Poularikas A. D. “Windows ”
The Handbook of Formulas and Tables for Signal Processing.
Ed. Alexander D. Poularikas
Boca Raton: CRC Press LLC, 1999
7
Windows
7.1
7.2
7.3
Introductory Material
Figures of Merit
Window (Filter) Descriptions
Rectangle (Dirichlet) Window • Triangle (Feyer, Bartlet)
Window • cosα(t) Windows • Hann • Hamming • Short Cosine
series • Blackman • Harris-Nutall • Sampled Kaiser-Bessel •
Parabolic (Riesz, Bochner, Parzen) • Riemann • de la VallePoussin (Jackson, Parzen) • Cosine taper (Tukey) • Bohman •
Poisson • Hann-Poisson • Cauchy (Abel, Poisson) • Gaussian
(Weierstrass) • Dolph-Chebyshev • Kaiser-Bessel BarcilonThemes • Highest Sidelobe Level versus Worst-Case
Processing Loss
References
7.1 Introductory Material
7.1.1 Introduction
N = number of samples
T = interval between samples
NT = total time duration of the signal
1
= minimum spectral resolution (s–1)
NT
DFT = discrete Fourier transform
Leakage = Spectral leakage takes place when the signal has frequencies other than those of the
basis set. These other frequencies will exhibit non-zero properties on the entire basis
set known as leakage.
7.2 Figures of Merit (see Table 7.1)
7.2.1 Equivalent Noise Bandwidth
∑ w (nT )
the width of an equivalent ideal
rectangular spectral response that will
2 = equivalent noise bandwidth =
pass the same noise power as the window
w(nT )
(filter) under test.
2
ENBW =
n
∑
n
w(nT ) = window samples
© 1999 by CRC Press LLC
TABLE 7.1 Figures of Merit for Shaped DFT Filters
Figure of Merit
Overlap correlation (%)
Weighting
Highest
Equivalent
sidelobe level Sidelobe falloff
noise BW
(dB)
(dB/octave) Coherent gain
(bins)
Rectangle
Triangle
cosα (x)
Hann
α
α
α
α
=
=
=
=
1.0
2.0
3.0
4.0
Hamming
Parabolic
Riemann
Cubic
Tukey
α = 0.25
α = 0.50
α = 0.75
Bohman
Poisson
Hamming
Poisson
Cauchy
© 1999 by CRC Press LLC
α
α
α
α
α
α
α
α
α
=
=
=
=
=
=
=
=
=
2.0
3.0
4.0
0.5
1.0
2.0
3.0
4.0
5.0
–13
–27
–23
–32
–39
–47
–43
–21
–26
–53
–14
–15
–19
–46
–19
–24
–31
–35
–39
none
–31
–35
–30
–6
–12
–12
–18
–24
–30
–6
–12
–12
–24
–18
–18
–18
–24
–6
–6
–6
–18
–18
–18
–6
–6
–6
1.00
0.50
0.64
0.50
0.42
0.38
0.54
0.67
0.59
0.38
0.88
0.75
0.63
0.41
0.44
0.32
0.25
0.43
0.38
0.29
0.42
0.33
0.28
1.00
1.33
1.23
1.50
1.73
1.94
1.36
1.20
1.30
1.92
1.10
1.22
1.36
1.79
1.30
1.65
2.08
1.61
1.73
2.02
1.48
1.76
2.06
3.0-dB BW
(bins)
Scallop loss
(dB)
0.89
1.28
1.20
1.44
1.66
1.86
1.30
1.16
1.26
1.82
1.01
1.15
1.31
1.71
1.21
1.45
1.75
1.54
1.64
1.87
1.34
1.50
1.68
3.92
1.82
2.10
1.42
1.08
0.86
1.78
2.22
1.89
0.90
2.96
2.24
1.73
1.02
2.09
1.46
1.03
1.26
1.11
0.87
1.71
1.36
1.13
Worst-case
process loss 6.0-dB BW
(dB)
(bins)
75% OL
3.92
3.07
3.01
3.18
3.47
3.75
3.10
3.01
3.03
3.72
3.39
3.11
3.07
3.54
3.23
3.64
4.21
3.33
3.50
3.94
3.40
3.83
4.28
1.21
1.78
1.65
2.00
2.32
2.59
1.81
1.59
1.74
2.55
1.38
1.57
1.80
2.38
1.69
2.08
2.58
2.14
2.30
2.65
1.90
2.20
2.53
75.0
71.9
75.5
65.9
56.7
48.6
70.7
76.5
73.4
49.3
74.1
72.7
70.5
54.5
69.9
54.8
40.4
61.3
56.0
44.6
61.6
48.8
38.3
50% OL
50.0
25.0
31.8
16.7
8.5
4.3
23.5
34.4
27.4
5.0
44.4
36.4
25.1
7.4
27.8
15.1
7.4
12.6
9.2
4.7
20.2
13.2
9.0
Taylor
Gaussian
DolphChebyshev
KaiserBessel
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
α
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
=
2.0
2.5
3.0
3.5
4.0
2.5
3.0
3.5
2.5
3.0
3.5
4.0
2.0
2.5
3.0
3.5
3.0
3.5
4.0
Barcilon
Temes
Exact Blackman
Blackman
Minimum 3-sample
Blackman-Harris
Minimum 4-sample
Blackman-Harris
62-dB 3-sample
Blackman-Harris
74-dB 4-sample
Blackman-Harris
4-sample
α = 3.0
Kaiser–Bessel
© 1999 by CRC Press LLC
–40
–50
–60
–70
–80
–42
–55
–69
–50
–60
–70
–80
–46
–57
–69
–82
–53
–58
–68
–68
–58
–71
–6
–6
–6
–6
–6
–6
–6
–6
0
0
0
0
–6
–6
–6
–6
–6
–6
–6
–6
–18
–6
0.57
0.51
0.47
0.44
0.41
0.51
0.43
0.37
0.53
0.48
0.45
0.42
0.49
0.44
0.40
0.37
0.47
0.43
0.41
0.46
0.42
0.42
1.30
1.43
1.55
1.66
1.76
1.39
1.64
1.90
1.39
1.51
1.62
1.73
1.50
1.65
1.80
1.93
1.56
1.67
1.77
1.57
1.73
1.71
1.25
1.36
1.47
1.58
1.67
1.33
1.55
1.79
1.33
1.44
1.55
1.65
1.43
1.57
1.71
1.83
1.49
1.59
1.69
1.52
1.68
1.66
1.91
1.60
1.37
1.20
1.06
1.69
1.25
0.94
1.70
1.44
1.25
1.10
1.46
1.20
1.02
0.89
1.34
1.18
1.05
1.33
1.10
1.13
3.06
3.15
3.26
3.40
3.52
3.14
3.40
3.73
3.12
3.23
3.35
3.48
3.20
3.38
3.56
3.74
3.27
3.40
3.52
3.29
3.47
3.45
1.74
1.90
2.06
2.21
2.35
1.86
2.18
2.52
1.85
2.01
2.17
2.31
1.99
2.20
2.39
2.57
2.07
2.23
2.36
2.13
2.35
1.81
75.7
71.3
67.0
62.9
59.1
67.7
57.5
47.2
69.6
64.7
60.2
55.9
65.7
59.5
53.9
48.8
63.0
58.6
54.4
62.7
56.7
57.2
28.3
21.4
16.1
12.1
9.1
20.0
10.6
4.9
22.3
16.3
11.9
8.7
16.9
11.2
7.4
4.8
14.2
10.4
7.6
14.0
9.0
9.6
–92
–6
0.36
2.00
1.90
0.83
3.85
2.72
46.0
3.8
–62
–6
0.45
1.61
1.56
1.27
3.34
2.19
61.0
12.6
–74
–6
0.40
1.79
1.74
1.03
3.56
2.44
53.9
7.4
–69
–6
0.40
1.80
1.74
1.02
3.56
2.44
53.9
7.4
7.2.2 Coherent Gain
CG = coherent gain =
1
N
N −1
∑ w(nT ) = zero frequency gain (dc gain) of the window
n=0
7.2.3 Processing Gain
PG =
1
output signal-to-noise ratio
=
ENBW
input signal-to-noise ratio
7.2.4 Scalloping Loss
π
∑ w(nT )exp − j N n
=
scalloping loss =
∑ w(nT )
n
n
=
coherent gain for a tone located half a bin from DFT sample point
coherent gain for a tone located at a DFT sample point
= maximum reduction in PG due to signal frequency
7.2.5 Mainlobe Spectral Response
mainlobe spectral response = spectral interval between the peak gain and the –3.0 dB and –6.0 dB
response level
7.2.6 Overlap Correlation
Correlation coefficients represent the degree of correlation of filter output points that are separated by
25% and 50% of the filter length. These terms are useful in quantifying the estimation uncertainty (or
variance reduction) related to incoherent averaging of filter (window) data.
7.3 Window (Filter) Descriptions
7.3.1 Introduction
•
•
•
•
•
•
T=1
–π ≤ ω ≤ π or 0 ≤ ω ≤ 2π
DFT bin = 2π/N
Windows are even (about the origin) sequences with an odd number of points.
The right-most point of the window will be discarded.
N will be taken to be even, and the total points will be odd, and hence
N = 2 × ( total points) = even
7.3.2 Rectangle (Dirichlet) Window
w(n) = 1.0
© 1999 by CRC Press LLC
n = − N2 ,L, −1, 0,1,L, N2
N/2
∑ w( n ) e
W (ω ) =
− jnω
n =− N / 2
To make it realizable shift the sequence by N/2 to the right. Hence we obtain
w( n ) = 1
n = 0,1,L, N − 1
N −1
W (ω ) =
∑e
− jnω
=e
− j N2−1 ω
n=0
N
sin ω
2 
ω
sin
2
Figure 7.1 shows the rectangular window and its amplitude spectrum W(ω ) .
FIGURE 7.1 a) Rectangular window. b) Amplitude spectrum of rectangular window.
7.3.3 Triangle (Fejer, Bartlet) Window
w(n) = 1.0 −
n
N /2
n = − N2 ,L, −1, 0,1,L, N2
For DFT the window is
n

N /2
w( n ) = 
N−n

 N /2
n = 0,1,L, N2
n=
N
2
+ 1,L, N − 1
and its DFT is
 N 
 sin 4 ω 
− j ( N2 −1)ω
W (ω ) = e


 sin ω 

2 
2
since the symmetric function w(n) is shifted by
7.2 shows the triangular window and its DFT.
© 1999 by CRC Press LLC
N
2
− 1 positions to produce the DFT sequence. Figure
FIGURE 7.2 a) Triangular window. b) Amplitude spectrum of triangular window.
7.3.4 cosα (t) Windows
 n 
w(n) = cos α   π 
 N  
n = − N2 ,L, −1, 0,1,L, N2
 n 
w(n) = sin α   π 
 N  
n = 0,1,L, N − 1
Common values of α : 1 ≤ α ≤ 4
 nπ  , cos 3  nπ  ,
 nπ 
Figures 7.3 through 7.5 show the cos 2
and cos 4
windows and their Fourier
 N
 N
 N
transform.
FIGURE 7.3 a) Cosine window with α = 1. b) Amplitude spectrum of cosine window with α = 1.
FIGURE 7.4 a) Cosine window with α = 3. b) Amplitude spectrum of cosine window with α = 3.
© 1999 by CRC Press LLC
FIGURE 7.5 a) Cosine window with α = 4. b) Amplitude spectrum of cosine window with α = 4.
7.3.5 Hann Window
n
1
2n 
w(n) = cos 2  π = 1 + cos π 
N  2
 N 
n = − N2 ,L, −1, 0,1,L, N2
 2n 
 n  1
w(n) = sin 2   π  = 1 − cos  π 


 N  
 N  2
n = 0,1,L, N − 1
DFT of the window is
W (ω ) =
1
1
2π 
2π  
D (ω ) +  D  ω −
+ Dω +



2
4
N
N  
D(ω ) = e
jω
2
N
sin ω
2 
= Dirichlet Kernel
ω
sin
2
−π≤ω≤π
See Figure 7.6 for the Hann window.
FIGURE 7.6 a) Hann window. b) Amplitude spectrum of Hann window.
7.3.6 Hamming Window
w(n) = α + (1 − α )cos
© 1999 by CRC Press LLC
2π
n
N
n = − N2 ,L, −1, 0,1,L, N2
W (ω ) = α D (ω ) +
1
2π 
2π  

(1 − α )  D  ω −
+ Dω +

2
N
N  
 
−π ≤ ω ≤ π
D = Dirichlet Kernel (see 7.3.5)
w(n) = 0.54 + 0.46 cos
2π
n
N
n = − N2 ,L, −1, 0,1,L, N2
w(n) = 0.54 − 0.46 cos
2π
n
N
n = 0,1,L, N − 1
Figure 7.7 depicts the Hamming window and its amplitude spectrum.
FIGURE 7.7 a) Hamming window. b) Amplitude spectrum of Hamming window.
7.3.7 Short Cosine Series Window
K/2
w( n ) =
2π
∑ a(k)cos N  kn
n = − N2 ,L, −1, 0,1,L, N2
k =0
K/2
∑ a( k ) = 1
constraint
k =0
K/2
W (ω ) = a(0) D(ω ) +
∑
k =0
a( k )
2
2π 
2π  
 

D ω − k N  + D ω + k N  


−π ≤ ω ≤ π
7.3.8 Blackman Window
a(0) = 0.42659071 ≅ 0.42, a(1) = 0.49656062 ≅ 0.50, a(2) = 0.07684867 ≅ 0.08
2π 
2π 
w(n) = 0.42 + 0.5 cos
n + 0.08 cos
2n
 N 
 N

n = − N2 ,L, −1, 0,1,L, N2
2π
2π
w(n) = 0.42 + 0.5 cos
(n − 25) + 0.08 cos
2(n − 25)
 N

 N

Figure 7.8 shows the characteristics of the Blackman window.
© 1999 by CRC Press LLC
n = 0,1,L, N − 1
FIGURE 7.8 a) Blackman window. b) Amplitude spectrum of Blackman window.
7.3.9 Harris-Nutall Window
Table 7.2 gives the coefficients for short cosine series windows.
TABLE 7.2 Coefficients of Three- and Four-Term Harris-Nutall Windows
a(0)
a(1)
a(2)
a(3)
3-Term
(–61 dB)
3-Term
(–67 dB)
4-Term
(–74 dB)
4-Term
(–94 dB)
0.44959
0.49364
0.05677
0
0.42323
0.49755
0.07922
0
0.40217
0.49703
0.09392
0.00183
0.35875
0.48829
0.14128
0.01168
Figures 7.9 and 7.10 show the Harris-Nutall window characteristics.
FIGURE 7.9 a) Harris-Nutall window (3-term). b) Amplitude spectrum of Harris-Nutall window (3-term).
FIGURE 7.10 a) Harris-Nutall window (4-term). b) Amplitude spectrum of Harris-Nutall window (4-term).
© 1999 by CRC Press LLC
7.3.10 Sampled Kaiser-Bessel Window
Kaiser-Bessel spectrum window = W (ω ) =
H1 ( m) =
sinh(π α 2 − m 2 )
sinh π 2 α 2 − (ωN / 2) 2
π 2 α 2 − (ωN / 2) 2
ω = m( 2 π / N )
π α 2 − m2
c = H1 (0) + 2 H1 (1) + 2 H1 (2) + [2 H1 (3)]
a(0) =
H1 (0)
,
c
a(0) = 0.40243,
a( m) =
2 H1 (0)
,
c
m = 1, 2, 3
a(1) = 0.49804 ,
a(2) = 0.09831,
a(3) = 0.00122
7.3.11 Parabolic (Riesz, Bochner, Parzen) Window
 n 
w(n) = 1.0 − 

 N / 2
2
n− N
2
w(n) = 1.0 − 

N
/
2




0≤ n ≤
N
2
2
n = 0,1, 2,L, N − 1
Figure 7.11 shows the parabolic window and its spectrum characteristics.
FIGURE 7.11 a) Parabolic window. b) Amplitude spectrum of Parabolic window.
7.3.12 Riemann Window
w( n ) =
© 1999 by CRC Press LLC
2 πn
N
2 πn
N
sin
0≤ n ≤
N
2
0≤α≤4


N
 2 π n −  
sin 
2


N


w( n ) =
N

2π n −

2
N
n = 0,1, 2,L, N − 1
Figure 7.12 shows the window’s characteristics.
FIGURE 7.12 a) Riemann window. b) Amplitude spectrum of Riemann window.
7.3.13 de la Vallé-Poussin (Jackson, Parzen) Window
2

n 
 n  
1−
1 − 6 




N /2  N /2
w( n ) = 
3
2 1 − n 
  N / 2 
0≤ n ≤
N
4
≤n≤
N
4
N
2
Figure 7.13 shows the window and its frequency response.
FIGURE 7.13 a) de la Vallé-Poussin window. b) Amplitude spectrum of de la Vallé-Poussin window.
7.3.14 Cosine Taper (Tukey) Window
The Tukey window is equal to one over (1 – α/2)N points, with a cosine taper from one to zero for the
remaining points (α/2)N.
© 1999 by CRC Press LLC
1

 
 n − α( N / 2) 
w( n ) = 
0.5 1 + cosπ

 
 (1 − α )( N / 2) 

0 ≤ n ≤ α N2
α N2 < n ≤
N
2
Figures 7.14 and 7.15 show the window and its frequency responses for α = 8/25 and α = 12/25.
FIGURE 7.14 a) Tukey window with α = 8/25. b) Amplitude spectrum of Tukey window with α = 8/25.
FIGURE 7.15 a) Tukey window with α = 12/25. b) Amplitude spectrum of Tukey window with α = 12/25.
7.3.15 Bohman Window
n 
n  1
n 



w( n ) =  1 −
,
 + sin π
 cos π





N /2
N /2
π
N / 2
0≤ n ≤
Figure 7.16 shows the window and its spectrum.
FIGURE 7.16 a) Bohman window. b) Amplitude spectrum of Bohman window.
© 1999 by CRC Press LLC
N
2
7.3.16 Poisson Window
n 

w(n) = exp −α
,

N / 2
0≤ n ≤
N
2
Figures 7.17 through 7.19 show the window and its spectrum with α = 1.5, α = 3, and α = 4.
FIGURE 7.17 a) Poisson window with α = 1.5. b) Amplitude spectrum of Poisson window with α = 1.5.
FIGURE 7.18 a) Poisson window with α = 3.0. b) Amplitude spectrum of Poisson window with α = 3.0.
FIGURE 7.19 a) Poisson window with α = 4.0. b) Amplitude spectrum of Poisson window with α = 4.0.
7.3.17 Hann-Poisson Window

n 
n 


w(n) = 0.51 + cos π
  exp −α
,
 N / 2

N / 2

© 1999 by CRC Press LLC
0≤ n ≤
N
2
FIGURE 7.20 a) Hann-Poisson window with α = 0.5. b) Amplitude spectrum of Hann-Poisson window with α = 0.5.
FIGURE 7.21 a) Hann-Poisson window with α = 1.0. b) Amplitude spectrum of Hann-Poisson window with α = 1.0.
Figures 7.20 and 7.21 show the window and its spectrum with α = 0.5 and α = 1.0, respectively.
7.3.18 Cauchy (Abel, Poisson) Window
w( n ) =
1
n 

1 + α

 N / 2
2
,
0≤ n ≤
N
2
Figures 7.22 through 7.24 show the window and its spectrum with α = 3.0, α = 4.0, and α = 6.0,
respectively.
FIGURE 7.22 a) Cauchy window with α = 3.0. b) Amplitude spectrum of Cauchy window with α = 3.0.
© 1999 by CRC Press LLC
FIGURE 7.23 a) Cauchy window with α = 4.0. b) Amplitude spectrum of Cauchy window with α = 4.0.
FIGURE 7.24 a) Cauchy window with α = 6.0. b) Amplitude spectrum of Cauchy window with α = 6.0.
7.3.19 Gaussian (Weierstrass) Window
 
n 
w(n) = exp − 12  α


N / 2

2

,

0≤ n ≤
N
2
Figures 7.25 and 7.26 show the window and its spectrum for α = 2.5 and α = 3.5.
FIGURE 7.25 a) Gaussian window with α = 2.5. b) Amplitude spectrum of Gaussian window with α = 2.5.
7.3.20 Dolph-Chebyshev Window
W (k ) =
© 1999 by CRC Press LLC
cosh( N cosh −1 (β cosh(πk / N )))
cosh( N cosh −1 (β))
FIGURE 7.26 a) Gaussian window with α = 3.5. b) Amplitude spectrum of Gaussian window with α = 3.5.
where
(
)
cosh −1 ( X ) = ln X + X 2 − 1.0 ,
W (k ) =
X > 1.0
cos( N cos −1 (β cos(πk / N )))
cosh( N cosh −1 (β))
where
cos −1 ( X ) =


π
X
− tan −1 
,
2
2
 X − 1.0 
X ≤ 1.0
where β satisfies
1
β = cosh cosh −1 10 α 
N

and
N −1
w( n ) =
∑ W (k)exp j
k =0
2π 
nk

N
7.3.21 Kaiser-Bessel Window
2

 n  
Io πα 1.0 − 

 N / 2 


w( n ) = 
Io [πα]
0≤ n ≤
N
2
2
Io ( x ) =
∞
∑
k =0


k
 x  
  2   = zero-order modified Bessel function


 k! 


Figures 7.27 and 7.28 show the window and its spectrum for α = 2 and α = 3.
© 1999 by CRC Press LLC
FIGURE 7.27 a) Kaiser-Bessel window with α = 3.0. b) Amplitude spectrum of Kaiser-Bessel window with α = 3.0.
FIGURE 7.28 a) Kaiser-Bessel window with α = 2.0. b) Amplitude spectrum of Kaiser-Bessel window with α = 2.0.
7.3.22 Barcilon-Themes Window
W (k ) =
A cos[ y(k )] + B[ y(k ) / C]sin[ y(k )]
(C + AB)[( y(k ) / C ) 2 + 1.0]
where
A = sinh C = 10 2 α − 1.0
B = cosh C = 10 α
C = cosh −1 (10 α )
β = cosh(C / N )
πk 

y(k ) = N cos −1 β cos  
 N 

7.3.23 Highest Sidelobe Level versus Worst-Case Processing Loss
Figure 7.29 shows the highest sidelobe level versus worst-case processing loss. Shaped DFT filters in
the lower left tend to perform well.
© 1999 by CRC Press LLC
FIGURE 7.29 Highest sidelobe level versus worst-case processing loss. Shaped DFT filters in the lower left tend
to perform well.
References
Harris, F. J., On the use of windows for harmonic analysis with the discrete fourier transforms, Proc.
IEEE, 66, 55-83, January 1978.
© 1999 by CRC Press LLC