Multi-Parameter Weighted Fractional Fourier Transform

• MP-WFRFT is a discrete transform that generalizes the DFT with multiple fractional-order parameters, ensuring unitarity and invertibility.
• Its algorithm splits signals into parallel DFT branches weighted with order-specific coefficients, enabling efficient FFT-style computation and stable inversion.
• In MIMO physical-layer security, MP-WFRFT enhances protection by injecting artificial noise for eavesdroppers while preserving reliable recovery for legitimate receivers.

The Multi-Parameter Weighted Fractional Fourier Transform (MP-WFRFT) is a family of unitary, invertible discrete signal transforms that generalize the traditional Discrete Fourier Transform (DFT) by introducing multiple fractional-order parameters. In the foundational four-parameter case (4-WFRFT), as detailed in "Secrecy Capacity Analysis of 4-WFRFT Based Physical Layer Security in MIMO System" (Xu et al., 2023), the transform produces a weighted superposition of DFT powers, distributed according to order-dependent coefficients. MP-WFRFT possesses crucial algebraic properties, including unitarity, invertibility (with the inverse parameter vector), and additivity under parameter variation, making it suitable for advanced applications in physical-layer security (PLS) for wireless systems, particularly in Multiple-Input Multiple-Output (MIMO) scenarios.

1. Formal Definition of MP-WFRFT

The 4-WFRFT acts on a length-$N$ complex sequence $x(n)$ ($n=0,1,\dots,N-1$), where the standard DFT and its powers are

$\DFT\{x\}(k) = X(k) = \sum_{n=0}^{N-1} x(n) e^{-j\frac{2\pi}{N}nk}$

$\IDFT\{X\}(n) = \frac{1}{N}\sum_{k=0}^{N-1} X(k) e^{+j\frac{2\pi}{N}nk}$

Let the set $\{X_p(n)\}$ denote successive DFT powers: $X_0(n) = x(n)$

$X_1(n) = \DFT\{x\}(n)$

$X_2(n) = \DFT^2\{x\}(n) = x(-n \bmod N)$

$X_3(n) = \DFT^3\{x\}(n) = X(-n \bmod N)$

For fractional "order" $a\in\mathbb{R}$, define the order-dependent weights: $$\omega_p(a) = \cos[(a-p)\frac{\pi}{4}] \cos[2(a-p)\frac{\pi}{4}] \cos[3(a-p)\frac{\pi}{4}],\quad p=0,1,2,3$$ The 4-WFRFT is then

$$\mathcal{F}^{a}\{x\}(n) = \sum_{p=0}^{3}\omega_{p}(a)X_{p}(n)$$

Extension to $N$ parameters leads to: $$\mathcal F_{\boldsymbol\alpha} = \sum_{p=0}^{N-1} \omega_p(\boldsymbol\alpha) {\bf F}^p$$ with $\boldsymbol\alpha\in\mathbb{R}^{N}$ and $\mathbf{F}$ denoting the DFT operator.

2. Algorithm and Computational Structure

Computation of $\mathcal{F}^{a}\{x\}(n)$ for 4-WFRFT proceeds as follows:

1. Serial-to-parallel convert $x(n)$ to a length-$N$ vector.
2. Branch into four parallel DFT modules producing $X_0(n)$, $X_1(n)$, $X_2(n)$, $X_3(n)$.
3. Multiply each branch $p$ by its corresponding $\omega_p(a)$.
4. Sum the four weighted branches.
5. Parallel-to-serial convert to form the output sequence $s(n)$.

The same approach generalizes naturally for $N$-parameter MP-WFRFT, with $N$ parallel transform branches. The orthonormality of the weights enables efficient implementation and numerically stable inversion (Xu et al., 2023).

3. Operator Properties and Mathematical Structure

The following algebraic properties are fundamental:

• Unitarity:

$$\langle\mathcal{F}^{a}x,\,\mathcal{F}^{a}y\rangle = \langle x,y\rangle$$

• Invertibility:

$(\mathcal{F}^{a})^{-1} = \mathcal{F}^{-a}$

• Additivity:

$$\mathcal{F}^{a}\circ\mathcal{F}^{b} = \mathcal{F}^{a+b}$$

For general $N$-parameter MP-WFRFT, with weight vector $\boldsymbol\alpha$, the weights $\{\omega_p(\boldsymbol\alpha)\}_{p=0}^{N-1}$ satisfy

$$\sum_{p=0}^{N-1} \omega_p(\boldsymbol\alpha)\omega_{(p-\ell)\bmod N}(\boldsymbol\alpha) = \delta_{\ell,0}$$

$\sum_{p=0}^{N-1}\omega_p(\boldsymbol\alpha) = 1$

These constraints guarantee the operator's unitarity, invertibility, and group structure under parameter (vector) addition.

4. Role in Physical-Layer Security for MIMO Systems

In wiretap MIMO scenarios, MP-WFRFT is used to transform baseband symbols prior to transmission. The legitimate receiver ("Bob") applies the exact inverse transform $\mathcal{F}^{-a}$ (or $-\boldsymbol\alpha$), enabling lossless recovery. An eavesdropper ("Eve") lacking knowledge of the correct parameter(s) is forced to guess a fractional order $\beta$, introducing bias $\Delta a = a - \beta$. Only the $\omega_0$-weighted DFT component remains informative for Eve, while the remaining components function as artificial noise: $$\gamma_{E} = \frac{|\omega_{0}(\Delta a)|^{2}P|h_{E}|^{2}}{N_{E}[1 - |\omega_{0}(\Delta a)|^{2}]}$$ where $P$ is transmit power and $N_E$ is noise variance at Eve's receiver.

Averaging over Rayleigh fading, Eve's mean SNR is

$$\bar\gamma_{E} = b c (1 - c e^{-c} \mathrm{Ei}(-c)), \quad \text{where } b=|\omega_{0}(\Delta a)|^{2}\tfrac{P}{N_{E}},\,c=\tfrac{1}{a-b}$$

By contrast, Bob's average SNR $\bar\gamma_{M}$ is unaffected by the transform. The secrecy capacity is defined as

$$C_{s}(x, y) = [\log_{2}(1 + x) - \log_{2}(1 + y)]^{+}$$

with ergodic secrecy capacity obtained by integrating over the joint SNR distribution [Eq. (46) in (Xu et al., 2023)].

5. Security Enhancement Mechanism

MP-WFRFT increases security by expanding the transform parameter space. Each additional fractional-order parameter multiplies the number of orthogonal transform “basis” states (i.e., $\DFT^p$) an eavesdropper must guess. A wrong guess projects the desired signal onto basis functions acting as artificial noise, directly degrading Eve’s SNR. In MIMO+AS (Antenna Selection) systems, this transform-domain secrecy compounds spatial security, as only a legitimate node knowing both the spatial pair and the correct fractional order can recover the information.

The practical consequence is that as $|\omega_0(\Delta a)|$ approaches zero (achievable for $\Delta a=1,2,3$ in the 4-WFRFT), Eve’s SNR vanishes and secrecy capacity attains its upper bound, $\log_2(1+\bar\gamma_M)$. When $\Delta a\to0$, the system reduces to ordinary DFT and provides no additional security ($C_s\to0$).

6. Performance Metrics and Closed-Form Results

The principal metric is the ergodic secrecy capacity $C_s$, expressible in closed form as

$$C_{s} = \sum_{k=0}^{N_{A}N_{B}-1}(-1)^{k}\binom{N_{A}N_{B}-1}{k} \Biggl\{ \frac{e^{(k+1)/\bar\gamma_{M}}}{(k+1)^{2}} \bigl[b\,c\,(1-c\,e^{-c}\mathrm{Ei}(-c))\bigr] Q(A_{p}\sqrt{k+1},B\sqrt{k+1}) \Biggr\}$$

where $N_A, N_B$ are transmitter and receiver antenna counts (with AS selection), $Q(\cdot,\cdot)$ is the Marcum-Q function, and all other parameters as above. Subordinate metrics include Bob’s and Eve’s joint SNR PDFs, numerical secrecy capacity curves, and explicit forms for artificial noise power.

Empirical evaluations (Figs. 7–11 in (Xu et al., 2023)) confirm substantial secrecy capacity gains even for modest parameter bias, with maxima at $\Delta a$ values where $\omega_0$ vanishes.

7. Generalizations and Implications

MP-WFRFT generalizes naturally to arbitrary parameter dimension $N$, maintaining the operator’s group structure and security-enhancing properties. Each added parameter further splits the information subspace, exponentially compounding the challenge for eavesdroppers to estimate the transform, especially under channel correlation. The structure of weighted DFT-powers ensures all such transforms can be computed via parallelized FFT-style modules, facilitating practical deployment in fast digital systems. A plausible implication is that broadening $N$ in MP-WFRFT could serve as a foundational component for scalable, low-complexity physical-layer security primitives in future MIMO-PLS architectures (Xu et al., 2023).