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Physical Audio Signal Processing
    Time-Varying Delay Effects
       Delay-Line Interpolation
          Linear Interpolation

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Linear Interpolation

Linear interpolation works by effectively drawing a straight line between two neighboring samples and returning the appropriate point along that line.

More specifically, let $ \eta$ be a number between 0 and 1 which represents how far we want to interpolate a signal $ y$ between time $ n$ and time $ n+1$. Then we can define the linearly interpolated value $ \hat y(n+\eta)$ as follows:

$\displaystyle \hat y(n+\eta) = (1-\eta) \cdot y(n) + \eta \cdot y(n+1) $

For $ \eta=0$, we get exactly $ \hat y(n)=y(n)$, and for $ \eta=1$, we get exactly $ \hat y(n+1)=y(n+1)$. In between, the interpolation error $ \left\vert\hat y(n+\eta)-y(n+\eta)\right\vert$ is nonzero, except when $ y(t)$ is a linear function between $ y(n)$ and $ y(n+1)$.

Note that by factoring out $ \eta$, we can obtain a one-multiply form,

$\displaystyle \hat y(n+\eta) = y(n) + \eta\cdot\left[y(n+1) - y(n)\right]. $

Thus, the computational complexity of linear interpolation is one multiply and two additions per sample of output.

A linearly interpolated delay line is depicted in Fig.3.2.

Figure 3.2: Linearly interpolated delay line.
\includegraphics[width=\twidth]{eps/delayli}

The C++ class implementing a linearly interpolated delay line in the Synthesis Tool Kit (STK) is called DelayL.

The frequency response of linear interpolation for fixed fractional delay ($ \eta$ fixed in Fig.3.2) is shown in Fig.3.3. From inspection of Fig.3.2, we see that linear interpolation is a one-zero FIR filter. When used to provide a fixed fractional delay, the filter is linear and time-invariant (LTI). When the delay provided changes over time, it is a linear time-varying filter.

Figure 3.3: Linear interpolation frequency responses for delays between 0 and 1. Note the higher accuracy at low frequencies, reaching zero error at