Expert-level signal processing covering Fourier analysis, sampling theory, digital filters, FFT, spectral estimation, and adaptive filtering.
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Fourier series: periodic signal decomposed into harmonics. Fourier transform: continuous signal to frequency domain representation. DTFT: discrete-time Fourier transform for discrete signals. DFT: discrete Fourier transform, finite length sequence, periodic in frequency. FFT: fast Fourier transform, O(N log N) algorithm for DFT computation.
Nyquist theorem: sample at twice the highest frequency to avoid aliasing. Aliasing: frequencies above Nyquist fold back, appear as lower frequencies. Anti-aliasing filter: lowpass filter before ADC to prevent aliasing. Interpolation: reconstruct continuous signal from samples using sinc function. Oversampling: sample faster than Nyquist, use decimation to reduce noise.
FIR: finite impulse response, always stable, linear phase possible. IIR: infinite impulse response, recursive, more efficient but can be unstable. Filter design: Parks-McClellan for FIR, bilinear transform for IIR from analog. Z-transform: discrete equivalent of Laplace transform, analyzes filter stability. Poles and zeros: determine filter frequency response and stability.
Periodogram: squared magnitude of FFT, basic power spectral density estimate. Welch method: average periodograms of overlapping segments, reduces variance. Windowing: reduce spectral leakage, trade resolution for dynamic range. Parametric methods: AR, ARMA models for spectral estimation.
| Pitfall | Fix |
|---|---|
| Aliasing from insufficient sampling rate | Sample at least twice highest frequency of interest |
| Spectral leakage distorting analysis | Apply appropriate window function |
| IIR filter instability | Check all poles inside unit circle |
| Circular convolution confusion | Use zero-padding for linear convolution via DFT |
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Expert-level abstract algebra knowledge. Use when working with groups, rings, fields, modules, Galois theory, representation theory, or algebraic structures. Also use when the user mentions 'group', 'ring', 'field', 'homomorphism', 'isomorphism', 'normal subgroup', 'quotient group', 'Sylow theorem', 'Galois theory', 'polynomial ring', 'ideal', 'module', 'vector space', or 'representation theory'.
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Expert-level acoustics knowledge. Use when working with sound waves, acoustic propagation, resonance, standing waves, Doppler effect, room acoustics, ultrasound, noise control, musical acoustics, or architectural acoustics. Also use when the user mentions 'sound wave', 'decibel', 'frequency', 'resonance', 'standing wave', 'Doppler effect', 'acoustic impedance', 'reverberation', 'ultrasound', 'noise control', 'harmonics', or 'sound intensity'.
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Expert-level aerodynamics covering subsonic and supersonic flow, lift and drag, airfoil theory, boundary layers, compressible flow, and CFD methods.
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Expert-level air quality engineering covering pollutant sources, atmospheric dispersion, air quality standards, emission controls, and air quality modeling.
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Expert-level aircraft structures covering structural design philosophy, wing and fuselage analysis, fatigue and damage tolerance, composite structures, and certification.
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Expert-level algorithms and complexity theory covering sorting, searching, graph algorithms, dynamic programming, greedy algorithms, NP-completeness, and algorithm design paradigms.
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