### EDR-Based Loop-Filter Design

This section discusses use of the Energy Decay Relief (EDR) (§3.2.2) 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 §6.7 above. Assume we have the synthesis model of Fig.6.12, where now the loss factor is replaced by the digital filter that we wish to design. Let denote the contents of the delay line as a vector at time , with denoting the initial contents of the delay line.

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

Applying the definition of the EDR (§3.2.2) to this short-time spectrum gives

We therefore have the following recursion for successive EDR
time-slices:^{7.13}

This analysis can be generalized to a time-varying model in which the
loop filter is allowed to change once per ``period''
.^{7.14}

An online laboratory exercise covering the practical details of measuring overtone decay-times and designing a corresponding loop filter is given in [280].

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