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Relationship between window function's bandwidth and response time
Relationship between window function's bandwidth and response time
Brick-wall Window
w_bw[T_,t_]=Assuming[T>0,UnitBox[t/T]]
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UnitBox
t
T
With[{T=1},Plot[w_bw[T,t],{t,-T,T},ExclusionsStyleDashed,AxesLabelAutomatic,PlotLabel"Time Domain Form"]]
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W_bw[T_,ω_]=Assuming[T>0,Simplify[FourierTransform[w_bw[T,t],t,ω,FourierParameters{0,-1}]]]
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TSinc
Tω
2
2π
With[{T=1},Plot[W_bw[T,ω],{ω,-2Pi*10/T,2Pi*10/T},PlotRangeFull,AxesLabelAutomatic,PlotLabel"Frequency Domain Form"]]
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tot_freq_wgt_bw=Assuming[T>0,Integrate[W_bw[T,ω],{ω,-Infinity,Infinity}]](*totalweightofbrickwallwindowinfrequencydomain*)
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2π
norm_freq_wgt_bw[T_,B_ω_]=Assuming[T>0&&B_ω≥0,Integrate[W_bw[T,ω],{ω,-B_ω/2,B_ω/2}]]/tot_freq_wgt_bw(*B_ωisbandwidth.*)
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2SinIntegral
B_ωT
4
π
Assuming[T>0,Plot[norm_freq_wgt_bw[T,x/T],{x,0,2Pi*20},GridLinesAutomatic,AxesLabel{"TB_ω"},PlotLabel"Normalized Frequency Cumulative Weight"]]
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As shown in the above figure, is sufficient for ≥ 95 % of total weight to be included in the range . So a given 2π, T should be ≈3.98 or more.
T≥50
B
ω
[-/2,/2]
B
ω
B
ω
B
ω
B
f
25
2π
B
f
B
f
Gaussian Window
w_G[σ_,t_]=Assuming[σ>0,Exp[-t^2/(2σ^2)]]
Out[]=
-
2
t
2
2
σ
With[{σ=1},Plot[w_G[σ,t],{t,-5σ,5σ},ExclusionsStyleDashed,AxesLabelAutomatic,PlotLabel"Time Domain Form"]]
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W_G[σ_,ω_]=Assuming[σ>0,Simplify[FourierTransform[w_G[σ,t],t,ω,FourierParameters{0,-1}]]]
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-
1
2
2
σ
2
ω
With[{σ=1},Plot[W_G[σ,ω],{ω,-5/σ,5/σ},PlotRangeFull,AxesLabelAutomatic,PlotLabel"Frequency Domain Form"]]
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tot_freq_wgt_G=Assuming[σ>0,Integrate[W_G[σ,ω],{ω,-Infinity,Infinity}]](*totalweightofbrickwallfilterinfrequencydomain*)
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2π
norm_freq_wgt_G[σ_,B_ω_]=Assuming[σ>0&&B_ω≥0,Integrate[W_G[σ,ω],{ω,-B_ω/2,B_ω/2}]]/tot_freq_wgt_G(*B_ωisbandwidth.*)
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Erf
B_ωσ
2
2
Assuming[σ>0,Plot[norm_freq_wgt_G[σ,x/σ],{x,0,5},GridLinesAutomatic,AxesLabel{"σB_ω"},PlotLabel"Normalized Frequency Cumulative Weight"]]
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As in the case of the Brick-wall Window, σ should be 4/(2π) or more to ensure that ≥95 % of weight is included in the range . The practical finite time length approximation of the window will be the [-3σ,3σ]≈[-1.91/,1.91/] part of the original function .
B
f
[-/2,/2]
B
ω
B
ω
B
f
B
f
w
G
Hann Window
w_Hann[T_,t_]=Assuming[T>0,Cos[(Pi*t)/T]^2]*UnitBox[t/T]
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2
Cos
πt
T
t
T
With[{T=1},Plot[w_Hann[T,t],{t,-T,T},ExclusionsStyleDashed,AxesLabelAutomatic,PlotLabel"Time Domain Form"]]
Out[]=