📻 Spectral Analysis

Decompose signals into frequency components — FFT, windowing, and the live spectrogram

Time Domain Signal x(t) = Σ Aₙ·cos(2π·fₙ·t + φₙ)

Magnitude Spectrum (FFT) |X[k]| — log scale

Spectrogram (STFT) Time → Frequency heatmap

Presets

Harmonics (up to 6)

FFT Settings

FFT Size 1024

Noise Level 0.00

Show Phase Spectrum

Power Spectral Density

Window:

Live Metrics

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Fund. (Hz)

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Δf (Hz)

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SNR (dB)

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Parseval err

About Spectral Analysis

The Fast Fourier Transform

The DFT decomposes any discrete signal into N complex sinusoids: X[k] = Σ x[n]·e^(−2πijk/N). The Cooley-Tukey radix-2 FFT computes this in O(N·log₂N) operations instead of O(N²), making real-time spectrum analysis practical. A 1024-point FFT needs only ~10,000 operations instead of one million.

Windowing & Spectral Leakage

A finite signal segment is implicitly multiplied by a rectangular window, whose sharp edges create spectral leakage — energy from one frequency bleeds into neighbours. The Hann window w[n] = 0.5(1−cos(2πn/N)) tapers the edges smoothly, trading frequency resolution for dramatically reduced leakage. Blackman windows suppress leakage further at the cost of wider main lobes.

The Spectrogram (STFT)

The Short-Time Fourier Transform runs an FFT on overlapping windows slid across time: X(m,k) = Σ x[n+m·hop]·w[n]·e^(−2πijk/N). The result is displayed as a 2D heatmap (time × frequency). There is an inherent resolution trade-off: long windows give fine frequency resolution but blur rapid changes; short windows capture transients but smear frequencies.