integrate exactly to unity, while the truncated CDF may take values outside $[0,1]$, as discussed in [42]. Similar consistency issues may also arise in approximate methods. For example, the approximation in [40] may yield CDF values greater than one in the right tail, as observed in [41].

• Limited analytical tractability: Some exact and approximate formulations are difficult to use in subsequent analytical derivations. In Raphaeli’s method [1], the recursively generated coefficients depend on the evaluation variable, complicating integration, differentiation, and transformation. The approximate solution in [40] relies on a one-dimensional integral and saddlepoint evaluation, so additional expectations generally require further numerical integration. Likewise, the approximate method in [41] may require high-order derivatives of the moment-generating function, which must be computed numerically when closed-form expressions are unavailable. These limitations hinder the derivation of moments, transforms, and average performance measures.

Our Contributions

Motivated by the drawbacks and limitations discussed previously, this paper derives analytically tractable and computationally efficient formulations for the exact characterization of definite noncentral CGQFs. The principal contributions are summarized as follows:

1. We derive novel exact Gamma-mixture representations for the PDF and CDF of positive- and negative-definite noncentral CGQFs. A key feature is that the mixture

weights are independent of the evaluation variable, so they can be computed once and reused across multiple points. Unlike Raphaeli’s formulation [1], whose coefficients depend on the evaluation variable, the proposed structure yields a simpler and more tractable formulation, while improving computational efficiency.

2. We establish the absolute and uniform convergence of the PDF and CDF series and derive uniform upper bounds on their absolute truncation errors. These bounds enable systematic selection of a sufficient truncation order to guarantee a prescribed accuracy.

3. We show that expectations of integrable functions of definite noncentral CGQFs can be expressed as a weighted sum of the corresponding expectations over Gamma random variables, enabling the direct derivation of moments, transforms, and performance measures. As a sample application, we derive an exact average bit error rate (ABER) expression for square $M$-ary quadrature amplitude modulation ($M$-QAM) in maximal-ratio combining (MRC) receivers over correlated Rician fading channels. To the best of our knowledge, this is the first exact result for an arbitrary number of diversity branches under arbitrary Rician correlation.

4. We derive the computational complexity orders of the proposed PDF and CDF representations and compare them with those of Raphaeli’s method. The analysis shows that the proposed solutions have lower computational complexity, with the advantage becoming particularly pronounced for multipoint evaluations, i.e., when generating complete PDF and CDF curves over their full support.

5. Capitalizing on the Gamma-mixture structure of the proposed representations, we derive finite-sum closed-form approximations by normalizing the retained mixture weights. Unlike direct truncations of existing exact series, the proposed approximations define valid probability distributions for every truncation order: the approximate PDF remains nonnegative and integrates to unity, while the corresponding CDF preserves the correct support, range, and limiting behavior.

6. We validate the proposed analytical representations through Monte Carlo simulations. The numerical results show that the proposed method outperforms Raphaeli’s formulation [1] in terms of computational complexity and execution time. For high-dimensional quadratic forms and a prescribed accuracy level, our method also surpasses the approximation proposed by Ramírez-Espinosa et al. [42] in terms of execution time. The approximate results of Al-Naffouri et al. [40] are also included as accuracy benchmarks.

Notation

Throughout the paper, $\Pr\{\cdot\}$ denotes probability; $f_{(\cdot)}(\cdot)$, PDF; $F_{(\cdot)}(\cdot)$, CDF; $\mathbb{E}\{\cdot\}$, expectation; $\Re\{\cdot\}$, the real part; $\Gamma(\cdot)$, the gamma function; $\Upsilon(\cdot,\cdot)$, the lower incomplete gamma function; $I_{(\cdot)}(\cdot,\cdot)$, the regularized incomplete beta function; $\Gamma(\cdot,\cdot)$, the upper incomplete gamma function; $\mathbb{N}$, the set of natural numbers; $\mathbb{N}_0$, the set of natural numbers including zero; $\mathbb{R}$, the set of real numbers; $\mathbb{C}$, the set of complex numbers; $u(\cdot)$, the unit-step function; $(\cdot)^\dagger$, the conjugate transpose; $\operatorname{sgn}(\cdot)$, the sign function; $\operatorname{card}(\mathcal{A})$, the cardinality of the set $\mathcal{A}$; $\mathcal{CN}_n(\boldsymbol{a},\boldsymbol{B})$, an $n$-dimensional complex Gaussian distribution with mean vector $\boldsymbol{a}$ and covariance matrix $\boldsymbol{B}$; $\chi_b^2(b)$, a noncentral chi-square distribution with a degrees of freedom and noncentrality parameter $b$; $\mathcal{L}\{\cdot\}(s)$, the Laplace transform evaluated at $s$; $O(\cdot)$, the asymptotic order of computational complexity; and $\sim$, “distributed as.” Moreover, $\max_i\{a_i\}$ and $\min_i\{a_i\}$ denote the maximum and minimum elements of the set $\{a_i\}_{i=1}^{n}$, respectively, while $\sup_{x\in\mathcal{A}}g(x)$ and $\inf_{x\in\mathcal{A}}g(x)$ denote the supremum and infimum of $g(x)$ over $\mathcal{A}$, respectively.

Preliminaries: Definite Noncentral Complex Gaussian Quadratic Forms

This section introduces definite noncentral CGQFs and establishes the mathematical foundation underpinning the subsequent analysis.

It was shown by Tzir itas [35] and Khuri [47] that the noncentral CGQF in (1) can be expressed as

where $\{Y_i\}_{i=1}^{n}$ are mutually independent random variables with $Y_i\sim\chi^2_2(2\mu_i)$, $\eta_i\triangleq\lambda_i/2$, $\{\lambda_i\}_{i=1}^{n}$ denote the eigenvalues of $\boldsymbol{L}\boldsymbol{A}$, and the parameters $\{\mu_i\}_{i=1}^{n}$ are given by

where $\boldsymbol{C}\in\mathbb{C}^{n\times n}$ is an invertible lower triangular matrix with non-negative diagonal entries obtained from the Cholesky factorization $\boldsymbol{L}=\boldsymbol{C}\boldsymbol{C}^{\dagger}$, and $\boldsymbol{U}$ is the unitary matrix that diagonalizes $\boldsymbol{C}^{\dagger}\boldsymbol{A}\boldsymbol{C}$. The representation in (2) holds without loss of generality for positive or negative semi-definite noncentral CGQFs. In these cases, one or more eigenvalues $\{\lambda_i\}$ may be equal to zero, in which case the corresponding terms in (2) vanish identically. Consequently, if $r$ eigenvalues are zero, after re-indexing the nonzero eigenvalues, the effective summation range in (2) reduces to $i=1,\ldots,n-r$.

More recently, García et al. [48] showed that the PDF and CDF of sums of independent squared Gaussian random variables with nonzero deterministic or random means admit Gamma-mixture representations. Combined with the canonical decomposition of definite noncentral CGQFs established by by Tziritas [35] and Khuri [47], this result suggests that the distribution of $Q$ in (2) may likewise admit a Gamma-mixture characterization. This observation motivates the analytical developments presented in the next section.

Fundamental Statistics

This section derives novel exact expressions for the fundamental statistics of definite noncentral CGQFs, analyzes the computational complexity of the proposed formulation, and compares it with Raphaeli’s exact method [1]. Uniform truncation-error bounds are subsequently established, followed by the development of closed-form approximations for the practical evaluation of the PDF and CDF.

Main Result

Proof. See Appendix A.

The absolute and uniform convergence of the PDF and CDF series in (4) and (5) is established in Appendix B.

Remark 1: To the best of our knowledge, the Gamma-mixture representations in (4) and (5) are novel and arguably constitute the most efficient exact solutions available for this class of definite noncentral CGQFs. Notably, the Gamma-mixture structure is derived directly from the bilateral Laplace transform of the QF, rather than being imposed through a prespecified basis expansion or assumed distributional form, as in [33, 34, 35, 36]. This probabilistic structure naturally yields valid closed-form approximations, as shown in Section III-D. Furthermore, as the mixture weights are independent of the evaluation variable, the proposed representation facilitates subsequent analytical derivations, as established in the following corollary.

Proof. See Appendix C.

Remark 2: The corollary establishes that the expectation of any integrable function of a definite noncentral CGQF can be expressed as weighted sums of expectations of the same function evaluated at univariate Gamma random variables. The absolute convergence of the resulting series ensures that this representation is well defined and permits term-by-term numerical evaluation. For a positive-definite CGQF, $\sigma=1$, so that the component terms reduce to $\mathbb{E}\{\psi(G_k)\}$. For a negative-definite CGQF, $\sigma=-1$, and the sign reversal is incorporated through $\mathbb{E}\{\psi(-G_k)\}$. Therefore, whenever the corresponding expectation involving a Gamma random variable admits a closed-form expression, the associated functional of $Q$ follows directly from (11). This representation thus provides a general mechanism for deriving moments, integral transforms, average error probabilities, detection probabilities, and other average performance measures involving definite noncentral CGQFs.

A particularly useful consequence arises in wireless communication applications. When system performance depends on a definite noncentral CGQF, metrics such as the ABER can be expressed as weighted sums of the corresponding ABER expressions associated with a single-link Nakagami-$m$ fading channel, as illustrated in § 4. This connection follows from the fact that the instantaneous power under Nakagami-$m$ fading is Gamma distributed.

Computational Complexity

Here, we analyze the computational complexity of the truncated PDF and CDF representations in (4) and (5), respectively, and compare their complexity with that of Raphaeli’s method [1]. The analysis is carried out by retaining the terms indexed by $k=0,\ldots,K$. Throughout this analysis, fixed-precision arithmetic operations and evaluations of standard built-in special functions are assumed to have constant computational cost.

Complexity of the Proposed Method

1. Complexity of the Proposed Method: The component-wise computational costs of the proposed Gamma-mixture representations are summarized as follows.

• Gamma-component evaluation: For a fixed value of $q$, evaluating either $f_G(\alpha_k,\theta,|q|)$ in (7) or $F_G(\alpha_k,\theta,|q|)$ in (8) requires a constant number of arithmetic operations and one evaluation of a standard Gamma or incomplete-Gamma function. Hence, each Gamma component has complexity $O(1)$. Since each truncated series contains $K+1$ components, evaluating either the PDF or the CDF, once the mixture weights are available, requires $O(K)$ operations at each value of $q$.

• Initial mixture weight: Computing $\beta_0$ requires one product and one sum over $n$ terms. Therefore, its computational complexity is $O(n)$.

• Recursive mixture weights: From (64), the recursion in (9b) can be written as

A key computational feature is that $S_m$ depends on $m$, but not on $k$. Therefore, the quantities $\{S_m\}_{m=1}^{K}$ can be computed once and subsequently reused in the recursion for all mixture weights. Thus, computing all $K$ quantities $S_m$ has complexity $C_S=O(nK)$. Once $\{S_m\}_{m=1}^{K}$ are available, evaluating $\beta_k$ through (11) requires $O(k)$ operations. Hence, generating all recursive mixture weights requires $\sum_{k=1}^{K}O(k)=O(K^2)$. Including the computation of $\beta_0$, the total complexity required to generate the mixture weights is therefore

• Overall complexity: After the mixture weights have been generated, evaluating and accumulating the $K+1$ Gamma components requires an additional $O(K)$ operations for either the PDF or the CDF. Therefore, the total computational complexities at a single evaluation point $q$ are

Therefore, the computational complexity of the proposed PDF and CDF representations grows linearly with $n$ in the pre-computation stage and quadratically with $K$ in the recursive generation of the mixture weights. Importantly, the quadratic term in $K$ is independent of the quadratic-form dimension $n$.

If the PDF or CDF is evaluated at $N_q$ evaluation points, both the quantities $\{S_m\}_{m=1}^{K}$ and the mixture weights $\{\beta_k\}_{k=0}^{K}$ are generated only once, since they are independent of $q$. Therefore, the corresponding total complexity is

Complexity of Raphaeli's Method

2. Complexity of Raphaeli's Method: For comparison, we analyze the computational complexity of Raphaeli’s PDF and CDF-related series in [1], Eqs. (25) and (32). Strictly speaking, [1], eq. (32) gives the upper-tail probability $\Pr\{Q\ge q\}$ for $q>0$. The conventional CDF is obtained by taking its complement, which does not alter its computational complexity.

Let $N_\lambda$ denote the number of distinct eigenvalues after grouping repeated eigenvalues, and let $\{\tilde{\lambda}_k\}_{k=1}^{N_\lambda}$ denote these distinct eigenvalues, with corresponding multiplicities $\{\varpi_k\}_{k=1}^{N_\lambda}$. Furthermore, define the set of indices associated with the distinct eigenvalues contributing on the selected half-line as

where $k\in\{1,\ldots,N_\lambda\}$, and let $N_s\triangleq\operatorname{card}(\mathcal{I}_s)$. Assume that Raphaeli’s infinite series is truncated by retaining $K_R+1$ terms for each contributing distinct eigenvalue. Since the series associated with $\tilde{\lambda}_k$ begins at derivative order $\varpi_k-1$, the largest required derivative order is $D_k\triangleq\varpi_k+K_R-1$.

The component-wise computational costs are summarized as follows.

• Logarithmic derivatives: For each derivative order $r$, the expression for $[\log g_k(0,q)]^{(r)}$ contains a summation over the distinct eigenvalues. Therefore, evaluating one logarithmic derivative requires $O(N_\lambda)$ operations. Computing all logarithmic derivatives through order $D_k$ requires $O(N_\lambda D_k)$ operations for the $k$th contributing eigenvalue.

• Recursive derivatives: Raphaeli’s recursion for $g_k^{(r)}(0,q)$ contains $r$ previously computed quantities. Hence, computing the derivative of order $r$ requires $O(r)$ operations. Generating all derivatives through order $D_k$ consequently requires

• Series accumulation: Once the required derivatives have been generated, evaluating and accumulating the $K_R+1$ retained terms requires $O(K_R)$ operations for each contributing eigenvalue.

Consequently, Raphaeli’s PDF and upper-tail-probability expressions have the same computational order:

Since the ordinary CDF is obtained by complementation, it follows that $C_{\mathrm{CDF}}^{(\mathrm{R})}=C_{\mathrm{CCDF}}^{(\mathrm{R})}$. If the eigenvalue multiplicities remain bounded independently of $K_R$, then $D_k=K_R+O(1)$, and (17) simplifies to

For a definite quadratic form evaluated on its nontrivial support half-line, all eigenvalues contribute and therefore $N_s=N_\lambda$. When all eigenvalues are distinct, one additionally has $N_s=N_\lambda=n$ and $\varpi_k=1$. Thus,

If the PDF and CDF are evaluated at $N_q$ distinct values of $q$, the recursively generated quantities that depend on $q$ must be recomputed at each evaluation point. Consequently, the computational complexities for $N_q$ evaluation points are

For a definite quadratic form with distinct eigenvalues, where $N_s=N_\lambda=n$, this reduces to

Complexity Comparison

For a single evaluation point and equal truncation orders, $K_{\mathrm{R}}=K$, the two methods satisfy

Thus, under equal truncation orders and the implementation assumptions adopted above, the proposed method removes an additional multiplicative dependence on the quadratic-form dimension $n$. In particular, when $K\gg n$, the proposed method scales as $O(K^2)$, whereas Raphaeli’s formulation scales as $O(nK^2)$. Conversely, when $n\gg K$, the proposed method scales as $O(nK)$, whereas Raphaeli’s formulation scales as $O(n^2K)$. Consequently, the computational advantage of the proposed representation becomes increasingly significant as either the truncation order or the quadratic-form dimension grows.

For equal truncation orders $K_{\mathrm{R}}=K$ and $N_q$ evaluation points, the corresponding computational complexities are

Hence, the proposed formulation is particularly advantageous for multipoint PDF and CDF evaluations. Its $q$-independent quantities are computed once and reused, whereas Raphaeli’s $q$-dependent recursive quantities must be recomputed at every evaluation point, making the proposed method particularly advantageous for complete PDF and CDF evaluations.

Finally, it should be emphasized that the truncation orders $K$ and $K_{\mathrm{R}}$ required to achieve the same prescribed accuracy are not necessarily equal. Therefore, the expressions above characterize the structural computational complexity of the two methods, whereas accuracy-matched execution-time comparisons must additionally account for their different convergence rates.

Truncation Error Analysis

Following the truncation procedure adopted in Section III-B3 assumes that the PDF and CDF series are truncated after the term $k=K$. Define the corresponding truncated series by

The associated truncation errors are

As shown in Appendix D, the uniform PDF and CDF truncation errors satisfy

Here, $R_\beta$ is defined in (58), and

Moreover,

The tightest bound in this family is obtained by minimizing $\mathcal{E}_K(\rho)$ over the admissible interval:

When numerical optimization is not desired, a simple admissible choice for $R_\beta<\infty$ is the midpoint

When $R_\beta=\infty$, any finite $\rho>1$ is admissible. Our truncation-error bounds in (30) and (31) are of practical importance as they can be used to determine a sufficient truncation order $K$ that guarantees a prescribed accuracy for the PDF and CDF, respectively.

Closed-Form Approximations

As will be demonstrated in § 5, the corresponding truncation errors decrease rapidly as the truncation order $K$ increases. This rapid convergence, together with the Gamma-mixture structure of the PDF and CDF expressions in (4) and (5), motivates the following finite-sum closed-form approximations for the PDF and CDF of $Q$.

Define the retained mixture mass as

where $T_K$ is the tail mass defined in (81). Since $\beta_0>0$ and $\beta_k\geq0$ for every $k\in\mathbb{N}_0$, one has $0<\Omega_K\leq1$. We then introduce the normalized mixture weights

By construction, these coefficients satisfy $\widetilde{\beta}_k^{(K)}\geq0$ for $k=0,\ldots,K$ and $\sum_{k=0}^{K}\widetilde{\beta}_k^{(K)}=1$. Accordingly, the PDF of $Q$ can be approximated by the finite Gamma-mixture representation

Similarly, the CDF of $Q$ can be approximated as

Since the normalized coefficients in (37) are nonnegative and sum to unity, (38) represents a valid PDF for every $K\in\mathbb{N}_0$. In particular, $\widetilde{f}_{Q,K}(q)\geq0$ and $\int_{-\infty}^{\infty}\widetilde{f}_{Q,K}(q)\,\mathrm{d}q=\sum_{k=0}^{K}\widetilde{\beta}_k^{(K)}=1$. In addition, the sign factor $\sigma$ and the additive term $(1-\sigma)/2$ ensure the correct CDF behavior in the positive- and negative-definite cases. In particular, the resulting CDF is nondecreasing, takes values in $[0,1]$, and satisfies the limiting conditions $\lim_{q\to-\infty}\widetilde{F}_{Q,K}(q)=0$ and $\lim_{q\to\infty}\widetilde{F}_{Q,K}(q)=1$.

Sample Application: Maximal Ratio Combining Over Correlated Rician Channels

This section illustrates the direct applicability of the Corollary by deriving an exact expression for the ABER of an MRC diversity receiver with an arbitrary number of branches operating over correlated Rician fading channels.

Consider an MRC diversity receiver with $P$ branches. The received signal vector can be expressed as

where $z$ is the complex transmitted symbol satisfying $\mathbb{E}\{|z|^2\}=E_s$, $\mathbf{w}\sim\mathcal{CN}_P(\mathbf{0},N_0\mathbf{I}_P)$ is the additive noise vector, and $\mathbf{g}\in\mathbb{C}^{P\times1}$ is the correlated Rician channel vector, modeled as $\mathbf{g}\sim\mathcal{CN}_P(\bar{\mathbf{g}},\mathbf{\Sigma})$, where $\bar{\mathbf{g}}\in\mathbb{C}^{P\times1}$ is the deterministic component and $\mathbf{\Sigma}\succ\mathbf{0}$ is the covariance matrix of the diffuse components.

Under this model, the $i$th channel coefficient is marginally Rician distributed with average power $\Omega_i\triangleq\mathbb{E}\{|g_i|^2\}=|\bar{g}_i|^2+\Sigma_{i,i}$, and Rician factor $\kappa_i=|\bar{g}_i|^2/\Sigma_{i,i}$. Thus, the diversity branches are not required to have identical or unit average powers.

Let the eigendecomposition of the covariance matrix be

where $\lambda_i>0$, $\mathbf{\Lambda}=\operatorname{diag}(\lambda_1,\ldots,\lambda_P)$, and $\mathbf{V}$ is unitary. Defining $\widetilde{\mathbf{g}}\triangleq\mathbf{V}^{\dagger}\bar{\mathbf{g}}$, the corresponding noncentrality parameters are $\mu_i\triangleq|\widetilde{g}_i|^2/\lambda_i$ with $i=1,\ldots,P$. Accordingly, the pairs $\{\lambda_i,\mu_i\}_{i=1}^{P}$ completely specify the canonical noncentral CGQF representation employed in the proposed framework.

Assuming perfect channel estimation and symbol synchronization, the normalized MRC output can be expressed as

Defining the reference SNR as $\gamma_0\triangleq E_s/N_0$, the instantaneous post-combining SNR is

Therefore, $Q\triangleq\gamma/\gamma_0=\mathbf{g}^{\dagger}\mathbf{I}_P\mathbf{g}$, which is precisely the positive-definite noncentral CGQF defined in (1), with $\boldsymbol{\nu}=\bar{\mathbf{g}}$, $\mathbf{A}=\mathbf{I}_P$, and $n=P$.

The Corollary can therefore be applied directly to any integrable performance measure that depends on the instantaneous SNR. In particular, for an integrable function $\psi(\cdot)$,

where $\gamma_k\triangleq\gamma_0G_k$ and $G_k\sim\operatorname{Gamma}(\alpha_k,\theta)$. Since $\alpha_k=P+k$, each $\gamma_k$ has the same distribution as the instantaneous SNR of a single-link Nakagami-$m$ fading channel with $m_k=P+k$ and $\bar{\gamma}_k=m_k\gamma_0\theta$. Moreover, since $\mathbf{A}=\mathbf{I}_P$, the scale parameter $\theta$ reduces to the smallest eigenvalue of $\mathbf{\Sigma}$, i.e., $\theta=\lambda_{\min}(\mathbf{\Sigma})>0$. Thus, the performance of the correlated Rician MRC receiver can be obtained as weighted sums of the corresponding performance measures of single-link Nakagami-$m$ fading channels.

Average Bit Error Rate for Square $M$-QAM

Consider coherent square $M$-QAM with Gray bit mapping, where $\nu\triangleq\log_2\sqrt{M}$. By applying (44), the ABER of the considered MRC receiver is given directly by

where $\overline{P}_b^{(\mathrm{Nak},M)}(m_k,\bar{\gamma}_k)$ denotes the exact ABER of Gray-coded square $M$-QAM over a single-link Nakagami-$m_k$ fading channel, given by

where $N_\ell\triangleq(1-2^{-\ell})\sqrt{M}-1$, $\xi_{r,k}\triangleq\left[1+\frac{3(2r+1)^2\bar{\gamma}_k}{2m_k(M-1)}\right]^{-1}$, and

Using $\overline{\gamma}_k=m_k\overline{\gamma}_{\theta}$, the argument $\xi_{r,k}$ becomes independent of $k$, namely, $\xi_r \triangleq\left[1+\frac{3(2r+1)^2\overline{\gamma}_{\theta}}{2(M-1)}\right]^{-1}$. Consequently, the exact ABER reduces to

Thus, the exact ABER of the correlated Rician MRC receiver is obtained as weighted sums of the corresponding single-link Nakagami-$m$ fading results.

Remark 3: Existing studies on MRC over correlated Rician fading provide approximate or asymptotic results for an arbitrary number of branches [42], [49], [50], whereas exact results are generally limited to the dual-branch case [51], [52]. To the best of our knowledge, (48) is the first reported exact ABER expression for arbitrary-branch MRC under an arbitrary positive-definite Rician correlation structure. This application also highlights how the proposed Gamma-mixture representation overcomes the limited analytical tractability of Raphaeli’s formulation [1] by reducing further performance measures to weighted sums of single-link Nakagami-$m$ results. Other modulation schemes and performance metrics (e.g., outage probability) can be treated analogously whenever the corresponding Nakagami-$m$ error expressions are available in closed form or can be evaluated tractably.

Numerical Results

In this section, we numerically corroborate the analytical developments and assess the accuracy, convergence behavior, and computational efficiency of the proposed Gamma-mixture representations. All results are generated using the parameter set in tbl:1, with the quadratic-form dimension ranging

from $n=1$ to $n=17$.³ The proposed PDF and CDF expressions are compared with the exact series formulation of Raphaeli [1], Eqs. (26) and (33), and the approximate formulations of Ramírez-Espinosa et al. [42], Eqs. (21) and (25) and Al-Naffouri et al. [40], Eq. (42). Since Al-Naffouri et al. derived a CDF-based formulation but no corresponding PDF expression, their results are included only in the CDF comparisons.

Figure 1 shows the PDF and CDF Eqs. (4) and (5)} for the positive-definite case, whereas Fig. 2` shows the corresponding results for the negative-definite case, both for $n=15$ and $n=17$. The proposed expressions in (4) and (5) are evaluated using a truncation order $K=350$, whereas Raphaeli’s series is truncated at $K_R=375$. For the approximation of Ramírez-Espinosa et al., the accuracy-controlling parameter is set to $m_E=30$. The Monte Carlo results are obtained from $10^7$ independent realizations. For both dimensions and both definiteness cases, the proposed PDF and CDF curves, Raphaeli’s exact formulation, and the approximation of Ramírez-Espinosa et al. are virtually indistinguishable from the Monte Carlo results. This close agreement provides direct numerical validation of the proposed Gamma-mixture representations in (4) and (5). A different behavior is observed for the approximation of Al-Naffouri et al. In the positive-definite case, it closely follows the reference CDF in the lowprobability region, begins to exhibit a noticeable discrepancy through the transition region, and progressively loses accuracy in the upper tail. Most importantly, its CDF is not constrained to remain within the interval $[0, 1]$. In the negative-definite case, the behavior is reversed: the largest discrepancy appears in the low-CDF region, whereas the approximation becomes increasingly accurate as the CDF approaches unity.

$i$	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15	16	17
$\lambda_i$	0.60	0.13	0.22	3.82	3.02	0.88	4.02	3.56	0.24	0.42	5.76	3.46	4.22	3.78	0.21	1.32	1.76
$\mu_i$	0.40	1.20	0.20	0.35	1.05	1.02	0.55	0.30	0.65	1.10	2.78	0.12	0.21	0.40	0.75	0.56	1.77

Fig. 3 investigates the convergence of the proposed series by plotting the uniform truncation error bounds for $\sup_{q \in\mathbb{R}} R_{f,K}(q)$ [Eq. (30)] and $\sup_{q \in\mathbb{R}} R_{F,K}(q)$ [Eq. (31)] as functions of the truncation order $K$, for $n = 10$. $\sup_{q \in\mathbb{R}} R_{f,K}(q)$ is approximated numerically over a sufficiently dense grid spanning $0 \leq\lvert q\rvert\leq463$, which covers the numerically relevant range of the distribution, whereas $\sup_{q \in\mathbb{R}} R_{F,K}(q)$ follows exactly from the remaining mixture mass. The corresponding uniform truncation-error bounds in (30) and (31) are shown using both the optimized choice in (34) and the explicit midpoint choice in (35). Observe that the actual uniform truncation errors and their corresponding uniform upper bounds decrease monotonically and rapidly with $K$, confirming the convergence predicted by the truncation-error analysis. For instance, the uniform PDF truncation error falls below $10^{-3}$ at $K = 43$ and below $10^{-4}$ at $K = 57$.

The corresponding uniform CDF truncation error falls below these levels at $K = 62$ and $K = 76$, respectively. As expected, the optimized uniform bound is consistently tighter than the explicit midpoint bound. For instance, the optimized PDF bound falls below $10^{-3}$ at $K = 60$, compared with $K = 43$ for the actual uniform error, while the optimized CDF bound reaches the same level at $K = 79$, compared with $K = 62$ for the actual uniform error. Hence, the bounds are conservative, but remain sufficiently tight to enable practical selection of the truncation order without requiring direct evaluation of the infinite-series remainder.

Figs. 4 and 5 compare the execution times of the proposed method, Raphaeli’s formulation, and the approximation of Ramírez-Espinosa et al. as the quadratic-form dimension $n$ increases. For a fair comparison, each method’s parameters are independently selected to achieve a pointwise absolute error below $10^{-3}$. In the positive-definite case, the proposed method uses $K = 190$, Raphaeli’s formulation uses $K_{\mathrm{R}} = 220$, and Ramírez-Espinosa et al. use $m_{\mathrm{E}} = 25$. For the negative-definite case, these parameters are $K = 190$, $K_{\mathrm{R}} = 375$, and $m_{\mathrm{E}} = 30$, respectively. Single-point PDF and CDF evaluations are performed at $q = 20$ and $q = -20$ for the positive- and negative-definite cases, respectively. In the positive-definite case, as $n$ increases from 1 to 17, the proposed method’s execution time grows modestly from approximately $1.9 \times10^{-2}$ to $3.4 \times10^{-2}$ s, compared with $4.6 \times10^{-2}$ to $1.5$ s for Raphaeli and $2.3 \times10^{-2}$ to $9.3$ s for Ramírez-Espinosa et al. Consequently, at $n = 17$, the proposed method is approximately 43 and 270 times faster, respectively. The computational advantage is even greater in the negative-definite case, where the proposed method’s execution time increases only from approximately $2.0 \times10^{-2}$ to $3.5 \times10^{-2}$ s, compared with $1.6 \times10^{-1}$ to $4.4$ s for Raphaeli and $2.0 \times10^{-2}$ to $14.8$ s for Ramírez-Espinosa et al. At $n = 17$, this corresponds to speedups of approximately 125 and more than 420, respectively. Overall, the proposed Gamma-mixture formulation exhibits substantially better execution-time scaling over $1 \leq n \leq17$, with its computational advantage becoming increasingly significant as $n$ grows. Since these comparisons involve only single-point evaluations, even greater gains are expected for multipoint PDF and CDF evaluations, where the proposed method reuses its evaluation-independent mixture weights.

Finally, Fig. 6 illustrates the ABER [Eq. (48)] as a function of the reference SNR for $P \in\{13, 15, 17\}$, using the parameter sets in Table I and Gray-coded 16-QAM. The analytical results are in excellent agreement with the Monte Carlo simulations and, as expected, the system performance improves as the number of diversity branches increases.

Conclusions

In this paper, we derived novel exact Gamma-mixture representations for definite noncentral CGQFs, yielding analytically tractable and computationally efficient formulations. As a direct application of the proposed framework, we obtained an exact ABER expression for MRC receivers with an arbitrary number of diversity branches under an arbitrary positive-definite Rician correlation structure. The absolute and uniform convergence of the resulting PDF and CDF series, together with explicit uniform truncation-error bounds, was established, providing rigorous control over the numerical accuracy. Moreover, the Gamma-mixture structure enables finite-sum approximations that, by construction, remain valid probability distributions for any truncation order. Numerical results confirm the rapid convergence of the proposed series and demonstrate substantial computational gains over the formulations of Raphaeli and Ramírez-Espinosa et al..

Appendix A: Proof of the Theorem

Define $X_i \triangleq\eta_iY_i$, $i \in\{1,\ldots,n\}$. Since $Y_i \sim\chi^2(2\mu_i)$, the random variable $X_i$ was shown to follow a Poisson-weighted Gamma mixture, whose PDF is given by [18]

where $f_G(\cdot,\cdot,\cdot)$ denotes the PDF of a Gamma distribution, as defined in (7). Moreover, $\{\phi_{k,i}\}$ are the Poisson mixture weights given by $\phi_{k,i}=\frac{\mu_i^k}{k!}\exp(-\mu_i)$. Taking the bilateral Laplace transform of $X_i$, we obtain

where $s\in\mathbb{C}$. Using the Taylor expansion of the second exponential term in (50), the bilateral Laplace transform can be written as

with the corresponding region of convergence (ROC) determined by

By the definition of $\eta_i^*$ in (6), it follows that $0<r_i\leq1$ and $0\leq a_i<1$.

Define $b_i\triangleq|\eta_i|$, $b_*\triangleq|\eta^*|=\min_i\{b_i\}$, $a_{\max}\triangleq\max_i a_i$, and $b_{\max}\triangleq\max_i\{b_i\}$. Since all coefficients $\eta_i$ have the same sign in the definite case, it follows from (52) that the intersection of the individual regions of convergence is

Hence, (53) provides a common region of convergence for the bilateral Laplace transforms of all the random variables $X_i$. Within this region,

where the second inequality follows from $b_i\leq b_{\max}$. Since $0\leq a_i<1$ and $\Re(1+2\eta^*s)>a_i\geq0$, we have

Therefore, the power-series expansions employed below converge absolutely throughout the common ROC. Defining $t\triangleq\frac{1}{1+2\eta^*s}$, together with $a_i=1-r_i$, we obtain $1+2\eta_i s=\frac{1-a_i t}{r_i t}$. Substituting this result into (51) and collecting the terms that depend on $t$, the marginal bilateral Laplace transform can be rewritten as $\mathcal{L}\{f_{X_i}\}(s)=r_i t\frac{\exp(-\mu_i)}{1-a_i t}\exp\left(\frac{\mu_i r_i t}{1-a_i t}\right)$. Since the random variables $\{X_i\}_{i=1}^n$ are mutually independent, the bilateral Laplace transform of $Q=\sum_{i=1}^n X_i$ is given by the product of the corresponding marginal transforms. Therefore,

where

The second equality defines $\{\beta_k\}_{k=0}^{\infty}$ as the Taylor coefficients of $B(t)$ about $t=0$.

From the definition in (57), the possible finite singularities of $B(t)$ occur at $t=\frac{1}{a_i}$, for $a_i>0$. Since $0\leq a_i\leq a_{\max}$, the singularity closest to the origin is located at $t=1/a_{\max}$ whenever $a_{\max}>0$. Therefore, $B(t)$ is analytic in the open disk $|t|<R_\beta$, where

When $a_{\max}=0$, all $a_i=0$, and hence $B(t)$ is an entire function. Consequently, the Taylor series of $B(t)$ about $t=0$ converges absolutely for every $t$ satisfying $|t|<R_\beta$.

Condition (55) guarantees that, for every $s$ in the common ROC, the corresponding value $t=\frac{1}{1+2\eta^*s}$ lies inside the disk of analyticity of $B(t)$. Therefore, the Taylor expansion of $B(t)$ in (57) can be evaluated at this value of $t$. Substituting the resulting expansion into (56) yields

Setting $t=0$ in (57) gives the zeroth-order coefficient,

To derive the remaining coefficients, we logarithmically differentiate $B(t)$. For $|t|<R_{\beta}$, this gives

Here, the prime symbol ($'$) denotes differentiation with respect to $t$. Using the power-series identities

we obtain

where

Multiplying both sides of (64) by $B(t)$ gives

Since $tB'(t)=\sum_{k=1}^{\infty}k\beta_k t^k$ and $B(t)\sum_{m=1}^{\infty}S_m t^m=\sum_{k=1}^{\infty}\left(\sum_{m=1}^{k}\beta_{k-m}S_m\right)t^k$, equating the coefficients of $t^k$ yields [48]

Finally, substituting (64) into (66) and using the initial coefficient in (60), we recover the recursion given in (9). Since $0\le a_i<1$, $0<r_i\le1$, and $\mu_i\ge0$, every term in (64) is nonnegative. Consequently, $S_m\ge0$ for $m\ge1$. Moreover, (60) implies that $\beta_0>0$. It then follows by induction from the recursion in (66) that

Furthermore, because $a_{\max}<1$, the radius of convergence satisfies $R_{\beta}>1$, and therefore the Taylor series of $B(t)$ converges absolutely at $t=1$. Evaluating (57) at $t=1$ and using the definition $a_i\coloneqq1-r_i$, we obtain

Hence, the sequence $\{\beta_k\}_{k=0}^{\infty}$ consists of nonnegative coefficients that sum to unity and therefore constitutes a valid sequence of mixture weights.

For $\alpha_k>0$, $\theta>0$, and $\sigma\in\{-1,1\}$, the following bilateral Laplace-transform pair holds:

with region of convergence $\sigma\Re(s)>-\frac{1}{\theta}$. Since $\eta^*=\sigma|\eta^*|$ and $\theta=2|\eta^*|$, it follows that $1+2\eta^*s=1+\sigma\theta s$. Therefore, each factor $(1+2\eta_i^*s)^{-\alpha_k}$ in (59) coincides with the bilateral Laplace transform in (69).

Consequently, taking the inverse bilateral Laplace transform term by term yields a Gamma density with shape parameter $\alpha_k$ and scale parameter $\theta$, supported on the half-line determined by the sign of $\eta^*$.

For real $s$ in the common region of convergence (ROC), the bilateral Laplace-transform kernel $e^{-sx}$ and all Gamma-mixture terms are nonnegative. Tonelli’s theorem therefore permits the interchange of the infinite summation and the bilateral Laplace integral, yielding a valid transform identity for all real $s$ in the common ROC. Both sides are analytic functions of $s$ in the interior of the ROC. Hence, by the identity theorem [53], the equality extends to all complex $s$ in the same region. This extension permits term-by-term inversion, and uniqueness of the bilateral Laplace transform establishes that the PDF of $Q$ is given by (4).

Moreover, since the summands in (4) are nonnegative, Tonelli’s theorem permits the interchange of the infinite summation and integration. Because $\beta_k\ge0$ for every $k\in\mathbb{N}_0$ and $\sum_{k=0}^{\infty}\beta_k=1$, term-by-term integration of the PDF series yields the CDF representation in (5). For the negative-definite case, the upper incomplete-Gamma terms are converted to the corresponding lower incomplete-Gamma form by using $\Upsilon(a,x)+\Gamma(a,x)=\Gamma(a)$ [54]. This completes the proof.

Appendix B: Absolute and Uniform Convergence of the PDF and CDF of Q

Define the component Gamma PDF and CDF as

and let $d_k\coloneqq\alpha_k-1=n+k-1\in\mathbb{N}_0$.

The maximum value of $g_k(x)$ over $x\ge0$ is given by [18]

where

For $d_k=0$, $g_k(x)$ reduces to an exponential density and attains its maximum at $x=0$. Using the standard bound $\Gamma(d_k+1)=d_k!>\left(\frac{d_k}{e}\right)^{d_k}$ for $d_k\ge1$ [55], we obtain $0<c_{d_k}\le1$ for $d_k\ge1$. Together with $c_0=1$, this implies the uniform bound

Using (67), (68), the bound in (76), and $0\le u(q\eta^*)\le1$, we obtain $\left|\beta_k g_k(|q|)u(q\eta^*)\right|\le\frac{\beta_k}{\theta}$ for $q\in\mathbb{R}$. Moreover, $\sum_{k=0}^{\infty}\frac{\beta_k}{\theta}=\frac{1}{\theta}\sum_{k=0}^{\infty}\beta_k=\frac{1}{\theta}$. Therefore, the Weierstrass $M$-test establishes that the PDF series converges absolutely and uniformly on $\mathbb{R}$.

Similarly, since each $G_k(x)$ is a Gamma CDF, $0\le G_k(x)\le1$, for $x\ge0$. Hence, $|\beta_kG_k(|q|)u(q\eta^*)|\le\beta_k$, for $q\in\mathbb{R}$. Since $\sum_{k=0}^{\infty}\beta_k=1$, the Weierstrass $M$-test likewise establishes the absolute and uniform convergence of the CDF series on $\mathbb{R}$. The finite additive constant and the sign factor appearing in (5) do not affect this convergence result. Consequently, both the PDF and CDF series converge absolutely and uniformly over their respective domains. This concludes the proof.

Appendix C: Proof of Corollary III-A

By the definition of expectation, we have $\mathbb{E}\{\psi(Q)\} = \int_{-\infty}^{\infty} \psi(q)f_Q(q)\,\mathrm{d}q$. Thus, substituting the PDF in (4), we obtain

To justify interchanging the infinite summation and the integral, first apply Tonelli’s theorem to the corresponding nonnegative integrand involving $|\psi(q)|$. Since $\beta_k \ge0\ \forall k$, we obtain

The left-hand side of (78) is finite by hypothesis. Therefore, Fubini’s theorem permits the interchange of the summation and integration in (77), yielding

Because $\eta^*$ and $\sigma$ have the same sign, the unit-step function restricts the integral to the half-line $H_\sigma\triangleq\{q \in\mathbb{R}:\sigma q \ge0\}$. Now, applying the change of variables $q=\sigma x$, for $x\ge0$, to each integral in (79) gives

Substituting (80) into (79) yields (11). It remains to establish the absolute convergence of the series. Using the inequality $|\mathbb{E}\{X\}| \le\mathbb{E}\{|X|\}$, together with the nonnegativity of the mixture weights and Tonelli’s theorem, we obtain

Hence, the series in (11) converges absolutely. If $\psi$ is nonnegative, Tonelli’s theorem applies directly, and the identity remains valid without requiring the expectation to be finite, with both sides interpreted in the extended nonnegative real numbers. This completes the proof.

Appendix D: Closed-Form Truncation-Error Bounds

Assume that the PDF and CDF series are truncated after the term $k=K$, and define the remaining mixture mass as

Let $\rho$ be any real number satisfying $1<\rho<R_\beta$. Since $\beta_k\ge0$ for all $k\in\mathbb{N}_0$ and $\rho^k\ge\rho^{K+1}$ for every $k\ge K+1$, we have

Consequently,

Using (58), the bound can be written explicitly as in (34). The sequence $\{c_m\}_{m\ge1}$ in (75) is strictly decreasing since

For every omitted term $k\ge K+1$, we have $d_k=n+k-1\ge n+K$. Therefore, by the monotonicity of $c_m$ and (73),

Using (85), $\beta_k\ge0$, $0\le u(q\eta^*)\le1$, and the definition of $T_K$, it follows that

Combining this result with (84), we obtain the uniform PDF truncation-error bound given in (31). Since $0\le G_k(x)\le1$ for $x\ge0$, and $|\sigma|=1$, it follows that

Combining this inequality with (84) yields the uniform CDF truncation-error bound in (32). This completes the proof.