SIGNALS & SPECTRA · VISUAL LAB

Change a tone. See its spectrum.

One frequency slider moves the sampled wave and its computed spectrum.

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Interactive experiment

Signal → spectrum

Bin width Δf0.500 Hz

Nyquist limit32.00 Hz

Record length T2.00 s

All tones are at or below the Nyquist limit.

Largest spectrum bins (frequency · amplitude): 3.00 Hz (1.00) · 9.00 Hz (0.55) · 17.00 Hz (0.30)

Phase advances automatically. The spectrum is recomputed from each displayed finite sample record.

Input tones:f₁ 3.00 Hzf₂ 9.00 Hzf₃ 17.00 Hz

Sampled waveform · first 0.5 s

Default sampled waveformComputed samples for tones at 3, 9, and 17 hertz, with a 64 hertz sample rate. The plot shows the first half-second.0 s0.50 s+2−2

Single-sided amplitude spectrum

Default single-sided amplitude spectrumRadix-2 FFT of 128 samples at 64 hertz. Peaks appear at 3, 9, and 17 hertz.0 Hz16 Hz32 Hzmax
Xk=∑n=0N−1xne−2πikn/N,Δf=FsN=1T,fNyquist=Fs2X_k=\sum_{n=0}^{N-1}x_n e^{-2\pi i kn/N},\qquad \Delta f=\frac{F_s}{N}=\frac{1}{T},\qquad f_{\mathrm{Nyquist}}=\frac{F_s}{2}
The dots are the actual samples. The spectrum is computed from those samples with a radix-2 FFT and single-sided amplitude normalization. Try f₁ = 3.25 Hz to see leakage, lower Fₛ below 34 Hz to see aliasing, or change N to alter the bin width.
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