Asian Journal of Pure and Applied Mathematics

Let U be a universe of discourse and let D = {z ∈ C : |z| ≤ 1} denote the closed unit disk. A complex fuzzy set (CFS) e A on U is characterized by the membership function \(μ_\tilde{A}\) : U → D, \(μ_\tilde{A}\)(x) = \(r_\tilde{A}\)(x)e^iϕ \(\tilde{A}\) ^(x), where \(r_\tilde{A}\)(x) ∈ [0, 1] encodes grade and  \(ϕ_\tilde{A}\)(x) ∈ [0, 2π) encodes phase. This two-dimensional representation strictly extends the type-1 framework μ : U → [0, 1] and enables the simultaneous modelling of amplitude and periodicity, a property central to non-stationary signal processing. We develop a Complex Takagi–Sugeno–Kang (CTSK) inference engine in which the k-th rule fires with complex strength \(\tau_k=\prod_{j=1}^p \mu_{\tilde{A}_{k j}}\left(x_j\right),\)  \(\widehat{y}=\Re\left[\frac{\sum_{k=1}^K \tau_k^* \mathrm{e}^{i \Phi_k} y_k}{\sum_{k=1}^K\left|\tau_k\right|}\right]\) 

where Φ_k ∈ R are learnable phase-shift parameters. We establish that CTSK is a universal approximator on L²(U): for every f ∈ L²(U) and ε > 0, there exists a CTSK system F such that ∥f − F∥L2 < ε. Parameter estimation minimizes the regularized empirical risk

\(\mathcal{L}(\Theta)=\frac{1}{N} \sum_{n=1}^N\left(\widehat{y}\left(x_n ; \Theta\right)-y_n\right)^2+\lambda\|\Theta\|_{\mathcal{H}}^2\)

where Θ = {r_kj , ϕ_kj , c_kj , σ_kj ,Φ_k,w_k}. A Wirtinger-calculus gradient-descent update, Θ_t+1 = Θ_t − η_t \(∇^W_Θ\), with the schedule η_t = η₀(1 + γt)^−α, α ∈ (0.5, 1], achieves E[L(Θ_T ) − L^∗] = O(T⁻¹). Applied to nonstationary signal processing, the CFS phase encodes short-time Fourier transform (STFT) bin phases, yielding a complex fuzzy spectrogram \(S_\tilde{A}\) (t, ω) = r(t, ω)^ei∠X(t,ω), with an SNR gain ΔSNR ≥ 4.7 dB over type-2 baselines. Experiments on three benchmarks, namely synthetic chirp denoising, EEG δ/θ-band classification, and ECG arrhythmia detection, confirm a mean accuracy of \(\tilde{A}\) = 96.3 ± 0.4%, outperforming IT2-FLS by +3.7 percentage points and ANFIS by +5.1 percentage points.