Linear interpolation works by effectively drawing a straight line between two neighboring samples and returning the appropriate point along that line.
More specifically, let be a number between 0 and 1 which represents how far we want to interpolate a signal between time and time . Then we can define the linearly interpolated value as follows:
For , we get exactly , and for , we get exactly . In between, the interpolation error is nonzero, except when happens to be a linear function between and .
Note that by factoring out , we can obtain a one-multiply form,
A linearly interpolated delay line is depicted in Fig.4.1. In contrast to Eq.(4.1), we interpolate linearly between times and , and is called the fractional delay in samples. The first-order (linear-interpolating) filter following the delay line in Fig.4.1 may be called a fractional delay filter . Equation (4.1), on the other hand, expresses the more general case of an interpolated table lookup, where is regarded as a table of samples and is regarded as an interpolated table-lookup based on the samples stored at indices and .
The difference between a fractional delay filter and an interpolated table lookup is that table-lookups can ``jump around,'' while fractional delay filters receive a sequential stream of input samples and produce a corresponding sequential stream of interpolated output values. As a result of this sequential access, fractional delay filters may be recursive IIR digital filters (provided the desired delay does not change too rapidly over time). In contrast, ``random-access'' interpolated table lookups are typically implemented using weighted linear combinations, making them equivalent to nonrecursive FIR filters in the sequential case.5.1
The frequency response of linear interpolation for fixed fractional delay ( fixed in Fig.4.1) is shown in Fig.4.2. From inspection of Fig.4.1, 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 fractional delay changes over time, it is a linear time-varying filter.
Linear interpolation sounds best when the signal is oversampled. Since natural audio spectra tend to be relatively concentrated at low frequencies, linear interpolation tends to sound very good at high sampling rates.
When interpolation occurs inside a feedback loop, such as in digital waveguide models for vibrating strings (see Chapter 6), errors in the amplitude response can be highly audible (particularly when the loop gain is close to 1, as it is for steel strings, for example). In these cases, it is possible to eliminate amplitude error (at some cost in delay error) by using an allpass filter for delay-line interpolation.
First-Order Allpass Interpolation