First-Order Allpass Interpolation

$$\begin{eqnarray*} {\hat x}(n-\Delta) \mathrel{\stackrel{\mathrm{\Delta}}{=}}y(n) &=& \eta \cdot x(n) + x(n-1) - \eta \cdot y(n-1) \\ &=& \eta \cdot \left[ x(n) - y(n-1)\right] + x(n-1) \end{eqnarray*}$$

$$H(z) = \frac{\eta + z^{-1}}{1 + \eta z^{-1}}$$

• Low frequency delay given by (exact at DC):
  $$\Delta \approx \frac{1-\eta}{1+\eta} \quad\Longleftrightarrow\quad \eta \approx \frac{1-\Delta}{1+\Delta}$$

• Same complexity as linear interpolation
• Good for delay-line interpolation, not random access
• Best used with fixed fractional delay $\Delta$
• To avoid pole near $z = -1$ , offset delay range, e.g.,
  $$\Delta\in[0.1,1.1] \;\leftrightarrow\; \eta\in[-0.05,0.82]$$

• Change delay slowly compared to $\tau\approx T/(1-\eta_{\mbox{max}})$

Intuitively, ramping the coefficients of the allpass gradually ``grows'' or ``hides'' one sample of delay. This tells us how to handle resets when crossing sample boundaries (sufficiently slowly).