Delay Line Interpolation: Allpass Interpolation

• An allpass filter has unity magnitude response but variable phase delay properties.

• The difference equation for a first-order allpass interpolation filter is given by
  

  	$$\hat{x}[n-\Delta] \ensuremath{\stackrel{\Delta}{=}}y[n]$$	$=$	$$a \cdot x[n] + x[n-1] - a \cdot y[n-1]$$
  		$=$	$$a \cdot (x[n] - y[n-1]) + x[n-1].$$

• A block diagram for a first-order allpass interpolation filter is shown in Fig. 16.

  Figure 16: A first-order allpass interpolated delay line.

  

• The frequency response of the first-order allpass interpolation filter is
  $$H(z) = \frac{a + z^{-1}}{1 + a z^{-1}}.$$
  At low frequencies (as $z \rightarrow 1)$, the delay becomes
  $$\Delta \approx \frac{1 - a}{1 + a}.$$

• Figure 17 shows the phase delay, calculated with the Matlab script interpolate.m, of the first-order allpass filter for fractional delay values between 0.1 and 2.

  Figure 17: Phase delay for allpass interpolation with fractional delays between 0.1 and 2.

  

• For a given desired fractional delay $\Delta$, the allpass coefficient $a$ is determined as
  $$a = \frac{1 - \Delta}{1 + \Delta},$$
  where $\Delta$ is best maintained in the range $0.3 \leq \Delta \leq 1.3$ to achieve maximally flat phase delay response together with the fastest decaying impulse response, a desired characteristic when dealing with dynamic delay values and their associated transient responses.

• The transient responses, calculated with the Matlab script aptransient.m, of first-order allpass filters used to implement several different fractional delay values are shown in Fig. 18. Note how values of $\Delta$ closer to 0.0 have significantly longer transient “tails”.

  Figure 18: Impulse response of first-order allpass filters with fractional delay settings of $\Delta$ = 0.1, 0.4, 0.7, and 1.0.

  

• First-order allpass interpolation is the most simple case of Thiran allpass interpolation.