4.1 Requirements

The requirements  for the estimation are basically the fulfillment of the properties of spectral envelopes as described in section 2.3, with some additions and precisations.

Exactness 

A precise hit on the point in the frequency-amplitude plane defined by a sinusoidal partial is desirable. In section 2.3 this was called the envelope fit property, in that the spectral envelope is an envelope of the spectrum, i.e. it wraps tightly around the magnitude spectrum, linking the peaks.

The degree of exactness is determined by the perceptual abilities of the human ear. In the lower frequency range, it can distinguish differences in amplitude as small as 1 dB. For higher frequencies, the sensitivity is a little lower.

Sometimes it is not possible to link every peak, e.g. when the additive analysis finds a group of peaks close to each other in the upper frequency range. Then, the spectral envelope should find a resonable intermediate path, e.g. through the center of gravity of each frequency slice of the cloud.

Robustness 

The estimation method has to be applicable to a wide range of signals with very different characteristics, from high pitched harmonic sounds with their wide spaced partials to noisy sounds or mixtures of harmonic and noisy sounds. Very often, the problems lie in additive analysis in that very low amplitude peaks are identified as sinusoidal partials although they pertain to the residual noise or even to the noise floor of the recording. This is also a question of choosing the right parameters for additive analysis, e.g. the threshold for partial amplitudes.

Regularity 

A certain smoothness or regularity is required. This means, the spectral envelope must not oscillate too much, but it should still give a general idea of the distribution of the signal's energy over frequency. This translates to a restriction on the slope of the envelope (given by its first derivative), which may be dependent on context.

Steadyness 

We want the curve to be steady (in the mathematical sense of a steady function), i.e. it has no corners (where the first derivative jumps).