Superoscillatory functions are globally bandlimited signals that, over a finite interval, oscillate faster than their highest Fourier frequency. This challenges the general assumption that a signal whose bandwidth is limited to b/2 Hz cannot oscillate at frequencies exceeding b Hz. The effect is not a violation of Fourier theory, nor a hidden high frequency — it arises from precise interference among the allowed spectral components.
Below, a product of slow, bandlimited sinusoids is constructed with its zero crossings clustered tightly together. Inside that cluster — the superoscillation region — the function oscillates at a chosen target frequency well above its global band limit, while its spectrum contains nothing above that limit. The local oscillation is never free: it carries a vanishingly small amplitude relative to the sidelobes outside, which is why it is only visible on a logarithmic scale. Drag the sliders to add zeros and raise the target frequency.
1 · The function SO(x) = 3∏j=1 sin(x − φj) — in linear scale
The is the target sine at the local frequency — notice how the product crosses the same zeros in the SO region. On this linear axis the SO oscillation is almost invisible.
2 · The same function but in log scale |SO(x)|
Now with a log scaling, the SO features appear: between the SO zeros, the function makes several small lobes (the fast local oscillation), orders of magnitude below the giant sidelobes outside. That amplitude gap is the cost of superoscillation. Energy flows out of the SO region as the number of zeros increases, and as the target frequency increases.
3 · Global Fourier Transform of SO(x) — |F{SO(x)}(ω)|
A product of sines is a sum of delta functions, with the maximum at the band limit. The target frequency the SO function oscillates at inside the SO region is not captured in the global Fourier spectrum — band-limited globally, fast locally.
4 · Short-time Fourier transform (STFT)
The STFT slides a short window along the signal and takes the Fourier transform of each segment, revealing how the frequency content changes with position. That window is a Hann window () : it tapers smoothly to zero at its edges, and its width sets the time–frequency tradeoff. Stacking each segment's transform side by side gives the spectrogram (bottom row): position runs left to right, frequency bottom to top, and brightness shows how much of each frequency is present at that location.
The STFT shows how frequency content varies across the signal. The pure target sine (left) carries its frequency everywhere — a uniform horizontal band. SO(x) (right) only reaches that frequency locally, inside the SO region, and stays below the band limit elsewhere. The superoscillation is invisible to the global Fourier transform but shows up here. As the Hann window widens to capture the sidelobes outside the SO region, the local high frequency washes out and the SO feature can no longer be picked out. As a sanity check, widening the window also makes the target's spectrogram approach its global Fourier transform — a single delta at the target frequency — so its band sharpens toward a thin line at that frequency.