Fixed Window Functions
Fixed Window Functions
• Using a tapered window causes the height
of the sidelobes to diminish, with a
corresponding increase in the main lobe
width resulting in a wider transition at the
discontinuity
• Hann:
w[ n] = 0.5 + 0.5 cos( 2π n ), − M ≤ n ≤ M
2M + 1
• Hamming:
w[ n] = 0.54 + 0.46 cos( 2π n ), − M ≤ n ≤ M
2M + 1
• Blackman:
w[n] = 0.42 + 0.5 cos( 2π n ) + 0.08 cos( 4π n )
2M + 1
2M + 1
• Plots of magnitudes of the DTFTs of these
windows for M = 25 are shown below:
Fixed Window Functions
Fixed Window Functions
• Magnitude spectrum of each window
characterized by a main lobe centered at
ω = 0 followed by a series of sidelobes with
decreasing amplitudes
• Parameters predicting the performance of a
window in filter design are:
• Main lobe width
• Relative sidelobe level
• Main lobe width ∆ ML - given by the
distance between zero crossings on both
sides of main lobe
• Relative sidelobe level Asl - given by the
difference in dB between amplitudes of
largest sidelobe and main lobe
Copyright © 2001, S. K. Mitra
Hanning window
0
-20
-20
Gain, dB
Gain, dB
Rectangular window
0
-40
-60
-60
-80
-80
-100
0
-40
0.2
0.4
0.6
0.8
-100
1
0
0.2
0
0.2
0
0
-20
-20
Gain, dB
Gain, dB
ω/π
Hamming window
-40
-60
-80
-100
0.4
0.6
0.8
ω/π
Blackman window
-40
-60
-80
0
0.2
0.4
0.6
0.8
1
-100
ω/π
Copyright © 2001, S. K. Mitra
Fixed Window Functions
1
0.4
0.6
0.8
1
ω/π
Copyright © 2001, S. K. Mitra
Copyright © 2001, S. K. Mitra
Fixed Window Functions
• Distance between the locations of the
maximum passband deviation and minimum
stopband value ≅ ∆ ML
• Observe H t (e j ( ωc + ∆ω) ) + H t (e j ( ωc − ∆ω) ) ≅ 1
• Thus,
H t (e jωc ) ≅ 0.5
• Passband and stopband ripples are the same
Copyright © 2001, S. K. Mitra
• Width of transition band
∆ω = ω s − ω p < ∆ ML
Copyright © 2001, S. K. Mitra
1
Fixed Window Functions
Fixed Window Functions
• To ensure a fast transition from passband to
stopband, window should have a very small
main lobe width
• To reduce the passband and stopband ripple
δ , the area under the sidelobes should be
very small
• Unfortunately, these two requirements are
contradictory
• In the case of rectangular, Hann, Hamming,
and Blackman windows, the value of ripple
does not depend on filter length or cutoff
frequency ωc , and is essentially constant
• In addition,
∆ω ≈ c
M
where c is a constant for most practical
purposes
Copyright © 2001, S. K. Mitra
Fixed Window Functions
• Rectangular window - ∆ ML = 4π /( 2 M + 1)
Asl = 13.3 dB, α s = 20.9 dB, ∆ω = 0.92π / M
• Hann window - ∆ ML = 8π /( 2 M + 1)
Asl = 31.5 dB, α s = 43.9 dB, ∆ω = 3.11π / M
• Hamming window - ∆ ML = 8π /( 2 M + 1)
Asl = 42.7 dB, α s = 54.5 dB, ∆ω = 3.32π / M
• Blackman window - ∆ ML = 12π /(2 M + 1)
Asl = 58.1 dB, α s = 75.3 dB, ∆ω = 5.56π / M
Copyright © 2001, S. K. Mitra
FIR Filter Design Example
• Lowpass filter of length 51 and ω c = π / 2
Lowpass Filter Designed Using Hamming window
0
Gain, dB
-50
-50
-100
-100
0
0.2
0.4
0.6
0.8
0
1
0.2
0.4
0.6
0.8
1
ω/π
ω/π
Lowpass Filter Designed Using Blackman window
0
Gain, dB
Gain, dB
Lowpass Filter Designed Using Hann window
0
Copyright © 2001, S. K. Mitra
Fixed Window Functions
• Filter Design Steps (1) Set
ω c = (ω p + ω s ) / 2
(2) Choose window based on specified α s
(3) Estimate M using
∆ω ≈ c
M
Copyright © 2001, S. K. Mitra
FIR Filter Design Example
• An increase in the main lobe width is
associated with an increase in the width of
the transition band
• A decrease in the sidelobe amplitude results
in an increase in the stopband attenuation
-50
-100
0
0.2
0.4
0.6
ω/π
0.8
1
Copyright © 2001, S. K. Mitra
Copyright © 2001, S. K. Mitra
2
Adjustable Window Functions
Adjustable Window Functions
• Dolph-Chebyshev Window M
w[n ] = 1 [γ1 + 2 å Tk ( β cos k ) cos 2nkπ ],
2M + 1
2M + 1
2M + 1
k =1
−M ≤n≤M
amplitude of sidelobe
where
γ=
main lobe amplitude
β = cosh( 1 cosh −1 γ1 )
2M
and
ì cos(l cos −1 x ),
x ≤1
Tl ( x) = í
−1
îcosh(l cosh x), x > 1
• Dolph-Chebyshev window can be designed
with any specified relative sidelobe level
while the main lobe width adjusted by
choosing length appropriately
• Filter order is estimated using
2.056α s − 16.4
N=
2.85(∆ω )
where ∆ω is the normalized transition
bandwidth, e.g, for a lowpass filter
∆ω = ωs − ω p
Copyright © 2001, S. K. Mitra
Copyright © 2001, S. K. Mitra
Adjustable Window Functions
Adjustable Window Functions
• Gain response of a Dolph-Chebyshev
window of length 51 and relative sidelobe
level of 50 dB is shown below
Properties of Dolph-Chebyshev window:
• All sidelobes are of equal height
• Stopband approximation error of filters
designed have essentially equiripple
behavior
• For a given window length, it has the
smallest main lobe width compared to other
windows resulting in filters with the
smallest transition band
Dolph-Chebyshev Window
Gain, dB
0
-20
-40
-60
-80
0
0.2
0.4
0.6
0.8
1
ω/π
Copyright © 2001, S. K. Mitra
Copyright © 2001, S. K. Mitra
Adjustable Window Functions
Adjustable Window Functions
• Kaiser Window I {β 1 − ( n / M ) 2 }
w[ n] = 0
, −M ≤n≤ M
I0 ( β )
where β is an adjustable parameter and I 0 (u )
is the modified zeroth-order Bessel function
of the first kind:
∞ (u / 2) r
I 0 (u ) = 1 + å [
]2
r!
r =1
• Note I 0 (u ) > 0 for u > 0
20 (u / 2) r
• In practice I 0 (u ) ≅ 1 + å [
]2
r!
r =1
• β controls the minimum stopband
attenuation of the windowed filter response
• β is estimated using
Copyright © 2001, S. K. Mitra
0.1102( α s −8.7 ),
for α s > 50
ìï
β=í0.5842( α s − 21)0.4 + 0.07886( α s − 21), for 21 ≤ α s ≤ 50
0,
ïî
for α s < 21
• Filter order is estimated using
N=
αs − 8
2.285(∆ω )
where ∆ω is the normalized transition
bandwidth
Copyright © 2001, S. K. Mitra
3
FIR Filter Design Example
• Choose N = 24 implying M =12
FIR Filter Design Example
sin(0.4π n)
• Hence ht [ n] =
⋅ w[n], − 12 ≤ n ≤ 12
πn
where w[n] is the n-th coefficient of a
length-25 Kaiser window with β = 3.3953
0
-20
-20
Gain, dB
0
-40
0.2
0.4
0.6
ω/π
Impulse Responses of FIR Filters
with a Smooth Transition
• First-order spline passband-to-stopband
transition
ω c = (ω p + ω s ) / 2
∆ω = ωs − ω p
ωc / π ,
ìï
hLP [ n] = í 2 sin( ∆ω n / 2) sin(ω c n)
⋅ πn
ïî
∆ω n
n=0
-40
-60
-60
-80
0
Copyright © 2001, S. K. Mitra
Lowpass filter designed with Kaiser window
Kaiser Window
Gain, dB
• Specifications: ω p = 0.3π , ω s = 0.5π ,
α s = 40 dB
• Thus ω c = (ω p + ω s ) / 2 = 0.4π
δ s = 10−α s / 20 = 0.01
β = 0.5842(19) 0.4 + 0.07886 × 19 = 3.3953
32
N=
= 22.2886
2.285(0.2π)
0.8
1
-80
0
0.2
0.4
0.6
0.8
1
ω/π
Copyright © 2001, S. K. Mitra
Impulse Responses of FIR Filters
with a Smooth Transition
• Pth-order spline passband-to-stopband
transition
ωc / π ,
ì
ï
hLP [ n] = íæ 2 sin( ∆ω n / 2 P ) ö P sin(ω c n)
ç
÷ ⋅ πn
ï
îè ∆ω n / 2 P ø
n=0
n >0
n >0
Copyright © 2001, S. K. Mitra
Copyright © 2001, S. K. Mitra
Lowpass FIR Filter Design
Example
• Example
Magnitude
1
P = 1, N = 40
P = 2, N = 60
0.8
0.6
0.4
0.2
0
0
0.2
0.4
0.6
0.8
1
ω/π
Copyright © 2001, S. K. Mitra
4