cutoff frequency of a FIR Filter

Rune,

I think we agree on all points - so I'll just address the areas where you 
had questions or where we seemed to depart in understanding:

"Rune Allnor" <allnor@tele.ntnu.no> wrote in message 
news:f56893ae.0410062316.57c6c842@posting.google.com...
> "Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message 
> news:<gIOdnQA6NcVBo_ncRVn-vQ@centurytel.net>...
>> "Rune Allnor" <allnor@tele.ntnu.no> wrote in message
>> news:f56893ae.0410052220.4fc1b189@posting.google.com...
>> > "Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote in message
>>
>> ...............................
>
>> But, because of the spacing of the spectral samples, there is no way to
>> define a transition over 1/2*NT Hz.  We can't construct a sequence that 
>> will
>> do that because we don't have samples to define at that spacing.
>
> I'm not sure I understand what you sy here. f=Fs/2 is a problem with
> real-valued signals due to aliasing and all that. The problem disappears
> at Fs/2 for complex-valued signals, but a similar problem occurs at Fs.
> This is a degenrate situation that I think we can leave out of the
> discussion.

***NB: I'm not talking about f=fs/2 at all here.  I'm talking about 
frequency intervals of 1/NT or 1/2*NT which are fs/N or fs/2*N.

...........................................
>
>> Example: if we compute the frequency response of a length 5 filter using
>> 5,000 points in frequency between zero and fs, then we'll see lots of
>> wiggles - but the fact is that we can only control things so much.  If we
>> design an optimum length 5 minimax filter then there will be 6 degrees of
>> freedom: the 5 coefficients and the magnitude of the peak error.  This is
>> very much related to our ability to achieve transitions of some width or
>> another.
>
> I'm not sure I follow you. I have done some tests wyth the type of
> filters I showed above. The frequency response of an order 4 filter as
> above, will be smooth when plotted after, say, 1024 pt zero padding.
> Except for plotting artifacts, I see no wiggle in the response?
>

***Later in my response I gave an example of a filter that has a zero in its 
continuous frequency response but has no zeros in its frequency samples. 
You have to interpolate to see the zero points.  Peaks can occur between 
samples as well - thus, "wiggles".  Some "stopband" peaks can be quite large 
but not seen in the samples.

.......................

>> It's an obscure fact that we can depart from using cosines and switch to
>> using a family of shifted sincs or Dirichlet kernels.  By superposition 
>> it's
>> exactly the same thing as a sum of cosines.  However, using the shifted 
>> and
>> equally spaced sincs gives us some interesting insights.  It's much 
>> easier
>> to visualize the construction of the filter response using shifted and
>> equally spaced sincs.  This is satisfying because it looks the same as
>> convolving the filter spectral samples with a sinc - which is the dual of
>> time-limiting the filter.  The convolution of course becomes a discrete 
>> sum
>> of shifted, equally spaced sincs.  Some will note the similarity to the
>> windowing method of filter design.
>
> All of this is very convenient, from a practical point of view. I can't
> see that any of this comes down to the fundamentals of FIR filters?

***Because FIR filter's frequency responses can be expressed in this way - 
and, I find it much easier to visualize the relationship between length and 
frequency response if I think of the frequency response as being made up of 
sincs rather than cosines.  I can't wrap my head around a sum of cosines 
very easily (except when doing half-band filters) but it's easy to visualize 
a sum of sincs.  The sums are the same - the visualization due to the basis 
function set is much different.

........................................
>
>> If we evaluate the filter polynomial over the z plane then we find zeros
>> placed wherever...  That doesn't take away from the fact that it's
>> constructed of real sincs (really Dirichlets) on the unit circle.  So,
>> transition widths remain limited by the width of those sincs which are
>> directly related to the spacing of the spectral samples.  (You may well
>> accept the notion that the practical width of the main lobe of a sinc is
>> equal to the spacing of the regular zeros in the sinc - it is 1/2 the
>> spacing between the first zeros adjacent to the peak).
>
> Well, yes, but the placements of the zeros are determined by the sampling
> interval, not the length of the FIR filter. The shortest sinc possible
> in any system, is the ...0,0,1,0,0,... sequence, that interpolates to
> a continuous sinc according to the sampling reconstruction theorem.
>
> But that has nothing to do with FIR filters, as far as I can see?

***When you say that placement of the zeros are determined by the sampling 
interval, are you referring to the time sampling interval or the frequency 
sampling interval?

***The frequency sinc's zeros are determined by the length of the FIR 
filter.  This is because their separation distance is the frequency sampling 
interval and the distance between adjacent zeros in the frequency sincs.
.......................
>
>> It's probably easier to visualize if we consider a brick wall transition
>> that is preceded by a whole sequence of spectral samples with value 1.0
>> followed by a sequence of spectral samples with value zero.  In that case 
>> we
>> force the stopband zeros to be equispaced - which may not be at all 
>> optimum.
>> Then, if we tweak the stopband peaks to push down the largest ones there 
>> is
>> a related widening of the transition - very much as the van der Maas, 
>> Dolph,
>> Taylor functions have wider main lobes than a uniformly weighted array 
>> with
>> larger sidelobes would have.  (Much of the early theory of this sort was
>> done by the antenna folks).  Then Bartlett, von Hann, harris, Kaiser, et 
>> al
>> did similar things with "windows" for DSP.  I don't make a great 
>> distinction
>> between continuous functions and discrete ones in my arm-waving
>> descriptions.
>
> I think it's intersting. Still, the starting point here was what the
> relation between the length of a FIR filter (N, including zero 
> coefficients)
> and key properties of the frequency response. I belive we agree that the
> one key property that is decided/controlled by N is the transition
> bandwidth. Otherwise, we have the freedom to manipulate the coefficients,
> as N degrees of freedom, in order to meet whatever requirements and
> constraints the filter will have to meet.


***We agree.  I think the starting point was about filter temporal length 
and its relation to cutoff frequency.  I said there is none and suggested 
the relationship in the frequency response to temporal length is in 
transition band widths.

***I think you introduced the issue of the placement of zeros in the 
continuous frequency response.  That had not been my intent because I don't 
think there's much that can be said other than there will be zeros 
somewhere - and closely related to the stop bands but not sppecifically 
located anywhere.  I surely agree that zeros could be close or double or 
whatever - but these aren't among the gross "features" I'd intended to 
discuss.  And, I see no direct relationship with the length of the filter - 
other than the number of zeros real or complex.

***To bring this back to the beginning, I was trying to provide a 
constructive approach to discussion of filter characteristics that *are* 
related to the filter's temporal length.  That's why superposition of 
frequency sincs is a key point.
That's where minimum transition band width is easily demonstrated to be 
dependent on temporal length.
That's where the relative cutoff frequency is shown to be independent of 
temporal length.

Fred