Sign in

username:

password:



Not a member?

Search Online Books



Search tips

Free Online Books

Chapters

Physical Audio Signal Processing
    Time-Varying Delay Effects
       Delay-Line Interpolation
          First-Order Allpass Interpolation

Chapter Contents:

Search Physical Audio Signal Processing

  

Book Index | Global Index

Would you like to be notified by email when Julius Orion Smith III publishes a new entry into his blog?

  

First-Order Allpass Interpolation

A delay line interpolated by a first-order allpass filter is drawn in Fig.3.4.

Figure 3.4: Allpass-interpolated delay line.
\includegraphics[width=\twidth]{eps/delayai}

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.

The difference equation is

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

Thus, like linear interpolation, first-order allpass interpolation requires only one multiply and two adds per sample of output.

The transfer function is

$\displaystyle H(z) = \frac{\eta + z^{-1}}{1 + \eta z^{-1}}. \protect$ (4.1)

At low frequencies ($ z\to 1$), the delay becomes

$\displaystyle \Delta \approx \frac{1-\eta}{1+\eta} \protect$ (4.2)

Figure 3.5 shows the phase delay of the first-order digital allpass filter for a variety of desired delays at dc. Since the amplitude response of any allpass is 1 at all frequencies, there is no need to plot it.

Figure 3.5: Allpass-interpolation phase delay for a variety of desired delays (exact at dc).
\includegraphics[width=\twidth]{eps/allpass1}

The first-order allpass interpolator is generally controlled by setting its dc delay to the desired delay. Thus, for a given desired delay $ \Delta $, the allpass coefficient is (from Eq.$ \,$(3.2))

$\displaystyle \eta \approx \frac{1-\Delta}{1+\Delta} $

From Eq.$ \,$(3.1), we see that the allpass filter's pole is at $ z=-\eta$, and its zero is at $ z=-1/\eta$. A pole-zero diagram for $ \Delta =0.1$ is given in Fig.3.6. Thus, zero delay is provided by means of a pole-zero cancellation! Due to inevitable round-off errors, pole-zero cancellations are to be avoided in practice. For this reason and others (discussed below), allpass interpolation is best used to provide a delay range lying wholly above zero, e.g.,