Tuning by Linear Interpolation

An overview of linear interpolation, among others, is given in §3.2. The transfer function of first-order linear-interpolation can be written as

where denotes the desired delay in samples at low frequencies compared with the sampling rate.

Faust Implementation.

Faust includes a function fdelay(n,d,x) defined in music.lib which provides fractional (non-integer) delay by means of linear interpolation:

  fdelay(n,d,x) = delay(n,int(d),x)*(1 - frac(d)) 
                + delay(n,int(d)+1,x)*frac(d);

Note that it also specifies a second delay line. However, a second delay-line is not implemented in the generated C++ code because Faust has an optimization rule that consolidates all delay-lines having the same input signal to one shared delay line.

A single-delay-line version can be defined as follows:

  linterp(d,x) = (1-d) * x + d * x';
  fdelay1(n,d,x) = delay(n,int(d),x) : linterp(frac(d));

Note that these definitions are not equivalent. While they are equivalent when the delay d is fixed, they diverge when int(d) changes from one sample to the next. In the dual-delay-line specification, the desired result is obtained, while in the single-delay-line case, x' becomes a one-sample delay of the old delay-line output instead of the new delay-line output. This inconsistency can potentially cause audible clicks when the tuning is rapidly varied.

A proper implementation of the single-delay-line case involves resetting the linear-interpolator state variable when the delay-length changes. Conceptually, the linear interpolator should be implemented as two delay-line taps with gains and , as indicated in Fig.. This structure is produced by the Faust optimization rules from the dual-delay-line specification as above.

Linear delay-line interpolation works well in a digital waveguide string model as long as the modeled string is sufficiently damped. Specifically, the string damping must be sufficient to mask the changing roll-off in the