Generalized MFTR Fading Model

• The generalized MFTR fading model is a statistical channel model that extends traditional two-ray paradigms by incorporating multiple specular and diffuse multipath clusters.
• It provides closed-form expressions for key metrics such as outage probability, channel capacity, and error rates, ensuring tractable performance analysis.
• Its flexible parameterization unifies classical fading models, making it valuable for designing and evaluating modern wireless, mmWave, and THz systems.

The generalized multicluster fluctuating two-ray (MFTR) fading model is a flexible, physically motivated statistical channel model that extends the twowave and fluctuating two-ray (FTR) paradigms by integrating multiple clusters of multipath components—each potentially featuring one or two dominant, amplitude-fluctuating specular rays along with diffuse scattering. The MFTR model unifies and generalizes several established fading models (including TWDP, FTR, κ–μ shadowed, Rician, and Rayleigh) via a small set of physically meaningful parameters and supports mathematically tractable closed-form expressions for its probability density, cumulative distribution, and moment-generating functions. This tractability enables efficient and unified analysis of a broad set of performance metrics for modern wireless systems, including outage probability, channel capacity, error statistics, and secrecy measures, across a wide range of propagation regimes.

1. Model Framework and Parameterization

The MFTR model is constructed as the superposition of a primary cluster with two specular components (subject to joint amplitude fluctuations and random phases) and multiple secondary clusters each contributing their own specular and diffuse components. The total received baseband signal can be expressed as: $$r = \sqrt{\zeta} V_1 e^{j\phi_1} + \sqrt{\zeta} V_2 e^{j\phi_2} + \sum_{i=2}^{\mu} V_i e^{j\phi_i} + W$$ where:

• $\zeta$ is a Gamma-distributed random variable (with shape %%%%1%%%%) modeling the common amplitude fluctuation (“shadowing”) of the dominant rays in the primary cluster,
• $V_1$, $V_2$ are the mean amplitudes of the primary specular components, with $\phi_1$, $\phi_2$ uniform over $[0,2\pi]$,
• $V_i$, $\phi_i$ represent the amplitudes and phases of additional specular rays (secondary clusters) with constant amplitude,
• $W$ is a zero-mean complex Gaussian variable modeling the cumulative diffuse multipath (its variance per quadrature is determined by the model’s total diffuse power),
• $\mu$ (typically $\mu \geq 1$ and possibly non-integer) counts the total number of resolved multipath clusters.

The chief parameters for the MFTR model are:

• $K$: ratio of the average power in the total specular rays to the diffuse power,
• $m$: fluctuation severity index for the dominant specular rays,
• $\Delta$: similarity/balance parameter between the two primary specular rays ($\Delta = 2V_1V_2/(V_1^2 + V_2^2)$, with $\Delta=1$ for equal amplitudes, $\Delta=0$ when only one is present),
• $\mu$: effective cluster number (can represent path diversity or sub-clustering effects).

Parameter selection governs the model’s structure:

• $m \to \infty$: removes amplitude fluctuations (recovers TWDP-like behavior);
• $\mu=1$: MFTR reduces to the FTR fading model;
• $\Delta=0$: collapses MFTR to the $\kappa$–$\mu$ shadowed model;
• $K=0$: degenerates to Rayleigh (purely diffuse) fading.

2. Closed-Form Statistical Characterization

The MFTR model’s statistics admit closed-form or analytically tractable forms, enabling rigorous performance analysis.

Moment Generating Function (MGF)

The MGF of the instantaneous SNR $\gamma$ for MFTR with positive integer $m$ is

$$\mathcal{M}_\gamma(s) = \frac{m^m \mu^{\mu} (1+K)^{\mu} [\mu(1+K)-\overline{\gamma}s]^{m-\mu} \left[\sqrt{\mathcal{R}(\mu,m,K,\Delta;s)}\right]^m} {P_{m-1}\left( \frac{m\mu(1+K)-(\mu K + m)\overline{\gamma} s}{\sqrt{\mathcal{R}(\mu,m,K,\Delta;s)}} \right)}$$

where

$$\mathcal{R}(\mu,m,K,\Delta;s) = \left[(m+\mu K)^2-(\mu K\Delta)^2\right] \overline{\gamma}^2 s^2 - 2m\mu(1+K)(m+\mu K)\overline{\gamma}s + [m\mu(1+K)]^2$$

and $P_{n}(z)$ denotes the Legendre polynomial of degree $n$. This form is structurally similar to that for FTR fading, with direct reduction to known models as special cases (Sánchez et al., 2022).

PDF and CDF

For positive integer $m$, the expressions for the PDF and CDF are obtained in closed form using confluent hypergeometric functions of several variables, e.g.,

$$f_\gamma(x) = (1+K)^\mu \mu^\mu 2^{m-1} \frac{\Gamma(\mu)}{\overline{\gamma}^\mu} \left(\frac{m}{\sqrt{(m+\mu K)^2 - (\mu K \Delta)^2}}\right)^m \sum_{q=0}^{\lfloor (m-1)/2 \rfloor} ... \ x^{\mu-1} \Phi_2^{(4)}(\cdots)$$

where $\Phi_2^{(4)}$ is a (type-two) confluent hypergeometric function in four variables. For arbitrary real $m$ or $\mu$, alternative continuous mixture or infinite sum representations are available (Sánchez et al., 2022, Zhang et al., 2017).

Alternative Formulations

• Continuous mixture: MFTR PDF/CDF is integrable as a finite-range mixture of $\kappa$–$\mu$ shadowed models:

$$f_\gamma(x) = \frac{1}{\pi}\int_0^\pi f_{\gamma|\theta}(x) d\theta$$

• Infinite gamma mixture: MFTR statistics can be expressed as infinite sums of gamma PDFs, enabling direct leveraging of results for Nakagami-$m$ fading (Sánchez et al., 2022, Olyaee et al., 2021).

3. Generalization and Relationship to Classical Models

MFTR unifies and extends established fading models:

• FTR/TWDP: Setting $\mu=1$ (single-cluster) produces the FTR or, with $m\to\infty$, the TWDP model (Sánchez et al., 2022, Romero-Jerez et al., 2016).
• $\kappa$–$\mu$ shadowed: Setting $\Delta=0$ yields the generalized shadowed-$\kappa$–$\mu$ model (Sánchez et al., 2022).
• Rayleigh/Rician/Nakagami-$m$: Specializing $K=0$ (no speculars), $\Delta=0$, $\mu=1$ or $m\to\infty$ produces these standard distributions.
• Multicluster structure: Arbitrary $\mu$ and $m$ allow the model to capture environments with rich multipath or multiple “specular” clusters—key for high-frequency communications (mmWave, sub-THz) where such channels can be empirically observed (Olyaee et al., 2023, Galeote-Cazorla et al., 2023).

Flexibility in MFTR parameterization permits accurate statistical fit to empirical fading distributions—especially where multimodality or heavy tail behavior emerges due to path clustering, strong reflections, or large-scale shadowing (Sánchez et al., 2022, Olyaee et al., 2023).

4. Analytical Performance Metrics

Via its mathematically tractable characterization, MFTR supports closed expression (exact and asymptotic) for critical communication metrics.

Outage Probability

The outage probability for an SNR threshold $\gamma_{\rm th}$, $P_{\rm out} = F_\gamma(\gamma_{\rm th})$, can be found by evaluating the MFTR CDF at the desired threshold. High-SNR analysis shows: $$P_{\rm out} \simeq \frac{\mu^{\mu-1}(1+K)^{\mu}m^m (2^{R_{\rm th}}-1)^{\mu}\Gamma(\mu)}{\overline{\gamma}^{\mu}\left(m-\left(\Delta-1\right)K\mu\right)^m} \; {}_2F_1\left(\frac{1}{2},m;1;\frac{2\Delta K\mu}{K\mu(1-\Delta)+m}\right)$$ with the diversity order $G_d = \mu$ (Sánchez et al., 2022).

Amount of Fading (AoF)

The normalized SNR variance is

$$\text{AoF} = \left(1-\frac{K^2}{(1+K)^2}\right)\left(1+\frac{1}{\mu}\right) + \frac{K^2}{(1+K)^2}\left(1+\frac{1}{m}\right)\left(1+\frac{\Delta^2}{2}\right) - 1$$

which explicitly shows the impact of specular/diffuse ratio ($K$), shadowing ($m$), specular similarity ($\Delta$), and cluster number ($\mu$) (Sánchez et al., 2022).

Ergodic Capacity and Bit/Symbol Error Rates

These are computed via standard integrals using the MFTR PDF or via MGF-based approaches, ensuring that quantities such as the channel capacity and symbol error rates can be expressed using elementary or special functions, depending on modulation format and fading parameters (Sánchez et al., 2022, Chun, 2018).

Secrecy Metrics in MFTR Channels

For physical layer security (PLS), recent work extends the MFTR model to quantify secrecy with diversity techniques. With maximal ratio combining (MRC) at the receiver and eavesdropper, the SNR sum’s distribution is a sum of weighted gamma-like terms, allowing efficient calculation of generalized secrecy outage probability (GSOP), average fractional equivocation (AFE), and average information leakage rate (AILR) (Mora et al., 15 Sep 2025).

5. Empirical Validation and Applications

Empirical studies at mmWave and sub-THz frequencies confirm the MFTR model’s ability to fit diverse and challenging channel statistics, notably outperforming classical single-cluster models in environments exhibiting multiple strong specular paths and path clustering (Olyaee et al., 2023, Galeote-Cazorla et al., 2023). The model parameters extracted from measurement data evidence environments with different degrees of path clustering, strength imbalance between specular components, and diffuse power contribution.

Applications include:

• Accurate performance prediction and optimization in urban macro/micro cellular, indoor, mmWave/THz, and vehicular channels.
• Flexible assessment of adaptive modulation, coding, diversity (MRC), and PLS strategies across topologies (Sánchez et al., 2022, Mora et al., 15 Sep 2025).
• Benchmarking for joint small-scale fading/large-scale shadowing via composite extensions (e.g., IG/MFTR channels) (Olyaee et al., 2022, Olyaee et al., 2023).

6. Alternative Representational and Composite Models

Tractable alternative representations facilitate simulation and analytical work:

• Mixtures: The MFTR statistics expressed as weighted integrals or sums over simpler PDFs (e.g., $\kappa$–$\mu$ shadowed, Nakagami-$m$) allow leveraging existing results for performance metrics (Sánchez et al., 2022, Olyaee et al., 2021).
• Composite shadowing: The MFTR model compounded with inverse gamma (IG) distributions models shadowed fading, and closed-form expressions for the resulting statistics remain accessible due to the gamma mixture structure (Olyaee et al., 2022, Olyaee et al., 2023).

7. Limitations and Outlook

The MFTR model, while flexible and broadly applicable, introduces additional parameters whose physical interpretation may depend on propagation environment and measurement granularity. Parameter estimation from data requires caution to avoid overfitting, particularly as the number of clusters or amplitude fluctuation indices increases (Chun, 2018). Further refinements (e.g., relaxing phase uniformity, accommodating non-Gamma-distributed amplitude variations, or including point processes for path arrival clustering) remain open for future investigation.

The MFTR model’s unification of classical and modern fading scenarios, together with solid empirical underpinning and wide analytical applicability, positions it as a strategic tool for next-generation wireless research, design, and performance evaluation.