🚀 go-pugleaf

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