Triangle Window ?

On 5/26/2011 4:32 PM, dbd wrote:
>> Consider that the sinc has the narrowest main lobe of all....

> Not really. Perhaps narrower than the common ones. Perhaps the
> narrowest useful one. You can get narrower mainlobe, but only at the
> expense of having less that the 13.6 dB sidelobe rejection that the
> rectangular window gives. The Taylor window, for example gives a
> narrower mainlobe when designed with a specified 10 dB sidelobe
> rejection.
>

Dale,

OK.  I would call that "supergaining" where stuff outside the main lobe 
is "unusual". I've not studied many cases like that - just a few.  I'd 
say that 10dB sidelobes are unusual.  But maybe useful.

I can but imagine that the main lobe width difference is tiny.  Is there 
a practical application of one designed this way (i.e. 10dB sidelobes)?

Fred

On May 26, 11:18=A0pm, Fred Marshall <fmarshallxremove_th...@acm.org>
wrote:
> On 5/26/2011 4:32 PM, dbd wrote:
>
> >> I notice that some treatments show the sinc2 =A0with the same zero
> >> > =A0locations as the sinc and with a narrower main lobe. =A0All well =
and good
> >> > =A0but this only happens if the window length is doubled.
> > But isn't that about what we get from actually convolving the
> > rectangle with itself (as you said)?
>
> Dale,
>
> Yes. =A0But those 'treatments" I refer to are using different window
> length for the two cases - which would be the temporal convolution of
> two rectangular windows. =A0In that case the zeros in frequency coincide.
>
> Whereas if one uses a triangular window of length N and compares it with
> a rectangular window of length N then the main lobe is wider for the
> triangular window and the zeros are further apart by a factor of 2.
> It's common to compare window functions having the same length.... and
> not all treatments do that.
>
> Fred

Fred

My point was that you describe a process that doubles the length of
the window and then act surprised that people who perform that process
use the double length window that it produces. If you want to produce
a window  size that will be constant, you should describe the process
as convolving half-length windows. It is common in DSP literature to
make the arbitrary assumption of constant window size without warning
to newcomers. Not all window literature makes that assumption. I think
that descriptions that make arbitrary assumptions but don't point them
out are broken.

Dale B. Dalrymple

On May 26, 11:36=A0pm, Fred Marshall <fmarshallxremove_th...@acm.org>
wrote:
> On 5/26/2011 4:32 PM, dbd wrote:
>
> >> Consider that the sinc has the narrowest main lobe of all....
> > Not really. Perhaps narrower than the common ones. Perhaps the
> > narrowest useful one. You can get narrower mainlobe, but only at the
> > expense of having less that the 13.6 dB sidelobe rejection that the
> > rectangular window gives. The Taylor window, for example gives a
> > narrower mainlobe when designed with a specified 10 dB sidelobe
> > rejection.
>
> Dale,
>
> OK. =A0I would call that "supergaining" where stuff outside the main lobe
> is "unusual".

That sounds backwards. Isn't "supergaining" normally used where
something unusual (higher gain) has occurred inside the main lobe?

>  I've not studied many cases like that - just a few. =A0I'd
> say that 10dB sidelobes are unusual. =A0But maybe useful.
>
> I can but imagine that the main lobe width difference is tiny. =A0Is ther=
e
> a practical application of one designed this way (i.e. 10dB sidelobes)?
>
> Fred

I didn't suggest it would be commonly useful. You quoted my disclaimer
to that.

The window design methods like convolving (shortened) shapes and
moving inner zeros to cancel sidelobe centers (Blackman) always widen
the main lobe. If this is all you think of it is easy to make the
false assumption that the rectangular window gives the narrowest main
lobe. Window design methods that control the position of all the
frequency domain zeros can often be used (like Taylor) to create more
arbitrary main lobe widths, with the usual trade-off in sidelobe
rejection.

Dale B. Dalrymple

On 5/27/2011 7:07 AM, dbd wrote:
>> OK.  I would call that "supergaining" where stuff outside the main lobe
>> >  is "unusual".

> That sounds backwards. Isn't "supergaining" normally used where
> something unusual (higher gain) has occurred inside the main lobe?
>

I think of supergaining as usually characterized by strange edges on the 
window function.  And, in turn, this usually means that something 
strange has happened in the sidelobes.  For example, one can manipulate 
the main lobe by letting distant sidelobes be big.

I think one prominent example of supergaining is where there are two 
distinct main lobes with decent separation (attentuation between them). 
  Maybe handy for separating sources like stars or lines in acoustics 
(probably less often used).

In the end, any window can be expressed as a sum of regularly-spaced 
Dirichlets (with zero locations related to the length of the window).
Given that this is the case then you have this one "main lobe" Dirichlet 
and can manipulate the sum by changing those removed from it - all but 
one pair of which whose main lobes lie in the sidelobe region.

Because those removed from the main lobe have zeros at the first zeros 
of the main Dirichlet and at it's peak AND switch sign from the right 
side to the left side (such that pairs of them tend to cancel) .. it's 
tough to make the main lobe much narrower without making these "sidelobe 
contributors" quite large.

Fred

On 5/27/2011 6:43 AM, dbd wrote:
> On May 26, 11:18 pm, Fred Marshall<fmarshallxremove_th...@acm.org>
> wrote:
>> On 5/26/2011 4:32 PM, dbd wrote:
>>
>>>> I notice that some treatments show the sinc2  with the same zero
>>>>>   locations as the sinc and with a narrower main lobe.  All well and good
>>>>>   but this only happens if the window length is doubled.
>>> But isn't that about what we get from actually convolving the
>>> rectangle with itself (as you said)?
>>
>> Dale,
>>
>> Yes.  But those 'treatments" I refer to are using different window
>> length for the two cases - which would be the temporal convolution of
>> two rectangular windows.  In that case the zeros in frequency coincide.
>>
>> Whereas if one uses a triangular window of length N and compares it with
>> a rectangular window of length N then the main lobe is wider for the
>> triangular window and the zeros are further apart by a factor of 2.
>> It's common to compare window functions having the same length.... and
>> not all treatments do that.
>>
>> Fred
>
> Fred
>
> My point was that you describe a process that doubles the length of
> the window and then act surprised that people who perform that process
> use the double length window that it produces. If you want to produce
> a window  size that will be constant, you should describe the process
> as convolving half-length windows. It is common in DSP literature to
> make the arbitrary assumption of constant window size without warning
> to newcomers. Not all window literature makes that assumption. I think
> that descriptions that make arbitrary assumptions but don't point them
> out are broken.
>
> Dale B. Dalrymple

Well, I do try to be clear.  I re-read what I posted and it seems clear 
enough to me - if somewhat awkward in the presentation.  You understand 
it quite well.

Fred

On May 28, 10:36=A0am, Fred Marshall <fmarshallxremove_th...@acm.org>
wrote:
> On 5/27/2011 7:07 AM, dbd wrote:
>
> >> OK. =A0I would call that "supergaining" where stuff outside the main l=
obe
> >> > =A0is "unusual".
> > That sounds backwards. Isn't "supergaining" normally used where
> > something unusual (higher gain) has occurred inside the main lobe?
>
> I think of supergaining as usually characterized by strange edges on the
> window function. =A0And, in turn, this usually means that something
> strange has happened in the sidelobes. ...=A0

This may be true. There will always be tradeoff between main lobe and
sidelobe. But it isn't the definition of supergain.

Dale B. Dalrymple