Re: human versus spectral resolution

[Yi-Wen Liu <jacobliu@xxxxxxxxx>]

Hi Machael and all,

>
> 1) Time/frequency resolution for digital signal spectral analysis , as (sort
> of) governed by Heisenberg's uncertainty principle with relation to the
> duration, sampling rate, frequency content and bandwidth of the sound.
>
> 2) The limits of the ability of the human ear to distinquish between
> frequency/pitch and the exact time location of sounds, more or less the same
> task as above.


I'd like to point out there are two notions of uncertainty regarding
frequency estimation.  One is the uncertainty due to the presence of
multiple sinusoids and limited observation time, and the other is the
uncertainly due to the presence of noise.  The first kind of
uncertainly is governed by Heisenberg's principle, the second kind can
be assessed by the theory of Fisher information and Cramer-Rao
inequality.

Heisenberg's principle here can be written as

(Delta f)(Delta t) >= 1  (maybe there is a factor of 2pi?)

which says that the best frequency resolution is in the order of
1/(Delta t).  In other words, two sinusoids of frequencies f1 and f2
are NOT well-resolved within

Delta t < 1/(f1-f2).

Fisher information considers the limit of parameter estimation under
noisy observation.  When applied to frequency estimation under
additive white Gaussian noise, the theory predicts that the variance
of frequency estimation error

(Delta f)^2 will always be greater than the Cramer-Rao lower bound,
which is proportional to SNR/(Delta t)^3.

For a thorough discussion of Cramer-Rao bound in sinusoidal parameter
estimation and some estimators to achieve the bound, check

Boaz Porat (1993). Digital Processing of Random Signals, chapter 9:
"Estimation of deterministic processes".

For a compact and elegant discussion of Fisher information, you can check

Cover and Thomas (1991) Elements of information theory, Chapter 12
"Information theory and statistics"

An early paper on this topic is

Single tone parameter estimation from discrete-time observations -
D Rife, R Boorstyn - Information Theory, IEEE Transactions on, 1974 -
Vol. 20, Page 591-98.

Best regards,

Yi-Wen