Chebyshev filter

This research area investigates the application of Chebyshev polynomials and wavelets — especially shifted forms like the second and fourth kind — as basis functions for numerical approximation, solution, and convergence analysis of high-order boundary value problems (BVPs) and inverse nodal problems for differential operators such as Dirac operators. These approaches transform differential problems into solvable algebraic systems that offer high accuracy and uniform convergence. The practical impact includes robust computational methods for solving complex physical and engineering models expressed as boundary or inverse spectral problems.

This theme covers the development and computational investigation of Chebyshev filters — including analog, digital, waveguide, and ultra-wideband bandpass variants — using modern numerical and simulation tools. It focuses on how Chebyshev polynomials and filter design methodologies contribute to filter characteristics such as sharper roll-off, wider bandwidth, physical miniaturization, and robustness against coefficient quantization effects. The synergy between classical theory and computational methods enables optimized filter design for telecommunications and DSP systems.

This theme encompasses the application of Chebyshev polynomial-based models combined with advanced signal processing methods such as swept sine excitation signals and wavelet transforms. The approaches produce accurate nonlinear system identifications and fault diagnostics for audio effects and mechanical faults through time-frequency analysis. Chebyshev polynomials provide a mathematically rigorous yet computationally efficient basis to capture nonlinearities and complex signal dynamics.