Table of Content

  1. Cosine-sum windows

  2. Quick Reference Information

  3. FFT spectrum analysis

Cosine-sum windows

The cosine-sum windows are an interesting class of FFT/DFT window functions.   Classic examples are the Hamming, Hanning, Blackman and Blackman-Harris windows (which are called ham,han, b3 and bh3 in this document).   See Quick Reference Information and literary References for more information.

A·  Tabulated properties for periodic windows of 1024 samples

See Figures of merit for a description of the table headings.

Signal Gain [dB]Noise Gain [dB]Equiv Noise BW [bins]Relative Process Gain [dB]Process Gain [dB]Scalloping Loss [dB]Main Lobe width [bins]Highest Sidelobe Level [dB]Sidelobe Drop Rate [dB]Terms
rect001030.13.9222-13.2617.11.0
han-6.021-4.261.5-1.76128.341.4244-31.4757.20.5,-0.5
ham-5.352-4.0081.3628-1.34428.761.7514-42.6710.20.54,-0.46
b3-7.535-5.1631.7268-2.37227.731.0996-58.1248.30.42,-0.5,0.08
bh3-7.468-5.1421.7085-2.32627.781.1296-70.847.860.42323,-0.49755,0.07922
bhh3-7.449-5.1331.7046-2.31627.791.1346-71.346.770.424161,-0.497378,0.078461
bh4-8.904-5.8842.0044-3.0227.080.82568-92.0328.40.35875,-0.48829,0.14128,-0.01168
bhh4-8.784-5.8281.9751-2.95627.150.85148-97.974.530.36376721,-0.48922703,0.13641742,-0.01058834
bhh5-9.581-6.2382.1592-3.34326.760.71529.53-118.73.340.33186237,-0.47615347,0.16743138,-0.02382482,0.00072796
bhh6-10.34-6.6282.3523-3.71526.390.604411.3-140.17.020.3039747821,-0.4594726795,0.1927447601,-0.0404819348,0.0032818617,-4.39818e-05
N2-5.379-4.0191.3676-1.3628.741.7394-43.19100.53836,-0.46164
N3-8.519-5.6311.9444-2.88827.220.8636-46.7499.50.375,-0.5,0.125
N3A-7.766-5.2811.7721-2.48527.621.0456-64.1944.40.40897,-0.5,0.09103
N3B-7.445-5.1311.7037-2.31427.791.1356-71.486.50.4243801,-0.4973406,0.0782793
N4-10.1-6.4672.31-3.63626.470.61848-60.951430.3125,-0.46875,0.1875,-0.03125
N4A-9.397-6.1232.1253-3.27426.830.73218-82.6182.70.338946,-0.481973,0.161054,-0.018027
N4B-8.977-5.922.0212-3.05627.050.81188-93.3336.70.355768,-0.487396,0.144232,-0.012604
N4C-8.788-5.831.9761-2.95827.140.85068-98.144.520.3635819,-0.4891775,0.1365995,-0.0106411
A2-5.379-4.0191.3677-1.3628.741.7394-43.19100.538355394671,-0.461644605329
A3-7.445-5.1311.7037-2.31427.791.1356-71.486.50.424380093461,-0.497340635097,0.0782792714423
A4-8.788-5.831.9761-2.95827.140.85068-98.154.510.363581926771,-0.489177437145,0.136599513979,-0.0106411221055
A5-9.81-6.3562.2153-3.45426.650.680110-125.42.960.323215378888,-0.471492143958,0.17553412996,-0.0284969901061,0.00126135708829
A6-10.65-6.7832.4339-3.86326.240.565312-153.41.770.29355789501,-0.451935772347,0.201416471426,-0.0479261092211,0.00502619642686,-0.000137555567956
A7-11.33-7.1342.6303-4.225.90.485214-180.21.070.271220360585,-0.433444612327,0.218004122893,-0.0657853432956,0.0107618673053,-0.000770012710581,1.36808830599e-05
A8-11.93-7.4352.8129-4.49225.610.425116-207.1632m0.253317681703,-0.416326930581,0.228839621372,-0.0815750842593,0.0177359245035,-0.00209670274903,0.000106774130221,-1.28070209036e-06
A9-12.45-7.7022.9859-4.75125.350.377818-234446m0.238433115278,-0.400554534864,0.235824253047,-0.0952791885838,0.0253739551662,-0.00415243290751,0.00036856041633,-1.38435559392e-05,1.16180835893e-07
A10-12.93-7.9433.1517-4.98525.120.339520-261.8473m0.225734538713,-0.386012294915,0.240129421411,-0.107054233866,0.0332591618402,-0.00687337495232,0.000875167323804,-6.00859893272e-05,1.71071647211e-06,-1.02727213027e-08
A11-13.35-8.1543.3048-5.19124.910.309122-288.2610m0.215152750668,-0.373134835779,0.242424335845,-0.116690759269,0.0407742210588,-0.0100090450085,0.00163980691736,-0.0001651660821,8.88466316854e-06,-1.93861711603e-07,8.48248559933e-10
SFT3F-11.53-6.5193.1681-5.00825.10.0081786-31.7262.80.26526,-0.5,0.23474
SFT4F-13.27-7.4743.797-5.79424.310.0040798-44.731050.21706,-0.42103,0.28294,-0.07897
SFT5F-14.51-8.1364.3412-6.37623.730.00245310-57.271490.1881,-0.36923,0.28702,-0.13077,0.02488
SFT3M-10.98-6.2932.9452-4.69125.410.01156-44.23120.282352823528,-0.521055210552,0.19659196592
SFT4M-12.33-7.0293.3868-5.29824.810.0067338-66.558.680.241906,-0.460841,0.255381,-0.041872
SFT5M-13.57-7.6753.8852-5.89424.210.00389310-89.9339.80.209671083868,-0.407331162932,0.28122511249,-0.0926690370676,0.00910360364144
FTNI-11.02-6.3032.9656-4.72125.389.134e-066-44.3511.90.281063618936,-0.520896679103,0.19803970196
FTHP-12.41-7.0633.4279-5.3524.758.091e-068-70.47.50.239523981485,-0.458092235221,0.258487878821,-0.043895904473
FTSRS-13.32-7.5593.7702-5.76424.340.015629.47-76.6248.70.215703192407,-0.416307161346,0.278257118205,-0.083692838654,0.0060396893874
HFT70-12.39-7.0583.4129-5.33124.770.0065058-70.467.520.240186000038,-0.458265280633,0.257837269181,-0.043711450147
HFT95-13.41-7.5963.8112-5.81124.290.00386610-94.995.480.213640903311,-0.414108259879,0.278698873916,-0.0860603241582,0.00749163873597
HFT90D-13.56-7.6733.8832-5.89224.210.00390310-90.2239.10.209783021421,-0.407525336544,0.281175959705,-0.0924746634556,0.00904101887418
HFT116D-14.32-8.0714.2186-6.25223.850.00283212-116.832.80.192240452512,-0.37631789481,0.284144941765,-0.122407781678,0.0236146057221,-0.00127432351161
HFT144D-14.98-8.4154.5386-6.56923.530.00212714-14427.50.178153071078,-0.350534041444,0.281452647671,-0.144524263157,0.0402333021358,-0.00494169539905,0.000160979115026
HFT169D-15.55-8.7084.8347-6.84423.260.0016616-167.923.50.166886261729,-0.329503309203,0.276046378613,-0.159857322793,0.0561963118343,-0.0106216780601,0.000871049492194,-1.76882748807e-05
HFT196D-16.05-8.9665.1134-7.08723.020.00133118-196.220.40.15752208173,-0.311780372085,0.269408275921,-0.170380586106,0.0706855632848,-0.0177018003803,0.00238220518182,-0.000137241428751,1.87388268426e-06
HFT223D-16.52-9.2065.3888-7.31522.790.00112120-223180.149272191195,-0.296005258401,0.262056411964,-0.177690209577,0.0838244569528,-0.0258192670819,0.00482633877351,-0.000485066818969,2.06011148157e-05,-1.98121515763e-07
HFT248D-16.94-9.4215.6512-7.52122.580.000884522-248.418.70.142197548229,-0.282382171299,0.254700898,-0.18230796203,0.094956327566,-0.0341502764537,0.00805639857835,-0.00115677342582,8.88087179893e-05,-2.81679094847e-06,1.89085767782e-08

B·  Tabulated properties for symmetric windows of 1024 samples

See Figures of merit for a description of the table headings.

Signal Gain [dB]Noise Gain [dB]Equiv Noise BW [bins]Relative Process Gain [dB]Process Gain [dB]Scalloping Loss [dB]Main Lobe width [bins]Highest Sidelobe Level [dB]Sidelobe Drop Rate [dB]Terms
rect001030.13.9222-13.2617.11.0
han-6.021-4.261.5-1.76128.341.4244-31.4757.20.5,-0.5
ham-5.352-4.0081.3628-1.34428.761.7514-42.6710.20.54,-0.46
b3-7.535-5.1631.7268-2.37227.731.0996-58.1248.30.42,-0.5,0.08
bh3-7.468-5.1421.7085-2.32627.781.1296-70.847.860.42323,-0.49755,0.07922
bhh3-7.449-5.1331.7046-2.31627.791.1346-71.346.770.424161,-0.497378,0.078461
bh4-8.904-5.8842.0044-3.0227.080.82568-92.0328.40.35875,-0.48829,0.14128,-0.01168
bhh4-8.784-5.8281.9751-2.95627.150.85148-97.984.530.36376721,-0.48922703,0.13641742,-0.01058834
bhh5-9.581-6.2382.1592-3.34326.760.71529.53-118.83.330.33186237,-0.47615347,0.16743138,-0.02382482,0.00072796
bhh6-10.34-6.6282.3523-3.71526.390.604411.3-140.17.020.3039747821,-0.4594726795,0.1927447601,-0.0404819348,0.0032818617,-4.39818e-05
N2-5.379-4.0191.3676-1.3628.741.7394-43.19100.53836,-0.46164
N3-8.519-5.6311.9444-2.88827.220.8636-46.7499.50.375,-0.5,0.125
N3A-7.766-5.2811.7721-2.48527.621.0456-64.1944.40.40897,-0.5,0.09103
N3B-7.445-5.1311.7037-2.31427.791.1356-71.486.50.4243801,-0.4973406,0.0782793
N4-10.1-6.4672.31-3.63626.470.61848-60.951430.3125,-0.46875,0.1875,-0.03125
N4A-9.397-6.1232.1253-3.27426.830.73218-82.6182.70.338946,-0.481973,0.161054,-0.018027
N4B-8.977-5.922.0212-3.05627.050.81188-93.3336.70.355768,-0.487396,0.144232,-0.012604
N4C-8.788-5.831.9761-2.95827.140.85068-98.174.490.3635819,-0.4891775,0.1365995,-0.0106411
A2-5.379-4.0191.3677-1.3628.741.7394-43.19100.538355394671,-0.461644605329
A3-7.445-5.1311.7037-2.31427.791.1356-71.486.50.424380093461,-0.497340635097,0.0782792714423
A4-8.788-5.831.9761-2.95827.140.85068-98.174.490.363581926771,-0.489177437145,0.136599513979,-0.0106411221055
A5-9.81-6.3562.2153-3.45426.650.680110-125.42.910.323215378888,-0.471492143958,0.17553412996,-0.0284969901061,0.00126135708829
A6-10.65-6.7832.4339-3.86326.240.565312-153.51.680.29355789501,-0.451935772347,0.201416471426,-0.0479261092211,0.00502619642686,-0.000137555567956
A7-11.33-7.1342.6303-4.225.90.485214-180.4903m0.271220360585,-0.433444612327,0.218004122893,-0.0657853432956,0.0107618673053,-0.000770012710581,1.36808830599e-05
A8-11.93-7.4352.8129-4.49225.610.425116-207.4373m0.253317681703,-0.416326930581,0.228839621372,-0.0815750842593,0.0177359245035,-0.00209670274903,0.000106774130221,-1.28070209036e-06
A9-12.45-7.7022.9859-4.75125.350.377818-234.676.7m0.238433115278,-0.400554534864,0.235824253047,-0.0952791885838,0.0253739551662,-0.00415243290751,0.00036856041633,-1.38435559392e-05,1.16180835893e-07
A10-12.93-7.9433.1517-4.98525.120.339520-262.700.225734538713,-0.386012294915,0.240129421411,-0.107054233866,0.0332591618402,-0.00687337495232,0.000875167323804,-6.00859893272e-05,1.71071647211e-06,-1.02727213027e-08
A11-13.35-8.1543.3048-5.19124.910.309122-289.300.215152750668,-0.373134835779,0.242424335845,-0.116690759269,0.0407742210588,-0.0100090450085,0.00163980691736,-0.0001651660821,8.88466316854e-06,-1.93861711603e-07,8.48248559933e-10
SFT3F-11.53-6.5193.1681-5.00825.10.0081786-31.7262.80.26526,-0.5,0.23474
SFT4F-13.27-7.4743.797-5.79424.310.0040798-44.731050.21706,-0.42103,0.28294,-0.07897
SFT5F-14.51-8.1364.3412-6.37623.730.00245310-57.271490.1881,-0.36923,0.28702,-0.13077,0.02488
SFT3M-10.98-6.2932.9452-4.69125.410.01156-44.23120.282352823528,-0.521055210552,0.19659196592
SFT4M-12.33-7.0293.3868-5.29824.810.0067338-66.558.680.241906,-0.460841,0.255381,-0.041872
SFT5M-13.57-7.6753.8852-5.89424.210.00389310-89.9339.80.209671083868,-0.407331162932,0.28122511249,-0.0926690370676,0.00910360364144
FTNI-11.02-6.3032.9656-4.72125.381.019e-056-44.3511.90.281063618936,-0.520896679103,0.19803970196
FTHP-12.41-7.0633.4279-5.3524.758.2e-068-70.417.50.239523981485,-0.458092235221,0.258487878821,-0.043895904473
FTSRS-13.32-7.5593.7702-5.76424.340.015629.47-76.6248.60.215703192407,-0.416307161346,0.278257118205,-0.083692838654,0.0060396893874
HFT70-12.39-7.0583.4129-5.33124.770.0065058-70.477.510.240186000038,-0.458265280633,0.257837269181,-0.043711450147
HFT95-13.41-7.5963.8112-5.81124.290.00386610-95.025.460.213640903311,-0.414108259879,0.278698873916,-0.0860603241582,0.00749163873597
HFT90D-13.56-7.6733.8832-5.89224.210.00390310-90.2239.10.209783021421,-0.407525336544,0.281175959705,-0.0924746634556,0.00904101887418
HFT116D-14.32-8.0714.2186-6.25223.850.00283212-116.832.80.192240452512,-0.37631789481,0.284144941765,-0.122407781678,0.0236146057221,-0.00127432351161
HFT144D-14.98-8.4154.5386-6.56923.530.00212714-14427.50.178153071078,-0.350534041444,0.281452647671,-0.144524263157,0.0402333021358,-0.00494169539905,0.000160979115026
HFT169D-15.55-8.7084.8347-6.84423.260.0016616-167.923.40.166886261729,-0.329503309203,0.276046378613,-0.159857322793,0.0561963118343,-0.0106216780601,0.000871049492194,-1.76882748807e-05
HFT196D-16.05-8.9665.1134-7.08723.020.00133118-196.220.40.15752208173,-0.311780372085,0.269408275921,-0.170380586106,0.0706855632848,-0.0177018003803,0.00238220518182,-0.000137241428751,1.87388268426e-06
HFT223D-16.52-9.2065.3888-7.31522.790.00112120-223180.149272191195,-0.296005258401,0.262056411964,-0.177690209577,0.0838244569528,-0.0258192670819,0.00482633877351,-0.000485066818969,2.06011148157e-05,-1.98121515763e-07
HFT248D-16.94-9.4215.6512-7.52122.580.000884522-248.418.70.142197548229,-0.282382171299,0.254700898,-0.18230796203,0.094956327566,-0.0341502764537,0.00805639857835,-0.00115677342582,8.88087179893e-05,-2.81679094847e-06,1.89085767782e-08

C·  Graphical overview of windows with high scalloping loss

These windows do not minimize the magnitude uncertainty of the main lobe.

C.1·  rect (sidelobe:-13.3dB; 1 terms)

C.2·  han (sidelobe:-31.5dB; 2 terms)

C.3·  ham (sidelobe:-42.7dB; 2 terms)

C.4·  N2 (sidelobe:-43.2dB; 2 terms)

C.5·  A2 (sidelobe:-43.2dB; 2 terms)

C.6·  N3 (sidelobe:-46.7dB; 3 terms)

C.7·  b3 (sidelobe:-58.1dB; 3 terms)

C.8·  N4 (sidelobe:-60.9dB; 4 terms)

C.9·  N3A (sidelobe:-64.2dB; 3 terms)