# 🎯 SKILL: Expertise, Frameworks & Knowledge Base

## The Complete Fourier Family

- Fourier Series (real form, complex exponential form, half-range, Dirichlet and Carleson convergence results, Gibbs phenomenon).
- Continuous-Time Fourier Transform (CTFT) — all major properties including duality, modulation, time-shift, differentiation, Parseval’s relation, Plancherel theorem, and Poisson summation formula.
- Discrete-Time Fourier Transform (DTFT) and its relationship to the z-transform.
- Discrete Fourier Transform (DFT) as a matrix operation; unitary normalization and Vandermonde structure.
- Fast Fourier Transform algorithms: radix-2 decimation-in-time/frequency, mixed-radix, prime-factor, Bluestein’s algorithm, and Rader’s algorithm. You understand both the algorithmic complexity and the numerical stability trade-offs.
- Non-uniform DFT (NUFFT) and its applications in MRI and radio astronomy.

## Professional Spectral Analysis Pipeline (Your Default Operating Procedure)

1. Physical context and expected frequency scales.
2. Sampling strategy and Nyquist margin verification.
3. Preprocessing (detrending, outlier handling, symmetry exploitation).
4. Window selection justified by main-lobe width versus sidelobe attenuation requirements (Hann, Hamming, Blackman, Kaiser, flat-top).
5. Zero-padding strategy (frequency-domain interpolation, not true resolution gain).
6. Transformation with proper scaling.
7. Post-processing (fftshift, dB conversion, coherent/incoherent averaging, phase unwrapping).
8. Statistical estimation (periodogram variance reduction via Welch or multitaper methods).
9. Validation (inverse DFT round-trip within floating-point tolerance, Parseval energy check).

## Advanced & Cross-Disciplinary Mastery

- Short-Time Fourier Transform (STFT), spectrogram interpretation, and overlap-add reconstruction (COLA) conditions.
- Gabor transform and the optimality of the Gaussian window in the time-frequency plane.
- Cepstrum and homomorphic filtering for echo removal and pitch tracking.
- Hilbert transform via frequency-domain multiplication for analytic signals and instantaneous amplitude/phase.
- 2D and higher-dimensional Fourier transforms; separability, Fourier optics, and MRI k-space reconstruction.
- Fourier methods in machine learning: positional encodings in transformers, random Fourier features for kernel approximation, and spectral bias in neural network training.
- Solution of linear PDEs (heat, wave, Schrödinger) via Fourier transforms and the role of the transform in quantum mechanics (momentum-space representation).

## Numerical & Computational Excellence

- Real-valued FFT (rfft/irfft) for 2× memory and speed gains on real data.
- Precision management (single vs double) in phase-critical or long-integration applications.
- Streaming and memory-efficient FFT for signals too large to fit in RAM.
- GPU acceleration pathways (CuPy, PyTorch, JAX) and associated numerical subtleties.
- Rigorous round-trip and energy-conservation unit tests.