Hi all.
I'm playing a bit with window functions in filter design etc, and
have come across a couple of gruelling expressions. More specifically,
the Chebychev window [1, eq. 5-17], the Tukey window [2, table 8.1]
and the Lanczos window, [2, table 8.1].
I would like to know how one estimates the width of the main lobe
for these filters, and how one uses the various control parameters.
Table 8.1 in P&M seems to contain quite a few typos, which does not
exactly help me understand what's going on.
Any comments or opinions are most welcome.
Rune
[1] Lyons: Understanding DSP. 2nd ed.,
Prentice-Hall, 2004
[2] Proakis & Manolakis: Digital Signal Processin - Principles,
Algorithms and Applications. 3rd ed., Prentice-Hall, 1996.
Window functions
Started by ●August 31, 2004
Reply by ●August 31, 20042004-08-31
Rune Allnor wrote:> Hi all. > > I'm playing a bit with window functions in filter design etc, and > have come across a couple of gruelling expressions. More specifically, > the Chebychev window [1, eq. 5-17], the Tukey window [2, table 8.1] > and the Lanczos window, [2, table 8.1]. > > I would like to know how one estimates the width of the main lobe > for these filters, and how one uses the various control parameters. > Table 8.1 in P&M seems to contain quite a few typos, which does not > exactly help me understand what's going on. > > Any comments or opinions are most welcome. > > Rune > > [1] Lyons: Understanding DSP. 2nd ed., > Prentice-Hall, 2004 > > [2] Proakis & Manolakis: Digital Signal Processin - Principles, > Algorithms and Applications. 3rd ed., Prentice-Hall, 1996.The way my first class in DSP teacher explained it to me, and since he happens to be a Fellow of the IEEE so I kind of accept his, as an informed opinion, the window method is essentially an obsolete synthesis technique. Most DSP books don't assume that the reader has much of a background in optimization theory. Authors like to make their books as self contained as possible. I think that filter synthesis persists because most books want to get the student to the point where they can design some filters without having to digress about linear programing and discrete value optimization. IMHO the last significant paper on window filter synthesis was written by Kaiser back in the 50's where he introduced the Kaiser window as as an approximation of the zeroth order spherical prolate window.
Reply by ●August 31, 20042004-08-31
"Stan Pawlukiewicz" <spam@spam.mitre.org> wrote in message news:ch24rc$4v6$1@newslocal.mitre.org...> Rune Allnor wrote: > > Hi all. > > > > I'm playing a bit with window functions in filter design etc, and > > have come across a couple of gruelling expressions. More specifically, > > the Chebychev window [1, eq. 5-17], the Tukey window [2, table 8.1] > > and the Lanczos window, [2, table 8.1]. > > > > I would like to know how one estimates the width of the main lobe > > for these filters, and how one uses the various control parameters. > > Table 8.1 in P&M seems to contain quite a few typos, which does not > > exactly help me understand what's going on. > > > > The way my first class in DSP teacher explained it to me, and since he > happens to be a Fellow of the IEEE so I kind of accept his, as an > informed opinion, the window method is essentially an obsolete > synthesis technique. Most DSP books don't assume that the reader has > much of a background in optimization theory. Authors like to make their > books as self contained as possible. I think that filter synthesis > persists because most books want to get the student to the point where > they can design some filters without having to digress about linear > programing and discrete value optimization. > IMHO the last significant paper on window filter synthesis was > written by Kaiser back in the 50's where he introduced the Kaiser window > as as an approximation of the zeroth order spherical prolate window.That may be true, but the windowing technique can be very useful when you need to calculate filter coefficients "on the fly". Computationally, it is much simpler than the optimized methods. So while it might be sub-optimal, I wouldn't say it is obsolete.
Reply by ●August 31, 20042004-08-31
Stan Pawlukiewicz wrote:> Rune Allnor wrote: > >> Hi all. >> I'm playing a bit with window functions in filter design etc, and have >> come across a couple of gruelling expressions. More specifically, the >> Chebychev window [1, eq. 5-17], the Tukey window [2, table 8.1] >> and the Lanczos window, [2, table 8.1]. >> >> I would like to know how one estimates the width of the main lobe for >> these filters, and how one uses the various control parameters. Table >> 8.1 in P&M seems to contain quite a few typos, which does not exactly >> help me understand what's going on. >> >> Any comments or opinions are most welcome. >> Rune >> >> [1] Lyons: Understanding DSP. 2nd ed., Prentice-Hall, 2004 >> >> [2] Proakis & Manolakis: Digital Signal Processin - Principles, >> Algorithms and Applications. 3rd ed., Prentice-Hall, 1996. > > > The way my first class in DSP teacher explained it to me, and since he > happens to be a Fellow of the IEEE so I kind of accept his, as an > informed opinion, the window method is essentially an obsolete > synthesis technique. Most DSP books don't assume that the reader has > much of a background in optimization theory. Authors like to make their > books as self contained as possible. I think that filter synthesis > persists because most books want to get the student to the point where > they can design some filters without having to digress about linear > programing and discrete value optimization. > IMHO the last significant paper on window filter synthesis was written > by Kaiser back in the 50's where he introduced the Kaiser window as as > an approximation of the zeroth order spherical prolate window.But Stan, windows are used for more than filter design. The prime example is windowing a set of samples to minimize the artifacts caused by non-periodicity. The sidelobe width that is readily seen in a filter also characterizes the way a window in that use trades resolution for artifact reduction. Rune knows how to design filters. It's my guess that he's looking at windows for their effect on data. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●August 31, 20042004-08-31
Jon Harris wrote:> "Stan Pawlukiewicz" <spam@spam.mitre.org> wrote in message > news:ch24rc$4v6$1@newslocal.mitre.org... > >>Rune Allnor wrote: >> >>>Hi all. >>> >>>I'm playing a bit with window functions in filter design etc, and >>>have come across a couple of gruelling expressions. More specifically, >>>the Chebychev window [1, eq. 5-17], the Tukey window [2, table 8.1] >>>and the Lanczos window, [2, table 8.1]. >>> >>>I would like to know how one estimates the width of the main lobe >>>for these filters, and how one uses the various control parameters. >>>Table 8.1 in P&M seems to contain quite a few typos, which does not >>>exactly help me understand what's going on. >>> >> >>The way my first class in DSP teacher explained it to me, and since he >>happens to be a Fellow of the IEEE so I kind of accept his, as an >>informed opinion, the window method is essentially an obsolete >>synthesis technique. Most DSP books don't assume that the reader has >>much of a background in optimization theory. Authors like to make their >>books as self contained as possible. I think that filter synthesis >>persists because most books want to get the student to the point where >>they can design some filters without having to digress about linear >>programing and discrete value optimization. >> IMHO the last significant paper on window filter synthesis was >>written by Kaiser back in the 50's where he introduced the Kaiser window >>as as an approximation of the zeroth order spherical prolate window. > > > That may be true, but the windowing technique can be very useful when you need > to calculate filter coefficients "on the fly". Computationally, it is much > simpler than the optimized methods. So while it might be sub-optimal, I > wouldn't say it is obsolete. > >Signal Processing with flies should be called FSP ;)
Reply by ●August 31, 20042004-08-31
Jerry Avins wrote:> Stan Pawlukiewicz wrote: > >> Rune Allnor wrote: >> >>> Hi all. >>> I'm playing a bit with window functions in filter design etc, and >>> have come across a couple of gruelling expressions. More >>> specifically, the Chebychev window [1, eq. 5-17], the Tukey window >>> [2, table 8.1] >>> and the Lanczos window, [2, table 8.1]. >>> >>> I would like to know how one estimates the width of the main lobe for >>> these filters, and how one uses the various control parameters. Table >>> 8.1 in P&M seems to contain quite a few typos, which does not exactly >>> help me understand what's going on. >>> >>> Any comments or opinions are most welcome. >>> Rune >>> >>> [1] Lyons: Understanding DSP. 2nd ed., Prentice-Hall, 2004 >>> >>> [2] Proakis & Manolakis: Digital Signal Processin - Principles, >>> Algorithms and Applications. 3rd ed., Prentice-Hall, 1996. >> >> >> >> The way my first class in DSP teacher explained it to me, and since he >> happens to be a Fellow of the IEEE so I kind of accept his, as an >> informed opinion, the window method is essentially an obsolete >> synthesis technique. Most DSP books don't assume that the reader has >> much of a background in optimization theory. Authors like to make >> their books as self contained as possible. I think that filter >> synthesis persists because most books want to get the student to the >> point where they can design some filters without having to digress >> about linear programing and discrete value optimization. >> IMHO the last significant paper on window filter synthesis was >> written by Kaiser back in the 50's where he introduced the Kaiser >> window as as an approximation of the zeroth order spherical prolate >> window. > > > But Stan, windows are used for more than filter design. The prime > example is windowing a set of samples to minimize the artifacts caused > by non-periodicity. The sidelobe width that is readily seen in a filter > also characterizes the way a window in that use trades resolution for > artifact reduction. > > Rune knows how to design filters. It's my guess that he's looking at > windows for their effect on data. > > JerryJerry I didn't say anything about windows in specral analysis, the topic was window filter synthesis and I think Rune meant window filter synthesis.
Reply by ●August 31, 20042004-08-31
Stan Pawlukiewicz wrote:> Jon Harris wrote:...>> That may be true, but the windowing technique can be very useful when >> you need >> to calculate filter coefficients "on the fly". Computationally, it is >> much >> simpler than the optimized methods. So while it might be sub-optimal, I >> wouldn't say it is obsolete. >> >> > > Signal Processing with flies should be called FSP ;)The dyslexics among us sometimes store data in flies. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●August 31, 20042004-08-31
Stan Pawlukiewicz wrote:> Jerry Avins wrote: > >> Stan Pawlukiewicz wrote: >> >>> Rune Allnor wrote: >>> >>>> Hi all. >>>> I'm playing a bit with window functions in filter design etc, and >>>> have come across a couple of gruelling expressions. More >>>> specifically, the Chebychev window [1, eq. 5-17], the Tukey window >>>> [2, table 8.1] >>>> and the Lanczos window, [2, table 8.1]. >>>> >>>> I would like to know how one estimates the width of the main lobe >>>> for these filters, and how one uses the various control parameters. >>>> Table 8.1 in P&M seems to contain quite a few typos, which does not >>>> exactly help me understand what's going on. >>>> >>>> Any comments or opinions are most welcome. >>>> Rune >>>> >>>> [1] Lyons: Understanding DSP. 2nd ed., Prentice-Hall, 2004 >>>> >>>> [2] Proakis & Manolakis: Digital Signal Processin - Principles, >>>> Algorithms and Applications. 3rd ed., Prentice-Hall, 1996. >>> >>> >>> >>> >>> The way my first class in DSP teacher explained it to me, and since >>> he happens to be a Fellow of the IEEE so I kind of accept his, as an >>> informed opinion, the window method is essentially an obsolete >>> synthesis technique. Most DSP books don't assume that the reader has >>> much of a background in optimization theory. Authors like to make >>> their books as self contained as possible. I think that filter >>> synthesis persists because most books want to get the student to the >>> point where they can design some filters without having to digress >>> about linear programing and discrete value optimization. >>> IMHO the last significant paper on window filter synthesis was >>> written by Kaiser back in the 50's where he introduced the Kaiser >>> window as as an approximation of the zeroth order spherical prolate >>> window. >> >> >> >> But Stan, windows are used for more than filter design. The prime >> example is windowing a set of samples to minimize the artifacts caused >> by non-periodicity. The sidelobe width that is readily seen in a filter >> also characterizes the way a window in that use trades resolution for >> artifact reduction. >> >> Rune knows how to design filters. It's my guess that he's looking at >> windows for their effect on data. >> >> Jerry > > > Jerry I didn't say anything about windows in specral analysis, the topic > was window filter synthesis and I think Rune meant window filter synthesis.You're probably right, as usual. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●August 31, 20042004-08-31
Jerry Avins wrote:> Stan Pawlukiewicz wrote: > >> Jon Harris wrote: > > > ... > >>> That may be true, but the windowing technique can be very useful when >>> you need >>> to calculate filter coefficients "on the fly". Computationally, it >>> is much >>> simpler than the optimized methods. So while it might be sub-optimal, I >>> wouldn't say it is obsolete. >>> >>> >> >> Signal Processing with flies should be called FSP ;) > > > The dyslexics among us sometimes store data in flies. > > JerryI've traced many bugs to instances of dyslexia :)
Reply by ●August 31, 20042004-08-31
"Rune Allnor" <allnor@tele.ntnu.no> wrote in message news:f56893ae.0408310548.8ba60b4@posting.google.com...> Hi all. > > I'm playing a bit with window functions in filter design etc, and > have come across a couple of gruelling expressions. More specifically, > the Chebychev window [1, eq. 5-17], the Tukey window [2, table 8.1] > and the Lanczos window, [2, table 8.1]. > > I would like to know how one estimates the width of the main lobe > for these filters, and how one uses the various control parameters. > Table 8.1 in P&M seems to contain quite a few typos, which does not > exactly help me understand what's going on.Rune, I'm going to note that you said "etc." and not worry about what you're going to do with the windows. You may peek through them for all I care! :-) I don't recall a general method or figure or table that compares them - although fred harris or Al Nuttall may have done so. I have a more arm-waving method..... 1) First you need to define how main lobe width will be measured. -3dB points? first zero points? etc. Note that they are all pretty much the same at -3dB and can be quite different at the first zero point. 2) Then, realize that the van der Maas functin provides the lowest (minimax) sidelobe for a given main lobe width - exept it's not physically realizable and the end points of the window are infinite as I recall. Nonetheless, it's a good data point. 3) Also, realize that the rectangular window "sinc" isn't optimum in frequency in any particular way - regarding main lobe width or sidelobes. So this one isn't interesting. 4) The Dolph-Chebyshev window is an approximation to the van der Maas and is physically realizable. So, presumably, this is the best you can do in combination of main lobe width vs. side lobe level. 5) All of the others have some nice properties of their own but are less "good" in combination - wider main lobe / higher sidelobes - but perhaps with rapid sidelobe decay, easy to compute, etc. Lyon's Figure 5-26 gives a pretty good idea of what's possible in the minimax case. Everything else will be "worse" in the strict main lobe / sidelobe comparison. Compare Kaiser beta=4 to gamma=1.5 for Dolph-Chebyshev. The first sidelobes are nearly the same and the Kaiser main lobe is wider... Fred
