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.*,

where . That is is simply the th successive snapshot of the delay-line contents, where the snapshots are taken once every samples. We may interpret as th short-time spectrum of the output signal shown in Fig.6.12. Due to the special structure of our synthesis model, we have

for each DFT bin number .

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}

Since we normally try to fit straight-line decays to the EDR on a log scale (typically a decibel scale), we will see the relation

where the common argument is dropped for notational simplicity. Since we require for stability of the filtered-delay loop, the EDR decays monotonically in this example. Thus, the measured slope of the partial overtone decays will be found to be proportional to .

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 [282].

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