Pick-Position Comb Filter

A natural model of string excitation is an input signal summed into a virtual physical location along the length of a digital waveguide string model, as described in §4.9. This model can then be factored into a pick-position comb filter in series with a filtered delay loop, as used in the EKS [213,437] and derived in §4.9.1.

The EKS pick-position comb filter has the transfer function

where is the period (total loop delay) in samples, and denotes normalized position along the string (0 being at the ``bridge'' and being at the ``nut''). The notation means the ``greatest integer less than or equal to ,'' also called truncation to the next lower integer.^D.6 This transfer-function model of pick position is easily derived by simply factoring the transfer-function of the string from the picking point to any other point along the string, such as the bridge point (see §4.9.1) [437].

In Faust, a feedforward comb filter is readily implemented using the delay function defined in music.lib:

  ppdel = beta*P; // pick position as fraction of period
  pickpos = _ <: delay(Pmax,ppdel) : - ;

where Pmax is some power of 2 larger than ppdel (see the definition of delay in music.lib to understand why a power of 2). In Faust, we can bring out a ``continuous'' pick-position control spanning half the string as follows:

  beta = hslider("pick_position", 0.13, 0, 0.5, 0.01); // 0-1/2

The block diagram generated by ``faust -svg -simple-names'' is shown in Fig.D.5. Pick position accuracy is normally not critical, hence the 1% slider steps and lack of