Hi All, I have searched high and low to no avail; I am trying to find the formula for the 5-term Blackman-Harris. Certainly there is plenty of material on the 3 and 4 terms but not the fifth. Any help would be greatly appreciated. Cheers Pikey
5-term Blackman-Harris window
Started by ●April 13, 2008
Reply by ●April 14, 20082008-04-14
Hi, Search for follwing paper. if you dont get the paper over net send me your personal email id. my email id is bharat\at\arithos\com A FAMILY OF COSINE-SUM WINDOWS FOR HIGH-RESOLUTION MEASUREMENTS Hans-Helge Albrecht Physikalisch-Technische Bundesanstalt Abbestraße 2-12, D-10587 Berlin, Germany Phone: +49 30 3481 311 Fax: +49 30 3481 490 E-mail: hans-helge.albrecht@ptb.de rgds bharat pathak Arithos Designs www.Arithos.com DSP design consultancy and Training Company.
Reply by ●April 14, 20082008-04-14
On Apr 13, 8:03 am, "pikey" <pi...@amor888.fsnet.co.uk> wrote:> Hi All, > > I have searched high and low to no avail; I am trying to find the formula > for the 5-term Blackman-Harris. Certainly there is plenty of material on > the 3 and 4 terms but not the fifth. > Any help would be greatly appreciated. > > Cheers > > PikeyPikey A better terminology might be "minimum sidelobe cosine-sum" window. harris doesn't seem to have published past 4 terms and his 4 term optimized window had 92 dB sidelobe rejection. Nuttall provided a correction to the 4 term coefficients to yield 98 dB sidelobe rejection. So, harris' 4 term wasn't minimum sidelobe. Nuttall doesn't seem to have published past 4 terms either. So, if you want a four term minimum sidelobe cosine-sum window you could use: acoef = [0.3232153788877343 0.4714921439576260 0.1755341299601972 ... 2.849699010614994e-2 1.261357088292677e-3]; from the Albrecht paper: A family of cosine-sum windows for high-resolution measurements Albrecht, H.H. Acoustics, Speech, and Signal Processing, 2001. Proceedings. (ICASSP apos;01). 2001 IEEE International Conference on Volume 5, Issue , 2001 Page(s):3081 - 3084 vol.5 The paper gives up to 11 terms and -289 dB sidelobe rejection. At 12 terms there were numerical problems with running the optimization in 64 bit floating point. Albrecht's 4 term coefficients agree with Nuttal's 4 term values. The 7 term window appears in some 24-bit A-D converter application notes published before Albrecht's paper. The values published were consistent with Albrecht's. A number of people have successfully run the optimization for various numbers of coefficients so I go with "minimum sidelobe cosine-sum" window for the family. Dale B. Dalrymple http://dbdimages.com
Reply by ●April 14, 20082008-04-14
"pikey" <pikie@amor888.fsnet.co.uk> wrote in message news:h9-dnYu56pc2vZ_VnZ2dnUVZ_oqhnZ2d@giganews.com...> Hi All, > > I have searched high and low to no avail; I am trying to find the formula > for the 5-term Blackman-Harris. Certainly there is plenty of material on > the 3 and 4 terms but not the fifth. > Any help would be greatly appreciated. > > Cheers > > PikeyThe whole point of windows of this type is that they're simple. I'm guessing but I believe the reason you don't see 5-term Blackman-Harris is because there never was one formulated. By simple I mean that there are just a few cosine terms summed to form the window. For each cosine term there is a pair of sincs in frequency. The sincs add to get the overall, low sidelobe, frequency response of the window. The higher the "order" the larger the number of nonzero sincs. Outside the range of nonzero sincs, the sidelobes have the same regular zeros as the sincs have. I developed a method similar to Parks-McClellan - at about the same time - that can design windows using the same Remez algorithm but using sincs as the basis set. This gets directly to what you're looking for. The sidelobes can be minimax or weighted minimax and, if you need, can have forced zeros. And, the sincs can be replaced with Dirichlets if you're working with a periodic function / discrete in the transformed domain. See: Temes, Barcilon & Marshall: "The Optimization of Bandlimited Sytems" Proc IEEE Vol 61 No. 2 Feb 1973 pp 196-234. Back to why simple windows? A window is useful to reduce sidelobes / leakage at a bit of expense of the main lobe width or resolution. Some "windows" are actually implemented in their transform domain as a convolution. For example the [1/4 1/2 /14] filter often used in video processing for interpolation. It can be much more efficient and convenient to convolve with very short filters than to multiply an entire array with a window function. Fred
Reply by ●April 14, 20082008-04-14
On Apr 13, 10:15 pm, dbd <d...@ieee.org> wrote:> On Apr 13, 8:03 am, "pikey" <pi...@amor888.fsnet.co.uk> wrote: > > > Hi All, > > > I have searched high and low to no avail; I am trying to find the formula > > for the 5-term Blackman-Harris. Certainly there is plenty of material on > > the 3 and 4 terms but not the fifth. > > Any help would be greatly appreciated. > > > Cheers > > > Pikey > > Pikey > > A better terminology might be "minimum sidelobe cosine-sum" window. > > harris doesn't seem to have published past 4 terms and his 4 term > optimized window had 92 dB sidelobe rejection. Nuttall provided a > correction to the 4 term coefficients to yield 98 dB sidelobe > rejection. So, harris' 4 term wasn't minimum sidelobe. Nuttall doesn't > seem to have published past 4 terms either. > > So, if you want a four term minimum sidelobe cosine-sum window you > could use: > > acoef = [0.3232153788877343 0.4714921439576260 > 0.1755341299601972 ... > 2.849699010614994e-2 1.261357088292677e-3]; > > from the Albrecht paper: > > A family of cosine-sum windows for high-resolution measurements > Albrecht, H.H. > Acoustics, Speech, and Signal Processing, 2001. Proceedings. (ICASSP > apos;01). 2001 IEEE International Conference on > Volume 5, Issue , 2001 Page(s):3081 - 3084 vol.5 > > The paper gives up to 11 terms and -289 dB sidelobe rejection. At 12 > terms there were numerical problems with running the optimization in > 64 bit floating point. Albrecht's 4 term coefficients agree with > Nuttal's 4 term values. > The 7 term window appears in some 24-bit A-D converter application > notes published before Albrecht's paper. The values published were > consistent with Albrecht's. A number of people have successfully run > the optimization for various numbers of coefficients so I go with > "minimum sidelobe cosine-sum" window for the family. > > Dale B. Dalrymplehttp://dbdimages.comPlease note, this is the 5-term coefficient set: acoef = [0.3232153788877343 0.4714921439576260 0.1755341299601972 ... 2.849699010614994e-2 1.261357088292677e-3]; not the 4-term set, from the paper. Dale B. Dalrymple
Reply by ●April 14, 20082008-04-14
On Apr 14, 4:45 pm, "Fred Marshall" <fmarshallx@remove_the_x.acm.org> wrote:> "pikey" <pi...@amor888.fsnet.co.uk> wrote in message > > news:h9-dnYu56pc2vZ_VnZ2dnUVZ_oqhnZ2d@giganews.com... > > > Hi All, > > > I have searched high and low to no avail; I am trying to find the formula > > for the 5-term Blackman-Harris. Certainly there is plenty of material on > > the 3 and 4 terms but not the fifth. > > Any help would be greatly appreciated. > > > Cheers > > > Pikey > > The whole point of windows of this type is that they're simple. I'm > guessing but I believe the reason you don't see 5-term Blackman-Harris is > because there never was one formulated. By simple I mean that there are > just a few cosine terms summed to form the window. > > For each cosine term there is a pair of sincs in frequency. The sincs add > to get the overall, low sidelobe, frequency response of the window. The > higher the "order" the larger the number of nonzero sincs. Outside the > range of nonzero sincs, the sidelobes have the same regular zeros as the > sincs have. > > I developed a method similar to Parks-McClellan - at about the same time - > that can design windows using the same Remez algorithm but using sincs as > the basis set. This gets directly to what you're looking for. The > sidelobes can be minimax or weighted minimax and, if you need, can have > forced zeros. And, the sincs can be replaced with Dirichlets if you're > working with a periodic function / discrete in the transformed domain. > > See: Temes, Barcilon & Marshall: "The Optimization of Bandlimited Sytems" > Proc IEEE Vol 61 No. 2 Feb 1973 pp 196-234. > > Back to why simple windows? > > A window is useful to reduce sidelobes / leakage at a bit of expense of the > main lobe width or resolution. > > Some "windows" are actually implemented in their transform domain as a > convolution. For example the [1/4 1/2 /14] filter often used in video > processing for interpolation. It can be much more efficient and convenient > to convolve with very short filters than to multiply an entire array with a > window function. > > FredThis discussion reminds me of a pulse shaping "window" that the group might be interested in to replace RC and RRC designs. It's called a Harris-Moerder filter, google "An Improved Square-Root Nyquist Shaping Filter" for details. John
Reply by ●April 14, 20082008-04-14
Fred Marshall wrote: ...> Some "windows" are actually implemented in their transform domain as a > convolution. For example the [1/4 1/2 /14] filter often used in video > processing for interpolation. It can be much more efficient and convenient > to convolve with very short filters than to multiply an entire array with a > window function.I notice that 1/4 1/2 1/4 is a line of Pascal's triangle, normalized. 1 1 1 1 2 1 2 1 4 1 3 3 1 8 1 4 6 4 1 16 1 5 10 10 5 1 32 They all work as filters; the best I could devise in half an hour before I knew any DSP theory. (Binomial; --> Gaussian) Do they also serve as a basis of good windows? Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●April 14, 20082008-04-14
On Apr 14, 2:09 pm, Jerry Avins <j...@ieee.org> wrote:> Fred Marshall wrote: > > ... > > > Some "windows" are actually implemented in their transform domain as a > > convolution. For example the [1/4 1/2 /14] filter often used in video > > processing for interpolation. It can be much more efficient and convenient > > to convolve with very short filters than to multiply an entire array with a > > window function. > > I notice that 1/4 1/2 1/4 is a line of Pascal's triangle, normalized. > > 1 1 > 1 1 2 > 1 2 1 4 > 1 3 3 1 8 > 1 4 6 4 1 16 > 1 5 10 10 5 1 32 > > They all work as filters; the best I could devise in half an hour before > I knew any DSP theory. (Binomial; --> Gaussian) Do they also serve as a > basis of good windows? > > Jerry > -- > Engineering is the art of making what you want from things you can get. > �����������������������������������������������������������������������Jerry The binomial coefficients used as cosine-sum window coefficients are the maximum sidelobe rolloff rate windows. Nuttall derives them for the lines in the pyramid that sum to even powers of 2: TITLE: Some windows with very good sidelobe behavior ISSUE: IEEE Trans. Acoust., Speech, Signal Processing, vol. 29, pp. 84 - 91, February 1981 AUTHORS: Albert H. Nuttall as the windows with the greatest number of derivatives constrained to zero at the endpoints of the time domain window weighting. Note that the lines summing to odd powers of two resample to points midway between the bin centers of the original DFT that the weights are applied to. The first line is the rectangular window, the third is the von Hann. The fifth and seventh are in Nuttall's paper. Each line you drop increases sidelobe rolloff rate by 6 dB per octave. Dale B. Dalrymple
Reply by ●April 14, 20082008-04-14
dbd wrote:> On Apr 14, 2:09 pm, Jerry Avins <j...@ieee.org> wrote: >> Fred Marshall wrote: >> >> ... >> >>> Some "windows" are actually implemented in their transform domain as a >>> convolution. For example the [1/4 1/2 /14] filter often used in video >>> processing for interpolation. It can be much more efficient and convenient >>> to convolve with very short filters than to multiply an entire array with a >>> window function. >> I notice that 1/4 1/2 1/4 is a line of Pascal's triangle, normalized. >> >> 1 1 >> 1 1 2 >> 1 2 1 4 >> 1 3 3 1 8 >> 1 4 6 4 1 16 >> 1 5 10 10 5 1 32 >> >> They all work as filters; the best I could devise in half an hour before >> I knew any DSP theory. (Binomial; --> Gaussian) Do they also serve as a >> basis of good windows? >> >> Jerry >> -- >> Engineering is the art of making what you want from things you can get. >> ����������������������������������������������������������������������� > > Jerry > > The binomial coefficients used as cosine-sum window coefficients are > the maximum sidelobe rolloff rate windows. Nuttall derives them for > the lines in the pyramid that sum to even powers of 2: > > TITLE: Some windows with very good sidelobe behavior > ISSUE: IEEE Trans. Acoust., Speech, Signal Processing, vol. 29, pp. 84 > - 91, February 1981 > AUTHORS: Albert H. Nuttall > > as the windows with the greatest number of derivatives constrained to > zero at the endpoints of the time domain window weighting. Note that > the lines summing to odd powers of two resample to points midway > between the bin centers of the original DFT that the weights are > applied to. > > The first line is the rectangular window, the third is the von Hann. > The fifth and seventh are in Nuttall's paper. Each line you drop > increases sidelobe rolloff rate by 6 dB per octave.Sometimes the unity of math overwhelms me. Jerry -- Engineering is the art of making what you want from things you can get. �����������������������������������������������������������������������
Reply by ●April 15, 20082008-04-15
"John" <sampson164@gmail.com> wrote in message news:62d45fcb-8216-46f5-b7b7-a50bf5da32b8@b1g2000hsg.googlegroups.com...> On Apr 14, 4:45 pm, "Fred Marshall" <fmarshallx@remove_the_x.acm.org>> This discussion reminds me of a pulse shaping "window" that the group > might be interested in to replace RC and RRC designs. It's called a > Harris-Moerder filter, google "An Improved Square-Root Nyquist Shaping > Filter" for details. > > JohnYes.. interesting. My original objective was to design a single filter (or window) that had the regular zero crossing attribute for ISI reduction purposes - so it was a similar objective. Using sincs as a basis set was quite handy because the only nonzero sincs were in the main lobe region of the pulse. This led to the need for forcing zeros that were closer in - and thus, a "modified Remez algorithm" that allows equality constraints to be included. An interesting note: Al Nuttall worked at the Navy lab in New London, CT while fred harris and I worked at the Navy lab in San Diego.
