Enter any expression in t, select a preset, and adjust harmonics. Time domain, magnitude spectrum, phase spectrum and harmonic synthesis update live.
Phasor animation, STFT spectrogram, aliasing demonstrator, cross-correlation — all computed in-browser, no microphone required.
Welch's PSD estimation, professional CSV export, BPF calculator and flutter margin estimator.
Every equation behind the analyser, derived from first principles.
Any periodic function decomposes into harmonically related sinusoids. aₙ, bₙ are real coefficients; cₙ carries amplitude and phase.
Maps N time samples to N complex frequency bins. Bin k → frequency f = k·Fs/N Hz.
Recursively divides into two N/2-point DFTs on even and odd samples until N=1.
Frequencies above f_N fold back as aliases. Turbine blade sensors must satisfy Fs > 2·f_BPF to avoid aliasing structural health signals.
Total energy is conserved between domains. The analyser reports ε ≈ 10⁻¹² for exact transforms.
Spectral leakage arises when period is not exact multiple of window. Windowing tapers edges, suppressing side-lobes at cost of slightly widened main lobes.
Adding harmonics narrows the spike but never eliminates it. Load "Gibbs Demo" preset and increase N.
Each rotor blade sweeps a stator passage at f_BPF. FFT of nacelle data reveals this tone — used for ICAO noise certification and SHM.
Welch averages K overlapping periodograms, reducing PSD variance. Spectrogram (STFT) tracks time-varying frequency content — used in flutter margin and run-up analysis.
Three development phases from core analysis engine to full aerospace research toolkit.
"The deep study of nature is the most fruitful source of mathematical discoveries."
Jean-Baptiste Joseph Fourier proved in his 1822 treatise Théorie analytique de la chaleur that any periodic function decomposes into an infinite sum of sines and cosines. The mathematical apparatus he built to solve the heat equation turned out to govern every oscillatory phenomenon in physics.
The Cooley–Tukey FFT (1965) cut the computational cost from O(N²) to O(N log N), making real-time spectrum analysis practical. It now underlies radar, MRI, audio codecs, wireless comms, and every structural health monitoring system in aerospace.
In aero: flutter margins are extracted from flight test accelerometer FFTs; nacelle microphones reveal turbofan BPF tones; pulse-Doppler radar builds range-Doppler maps via 2D FFT; composite SHM tracks modal frequency shifts.
Submitted to the Institut de France — rejected by Lagrange who disputed convergence. The work sits unpublished for 15 years.
Fourier formally proves any periodic function can be expressed as a trigonometric series and derives the integral coefficient formulae.
First rigorous proof of convergence for piecewise smooth, bounded functions — establishing the mathematical foundations Fourier assumed.
Josiah Willard Gibbs characterises the ~8.9% overshoot at discontinuities that persists regardless of harmonic count.
Harry Nyquist establishes that a signal can be perfectly reconstructed from samples at more than twice the highest frequency.
Reduces DFT from O(N²) to O(N log N). Originally to detect Soviet nuclear tests — becomes the most important numerical algorithm of the 20th century.
Peter Welch introduces the averaged overlapping periodogram method, dramatically reducing PSD variance — now standard in structural health monitoring and vibration analysis.