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
rect	0	0	1	0	30.1	3.922	2	-13.26	17.1	1.0
han	-6.021	-4.26	1.5	-1.761	28.34	1.424	4	-31.47	57.2	0.5,-0.5
ham	-5.352	-4.008	1.3628	-1.344	28.76	1.751	4	-42.67	10.2	0.54,-0.46
b3	-7.535	-5.163	1.7268	-2.372	27.73	1.099	6	-58.12	48.3	0.42,-0.5,0.08
bh3	-7.468	-5.142	1.7085	-2.326	27.78	1.129	6	-70.84	7.86	0.42323,-0.49755,0.07922
bhh3	-7.449	-5.133	1.7046	-2.316	27.79	1.134	6	-71.34	6.77	0.424161,-0.497378,0.078461
bh4	-8.904	-5.884	2.0044	-3.02	27.08	0.8256	8	-92.03	28.4	0.35875,-0.48829,0.14128,-0.01168
bhh4	-8.784	-5.828	1.9751	-2.956	27.15	0.8514	8	-97.97	4.53	0.36376721,-0.48922703,0.13641742,-0.01058834
bhh5	-9.581	-6.238	2.1592	-3.343	26.76	0.7152	9.53	-118.7	3.34	0.33186237,-0.47615347,0.16743138,-0.02382482,0.00072796
bhh6	-10.34	-6.628	2.3523	-3.715	26.39	0.6044	11.3	-140.1	7.02	0.3039747821,-0.4594726795,0.1927447601,-0.0404819348,0.0032818617,-4.39818e-05
N2	-5.379	-4.019	1.3676	-1.36	28.74	1.739	4	-43.19	10	0.53836,-0.46164
N3	-8.519	-5.631	1.9444	-2.888	27.22	0.863	6	-46.74	99.5	0.375,-0.5,0.125
N3A	-7.766	-5.281	1.7721	-2.485	27.62	1.045	6	-64.19	44.4	0.40897,-0.5,0.09103
N3B	-7.445	-5.131	1.7037	-2.314	27.79	1.135	6	-71.48	6.5	0.4243801,-0.4973406,0.0782793
N4	-10.1	-6.467	2.31	-3.636	26.47	0.6184	8	-60.95	143	0.3125,-0.46875,0.1875,-0.03125
N4A	-9.397	-6.123	2.1253	-3.274	26.83	0.7321	8	-82.61	82.7	0.338946,-0.481973,0.161054,-0.018027
N4B	-8.977	-5.92	2.0212	-3.056	27.05	0.8118	8	-93.33	36.7	0.355768,-0.487396,0.144232,-0.012604
N4C	-8.788	-5.83	1.9761	-2.958	27.14	0.8506	8	-98.14	4.52	0.3635819,-0.4891775,0.1365995,-0.0106411
A2	-5.379	-4.019	1.3677	-1.36	28.74	1.739	4	-43.19	10	0.538355394671,-0.461644605329
A3	-7.445	-5.131	1.7037	-2.314	27.79	1.135	6	-71.48	6.5	0.424380093461,-0.497340635097,0.0782792714423
A4	-8.788	-5.83	1.9761	-2.958	27.14	0.8506	8	-98.15	4.51	0.363581926771,-0.489177437145,0.136599513979,-0.0106411221055
A5	-9.81	-6.356	2.2153	-3.454	26.65	0.6801	10	-125.4	2.96	0.323215378888,-0.471492143958,0.17553412996,-0.0284969901061,0.00126135708829
A6	-10.65	-6.783	2.4339	-3.863	26.24	0.5653	12	-153.4	1.77	0.29355789501,-0.451935772347,0.201416471426,-0.0479261092211,0.00502619642686,-0.000137555567956
A7	-11.33	-7.134	2.6303	-4.2	25.9	0.4852	14	-180.2	1.07	0.271220360585,-0.433444612327,0.218004122893,-0.0657853432956,0.0107618673053,-0.000770012710581,1.36808830599e-05
A8	-11.93	-7.435	2.8129	-4.492	25.61	0.4251	16	-207.1	632m	0.253317681703,-0.416326930581,0.228839621372,-0.0815750842593,0.0177359245035,-0.00209670274903,0.000106774130221,-1.28070209036e-06
A9	-12.45	-7.702	2.9859	-4.751	25.35	0.3778	18	-234	446m	0.238433115278,-0.400554534864,0.235824253047,-0.0952791885838,0.0253739551662,-0.00415243290751,0.00036856041633,-1.38435559392e-05,1.16180835893e-07
A10	-12.93	-7.943	3.1517	-4.985	25.12	0.3395	20	-261.8	473m	0.225734538713,-0.386012294915,0.240129421411,-0.107054233866,0.0332591618402,-0.00687337495232,0.000875167323804,-6.00859893272e-05,1.71071647211e-06,-1.02727213027e-08
A11	-13.35	-8.154	3.3048	-5.191	24.91	0.3091	22	-288.2	610m	0.215152750668,-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	-6.519	3.1681	-5.008	25.1	0.008178	6	-31.72	62.8	0.26526,-0.5,0.23474
SFT4F	-13.27	-7.474	3.797	-5.794	24.31	0.004079	8	-44.73	105	0.21706,-0.42103,0.28294,-0.07897
SFT5F	-14.51	-8.136	4.3412	-6.376	23.73	0.002453	10	-57.27	149	0.1881,-0.36923,0.28702,-0.13077,0.02488
SFT3M	-10.98	-6.293	2.9452	-4.691	25.41	0.0115	6	-44.23	12	0.282352823528,-0.521055210552,0.19659196592
SFT4M	-12.33	-7.029	3.3868	-5.298	24.81	0.006733	8	-66.55	8.68	0.241906,-0.460841,0.255381,-0.041872
SFT5M	-13.57	-7.675	3.8852	-5.894	24.21	0.003893	10	-89.93	39.8	0.209671083868,-0.407331162932,0.28122511249,-0.0926690370676,0.00910360364144
FTNI	-11.02	-6.303	2.9656	-4.721	25.38	9.134e-06	6	-44.35	11.9	0.281063618936,-0.520896679103,0.19803970196
FTHP	-12.41	-7.063	3.4279	-5.35	24.75	8.091e-06	8	-70.4	7.5	0.239523981485,-0.458092235221,0.258487878821,-0.043895904473
FTSRS	-13.32	-7.559	3.7702	-5.764	24.34	0.01562	9.47	-76.62	48.7	0.215703192407,-0.416307161346,0.278257118205,-0.083692838654,0.0060396893874
HFT70	-12.39	-7.058	3.4129	-5.331	24.77	0.006505	8	-70.46	7.52	0.240186000038,-0.458265280633,0.257837269181,-0.043711450147
HFT95	-13.41	-7.596	3.8112	-5.811	24.29	0.003866	10	-94.99	5.48	0.213640903311,-0.414108259879,0.278698873916,-0.0860603241582,0.00749163873597
HFT90D	-13.56	-7.673	3.8832	-5.892	24.21	0.003903	10	-90.22	39.1	0.209783021421,-0.407525336544,0.281175959705,-0.0924746634556,0.00904101887418
HFT116D	-14.32	-8.071	4.2186	-6.252	23.85	0.002832	12	-116.8	32.8	0.192240452512,-0.37631789481,0.284144941765,-0.122407781678,0.0236146057221,-0.00127432351161
HFT144D	-14.98	-8.415	4.5386	-6.569	23.53	0.002127	14	-144	27.5	0.178153071078,-0.350534041444,0.281452647671,-0.144524263157,0.0402333021358,-0.00494169539905,0.000160979115026
HFT169D	-15.55	-8.708	4.8347	-6.844	23.26	0.00166	16	-167.9	23.5	0.166886261729,-0.329503309203,0.276046378613,-0.159857322793,0.0561963118343,-0.0106216780601,0.000871049492194,-1.76882748807e-05
HFT196D	-16.05	-8.966	5.1134	-7.087	23.02	0.001331	18	-196.2	20.4	0.15752208173,-0.311780372085,0.269408275921,-0.170380586106,0.0706855632848,-0.0177018003803,0.00238220518182,-0.000137241428751,1.87388268426e-06
HFT223D	-16.52	-9.206	5.3888	-7.315	22.79	0.001121	20	-223	18	0.149272191195,-0.296005258401,0.262056411964,-0.177690209577,0.0838244569528,-0.0258192670819,0.00482633877351,-0.000485066818969,2.06011148157e-05,-1.98121515763e-07
HFT248D	-16.94	-9.421	5.6512	-7.521	22.58	0.0008845	22	-248.4	18.7	0.142197548229,-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
rect	0	0	1	0	30.1	3.922	2	-13.26	17.1	1.0
han	-6.021	-4.26	1.5	-1.761	28.34	1.424	4	-31.47	57.2	0.5,-0.5
ham	-5.352	-4.008	1.3628	-1.344	28.76	1.751	4	-42.67	10.2	0.54,-0.46
b3	-7.535	-5.163	1.7268	-2.372	27.73	1.099	6	-58.12	48.3	0.42,-0.5,0.08
bh3	-7.468	-5.142	1.7085	-2.326	27.78	1.129	6	-70.84	7.86	0.42323,-0.49755,0.07922
bhh3	-7.449	-5.133	1.7046	-2.316	27.79	1.134	6	-71.34	6.77	0.424161,-0.497378,0.078461
bh4	-8.904	-5.884	2.0044	-3.02	27.08	0.8256	8	-92.03	28.4	0.35875,-0.48829,0.14128,-0.01168
bhh4	-8.784	-5.828	1.9751	-2.956	27.15	0.8514	8	-97.98	4.53	0.36376721,-0.48922703,0.13641742,-0.01058834
bhh5	-9.581	-6.238	2.1592	-3.343	26.76	0.7152	9.53	-118.8	3.33	0.33186237,-0.47615347,0.16743138,-0.02382482,0.00072796
bhh6	-10.34	-6.628	2.3523	-3.715	26.39	0.6044	11.3	-140.1	7.02	0.3039747821,-0.4594726795,0.1927447601,-0.0404819348,0.0032818617,-4.39818e-05
N2	-5.379	-4.019	1.3676	-1.36	28.74	1.739	4	-43.19	10	0.53836,-0.46164
N3	-8.519	-5.631	1.9444	-2.888	27.22	0.863	6	-46.74	99.5	0.375,-0.5,0.125
N3A	-7.766	-5.281	1.7721	-2.485	27.62	1.045	6	-64.19	44.4	0.40897,-0.5,0.09103
N3B	-7.445	-5.131	1.7037	-2.314	27.79	1.135	6	-71.48	6.5	0.4243801,-0.4973406,0.0782793
N4	-10.1	-6.467	2.31	-3.636	26.47	0.6184	8	-60.95	143	0.3125,-0.46875,0.1875,-0.03125
N4A	-9.397	-6.123	2.1253	-3.274	26.83	0.7321	8	-82.61	82.7	0.338946,-0.481973,0.161054,-0.018027
N4B	-8.977	-5.92	2.0212	-3.056	27.05	0.8118	8	-93.33	36.7	0.355768,-0.487396,0.144232,-0.012604
N4C	-8.788	-5.83	1.9761	-2.958	27.14	0.8506	8	-98.17	4.49	0.3635819,-0.4891775,0.1365995,-0.0106411
A2	-5.379	-4.019	1.3677	-1.36	28.74	1.739	4	-43.19	10	0.538355394671,-0.461644605329
A3	-7.445	-5.131	1.7037	-2.314	27.79	1.135	6	-71.48	6.5	0.424380093461,-0.497340635097,0.0782792714423
A4	-8.788	-5.83	1.9761	-2.958	27.14	0.8506	8	-98.17	4.49	0.363581926771,-0.489177437145,0.136599513979,-0.0106411221055
A5	-9.81	-6.356	2.2153	-3.454	26.65	0.6801	10	-125.4	2.91	0.323215378888,-0.471492143958,0.17553412996,-0.0284969901061,0.00126135708829
A6	-10.65	-6.783	2.4339	-3.863	26.24	0.5653	12	-153.5	1.68	0.29355789501,-0.451935772347,0.201416471426,-0.0479261092211,0.00502619642686,-0.000137555567956
A7	-11.33	-7.134	2.6303	-4.2	25.9	0.4852	14	-180.4	903m	0.271220360585,-0.433444612327,0.218004122893,-0.0657853432956,0.0107618673053,-0.000770012710581,1.36808830599e-05
A8	-11.93	-7.435	2.8129	-4.492	25.61	0.4251	16	-207.4	373m	0.253317681703,-0.416326930581,0.228839621372,-0.0815750842593,0.0177359245035,-0.00209670274903,0.000106774130221,-1.28070209036e-06
A9	-12.45	-7.702	2.9859	-4.751	25.35	0.3778	18	-234.6	76.7m	0.238433115278,-0.400554534864,0.235824253047,-0.0952791885838,0.0253739551662,-0.00415243290751,0.00036856041633,-1.38435559392e-05,1.16180835893e-07
A10	-12.93	-7.943	3.1517	-4.985	25.12	0.3395	20	-262.7	0	0.225734538713,-0.386012294915,0.240129421411,-0.107054233866,0.0332591618402,-0.00687337495232,0.000875167323804,-6.00859893272e-05,1.71071647211e-06,-1.02727213027e-08
A11	-13.35	-8.154	3.3048	-5.191	24.91	0.3091	22	-289.3	0	0.215152750668,-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	-6.519	3.1681	-5.008	25.1	0.008178	6	-31.72	62.8	0.26526,-0.5,0.23474
SFT4F	-13.27	-7.474	3.797	-5.794	24.31	0.004079	8	-44.73	105	0.21706,-0.42103,0.28294,-0.07897
SFT5F	-14.51	-8.136	4.3412	-6.376	23.73	0.002453	10	-57.27	149	0.1881,-0.36923,0.28702,-0.13077,0.02488
SFT3M	-10.98	-6.293	2.9452	-4.691	25.41	0.0115	6	-44.23	12	0.282352823528,-0.521055210552,0.19659196592
SFT4M	-12.33	-7.029	3.3868	-5.298	24.81	0.006733	8	-66.55	8.68	0.241906,-0.460841,0.255381,-0.041872
SFT5M	-13.57	-7.675	3.8852	-5.894	24.21	0.003893	10	-89.93	39.8	0.209671083868,-0.407331162932,0.28122511249,-0.0926690370676,0.00910360364144
FTNI	-11.02	-6.303	2.9656	-4.721	25.38	1.019e-05	6	-44.35	11.9	0.281063618936,-0.520896679103,0.19803970196
FTHP	-12.41	-7.063	3.4279	-5.35	24.75	8.2e-06	8	-70.41	7.5	0.239523981485,-0.458092235221,0.258487878821,-0.043895904473
FTSRS	-13.32	-7.559	3.7702	-5.764	24.34	0.01562	9.47	-76.62	48.6	0.215703192407,-0.416307161346,0.278257118205,-0.083692838654,0.0060396893874
HFT70	-12.39	-7.058	3.4129	-5.331	24.77	0.006505	8	-70.47	7.51	0.240186000038,-0.458265280633,0.257837269181,-0.043711450147
HFT95	-13.41	-7.596	3.8112	-5.811	24.29	0.003866	10	-95.02	5.46	0.213640903311,-0.414108259879,0.278698873916,-0.0860603241582,0.00749163873597
HFT90D	-13.56	-7.673	3.8832	-5.892	24.21	0.003903	10	-90.22	39.1	0.209783021421,-0.407525336544,0.281175959705,-0.0924746634556,0.00904101887418
HFT116D	-14.32	-8.071	4.2186	-6.252	23.85	0.002832	12	-116.8	32.8	0.192240452512,-0.37631789481,0.284144941765,-0.122407781678,0.0236146057221,-0.00127432351161
HFT144D	-14.98	-8.415	4.5386	-6.569	23.53	0.002127	14	-144	27.5	0.178153071078,-0.350534041444,0.281452647671,-0.144524263157,0.0402333021358,-0.00494169539905,0.000160979115026
HFT169D	-15.55	-8.708	4.8347	-6.844	23.26	0.00166	16	-167.9	23.4	0.166886261729,-0.329503309203,0.276046378613,-0.159857322793,0.0561963118343,-0.0106216780601,0.000871049492194,-1.76882748807e-05
HFT196D	-16.05	-8.966	5.1134	-7.087	23.02	0.001331	18	-196.2	20.4	0.15752208173,-0.311780372085,0.269408275921,-0.170380586106,0.0706855632848,-0.0177018003803,0.00238220518182,-0.000137241428751,1.87388268426e-06
HFT223D	-16.52	-9.206	5.3888	-7.315	22.79	0.001121	20	-223	18	0.149272191195,-0.296005258401,0.262056411964,-0.177690209577,0.0838244569528,-0.0258192670819,0.00482633877351,-0.000485066818969,2.06011148157e-05,-1.98121515763e-07
HFT248D	-16.94	-9.421	5.6512	-7.521	22.58	0.0008845	22	-248.4	18.7	0.142197548229,-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)

C.10·  bh3 (sidelobe:-70.8dB; 3 terms)

C.11·  bhh3 (sidelobe:-71.3dB; 3 terms)

C.12·  N3B (sidelobe:-71.5dB; 3 terms)

C.13·  A3 (sidelobe:-71.5dB; 3 terms)

C.14·  N4A (sidelobe:-82.6dB; 4 terms)

C.15·  bh4 (sidelobe:-92dB; 4 terms)

C.16·  N4B (sidelobe:-93.3dB; 4 terms)

C.17·  bhh4 (sidelobe:-98dB; 4 terms)

C.18·  A4 (sidelobe:-98.1dB; 4 terms)

C.19·  N4C (sidelobe:-98.2dB; 4 terms)

C.20·  bhh5 (sidelobe:-119dB; 5 terms)

C.21·  A5 (sidelobe:-125dB; 5 terms)

C.22·  bhh6 (sidelobe:-140dB; 6 terms)

C.23·  A6 (sidelobe:-153dB; 6 terms)

C.24·  A7 (sidelobe:-180dB; 7 terms)

C.25·  A8 (sidelobe:-207dB; 8 terms)

C.26·  A9 (sidelobe:-234dB; 9 terms)

C.27·  A10 (sidelobe:-262dB; 10 terms)

C.28·  A11 (sidelobe:-288dB; 11 terms)

D·  Graphical overview of windows with low scalloping loss

These windows minimize magnitude uncertainty of the main lobe.

D.1·  SFT3F (sidelobe:-31.7dB; 3 terms)

D.2·  SFT3M (sidelobe:-44.2dB; 3 terms)

D.3·  FTNI (sidelobe:-44.3dB; 3 terms)

D.4·  SFT4F (sidelobe:-44.7dB; 4 terms)

D.5·  SFT5F (sidelobe:-57.3dB; 5 terms)

D.6·  SFT4M (sidelobe:-66.5dB; 4 terms)

D.7·  FTHP (sidelobe:-70.4dB; 4 terms)

D.8·  HFT70 (sidelobe:-70.5dB; 4 terms)

D.9·  FTSRS (sidelobe:-76.6dB; 5 terms)

D.10·  SFT5M (sidelobe:-89.9dB; 5 terms)

D.11·  HFT90D (sidelobe:-90.2dB; 5 terms)

D.12·  HFT95 (sidelobe:-95dB; 5 terms)

D.13·  HFT116D (sidelobe:-117dB; 6 terms)

D.14·  HFT144D (sidelobe:-144dB; 7 terms)

D.15·  HFT169D (sidelobe:-168dB; 8 terms)

D.16·  HFT196D (sidelobe:-196dB; 9 terms)

D.17·  HFT223D (sidelobe:-223dB; 10 terms)

D.18·  HFT248D (sidelobe:-248dB; 11 terms)

Quick Reference Information

A·  Generating cosine-sum windows

Parameters used in the formulas below:

• w = array of window samples
• N = number of window samples
• n = index to window samples, 0 to N-1
• m = number of terms
• i = index to terms, 0 to m-1
• C = array of terms (see Tabulated properties)

A.1·  Periodic

Periodic windows (also known as DFT even) are generally best for spectrum analysis.

Formula for generating periodic windows:

      m-1
      ---
      \    /           / 2.pi.i.n \ \   
w  =  /   (  C  . cos (  --------  ) )
 n    ---  \  i        \     N    / /
      i=0

A periodic window as generated by this formula starts with a zero sample.

A.2·  Symmetric

Symmetric windows are generally best for filter design [*].   Formula for generating symmetric windows:

      m-1
      ---
      \    /           / 2.pi.i.(n+1/2) \ \   
w  =  /   (  C  . cos (  --------------  ) )
 n    ---  \  i        \       N        / /
      i=0

A symmetric window as generated with this formula has no zero samples.

A.3·  Alternative formulas

Tektronix™ windows are identical to the above symmetric windows:

      m-1
      ---
      \    / |  |        / 2·pi·i   /     N-1 \ \ \
w  =  /   (  |C | · cos (  ------ ·(  n - ---  ) ) )
 n    ---  \ | i|        \   N      \      2  / / /
      i=0

Matlab™ symmetric windows are not used in this document, these start with a zero sample and also end with a zero sample:

      m-1
      ---
      \    /           / 2·pi·i·n \ \
w  =  /   (  C  · cos (  --------  ) )
 n    ---  \  i        \    N-1   / /
      i=0

[*] F.   J.   Harris, "On the use of windows for harmonic analysis with the Discrete Fourier Transform", Proc.   IEEE, vol.66, jan. 1978.

B·  Advantages of cosine-sum windows

All cosine-sum windows have unique advantages over other windows:

• Very easy to implement and fast to compute.   This can be important for embedded software (like oscilloscope firmware).
• Compact support in the transformed domain where the window has only a few non-zero terms.   Because of this the windowing operation can be efficiently performed by convolution in the frequency domain when an unwindowed FFT/DFT is available.   This can be important for running FFT/DFT implementations.
• Properly overlapped windows sum to a constant.   This can be important for block processing of long waveforms, because it enables perfect reconstruction.

C·  Figures of merit list

• Signal gain [dB] (also called coherent gain)

This is the power gain of the window for a tone located exactly at a DFT bin.   It is normalised, so it is relative to a rectangular window of identical length.

• Noise gain [dB] (also called incoherent gain)

This is the power gain of the window for noise.   It is normalised, so it is relative to a rectangular window of identical length.

• Equivalent noise bandwidth [Hz]

This is the width in DFT bins of a rectangular filter with the same peak power gain that would accumulate the same noise power.   The enbw is also equal to the ratio of noise gain to signal gain.

• Scalloping loss [dB]

This is the ratio of signal gain for a tone located halfway between DFT bins to the signal gain for a tone located exactly at a DFT bin.

• Processing gain [dB]

This is the improvement of signal to noise ratio produced by the window: the output signal to noise ratio relative to input signal to noise ratio.

Note: This increases by 10dB for every 10 fold increase of window size.

• Relative processing gain [dB]

This is the normalized improvement of signal to noise ratio produced by the window, so it is relative to a rectangular window of identical length.

• Highest Sidelobe Level [dB]

Maximum spectral leakage, ignoring the main lobe, for a tone located halfway between DFT bins.

• Main Lobe Width [bins]:

Full width of main lobe at the highest sidelobe level, including negative frequencies.

Note: In this document only positive frequencies (DFT bins) are plotted, so only half of the main lobe is shown.

• Sidelobe Drop Rate [dB]:

Drop of sidelobe level beyond 10x main lobe width, relative to highest sidelobe level.

If for example the main lobe ranges from -3 to +3 bins, then this figure gives how much lower the highest sidelobe level outside of -30 to +30 bin range is.

Note: The calculation of most of these figures of merit is described in:

[*] F.   J.   Harris, "On the use of windows for harmonic analysis with the Discrete Fourier Transform", Proc.   IEEE, vol.66, jan. 1978.

D·  References

List of sources of the windows in this document (chronological order).

1. Classic cosine-sum windows:
• rect = Rectangular (no windowing)
• han = Hanning (originally: Hann)
• ham = Hamming
• b = Blackman
• bh = Blackman-Harris

F.J.Harris, "On the use of windows for harmonic analysis with the
Discrete Fourier Transform", Proc. IEEE, vol.66, jan. 1978.

2. Low sidelobe level cosine-sum windows:
• bhh = Blackman-Harris-Heylen

By myself, 1998. Previously unpublished.

• A = Albrecht

"A family of cosine-sum windows for high-resolution measurements"
Albrecht, H.H.
(ICASSP apos;01). 2001 IEEE International Conference on
Acoustics, Speech, and Signal Processing, 2001. Proceedings. 
Volume 5, Issue , 2001 Page(s):3081 - 3084 vol.5

3. Flat-top cosine-sum windows:
• FT = flat-top by instrument makers
• SFT = Salvatore flat-top
• HFT = Heinzel flat-top

And many others cosine-sum windows, such as:

• rect = Rectangular (no windowing)
• han = Hanning (originally: Hann)
• ham = Hamming
• b = Blackman
• bh = Blackman-Harris
• N = Nuttall

"Spectrum and spectral density estimation by the Discrete Fourier
  transform (DFT), including a comprehensive list of window
  functions and some new flat-top windows."
 Heinzel, Rudiger and Schilling, 2002

FFT spectrum analysis

A·  Test waveform

The time domain signal that will be analyzed is the sum of two cosine waves (or tones): 6Hz (exactly on a DFT bin) and 33.5Hz .   The first is exactly harmonically related to the time window (1 second), or exactly circular, in other words there is no discontinuity when multiple sequences are concatenated.   The second tone does not map to a DFT bin, if falls exactly halfway between two DFT bins.   The sequency size is (FFT size) 128 samples.

Like elsewhere in this document, ony the real part of the FFT spectrum will be shown, corrected with the window gain.   Elsewhere, spectrum interpolation was achieved by zero-padding the window to 30 times its original length.   However, no interpolation is used here, individual samples have a marker, and the phase of the spectrum is plotted as well as the magnitude.

B·  FFT without window

The harmonically related tone is resolved perfectly: amplitude is correct and all energy is concentrated in one bin only.   The energy of the other tone is distributed over many DFT bins.   This is called spectral leakage.

C·  FFT with window

Leakage of the non-harmonic tone is much reduced by the window.

Amplitude differences as well as shape differences are further reduced.

Notice that the periodic and symmetric windows produce a similar amplitude response but the symmetric window exhibits a phase error.

D·  The symmetric window phase error explained

Comparing an 8 sample Blackman window (see Generating cosine-sum windows):

dec bin	Periodic	Symmetric	Matlab™ symmetric
-4 100	0.0	0.014629	0.0
-3 101	0.066447	0.17209	0.090453
-2 110	0.34	0.55477	0.45918
-1 111	0.77355	0.93851	0.92036
+0 000	1	0.93851	0.92036
+1 001	0.77355	0.55477	0.45918
+2 010	0.34	0.17209	0.090453
+3 011	0.066447	0.014629	0.0

The first column shows a rotated sequence sample numbering (in decimal and two's complement binary).   The DFT considers all input sequences to be periodic.   For an 8 sample periodic sequence, it is evident that the sample at -4 will be identical to the sample at +4, which is actually the -4 sample of the next period.   When a 8 sample window is applied to a an 8 sample periodic sequence then the window should preserve this property.   A periodic Window will perfectly align with the fundamental frequency of the sequence, while a symmetric window is a halve sample shifted.