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Home/Math Visualization/STFT & Spectrogram

STFT & Spectrogram

This interactive simulator explores STFT & Spectrogram in Math Visualization. Slide a windowed FFT across the signal: chirps, two-tones, bursts. Tune window M, hop, type — see the time–frequency trade-off live. Use the controls to change the scenario; watch the visualization and any graphs or readouts to connect the model with lectures, labs, and homework.

Who it's for: For learners comfortable with heavier math or second-level detail. Typical context: Math Visualization.

Key terms

  • stft
  • spectrogram
  • stft spectrogram
  • math
  • visualization

Signal & STFT

Signal preset

8
0.25
60dB
2.048s

Shortcuts

  • •Drag on the spectrogram to move the time cursor

Measured values

time resolution Δt64.00ms
freq resolution Δf3.91Hz
overlap75%
frames61
peak f at cursor226.6Hz

How it works

Short-Time Fourier Transform: slide a window w[n] of length M across the signal in steps of hop, FFT each frame, plot |X(t,f)| in dB as a heatmap. There is a hard time–frequency trade-off: large M sharpens Δf = f_s/M but blurs Δt ≈ M/f_s; small M does the opposite. Try the up-chirp with M = 32 vs M = 1024, or compare windows on the two-tone preset to see leakage on a rectangular window vs Hann/Blackman side-lobes.

Key equations

X(m, k) = Σₙ x[n + m·H] · w[n] · e⁻²πⁱᵏⁿᐟᴹ
Δt = H/f_s, Δf = f_s/M (Heisenberg Δt·Δf ≥ 1/(4π))