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
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
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].