#### Envelope Compression

Once we have our data in the form of amplitude and frequency envelopes for each filter-bank channel, we can compress them by a large factor. If there are channels, we nominally expect to be able to downsample by a factor of , as discussed initially in Chapter 8 and more extensively in Chapter 10.

In early computer music [93,175], amplitude and
frequency envelopes were ``downsampled'' by means of *piecewise
linear approximation*. That is, a set of *breakpoints* were
defined in time between which linear segments were used. These
breakpoints correspond to ``knot points'' in the context of polynomial
spline interpolation. Piecewise linear approximation yielded large
compression ratios for relatively steady tonal signals. For example,
compression ratios of 100:1 were not uncommon for isolated ``toots''
on tonal orchestral instruments [93].

A more straightforward method is to simply downsample each envelope by
some factor. Since each subband is bandlimited to the channel
bandwidth, we expect a downsampling factor on the order of the number
of channels in the filter bank. Using a hop size in the STFT
results in downsampling by the factor (discussed
in §8.8). If channels are downsampled by , then the
total number of samples coming out of the filter bank equals the
number of samples going into the filter bank. This may be called
*critical downsampling*, which is invariably used in filter banks
for *audio compression*, as discussed in Chapter 10. A benefit
of converting a signal to critically sampled filter-bank form is that
bits can be allocated based on the amount of energy in each subband
relative to the psychoacoustic masking threshold in that band.
Bit-allocation is typically different for tonal and noise signals in a
band [108,24,16].

**Next Section:**

Vocoder-Based Additive-Synthesis Limitations

**Previous Section:**

Frequency Envelopes