🚀 go-pugleaf

Michael Hucka wrote:
> 
> In the field I'm studying (biological visual processing), it is common to
> define the bandwidth of a (real) filter by the width at half-amplitude of the
> response function, i.e.,
> 
>          Log[f_high] - Log[f_low]          where f_high is the high-
>     BW = ------------------------,         frequency 1/2 amplitude point,
>                   Log[2]                   f_low is the low-freq. point.
> 
> This works fine for bandpass filters, such as derivatives of Gaussians, but
> doesn't work if the filter is low-pass, with a response centered at frequency
> = 0.  An example of the kind of filter giving me this problem is a plain
> Gaussian, which is real-valued and even-symmetric.  From my reading of the
> literature, it appears that bandwidth for low-pass filters must be defined in
> another way.  The definition that seems to be used most commonly is sometimes
> called the "equivalent bandwidth", and defined as the width of a rectangle of
> height equal to the height of the response function at 0 and which has an
> area equal to the area under the function.  I.e.,
> 
>              .- oo
>             /      F(s) ds                 where F(s) is the Fourier
>           -' -oo                           spectrum of a real-valued
>     BW = ------------------------.         function f(x).
>                   F(0)
> 
> Basically, I'm unsure whether my understanding of all this is correct.  Can
> someone answer the following questions for me?
> 
> 1) Is this the generally-accepted approach to defining the bandwidth of
>    bandpass and lowpass filters?
> 
> 2) When computing bandwidths this way, should the amplitude spectrum be used,
>    or the Fourier spectrum?
> 
> 3) In the case of a bandpass filter defined by a real-valued function, the
>    Fourier transform will have two "humps", one on the positive axis and one
>    on the negative axis.  In that case, the negative portion is usually
>    ignored and the bandwidth is computed using the upper and lower points on
>    the positive hump.  However, a lowpass function which is even-symmetric
>    like a Gaussian will have a spectrum with one hump, centered about 0,
>    extending both in the positive and negative frequency directions.  The
>    second bandwidth definition above thus will include the negative frequency
>    component in the calculation.  This seems inconsistent with the bandpass
>    case.  Is it a problem?
> 
> I appreciate any help and advice you can offer.
> 
> --
> Mike Hucka    hucka@umich.edu    http://www.eecs.umich.edu/~hucka    University
>  PhD to be, computational models of human visual processing (AI Lab)     of
>    UNIX systems administrator & programmer/analyst (EECS DCO)         Michigan



I'll give it a shot....

1. In general, this formula can be use for either a low pass or bandpass process.
   If you have a bandpass process, then you can get the low pass equivalent via
   analytic (pre-envelope) signal representation. If you stay with the bandpass 
   representation then you must use the F(f_center) instead of F(0). Usually, the 
   definition of a bandpass signal is F(0)=0 (or a very small number).

2. For Eq noise bandwidth, if you have its spectral density then use the F(f). By def,
   F(f) is the PSD which relates to the autocorrelation function of the rand process.

   If you use the amp spectrum then you should take the mag-square of the transfer
   function of the filter, this assumes the model is a linear (time invataint) system 
   with a stochastic input, refer to Papoulis; Prob, Rand Var ... In this case, the 
   prefered method is to use the low pass equivalent of the bandpass process. 

3. For the low pass process, (if you assume even symmetry) then you can integrate 
   from 0 to positive inf. This would be the same as ignoring the neg freq for the 
   bandpass process. Again, this can be seen by going from the bandpass process to its          
   equivalent low pass representation. 


Note that the 'bandwidth' and the 'eq noise bandwith' have different definitions,
the later relates to the power spectral densities of a stochastic process. The half amp
bandwith traditionally has been use to characterized filter structures and in general 
has very little meaning in stochastic processes. 


For more material on low pass eq & analytic signal, try the following;
  Proakis 'Dig Comm',  Papoulis 'Signal Theory', Whalen 'Det of Signal in Noise'
  
For caveats such as non symmetrical bandpass filters to low pass eq; 
  Papoulis 'Signal Theory', Papoulis ' Fourier Transform & apps'
 

Good Luck
jack