Loop-Filter Design Methods

One of the earliest methods was ``periodic linear prediction'' -- a linear combination of a small group of samples predicts a sample one period away from the midpoint of the group.

More recently, methods have been developed based on measurements of the time-constant of decay for each partial overtone, since this is what matters the most.

• Record a plucked (or struck) string.
• Measure frequency $f_k$ and decay time $\tau_k$ for each overtone.
  • Energy Decay Relief (EDR) useful for this purpose
  • Sinusoidal modeling has been used as well
  • In either case:
    • Fit a straight line to the log of the amplitude envelope (e.g., using polyfit in Matlab or Octave).
    • Decay time $\tau_k$ is simply related to the slope of this line.
• Translate each decay-time $\tau_k$ into a desired loop-gain at each frequency $f_k$ , thus determining $\left\vert H_l(e^{j\omega_k })\right\vert$ for each partial.
• If desired, use the inharmonic spacing of the $f_k$ to compute samples of the desired phase of the loop filter, $\angle{H_l(e^{j\omega_k })}$ .
• Use a general purpose digital filter design routine to design the loop filter which best approximates the desired frequency-response samples $H_l(e^{j\omega_k })$ .
• Phase-matching filter design utilities:
  • Steiglitz-McBride algorithm (IIR or FIR)
    See stmcb in Matlab Signal Processing Tool Box
  • Equation-Error Minimization (IIR or FIR)
    See invfreqz in matlab
  • Hankel-Norm Minimization (IIR)
    Also called a ``balanced truncation model'' (state space)
  • Linear-phase FIR (fir1, firls, remez)
• Phase-insenstive filter design:
  • Linear Prediction
    See lpc or yulewalk in the Matlab SPTB
  • Create a desired linear or minimum phase and use a phase-sensitive design method.
• The fit at low frequencies should be weighted higher.
• A simple $1/f$ weighting usually works well.
• Bark spectral warping is commonly used. It implicitly weights low frequencies more by ``stretching out'' the low-freqency frequency axis and compressing the high-frequency frequency axis.
• Piano strings are highly dispersive:
  • Recall that every LTI filter can be factored into a linear phase filter in series with an allpass filter
  • The allpass part for a low-pitched string requires on the order of 100 poles (and 100 zeros) to provide a closely matched impulse response
  • Ten poles is a typical number used in practice (can be very good perceptually)
  • Most methods for allpass filter design fail numerically at this high order
  • The linear-phase part, in contrast, is well approximated using only an 8-tap FIR filter, and can be designed by any good method for FIR filter design (e.g., invfreqz in Matlab/Octave)