Reconstruction & sinc

The sampling theorem promises the dots contain everything. But your speaker needs a continuous voltage, not dots. Reconstruction is the keeping of the promise — and the keeper is a curious wave-shaped function called sinc.

Build the intuition

Connect-the-dots is the wrong instinct

Straight lines between samples create corners, and corners (Fourier series taught you) require high frequencies — frequencies above Nyquist that the true band-limited signal cannot contain. Honest reconstruction must thread the dots using only legal frequencies: it must be the smoothest possible interpolation, not the laziest.

Sinc: the one legal brushstroke

sinc(t) = sin(πt)/πt is the unique band-limited curve that equals 1 at its own sample instant and 0 at every other. Plant one scaled sinc on each sample and add: each dot is hit exactly (all other sincs are zero there), and the sum contains no illegal frequencies. That superposition is the perfect reconstruction the theorem promised.

x(t) = \sum_n x[n]\, \mathrm{sinc}\!\left(\frac{t - nT}{T}\right)

Why sinc, intuitively — and its price

In frequency, ideal reconstruction is a brick-wall low-pass: keep everything below Nyquist, kill everything above. The time-domain shape of that brick wall is exactly sinc — the duality at work. The price: sinc's ripples extend forever, so perfection needs all samples (including future ones). Real DACs use short, causal approximations — engineering's polite compromise with infinity.

See it move

InteractiveThe sampling theorem, felt

Signal f3 Hz

Sample rate8 Hz

Sampling at 8 Hz — more than twice the signal's 3 Hz — the dots pin down exactly one sine. The signal survives digitization.

Above the Nyquist rate there is exactly one band-limited curve through the dots — the one sinc interpolation rebuilds. Below it, reconstruction faithfully rebuilds… the impostor.

A worked example

Resize a photo, honestly

Doubling an image's size requires inventing pixels between pixels — interpolation, i.e. reconstruction.
Nearest-neighbor (copy the dot): blocky — it reconstructs with rectangles, spraying high-frequency edges.
Bilinear (connect the dots): smoother but soft, with subtle artifacts — straight lines still aren't band-limited.
Lanczos (a windowed sinc): crisp and clean — closest to the legal ideal. Photo editors' “best quality” setting is sinc wearing a stage name.

Out in the world

Inside every DAC

Your phone's audio chip converts samples to a staircase voltage, then an analog filter smooths the stairs — a physical approximation of sinc interpolation. “Oversampling” DACs first interpolate digitally to push the staircase corners far above hearing, making the analog filter's job easy. Both stages are this lesson in silicon.

Common confusion, cleared

“Digital audio outputs are stairsteps, so digital sounds “steppy.””

The steps exist only inside the converter; the reconstruction filter removes them (they're all above Nyquist). What reaches the speaker is the smooth band-limited curve — the staircase critique misunderstands reconstruction.

“Better interpolation could beat the sampling theorem.”

No algorithm recovers frequencies the samples never captured. Sinc is already optimal for band-limited signals; everything else approximates it. The theorem is a wall, not a hurdle.

Recap

Reconstruction must use only sub-Nyquist frequencies — no corners allowed.
Sinc hits its own sample, zeroes all others: superpose for perfect recovery.
Ideal sinc is infinite; real DACs and resizers ship faithful approximations.

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