SincInterpolation

Sinc interpolation is an ideal method for reconstructing a continuous-time signal from its discrete samples, under the assumption that the original signal is bandlimited. It arises from the Nyquist–Shannon sampling theorem, which states that a signal with maximum frequency B can be perfectly reconstructed from samples taken at a rate Fs greater than 2B.

The interpolation uses the sinc function, defined as sinc(x) = sin(pi x)/(pi x). If the samples are x[n] = x(nT) with sampling interval T = 1/Fs, the interpolated value at time t is given by

x(t) = sum_{n=-∞}^{∞} x[n] · sinc((t - nT)/T).

This kernel acts as an ideal low-pass reconstruction filter, preserving frequencies below the bandwidth limit.

In theory, sinc interpolation yields exact reconstruction for any bandlimited signal where the sampling rate satisfies the Nyquist criterion. In practice, however, the sinc function has infinite support and is non-causal, making exact implementation impossible. Real systems use approximations such as windowed sinc filters (truncating the kernel with a window like Hann or Hamming) or other finite-impulse-response/interpolation methods for resampling and up/down-conversion. The quality of the reconstruction depends on the kernel length and the window chosen.

Applications include reconstructing analog signals from digital samples, upsampling and fractional-decimation, and general resampling in digital audio and video processing. While sinc interpolation offers an ideal theoretical basis for perfect reconstruction, practical implementations balance accuracy against computational cost and latency through approximations.