EDR-Based Loop-Filter Design

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EDR-Based Loop-Filter Design

This section discusses use of the Energy Decay Relief (EDR) (§2.2.2) to measure the decay times of the partial overtones in a recorded vibrating string.

First we derive what to expect in the case of a simplified string model along the lines discussed in §4.6 above. Assume we have the synthesis model of Fig.4.11, where now the loss factor is replaced by the digital filter that we wish to design. Let $\underline{x}(n)$ denote the contents of the delay line as a vector at time , with $\underline{x}(0)$ denoting the initial contents of the delay line.

For simplicity, we define the EDR based on a length DFT of the delay-line vector $\underline{x}$ , and use a rectangular window with a ``hop size'' of samples, i.e.,

$\displaystyle \underline{X}_m(\omega_k) \isdef \dft _{N,\omega_k}\{\underline{x}_m\}, \quad m=0,1,2,\ldots,$

where $\underline{x}_m(n)\isdef \underline{x}(mN)$ . That is $\underline{x}_m$ is simply the

th successive snapshot of the delay-line contents, where the snapshots are taken once every

samples. We may interpret $\underline{X}_m$ as

th short-time spectrum of the output signal $y^{+}(n)$ shown in Fig.4.11. Due to the special structure of our synthesis model, we have

$\displaystyle \underline{X}_m(\omega_k) = H_l^m(\omega_k) \underline{X}_0(\omega_k)$

for each DFT

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Chapters

Physical Audio Signal Processing
    Elementary String Instruments
       Loop Filter Identification
          EDR-Based Loop-Filter Design