Re: Hilbert envelope bandwidth

[Eckard Blumschein <Eckard.Blumschein@xxxxxxxxxxxxxxxxxxxxxxxxxx>]

Dear Yadong Wang,

Thank you for pointing to history of MIT. Despite of deeply respecting Hilbert,
Gabor, and Wiener, I would prefer using the term temporal envelope instead of
Hilbert envelope just in order to avoid confusion with the spectral envelope.

> The name Hilbert envelope was given by Dennis Gabor.

Perhaps in 1946? Hilbert died in 1943 in Germany. Does Therrien's 2002 article
in IEEE Signal Processing Magazine tell how Lee relates to Hilbert?

You added and already answered two questions:

> 'How to get it? Will it work in ASR?'
> To get the Hilbert envelope, the Hilbert operator in the time domain or
> halfway rectifier followed by low pass filtering would normally be used.

Indeed, the inner ear performs halfway rectification after real-valued cosine
transform. In contrast to other integral transforms like Fourier, Laplace or
cosine transform, Hilbert transform is not a transform between domains. It
rather assigns a complementary imaginary part to a given real part or vice versa
by shifting each component of the signal by a quarter of period. Temporal
envelopes are the magnitudes of analytical, i.e. complex, frequency components
of the fictitious analytical signal. They are real-valued and always positive.
Do not worried about  terms like complex envelope. While the spectral envelope
is a function of time, temporal envelopes are different for each frequency.

> A more interesting and practical way is to model the envelope by an all-pole
> model.

Do not, in principle, all LP methods relate to autocorrelation?

> In order to understand the relative importance of temporal Hilbert envelope and
> fine structure, Smith, Delgutte, and Oxenham (letter to nature, 2002) performed
> the famous chimaeric sounds experiments and concluded that the envelope is most
> important for speech perception with increasing number of filters.
> However, some of the technical issues in the design of the experiments
> make their conclusion quite dubious.

When Delgutte reported these experiments in Magdeburg, I did not get aware of
dubious conclusions. Of course, I would like to stress that it might no be
justified to speak of “ t h e   temporal Hilbert envelope”. Aren’t there of at
least as many temporal envelopes as critical bands, and aren’t the bands
flexible on partition?

Regards,
Eckard Blumschein

I cannot confirm that so much confusing redundancy in theory is really
justified. Let's tear down a lot of unnecessary sophistication after restricting
to either really elapsed time or time to come after a given point. In other
words, let's abandon the wrong belief that frequency analysis must be
immediately merged with complex calculus.