Windowed Sinc Interpolation

• Sinc function can be windowed more generally to yield
  $${{\hat h}_\Delta}(n) = \left\{\begin{array}{ll} w(n-\Delta)\mbox{sinc}[\alpha(n-\Delta)], & 0\leq n \leq L-1 \\ [5pt] 0, & \hbox{otherwise} \\ \end{array} \right.$$

• Example of window method for FIR lowpass filter design applied to sinc functions (ideal lowpass filters) sampled at various phases (corresponding to desired delay between samples)

• For best results, $\Delta\approx L/2$ (causal length $L$ window)

• $w(n)$ is any real symmetric window (e.g., Hamming, Blackman, Kaiser)

• Non-rectangular windows taper truncation which reduces Gibbs phenomenon, as in FFT analysis

• $\alpha<1$ provides for a nonzero transition band