HPC Tier Module
FFT Spectral Analyzer with GeoNum Precision
Radix-2 FFT spectra with windowing, a Parseval energy check, and a GeoNum drift gauge on the butterflies.
This tool runs inside GDBS HPC. What it computes and the authorities it applies are below.
The problem
Spectral analysis tools often hide their assumptions: the sample-rate labeling, the leakage from windowing, and the round-off accumulated through the transform are easy to get silently wrong.
The local-first solution
This analyzer runs a cited radix-2 Cooley-Tukey FFT/IFFT in your browser, surfaces the sample rate it used, applies named windows with their gains, verifies energy with a Parseval residual, cross-checks against a naive DFT, and reports a GeoNum drift gauge on the butterfly sums. Deterministic f64 math, nothing uploaded.
What it does
Radix-2 decimation-in-time FFT and inverse FFT (FFT(IFFT) identity to round-off)
Magnitude, phase, power spectrum, and one-sided PSD with Welch normalization
Bin-frequency mapping f[k]=k*fs/N with the sample rate always echoed back
Rectangular, Hann, and Hamming windows with coherent and power gains reported
Parseval energy identity reported as a near-epsilon residual
Naive O(N^2) DFT cross-check plus a GeoNum drift verdict on the butterfly accumulations
Honest scope
EXACT: the radix-2 FFT/IFFT, the spectral formulas, the window gains, and the Parseval identity, all cited f64 arithmetic. The sample rate fs is a user input that the bin frequencies depend on - it is always echoed back. NOT modeled: non-power-of-2 lengths (zero-padded and flagged), multi-channel transforms, RFFT packing, overlap-averaged Welch segmenting, or anti-alias filtering; frequencies above fs/2 alias. A signal-processing tool, not measurement-grade metrology.
Authorities cited
- Cooley, J. W. & Tukey, J. W. (1965). An Algorithm for the Machine Calculation of Complex Fourier Series. Mathematics of Computation 19(90), 297-301. DOI 10.1090/S0025-5718-1965-0178586-1. (The radix-2 FFT factorization.)
- Oppenheim, A. V. & Schafer, R. W. (2010). Discrete-Time Signal Processing, 3rd ed., Prentice Hall. Ch. 8-9 - the DFT X[k]=sum x[n] e^{-i 2 pi k n / N}, its inverse, and FFT algorithms.
- Harris, F. J. (1978). On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform. Proceedings of the IEEE 66(1), 51-83. DOI 10.1109/PROC.1978.10837. (Hann/Hamming windows; coherent and power gain.)
- Parseval/Rayleigh energy theorem: sum_{n} |x[n]|^2 = (1/N) sum_{k} |X[k]|^2 (DFT form) - Oppenheim & Schafer Eq. for the DFT, conservation of energy between time and frequency domains.
- Welch, P. D. (1967). The Use of Fast Fourier Transform for the Estimation of Power Spectra. IEEE Trans. Audio Electroacoustics 15(2), 70-73. DOI 10.1109/TAU.1967.1161901. (One-sided PSD normalization S=2|X|^2/(fs*sum w^2).)
- Higham, N. J. (2002). Accuracy and Stability of Numerical Algorithms, 2nd ed., SIAM. Sec. 24 - the O(eps log2 N) round-off error bound of the radix-2 FFT (the basis for the GeoNum drift gauge). DOI 10.1137/1.9780898718027.
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Analyze a spectrum you can trust
Run the transform in your browser with nothing uploaded, then save the spectra and Parseval/drift report to a Sandbox workspace or attach them to a Worklog case.