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Re: Need to calculate Nyquist frequency from data

• To: mathgroup at smc.vnet.net
• Subject: [mg41251] Re: Need to calculate Nyquist frequency from data
• From: Sseziwa Mukasa <mukasa at jeol.com>
• Date: Sat, 10 May 2003 04:01:34 -0400 (EDT)
• Sender: owner-wri-mathgroup at wolfram.com

On Friday, May 9, 2003, at 03:20 AM, Bob Buchanan wrote:

> Hello,
>
> I have a time series of real numbers which I can treat as representing
> a signal. How do I estimate the Nyquist frequency from the time
> series? If the previous question does not make sense, is it the case
> that the Nyquist frequency can only be estimated if one has a
> continuous function of time to work with?
>

Once you sample information is lost.  In particular, any signal energy 
at frequencies higher than half the sample rate will be aliased into 
sampled bandwidth.  Without a model of the original signal it is 
impossible to exactly reconstruct the original signal.  That said you 
still have a few options for estimating the original spectrum.  You can 
attempt to fit a model function to your time series or interpolate and 
resample at a higher rate.  Another approach is to try some kind of 
harmonic inversion approach like Maximum Entropy (as described in 
Skilling, J. & Bryan, R.K., (1984),  Mon. Not. R. astr. Soc.,  211, 
111-124 not the method ascribed to Burg found in Numerical Recipes) or 
some kind of Bayesian approach.  You can find descriptions of the 
latter methods by searching for the term MaxEnt and following the 
links.  If after doing this you have some reason to believe your 
spectral estimate is accurate you can search for the largest nonzero 
frequency component and call it half the Nyquist frequency.

Note that for any nonstationary signal (ie. one whose frequency 
components change with time) or a signal which is finite in duration 
there is no Nyquist frequency; the signal is by definition not 
bandwidth limited.  If you're lucky the spectrum decays in intensity as 
frequency goes to infinity and you can cut off at some value and still 
have a reasonable approximation of the continuous spectrum in the sense 
that the discrete power spectrum will have large values near the 
extrema of the continuous power spectrum.  Note that regardless of 
aliasing, the discrete spectrum is not exactly equivalent to the 
continuous spectrum, but for bandwidth limited signals there is a 
precise relationship between the two.

Regards,

Ssezi

• Follow-Ups:
  • Re: Re: Need to calculate Nyquist frequency from data
    • From: Stepan Yakovenko <yakovenko@ngs.ru>