6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
DSP and Digital Filters (2016-8872)
6: Window Filter Design
Windows: 6 – 1 / 12
Inverse DTFT
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
For any BIBO stable filter, H(ejω ) is the DTFT of h[n]
DSP and Digital Filters (2016-8872)
Windows: 6 – 2 / 12
Inverse DTFT
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
For any BIBO stable filter, H(ejω ) is the DTFT of h[n]
H(ejω ) =
DSP and Digital Filters (2016-8872)
P∞
−∞
h[n]e−jωn
Windows: 6 – 2 / 12
Inverse DTFT
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
For any BIBO stable filter, H(ejω ) is the DTFT of h[n]
jω
H(e ) =
DSP and Digital Filters (2016-8872)
P∞
−∞
−jωn
h[n]e
⇔
h[n] =
1
2π
Rπ
−π
H(ejω )ejωn dω
Windows: 6 – 2 / 12
Inverse DTFT
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
For any BIBO stable filter, H(ejω ) is the DTFT of h[n]
jω
H(e ) =
P∞
−∞
−jωn
h[n]e
⇔
h[n] =
1
2π
Rπ
−π
H(ejω )ejωn dω
If we know H(ejω ) exactly, the IDTFT gives the ideal h[n]
DSP and Digital Filters (2016-8872)
Windows: 6 – 2 / 12
Inverse DTFT
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
For any BIBO stable filter, H(ejω ) is the DTFT of h[n]
jω
H(e ) =
P∞
−∞
−jωn
h[n]e
⇔
h[n] =
1
2π
Rπ
−π
H(ejω )ejωn dω
If we know H(ejω ) exactly, the IDTFT gives the ideal h[n]
Example: Ideal Lowpass filter
H(ejω ) =
(
1 |ω| ≤ ω0
0 |ω| > ω0
1
2ω0
0.5
0
DSP and Digital Filters (2016-8872)
-2
0
ω
2
Windows: 6 – 2 / 12
Inverse DTFT
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
For any BIBO stable filter, H(ejω ) is the DTFT of h[n]
jω
H(e ) =
P∞
−∞
−jωn
h[n]e
⇔
h[n] =
1
2π
Rπ
−π
H(ejω )ejωn dω
If we know H(ejω ) exactly, the IDTFT gives the ideal h[n]
Example: Ideal Lowpass filter
H(ejω ) =
(
1 |ω| ≤ ω0
0 |ω| > ω0
⇔
h[n] =
2π/ω0
1
2ω0
0.5
0
DSP and Digital Filters (2016-8872)
sin ω0 n
πn
-2
0
ω
2
0
Windows: 6 – 2 / 12
Inverse DTFT
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
For any BIBO stable filter, H(ejω ) is the DTFT of h[n]
jω
H(e ) =
P∞
−∞
−jωn
h[n]e
⇔
h[n] =
1
2π
Rπ
−π
H(ejω )ejωn dω
If we know H(ejω ) exactly, the IDTFT gives the ideal h[n]
Example: Ideal Lowpass filter
H(ejω ) =
(
1 |ω| ≤ ω0
0 |ω| > ω0
⇔
h[n] =
2π/ω0
1
2ω0
0.5
0
sin ω0 n
πn
-2
0
ω
0
2
2π
Note: Width in ω is 2ω0 , width in n is ω
0
DSP and Digital Filters (2016-8872)
Windows: 6 – 2 / 12
Inverse DTFT
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
For any BIBO stable filter, H(ejω ) is the DTFT of h[n]
jω
H(e ) =
P∞
−∞
−jωn
h[n]e
⇔
h[n] =
1
2π
Rπ
−π
H(ejω )ejωn dω
If we know H(ejω ) exactly, the IDTFT gives the ideal h[n]
Example: Ideal Lowpass filter
H(ejω ) =
(
1 |ω| ≤ ω0
0 |ω| > ω0
⇔
h[n] =
2π/ω0
1
2ω0
0.5
0
sin ω0 n
πn
-2
0
ω
0
2
2π
: product is 4π always
Note: Width in ω is 2ω0 , width in n is ω
0
DSP and Digital Filters (2016-8872)
Windows: 6 – 2 / 12
Inverse DTFT
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
For any BIBO stable filter, H(ejω ) is the DTFT of h[n]
jω
H(e ) =
P∞
−∞
−jωn
h[n]e
⇔
h[n] =
1
2π
Rπ
−π
H(ejω )ejωn dω
If we know H(ejω ) exactly, the IDTFT gives the ideal h[n]
Example: Ideal Lowpass filter
H(ejω ) =
(
1 |ω| ≤ ω0
0 |ω| > ω0
⇔
h[n] =
2π/ω0
1
2ω0
0.5
0
sin ω0 n
πn
-2
0
ω
2
0
2π
: product is 4π always
Note: Width in ω is 2ω0 , width in n is ω
0
Sadly h[n] is infinite and non-causal.
DSP and Digital Filters (2016-8872)
Windows: 6 – 2 / 12
Inverse DTFT
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
For any BIBO stable filter, H(ejω ) is the DTFT of h[n]
jω
H(e ) =
P∞
−∞
−jωn
h[n]e
⇔
h[n] =
1
2π
Rπ
−π
H(ejω )ejωn dω
If we know H(ejω ) exactly, the IDTFT gives the ideal h[n]
Example: Ideal Lowpass filter
H(ejω ) =
(
1 |ω| ≤ ω0
0 |ω| > ω0
⇔
h[n] =
2π/ω0
1
2ω0
0.5
0
sin ω0 n
πn
-2
0
ω
2
0
2π
: product is 4π always
Note: Width in ω is 2ω0 , width in n is ω
0
Sadly h[n] is infinite and non-causal. Solution: multiply h[n] by a window
DSP and Digital Filters (2016-8872)
Windows: 6 – 2 / 12
Rectangular window
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncate to ± M
2 to make finite
2π/ω0
0
DSP and Digital Filters (2016-8872)
Windows: 6 – 3 / 12
Rectangular window
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncate to ± M
2 to make finite; h1 [n] is now of length M + 1
h1[n]
M=14
0
DSP and Digital Filters (2016-8872)
Windows: 6 – 3 / 12
Rectangular window
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncate to ± M
2 to make finite; h1 [n] is now of length M + 1
MSE Optimality:
Define mean square error (MSE) in frequency domain
E=
1
2π
Rπ H(ejω ) − H1 (ejω )2 dω
−π
h1[n]
1
M=14
M=14
0.5
0
DSP and Digital Filters (2016-8872)
0
0
1
ω
2
3
Windows: 6 – 3 / 12
Rectangular window
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncate to ± M
2 to make finite; h1 [n] is now of length M + 1
MSE Optimality:
Define mean square error (MSE) in frequency domain
E=
=
1
2π
1
2π
Rπ H(ejω ) − H1 (ejω )2 dω
−π
2
M
R π P
−jωn jω
2
h
[n]e
H(e
)
−
dω
−π
−M 1
2
h1[n]
1
M=14
M=14
0.5
0
DSP and Digital Filters (2016-8872)
0
0
1
ω
2
3
Windows: 6 – 3 / 12
Rectangular window
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncate to ± M
2 to make finite; h1 [n] is now of length M + 1
MSE Optimality:
Define mean square error (MSE) in frequency domain
E=
=
1
2π
1
2π
Rπ H(ejω ) − H1 (ejω )2 dω
−π
2
M
R π P
−jωn jω
2
h
[n]e
H(e
)
−
dω
−π
−M 1
2
Minimum E is when h1 [n] = h[n].
h1[n]
1
M=14
M=14
0.5
0
DSP and Digital Filters (2016-8872)
0
0
1
ω
2
3
Windows: 6 – 3 / 12
Rectangular window
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncate to ± M
2 to make finite; h1 [n] is now of length M + 1
MSE Optimality:
Define mean square error (MSE) in frequency domain
E=
=
1
2π
1
2π
Rπ H(ejω ) − H1 (ejω )2 dω
−π
2
M
R π P
−jωn jω
2
h
[n]e
H(e
)
−
dω
−π
−M 1
2
Minimum E is when h1 [n] = h[n].
Proof: From Parseval: E =
h1[n]
P M2
−M
2
2
|h[n] − h1 [n]| +
P
|n|> M
2
|h[n]|
2
1
M=14
M=14
0.5
0
DSP and Digital Filters (2016-8872)
0
0
1
ω
2
3
Windows: 6 – 3 / 12
Rectangular window
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncate to ± M
2 to make finite; h1 [n] is now of length M + 1
MSE Optimality:
Define mean square error (MSE) in frequency domain
E=
=
1
2π
1
2π
Rπ H(ejω ) − H1 (ejω )2 dω
−π
2
M
R π P
−jωn jω
2
h
[n]e
H(e
)
−
dω
−π
−M 1
2
Minimum E is when h1 [n] = h[n].
Proof: From Parseval: E =
P M2
−M
2
2
|h[n] − h1 [n]| +
P
|n|> M
2
|h[n]|
2
However: 9% overshoot at a discontinuity even for large n.
h1[n]
1
M=14
M=14
0.5
0
DSP and Digital Filters (2016-8872)
0
0
1
ω
2
3
Windows: 6 – 3 / 12
Rectangular window
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncate to ± M
2 to make finite; h1 [n] is now of length M + 1
MSE Optimality:
Define mean square error (MSE) in frequency domain
E=
=
1
2π
1
2π
Rπ H(ejω ) − H1 (ejω )2 dω
−π
2
M
R π P
−jωn jω
2
h
[n]e
H(e
)
−
dω
−π
−M 1
2
Minimum E is when h1 [n] = h[n].
Proof: From Parseval: E =
P M2
−M
2
2
|h[n] − h1 [n]| +
P
|n|> M
2
|h[n]|
2
However: 9% overshoot at a discontinuity even for large n.
h1[n]
1
M=14
M=28
M=14
0.5
0
DSP and Digital Filters (2016-8872)
0
0
1
ω
2
3
Windows: 6 – 3 / 12
Rectangular window
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncate to ± M
2 to make finite; h1 [n] is now of length M + 1
MSE Optimality:
Define mean square error (MSE) in frequency domain
E=
=
1
2π
1
2π
Rπ H(ejω ) − H1 (ejω )2 dω
−π
2
M
R π P
−jωn jω
2
h
[n]e
H(e
)
−
dω
−π
−M 1
2
Minimum E is when h1 [n] = h[n].
Proof: From Parseval: E =
P M2
−M
2
2
|h[n] − h1 [n]| +
P
|n|> M
2
|h[n]|
2
However: 9% overshoot at a discontinuity even for large n.
h2[n]
1
M=14
M=28
M=14
0.5
0
0
0
Normal to delay by
DSP and Digital Filters (2016-8872)
M
2
1
ω
jω
to make causal. Multiplies H(e
2
3
−j M
2 ω
) by e
.
Windows: 6 – 3 / 12
Dirichlet Kernel
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncation ⇔ Multiply h[n] by a rectangular window, w[n] = δ− M ≤n≤ M
2
2
DSP and Digital Filters (2016-8872)
Windows: 6 – 4 / 12
Dirichlet Kernel
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncation ⇔ Multiply h[n] by a rectangular window, w[n] = δ− M ≤n≤ M
2
2
1
jω
jω
jω
⇔ Circular Convolution HM +1 (e ) = 2π H(e ) ⊛ W (e )
DSP and Digital Filters (2016-8872)
Windows: 6 – 4 / 12
Dirichlet Kernel
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncation ⇔ Multiply h[n] by a rectangular window, w[n] = δ− M ≤n≤ M
2
2
1
jω
jω
jω
⇔ Circular Convolution HM +1 (e ) = 2π H(e ) ⊛ W (e )
jω
W (e ) =
DSP and Digital Filters (2016-8872)
P M2
−M
2
e−jωn
Windows: 6 – 4 / 12
Dirichlet Kernel
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncation ⇔ Multiply h[n] by a rectangular window, w[n] = δ− M ≤n≤ M
2
2
1
jω
jω
jω
⇔ Circular Convolution HM +1 (e ) = 2π H(e ) ⊛ W (e )
jω
W (e ) =
DSP and Digital Filters (2016-8872)
P M2
−M
2
(i)
e−jωn = 1 + 2
P0.5M
1
cos (nω)
Windows: 6 – 4 / 12
Dirichlet Kernel
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncation ⇔ Multiply h[n] by a rectangular window, w[n] = δ− M ≤n≤ M
2
2
1
jω
jω
jω
⇔ Circular Convolution HM +1 (e ) = 2π H(e ) ⊛ W (e )
jω
W (e ) =
−M
2
−jω(−n)
Proof: (i) e
DSP and Digital Filters (2016-8872)
P M2
(i)
e−jωn = 1 + 2
P0.5M
1
cos (nω)
+ e−jω(+n) = 2 cos (nω)
Windows: 6 – 4 / 12
Dirichlet Kernel
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncation ⇔ Multiply h[n] by a rectangular window, w[n] = δ− M ≤n≤ M
2
2
1
jω
jω
jω
⇔ Circular Convolution HM +1 (e ) = 2π H(e ) ⊛ W (e )
jω
W (e ) =
−M
2
−jω(−n)
Proof: (i) e
DSP and Digital Filters (2016-8872)
P M2
(i)
e−jωn = 1 + 2
P0.5M
1
(ii) sin 0.5(M +1)ω
cos (nω)=
sin 0.5ω
+ e−jω(+n) = 2 cos (nω)
Windows: 6 – 4 / 12
Dirichlet Kernel
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncation ⇔ Multiply h[n] by a rectangular window, w[n] = δ− M ≤n≤ M
2
2
1
jω
jω
jω
⇔ Circular Convolution HM +1 (e ) = 2π H(e ) ⊛ W (e )
jω
W (e ) =
−M
2
−jω(−n)
Proof: (i) e
DSP and Digital Filters (2016-8872)
P M2
(i)
e−jωn = 1 + 2
P0.5M
1
(ii) sin 0.5(M +1)ω
cos (nω)=
sin 0.5ω
+ e−jω(+n) = 2 cos (nω) (ii) Sum geom. progression
Windows: 6 – 4 / 12
Dirichlet Kernel
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncation ⇔ Multiply h[n] by a rectangular window, w[n] = δ− M ≤n≤ M
2
2
1
jω
jω
jω
⇔ Circular Convolution HM +1 (e ) = 2π H(e ) ⊛ W (e )
jω
W (e ) =
P M2
−M
2
−jω(−n)
Proof: (i) e
(i)
e−jωn = 1 + 2
P0.5M
1
(ii) sin 0.5(M +1)ω
cos (nω)=
sin 0.5ω
+ e−jω(+n) = 2 cos (nω) (ii) Sum geom. progression
Effect: convolve ideal freq response with Dirichlet kernel (aliassed sinc)
DSP and Digital Filters (2016-8872)
Windows: 6 – 4 / 12
Dirichlet Kernel
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncation ⇔ Multiply h[n] by a rectangular window, w[n] = δ− M ≤n≤ M
2
2
1
jω
jω
jω
⇔ Circular Convolution HM +1 (e ) = 2π H(e ) ⊛ W (e )
jω
W (e ) =
P M2
(i)
−M
2
−jω(−n)
Proof: (i) e
e−jωn = 1 + 2
P0.5M
1
(ii) sin 0.5(M +1)ω
cos (nω)=
sin 0.5ω
+ e−jω(+n) = 2 cos (nω) (ii) Sum geom. progression
Effect: convolve ideal freq response with Dirichlet kernel (aliassed sinc)
1
0.5
0
-2
1
0
ω
2
0
ω
2
M=14
0.5
0
-2
DSP and Digital Filters (2016-8872)
Windows: 6 – 4 / 12
Dirichlet Kernel
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncation ⇔ Multiply h[n] by a rectangular window, w[n] = δ− M ≤n≤ M
2
2
1
jω
jω
jω
⇔ Circular Convolution HM +1 (e ) = 2π H(e ) ⊛ W (e )
jω
W (e ) =
P M2
(i)
−M
2
−jω(−n)
Proof: (i) e
e−jωn = 1 + 2
P0.5M
1
(ii) sin 0.5(M +1)ω
cos (nω)=
sin 0.5ω
+ e−jω(+n) = 2 cos (nω) (ii) Sum geom. progression
Effect: convolve ideal freq response with Dirichlet kernel (aliassed sinc)
1
1
0.5
0.5
0
0
-2
1
0
ω
2
0
ω
2
4π/(M+1)
-2
0
ω
2
M=14
0.5
0
-2
DSP and Digital Filters (2016-8872)
Windows: 6 – 4 / 12
Dirichlet Kernel
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncation ⇔ Multiply h[n] by a rectangular window, w[n] = δ− M ≤n≤ M
2
2
1
jω
jω
jω
⇔ Circular Convolution HM +1 (e ) = 2π H(e ) ⊛ W (e )
jω
W (e ) =
P M2
(i)
−M
2
−jω(−n)
Proof: (i) e
e−jωn = 1 + 2
P0.5M
1
(ii) sin 0.5(M +1)ω
cos (nω)=
sin 0.5ω
+ e−jω(+n) = 2 cos (nω) (ii) Sum geom. progression
Effect: convolve ideal freq response with Dirichlet kernel (aliassed sinc)
1
1
0.5
0.5
0
0
-2
1
0
ω
2
0
ω
2
1
4π/(M+1)
0.5
0
-2
0
ω
2
-2
0
ω
2
M=14
0.5
0
-2
DSP and Digital Filters (2016-8872)
Windows: 6 – 4 / 12
Dirichlet Kernel
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncation ⇔ Multiply h[n] by a rectangular window, w[n] = δ− M ≤n≤ M
2
2
1
jω
jω
jω
⇔ Circular Convolution HM +1 (e ) = 2π H(e ) ⊛ W (e )
jω
W (e ) =
P M2
(i)
−M
2
−jω(−n)
Proof: (i) e
e−jωn = 1 + 2
P0.5M
1
(ii) sin 0.5(M +1)ω
cos (nω)=
sin 0.5ω
+ e−jω(+n) = 2 cos (nω) (ii) Sum geom. progression
Effect: convolve ideal freq response with Dirichlet kernel (aliassed sinc)
1
1
0.5
0.5
0
0
-2
1
0
ω
2
1
4π/(M+1)
0.5
0
-2
0
ω
2
-2
0
ω
2
2π
Provided that M4π
≪
2ω
⇔
M
+
1
≫
0
+1
ω0 :
M=14
0.5
0
-2
DSP and Digital Filters (2016-8872)
0
ω
2
Windows: 6 – 4 / 12
Dirichlet Kernel
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncation ⇔ Multiply h[n] by a rectangular window, w[n] = δ− M ≤n≤ M
2
2
1
jω
jω
jω
⇔ Circular Convolution HM +1 (e ) = 2π H(e ) ⊛ W (e )
jω
W (e ) =
P M2
(i)
−M
2
−jω(−n)
Proof: (i) e
e−jωn = 1 + 2
P0.5M
1
(ii) sin 0.5(M +1)ω
cos (nω)=
sin 0.5ω
+ e−jω(+n) = 2 cos (nω) (ii) Sum geom. progression
Effect: convolve ideal freq response with Dirichlet kernel (aliassed sinc)
1
1
0.5
0.5
0
0
-2
1
0
ω
2
1
4π/(M+1)
0.5
0
-2
0
ω
2
-2
0
ω
2
2π
Provided that M4π
≪
2ω
⇔
M
+
1
≫
0
+1
ω0 :
Passband ripple: ∆ω ≈ M4π
+1
M=14
0.5
0
-2
DSP and Digital Filters (2016-8872)
0
ω
2
Windows: 6 – 4 / 12
Dirichlet Kernel
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncation ⇔ Multiply h[n] by a rectangular window, w[n] = δ− M ≤n≤ M
2
2
1
jω
jω
jω
⇔ Circular Convolution HM +1 (e ) = 2π H(e ) ⊛ W (e )
jω
W (e ) =
P M2
(i)
−M
2
−jω(−n)
Proof: (i) e
e−jωn = 1 + 2
P0.5M
1
(ii) sin 0.5(M +1)ω
cos (nω)=
sin 0.5ω
+ e−jω(+n) = 2 cos (nω) (ii) Sum geom. progression
Effect: convolve ideal freq response with Dirichlet kernel (aliassed sinc)
1
1
0.5
0.5
0
0
-2
1
0
ω
2
1
4π/(M+1)
0.5
0
-2
0
ω
2
-2
0
ω
2
2π
Provided that M4π
≪
2ω
⇔
M
+
1
≫
0
+1
ω0 :
2π
Passband ripple: ∆ω ≈ M4π
+1 , stopband M +1
M=14
0.5
0
-2
DSP and Digital Filters (2016-8872)
0
ω
2
Windows: 6 – 4 / 12
Dirichlet Kernel
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncation ⇔ Multiply h[n] by a rectangular window, w[n] = δ− M ≤n≤ M
2
2
1
jω
jω
jω
⇔ Circular Convolution HM +1 (e ) = 2π H(e ) ⊛ W (e )
jω
W (e ) =
P M2
(i)
−M
2
−jω(−n)
Proof: (i) e
e−jωn = 1 + 2
P0.5M
1
(ii) sin 0.5(M +1)ω
cos (nω)=
sin 0.5ω
+ e−jω(+n) = 2 cos (nω) (ii) Sum geom. progression
Effect: convolve ideal freq response with Dirichlet kernel (aliassed sinc)
1
1
0.5
0.5
0
0
-2
1
0
ω
2
4π/(M+1)
0.5
0
-2
0
ω
2
-2
0
ω
2
2π
Provided that M4π
≪
2ω
⇔
M
+
1
≫
0
+1
ω0 :
2π
Passband ripple: ∆ω ≈ M4π
+1 , stopband M +1
M=14
0.5
Transition pk-to-pk: ∆ω ≈ M4π
+1
0
-2
DSP and Digital Filters (2016-8872)
1
0
ω
2
Windows: 6 – 4 / 12
Dirichlet Kernel
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Truncation ⇔ Multiply h[n] by a rectangular window, w[n] = δ− M ≤n≤ M
2
2
1
jω
jω
jω
⇔ Circular Convolution HM +1 (e ) = 2π H(e ) ⊛ W (e )
jω
W (e ) =
P M2
(i)
−M
2
−jω(−n)
Proof: (i) e
e−jωn = 1 + 2
P0.5M
1
(ii) sin 0.5(M +1)ω
cos (nω)=
sin 0.5ω
+ e−jω(+n) = 2 cos (nω) (ii) Sum geom. progression
Effect: convolve ideal freq response with Dirichlet kernel (aliassed sinc)
1
1
0.5
0.5
0
0
-2
1
0
ω
2
0.5
0
DSP and Digital Filters (2016-8872)
4π/(M+1)
0.5
0
-2
0
ω
2
-2
0
ω
2
2π
Provided that M4π
≪
2ω
⇔
M
+
1
≫
0
+1
ω0 :
2π
Passband ripple: ∆ω ≈ M4π
+1 , stopband M +1
M=14
-2
1
0
ω
2
Transition pk-to-pk: ∆ω ≈ M4π
+1
Transition Gradient:
d|H| dω ω=ω0
≈
M +1
2π
Windows: 6 – 4 / 12
Window relationships
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
When you multiply an impulse response by a window M + 1 long
DSP and Digital Filters (2016-8872)
HM +1 (ejω ) =
1
jω
2π H(e )
⊛ W (ejω )
Windows: 6 – 5 / 12
Window relationships
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
When you multiply an impulse response by a window M + 1 long
HM +1 (ejω ) =
1
jω
2π H(e )
⊛ W (ejω )
1
0.5
0
DSP and Digital Filters (2016-8872)
-2
0
ω
2
Windows: 6 – 5 / 12
Window relationships
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
When you multiply an impulse response by a window M + 1 long
HM +1 (ejω ) =
1
1
jω
2π H(e )
⊛ W (ejω )
20 M=20
10
0.5
0
0
DSP and Digital Filters (2016-8872)
-2
0
ω
2
-2
0
ω
2
Windows: 6 – 5 / 12
Window relationships
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
When you multiply an impulse response by a window M + 1 long
HM +1 (ejω ) =
1
1
jω
2π H(e )
⊛ W (ejω )
20 M=20
1
10
0.5
1
6: Window Filter Design
0.5
0
0
DSP and Digital Filters (2016-8872)
0
-2
0
ω
2
-2
0
ω
2
-2
0
ω
2
Windows: 6 – 5 / 12
Window relationships
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
When you multiply an impulse response by a window M + 1 long
HM +1 (ejω ) =
1
1
jω
2π H(e )
⊛ W (ejω )
20 M=20
1
10
0.5
1
6: Window Filter Design
0.5
0
0
0
-2
0
ω
2
-2
(a) passband gain ≈ w[0]; peak≈
DSP and Digital Filters (2016-8872)
0
ω
w[0]
2
2
+
-2
0.5
2π
R
0
ω
2
jω
W
(e
)dω
mainlobe
Windows: 6 – 5 / 12
Window relationships
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
When you multiply an impulse response by a window M + 1 long
HM +1 (ejω ) =
1
1
jω
2π H(e )
⊛ W (ejω )
20 M=20
1
10
0.5
1
6: Window Filter Design
0.5
0
0
0
-2
0
ω
2
-2
0
ω
w[0]
2
2
-2
0.5
2π
0
ω
2
jω
+
W
(e
)dω
(a) passband gain ≈ w[0]; peak≈
mainlobe
rectangular window: passband gain = 1; peak gain = 1.09
DSP and Digital Filters (2016-8872)
R
Windows: 6 – 5 / 12
Window relationships
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
When you multiply an impulse response by a window M + 1 long
HM +1 (ejω ) =
1
1
jω
2π H(e )
⊛ W (ejω )
20 M=20
1
10
0.5
1
6: Window Filter Design
0.5
0
0
0
-2
0
ω
2
-2
0
ω
w[0]
2
2
-2
0.5
2π
0
ω
2
jω
+
W
(e
)dω
(a) passband gain ≈ w[0]; peak≈
mainlobe
rectangular window: passband gain = 1; peak gain = 1.09
R
(b) transition bandwidth, ∆ω = width of the main lobe
transition amplitude, ∆H = integral of main lobe÷2π
DSP and Digital Filters (2016-8872)
Windows: 6 – 5 / 12
Window relationships
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
When you multiply an impulse response by a window M + 1 long
HM +1 (ejω ) =
1
1
jω
2π H(e )
⊛ W (ejω )
20 M=20
1
10
0.5
1
6: Window Filter Design
0.5
0
0
0
-2
0
ω
2
-2
0
ω
w[0]
2
2
-2
0.5
2π
0
ω
2
jω
+
W
(e
)dω
(a) passband gain ≈ w[0]; peak≈
mainlobe
rectangular window: passband gain = 1; peak gain = 1.09
R
(b) transition bandwidth, ∆ω = width of the main lobe
transition amplitude, ∆H = integral of main lobe÷2π
rectangular window: ∆ω = M4π
+1 , ∆H ≈ 1.18
DSP and Digital Filters (2016-8872)
Windows: 6 – 5 / 12
Window relationships
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
When you multiply an impulse response by a window M + 1 long
HM +1 (ejω ) =
1
1
jω
2π H(e )
⊛ W (ejω )
20 M=20
1
10
0.5
1
6: Window Filter Design
0.5
0
0
0
-2
0
ω
2
-2
0
ω
w[0]
2
2
-2
0.5
2π
0
ω
2
jω
+
W
(e
)dω
(a) passband gain ≈ w[0]; peak≈
mainlobe
rectangular window: passband gain = 1; peak gain = 1.09
R
(b) transition bandwidth, ∆ω = width of the main lobe
transition amplitude, ∆H = integral of main lobe÷2π
rectangular window: ∆ω = M4π
+1 , ∆H ≈ 1.18
(c) stopband gain is an integral over oscillating sidelobes of W (ejω )
DSP and Digital Filters (2016-8872)
Windows: 6 – 5 / 12
Window relationships
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
When you multiply an impulse response by a window M + 1 long
HM +1 (ejω ) =
1
1
jω
2π H(e )
⊛ W (ejω )
20 M=20
1
10
0.5
1
6: Window Filter Design
0.5
0
0
0
-2
0
ω
2
-2
0
ω
w[0]
2
2
-2
0.5
2π
0
ω
2
jω
+
W
(e
)dω
(a) passband gain ≈ w[0]; peak≈
mainlobe
rectangular window: passband gain = 1; peak gain = 1.09
R
(b) transition bandwidth, ∆ω = width of the main lobe
transition amplitude, ∆H = integral of main lobe÷2π
rectangular window: ∆ω = M4π
+1 , ∆H ≈ 1.18
(c) stopband gain is an integral over
oscillatingsidelobes of W
(ejω )
+1
rect window: min H(ejω ) = 0.09 ≪ min W (ejω ) = M
1.5π
DSP and Digital Filters (2016-8872)
Windows: 6 – 5 / 12
Window relationships
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
When you multiply an impulse response by a window M + 1 long
HM +1 (ejω ) =
1
1
jω
2π H(e )
⊛ W (ejω )
20 M=20
1
10
0.5
1
6: Window Filter Design
0.5
0
0
0
-2
0
ω
2
-2
0
ω
w[0]
2
2
-2
0.5
2π
0
ω
2
jω
+
W
(e
)dω
(a) passband gain ≈ w[0]; peak≈
mainlobe
rectangular window: passband gain = 1; peak gain = 1.09
R
(b) transition bandwidth, ∆ω = width of the main lobe
transition amplitude, ∆H = integral of main lobe÷2π
rectangular window: ∆ω = M4π
+1 , ∆H ≈ 1.18
(c) stopband gain is an integral over
oscillatingsidelobes of W
(ejω )
+1
rect window: min H(ejω ) = 0.09 ≪ min W (ejω ) = M
1.5π
(d) features narrower than the main lobe will be broadened and
attenuated
DSP and Digital Filters (2016-8872)
Windows: 6 – 5 / 12
Common Windows
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Rectangular: w[n] ≡ 1
DSP and Digital Filters (2016-8872)
0
-13 dB
6.27/(M+1)
-50
0
1
ω
2
3
Windows: 6 – 6 / 12
Common Windows
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Rectangular: w[n] ≡ 1
don’t use
DSP and Digital Filters (2016-8872)
0
-13 dB
6.27/(M+1)
-50
0
1
ω
2
3
Windows: 6 – 6 / 12
Common Windows
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Rectangular: w[n] ≡ 1
don’t use
0
-50
0
Hanning: 0.5 + 0.5c1
ck = cos
DSP and Digital Filters (2016-8872)
2πkn
M +1
-13 dB
6.27/(M+1)
0
1
ω
2
3
12.56/(M+1)
-31 dB
-50
0
1
ω
2
3
Windows: 6 – 6 / 12
Common Windows
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Rectangular: w[n] ≡ 1
don’t use
0
-50
0
Hanning: 0.5 + 0.5c1
ck = cos
2πkn
M +1
rapid sidelobe decay
DSP and Digital Filters (2016-8872)
-13 dB
6.27/(M+1)
0
1
ω
2
3
12.56/(M+1)
-31 dB
-50
0
1
ω
2
3
Windows: 6 – 6 / 12
Common Windows
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Rectangular: w[n] ≡ 1
don’t use
0
-50
0
Hanning: 0.5 + 0.5c1
ck = cos
2πkn
M +1
rapid sidelobe decay
Hamming: 0.54 + 0.46c1
0
1
ω
2
3
12.56/(M+1)
-31 dB
-50
0
0
1
ω
2
3
12.56/(M+1)
-40 dB
-50
0
DSP and Digital Filters (2016-8872)
-13 dB
6.27/(M+1)
1
ω
2
3
Windows: 6 – 6 / 12
Common Windows
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Rectangular: w[n] ≡ 1
don’t use
0
-50
0
Hanning: 0.5 + 0.5c1
ck = cos
2πkn
M +1
rapid sidelobe decay
Hamming: 0.54 + 0.46c1
best peak sidelobe
0
1
ω
2
3
12.56/(M+1)
-31 dB
-50
0
0
1
ω
2
3
12.56/(M+1)
-40 dB
-50
0
DSP and Digital Filters (2016-8872)
-13 dB
6.27/(M+1)
1
ω
2
3
Windows: 6 – 6 / 12
Common Windows
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Rectangular: w[n] ≡ 1
don’t use
0
-50
0
Hanning: 0.5 + 0.5c1
ck = cos
2πkn
M +1
rapid sidelobe decay
Hamming: 0.54 + 0.46c1
best peak sidelobe
0
0.42 + 0.5c1 + 0.08c2
ω
2
3
12.56/(M+1)
-50
0
0
1
ω
2
3
12.56/(M+1)
-40 dB
-50
1
0
ω
2
3
18.87/(M+1)
-50
0
DSP and Digital Filters (2016-8872)
1
-31 dB
0
Blackman-Harris 3-term:
-13 dB
6.27/(M+1)
-70 dB
1
ω
2
3
Windows: 6 – 6 / 12
Common Windows
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Rectangular: w[n] ≡ 1
don’t use
0
-50
0
Hanning: 0.5 + 0.5c1
ck = cos
2πkn
M +1
rapid sidelobe decay
Hamming: 0.54 + 0.46c1
best peak sidelobe
0
0.42 + 0.5c1 + 0.08c2
best peak sidelobe
DSP and Digital Filters (2016-8872)
1
ω
2
3
12.56/(M+1)
-31 dB
-50
0
0
1
ω
2
3
12.56/(M+1)
-40 dB
-50
0
Blackman-Harris 3-term:
-13 dB
6.27/(M+1)
1
0
ω
2
18.87/(M+1)
-50
0
3
-70 dB
1
ω
2
3
Windows: 6 – 6 / 12
Common Windows
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Rectangular: w[n] ≡ 1
don’t use
0
-50
0
Hanning: 0.5 + 0.5c1
ck = cos
2πkn
M +1
rapid sidelobe decay
Hamming: 0.54 + 0.46c1
best peak sidelobe
0
0.42 + 0.5c1 + 0.08c2
best peak sidelobe
Kaiser:
q
2
I0 β 1−( 2n
M )
I0 (β)
β controls width v sidelobes
ω
2
3
-31 dB
-50
0
0
1
ω
2
3
12.56/(M+1)
-40 dB
-50
1
0
ω
2
-70 dB
0
0
-50
3
18.87/(M+1)
-50
1
ω
2
3
13.25/(M+1)
-40 dB
β = 5.3
0
1
0
-50
ω
2
3
21.27/(M+1)
β = 9.5
0
DSP and Digital Filters (2016-8872)
1
12.56/(M+1)
0
Blackman-Harris 3-term:
-13 dB
6.27/(M+1)
-70 dB
1
ω
2
3
Windows: 6 – 6 / 12
Common Windows
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Rectangular: w[n] ≡ 1
don’t use
0
-50
0
Hanning: 0.5 + 0.5c1
ck = cos
2πkn
M +1
rapid sidelobe decay
Hamming: 0.54 + 0.46c1
best peak sidelobe
0
0.42 + 0.5c1 + 0.08c2
best peak sidelobe
Kaiser:
q
2
I0 β 1−( 2n
M )
I0 (β)
β controls width v sidelobes
Good compromise:
Width v sidelobe v decay
DSP and Digital Filters (2016-8872)
1
ω
2
3
12.56/(M+1)
-31 dB
-50
0
0
1
ω
2
3
12.56/(M+1)
-40 dB
-50
0
Blackman-Harris 3-term:
-13 dB
6.27/(M+1)
1
0
ω
2
18.87/(M+1)
-50
-70 dB
0
0
-50
1
ω
2
3
13.25/(M+1)
-40 dB
β = 5.3
0
1
0
-50
3
ω
2
3
21.27/(M+1)
β = 9.5
0
-70 dB
1
ω
2
3
Windows: 6 – 6 / 12
Uncertainty principle
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
CTFT uncertainty principle:
DSP and Digital Filters (2016-8872)
R
t2 |x(t)|2 dt
R
|x(t)|2 dt
21 R
ω 2 |X(jω)|2 dω
R
|X(jω)|2 dω
12
≥
1
2
Windows: 6 – 7 / 12
Uncertainty principle
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
CTFT uncertainty principle:
R
t2 |x(t)|2 dt
R
|x(t)|2 dt
21 R
ω 2 |X(jω)|2 dω
R
|X(jω)|2 dω
The first term measures the “width” of x(t) around t = 0.
DSP and Digital Filters (2016-8872)
12
≥
1
2
Windows: 6 – 7 / 12
Uncertainty principle
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
CTFT uncertainty principle:
R
t2 |x(t)|2 dt
R
|x(t)|2 dt
21 R
ω 2 |X(jω)|2 dω
R
|X(jω)|2 dω
The first term measures the “width” of x(t) around t = 0.
DSP and Digital Filters (2016-8872)
12
≥
1
2
2
It is like σ if |x(t)| was a zero-mean probability distribution.
Windows: 6 – 7 / 12
Uncertainty principle
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
CTFT uncertainty principle:
R
t2 |x(t)|2 dt
R
|x(t)|2 dt
21 R
ω 2 |X(jω)|2 dω
R
|X(jω)|2 dω
The first term measures the “width” of x(t) around t = 0.
12
≥
1
2
2
It is like σ if |x(t)| was a zero-mean probability distribution.
The second term is similarly the “width” of X(jω) in frequency.
DSP and Digital Filters (2016-8872)
Windows: 6 – 7 / 12
Uncertainty principle
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
CTFT uncertainty principle:
R
t2 |x(t)|2 dt
R
|x(t)|2 dt
21 R
ω 2 |X(jω)|2 dω
R
|X(jω)|2 dω
The first term measures the “width” of x(t) around t = 0.
12
≥
1
2
2
It is like σ if |x(t)| was a zero-mean probability distribution.
The second term is similarly the “width” of X(jω) in frequency.
A signal cannot be concentrated in both time and frequency.
DSP and Digital Filters (2016-8872)
Windows: 6 – 7 / 12
Uncertainty principle
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
CTFT uncertainty principle:
R
t2 |x(t)|2 dt
R
|x(t)|2 dt
21 R
ω 2 |X(jω)|2 dω
R
|X(jω)|2 dω
The first term measures the “width” of x(t) around t = 0.
12
≥
1
2
2
It is like σ if |x(t)| was a zero-mean probability distribution.
The second term is similarly the “width” of X(jω) in frequency.
A signal cannot be concentrated in both time and frequency.
So a short window cannot have a narrow main lobe.
DSP and Digital Filters (2016-8872)
Windows: 6 – 7 / 12
Uncertainty principle
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
CTFT uncertainty principle:
R
t2 |x(t)|2 dt
R
|x(t)|2 dt
21 R
ω 2 |X(jω)|2 dω
R
|X(jω)|2 dω
The first term measures the “width” of x(t) around t = 0.
12
≥
1
2
2
It is like σ if |x(t)| was a zero-mean probability distribution.
The second term is similarly the “width” of X(jω) in frequency.
A signal cannot be concentrated in both time and frequency.
So a short window cannot have a narrow main lobe.
Proof Outline:
R
R
2
2
Assume |x(t)| dt = 1⇒ |X(jω)| dω = 2π
DSP and Digital Filters (2016-8872)
[Parseval]
Windows: 6 – 7 / 12
Uncertainty principle
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
CTFT uncertainty principle:
R
t2 |x(t)|2 dt
R
|x(t)|2 dt
21 R
ω 2 |X(jω)|2 dω
R
|X(jω)|2 dω
The first term measures the “width” of x(t) around t = 0.
12
≥
1
2
2
It is like σ if |x(t)| was a zero-mean probability distribution.
The second term is similarly the “width” of X(jω) in frequency.
A signal cannot be concentrated in both time and frequency.
So a short window cannot have a narrow main lobe.
Proof Outline:
R
R
2
2
Assume |x(t)| dt = 1⇒ |X(jω)| dω = 2π
Set v(t) = dx
dt ⇒ V (jω) = jωX(jω) [by parts]
DSP and Digital Filters (2016-8872)
[Parseval]
Windows: 6 – 7 / 12
Uncertainty principle
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
CTFT uncertainty principle:
R
t2 |x(t)|2 dt
R
|x(t)|2 dt
21 R
ω 2 |X(jω)|2 dω
R
|X(jω)|2 dω
The first term measures the “width” of x(t) around t = 0.
12
≥
1
2
2
It is like σ if |x(t)| was a zero-mean probability distribution.
The second term is similarly the “width” of X(jω) in frequency.
A signal cannot be concentrated in both time and frequency.
So a short window cannot have a narrow main lobe.
Proof Outline:
R
R
2
2
Assume |x(t)| dt = 1⇒ |X(jω)| dω = 2π
Set v(t) = dx
dt ⇒ V (jω) = jωX(jω) [by parts]
Now
DSP and Digital Filters (2016-8872)
R
∞
dx
1
2
tx dt dt= 2 tx (t)t=−∞
−
R
1 2
2 x dt
[Parseval]
= − 12 [by parts]
Windows: 6 – 7 / 12
Uncertainty principle
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
CTFT uncertainty principle:
R
t2 |x(t)|2 dt
R
|x(t)|2 dt
21 R
ω 2 |X(jω)|2 dω
R
|X(jω)|2 dω
The first term measures the “width” of x(t) around t = 0.
12
≥
1
2
2
It is like σ if |x(t)| was a zero-mean probability distribution.
The second term is similarly the “width” of X(jω) in frequency.
A signal cannot be concentrated in both time and frequency.
So a short window cannot have a narrow main lobe.
Proof Outline:
R
R
2
2
Assume |x(t)| dt = 1⇒ |X(jω)| dω = 2π
Set v(t) = dx
dt ⇒ V (jω) = jωX(jω) [by parts]
∞
R 1 2
dx
1
2
Now tx dt dt= 2 tx (t) t=−∞ − 2 x dt = − 12
R 2 R
R
2
2 2
dx dt
≤
dt
t
x
dt
So 41 = tx dx
dt
dt
R
DSP and Digital Filters (2016-8872)
[Parseval]
[by parts]
[Schwartz]
Windows: 6 – 7 / 12
Uncertainty principle
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
CTFT uncertainty principle:
R
t2 |x(t)|2 dt
R
|x(t)|2 dt
21 R
ω 2 |X(jω)|2 dω
R
|X(jω)|2 dω
The first term measures the “width” of x(t) around t = 0.
12
≥
1
2
2
It is like σ if |x(t)| was a zero-mean probability distribution.
The second term is similarly the “width” of X(jω) in frequency.
A signal cannot be concentrated in both time and frequency.
So a short window cannot have a narrow main lobe.
Proof Outline:
R
R
2
2
Assume |x(t)| dt = 1⇒ |X(jω)| dω = 2π
Set v(t) = dx
dt ⇒ V (jω) = jωX(jω) [by parts]
∞
R 1 2
dx
1
2
Now tx dt dt= 2 tx (t) t=−∞ − 2 x dt = − 12
R 2 R
R
2
2 2
dx dt
≤
dt
t
x
dt
So 41 = tx dx
dt
dt
R
=
DSP and Digital Filters (2016-8872)
R
2 2
t x dt
R
2
|v(t)| dt
[Parseval]
[by parts]
[Schwartz]
Windows: 6 – 7 / 12
Uncertainty principle
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
CTFT uncertainty principle:
R
t2 |x(t)|2 dt
R
|x(t)|2 dt
21 R
ω 2 |X(jω)|2 dω
R
|X(jω)|2 dω
The first term measures the “width” of x(t) around t = 0.
12
≥
1
2
2
It is like σ if |x(t)| was a zero-mean probability distribution.
The second term is similarly the “width” of X(jω) in frequency.
A signal cannot be concentrated in both time and frequency.
So a short window cannot have a narrow main lobe.
Proof Outline:
R
R
2
2
Assume |x(t)| dt = 1⇒ |X(jω)| dω = 2π
Set v(t) = dx
dt ⇒ V (jω) = jωX(jω) [by parts]
∞
R 1 2
dx
1
2
Now tx dt dt= 2 tx (t) t=−∞ − 2 x dt = − 12
R 2 R
R
2
2 2
dx dt
≤
dt
t
x
dt
So 41 = tx dx
dt
dt
R
=
DSP and Digital Filters (2016-8872)
R
2 2
t x dt
R
[Parseval]
[by parts]
[Schwartz]
R 2 2 1 R
2
|v(t)| dt =
t x dt 2π |V (jω)| dω
2
Windows: 6 – 7 / 12
Uncertainty principle
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
CTFT uncertainty principle:
R
t2 |x(t)|2 dt
R
|x(t)|2 dt
21 R
ω 2 |X(jω)|2 dω
R
|X(jω)|2 dω
The first term measures the “width” of x(t) around t = 0.
12
≥
1
2
2
It is like σ if |x(t)| was a zero-mean probability distribution.
The second term is similarly the “width” of X(jω) in frequency.
A signal cannot be concentrated in both time and frequency.
So a short window cannot have a narrow main lobe.
Proof Outline:
R
R
2
2
Assume |x(t)| dt = 1⇒ |X(jω)| dω = 2π
Set v(t) = dx
dt ⇒ V (jω) = jωX(jω) [by parts]
∞
R 1 2
dx
1
2
Now tx dt dt= 2 tx (t) t=−∞ − 2 x dt = − 12
R 2 R
R
2
2 2
dx dt
≤
dt
t
x
dt
So 41 = tx dx
dt
dt
R
[Parseval]
[by parts]
[Schwartz]
R 2 2 1 R
2
=
t x dt
|v(t)| dt =
t x dt 2π |V (jω)| dω
R 2 2 1 R 2
2
=
t x dt 2π ω |X(jω)| dω
R
DSP and Digital Filters (2016-8872)
2 2
R
2
Windows: 6 – 7 / 12
Uncertainty principle
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
CTFT uncertainty principle:
R
t2 |x(t)|2 dt
R
|x(t)|2 dt
21 R
ω 2 |X(jω)|2 dω
R
|X(jω)|2 dω
The first term measures the “width” of x(t) around t = 0.
12
≥
1
2
2
It is like σ if |x(t)| was a zero-mean probability distribution.
The second term is similarly the “width” of X(jω) in frequency.
A signal cannot be concentrated in both time and frequency.
So a short window cannot have a narrow main lobe.
Proof Outline:
R
R
2
2
Assume |x(t)| dt = 1⇒ |X(jω)| dω = 2π
Set v(t) = dx
dt ⇒ V (jω) = jωX(jω) [by parts]
∞
R 1 2
dx
1
2
Now tx dt dt= 2 tx (t) t=−∞ − 2 x dt = − 12
R 2 R
R
2
2 2
dx dt
≤
dt
t
x
dt
So 41 = tx dx
dt
dt
R
[Parseval]
[by parts]
[Schwartz]
R 2 2 1 R
2
=
t x dt
|v(t)| dt =
t x dt 2π |V (jω)| dω
R 2 2 1 R 2
2
=
t x dt 2π ω |X(jω)| dω
R
2 2
R
2
No exact equivalent for DTFT/DFT but a similar effect is true
DSP and Digital Filters (2016-8872)
Windows: 6 – 7 / 12
Order Estimation
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Several formulae estimate the required order of a filter, M .
DSP and Digital Filters (2016-8872)
Windows: 6 – 8 / 12
Order Estimation
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Several formulae estimate the required order of a filter, M .
E.g. for lowpass filter
DSP and Digital Filters (2016-8872)
Windows: 6 – 8 / 12
Order Estimation
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Several formulae estimate the required order of a filter, M .
E.g. for lowpass filter
Estimated order is
M≈
DSP and Digital Filters (2016-8872)
−5.6−4.3 log10 (δǫ)
ω2 −ω1
Windows: 6 – 8 / 12
Order Estimation
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Several formulae estimate the required order of a filter, M .
E.g. for lowpass filter
Estimated order is
M≈
DSP and Digital Filters (2016-8872)
−5.6−4.3 log10 (δǫ)
ω2 −ω1
≈
−8−20 log10 ǫ
2.2∆ω
Windows: 6 – 8 / 12
Order Estimation
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Several formulae estimate the required order of a filter, M .
E.g. for lowpass filter
Estimated order is
M≈
−5.6−4.3 log10 (δǫ)
ω2 −ω1
≈
−8−20 log10 ǫ
2.2∆ω
Required M increases as either the
transition width, ω2 − ω1 , or the gain
tolerances δ and ǫ get smaller.
DSP and Digital Filters (2016-8872)
Windows: 6 – 8 / 12
Order Estimation
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Several formulae estimate the required order of a filter, M .
E.g. for lowpass filter
Estimated order is
M≈
−5.6−4.3 log10 (δǫ)
ω2 −ω1
≈
−8−20 log10 ǫ
2.2∆ω
Required M increases as either the
transition width, ω2 − ω1 , or the gain
tolerances δ and ǫ get smaller.
Example:
Transition band: f1 = 1.8 kHz, f2 = 2.0 kHz, fs = 12 kHz,.
DSP and Digital Filters (2016-8872)
Windows: 6 – 8 / 12
Order Estimation
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Several formulae estimate the required order of a filter, M .
E.g. for lowpass filter
Estimated order is
M≈
−5.6−4.3 log10 (δǫ)
ω2 −ω1
≈
−8−20 log10 ǫ
2.2∆ω
Required M increases as either the
transition width, ω2 − ω1 , or the gain
tolerances δ and ǫ get smaller.
Example:
Transition band: f1 = 1.8 kHz, f2 = 2.0 kHz, fs = 12 kHz,.
2πf2
1
ω1 = 2πf
=
0.943
,
ω
=
= 1.047
2
f
f
s
DSP and Digital Filters (2016-8872)
s
Windows: 6 – 8 / 12
Order Estimation
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Several formulae estimate the required order of a filter, M .
E.g. for lowpass filter
Estimated order is
M≈
−5.6−4.3 log10 (δǫ)
ω2 −ω1
≈
−8−20 log10 ǫ
2.2∆ω
Required M increases as either the
transition width, ω2 − ω1 , or the gain
tolerances δ and ǫ get smaller.
Example:
Transition band: f1 = 1.8 kHz, f2 = 2.0 kHz, fs = 12 kHz,.
2πf2
1
ω1 = 2πf
=
0.943
,
ω
=
= 1.047
2
f
f
s
s
Ripple: 20 log10 (1 + δ) = 0.1 dB, 20 log10 ǫ = −35 dB
DSP and Digital Filters (2016-8872)
Windows: 6 – 8 / 12
Order Estimation
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Several formulae estimate the required order of a filter, M .
E.g. for lowpass filter
Estimated order is
M≈
−5.6−4.3 log10 (δǫ)
ω2 −ω1
≈
−8−20 log10 ǫ
2.2∆ω
Required M increases as either the
transition width, ω2 − ω1 , or the gain
tolerances δ and ǫ get smaller.
Example:
Transition band: f1 = 1.8 kHz, f2 = 2.0 kHz, fs = 12 kHz,.
2πf2
1
ω1 = 2πf
=
0.943
,
ω
=
= 1.047
2
f
f
s
s
Ripple: 20 log10 (1 + δ) = 0.1 dB, 20 log10 ǫ = −35 dB
0.1
δ = 10 20 − 1 = 0.0116, ǫ = 10
DSP and Digital Filters (2016-8872)
−35
20
= 0.0178
Windows: 6 – 8 / 12
Order Estimation
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Several formulae estimate the required order of a filter, M .
E.g. for lowpass filter
Estimated order is
M≈
−5.6−4.3 log10 (δǫ)
ω2 −ω1
≈
−8−20 log10 ǫ
2.2∆ω
Required M increases as either the
transition width, ω2 − ω1 , or the gain
tolerances δ and ǫ get smaller.
Example:
Transition band: f1 = 1.8 kHz, f2 = 2.0 kHz, fs = 12 kHz,.
2πf2
1
ω1 = 2πf
=
0.943
,
ω
=
= 1.047
2
f
f
s
s
Ripple: 20 log10 (1 + δ) = 0.1 dB, 20 log10 ǫ = −35 dB
0.1
δ = 10 20 − 1 = 0.0116, ǫ = 10
M≈
DSP and Digital Filters (2016-8872)
−5.6−4.3 log10 (2×10−4 )
1.047−0.943
=
10.25
0.105
−35
20
= 0.0178
= 98
Windows: 6 – 8 / 12
Order Estimation
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Several formulae estimate the required order of a filter, M .
E.g. for lowpass filter
Estimated order is
M≈
−5.6−4.3 log10 (δǫ)
ω2 −ω1
≈
−8−20 log10 ǫ
2.2∆ω
Required M increases as either the
transition width, ω2 − ω1 , or the gain
tolerances δ and ǫ get smaller.
Example:
Transition band: f1 = 1.8 kHz, f2 = 2.0 kHz, fs = 12 kHz,.
2πf2
1
ω1 = 2πf
=
0.943
,
ω
=
= 1.047
2
f
f
s
s
Ripple: 20 log10 (1 + δ) = 0.1 dB, 20 log10 ǫ = −35 dB
0.1
δ = 10 20 − 1 = 0.0116, ǫ = 10
M≈
DSP and Digital Filters (2016-8872)
−5.6−4.3 log10 (2×10−4 )
1.047−0.943
=
10.25
0.105
−35
20
= 98
= 0.0178
35−8
or 2.2∆ω
= 117
Windows: 6 – 8 / 12
Order Estimation
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Several formulae estimate the required order of a filter, M .
E.g. for lowpass filter
Estimated order is
M≈
−5.6−4.3 log10 (δǫ)
ω2 −ω1
≈
−8−20 log10 ǫ
2.2∆ω
Required M increases as either the
transition width, ω2 − ω1 , or the gain
tolerances δ and ǫ get smaller.
Only approximate.
Example:
Transition band: f1 = 1.8 kHz, f2 = 2.0 kHz, fs = 12 kHz,.
2πf2
1
ω1 = 2πf
=
0.943
,
ω
=
= 1.047
2
f
f
s
s
Ripple: 20 log10 (1 + δ) = 0.1 dB, 20 log10 ǫ = −35 dB
0.1
δ = 10 20 − 1 = 0.0116, ǫ = 10
M≈
DSP and Digital Filters (2016-8872)
−5.6−4.3 log10 (2×10−4 )
1.047−0.943
=
10.25
0.105
−35
20
= 98
= 0.0178
35−8
or 2.2∆ω
= 117
Windows: 6 – 8 / 12
Example Design
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Specifications:
Bandpass: ω1 = 0.5, ω2 = 1
DSP and Digital Filters (2016-8872)
1
0.5
0
0
1
ω
2
3
Windows: 6 – 9 / 12
Example Design
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Specifications:
Bandpass: ω1 = 0.5, ω2 = 1
Transition bandwidth: ∆ω = 0.1
DSP and Digital Filters (2016-8872)
1
0.5
0
0
1
ω
2
3
Windows: 6 – 9 / 12
Example Design
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Specifications:
Bandpass: ω1 = 0.5, ω2 = 1
Transition bandwidth: ∆ω = 0.1
Ripple: δ = ǫ = 0.02
DSP and Digital Filters (2016-8872)
1
0.5
0
0
1
ω
2
3
Windows: 6 – 9 / 12
Example Design
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Specifications:
Bandpass: ω1 = 0.5, ω2 = 1
Transition bandwidth: ∆ω = 0.1
Ripple: δ = ǫ = 0.02
20 log10 ǫ = −34 dB
DSP and Digital Filters (2016-8872)
1
0.5
0
0
1
ω
2
3
Windows: 6 – 9 / 12
Example Design
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Specifications:
Bandpass: ω1 = 0.5, ω2 = 1
Transition bandwidth: ∆ω = 0.1
Ripple: δ = ǫ = 0.02
20 log10 ǫ = −34 dB
20 log10 (1 + δ) = 0.17 dB
DSP and Digital Filters (2016-8872)
1
0.5
0
0
1
ω
2
3
Windows: 6 – 9 / 12
Example Design
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Specifications:
Bandpass: ω1 = 0.5, ω2 = 1
Transition bandwidth: ∆ω = 0.1
Ripple: δ = ǫ = 0.02
20 log10 ǫ = −34 dB
20 log10 (1 + δ) = 0.17 dB
1
0.5
0
0
1
ω
2
3
Order:
M≈
DSP and Digital Filters (2016-8872)
−5.6−4.3 log10 (δǫ)
ω2 −ω1
= 92
Windows: 6 – 9 / 12
Example Design
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Specifications:
Bandpass: ω1 = 0.5, ω2 = 1
Transition bandwidth: ∆ω = 0.1
Ripple: δ = ǫ = 0.02
20 log10 ǫ = −34 dB
20 log10 (1 + δ) = 0.17 dB
1
0.5
0
0
1
ω
2
3
Order:
M≈
−5.6−4.3 log10 (δǫ)
ω2 −ω1
= 92
Ideal Impulse Response:
Difference of two lowpass filters
DSP and Digital Filters (2016-8872)
Windows: 6 – 9 / 12
Example Design
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Specifications:
Bandpass: ω1 = 0.5, ω2 = 1
Transition bandwidth: ∆ω = 0.1
Ripple: δ = ǫ = 0.02
20 log10 ǫ = −34 dB
20 log10 (1 + δ) = 0.17 dB
1
0.5
0
0
1
ω
2
3
Order:
M≈
−5.6−4.3 log10 (δǫ)
ω2 −ω1
= 92
Ideal Impulse Response:
Difference of two lowpass filters
h[n] =
DSP and Digital Filters (2016-8872)
sin ω2 n
πn
−
sin ω1 n
πn
Windows: 6 – 9 / 12
Example Design
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Specifications:
Bandpass: ω1 = 0.5, ω2 = 1
Transition bandwidth: ∆ω = 0.1
Ripple: δ = ǫ = 0.02
20 log10 ǫ = −34 dB
20 log10 (1 + δ) = 0.17 dB
1
0.5
0
0
1
ω
2
3
Order:
M≈
−5.6−4.3 log10 (δǫ)
ω2 −ω1
= 92
Ideal Impulse Response:
Difference of two lowpass filters
h[n] =
sin ω2 n
πn
−
sin ω1 n
πn
Kaiser Window: β = 2.5
DSP and Digital Filters (2016-8872)
Windows: 6 – 9 / 12
Example Design
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Specifications:
Bandpass: ω1 = 0.5, ω2 = 1
Transition bandwidth: ∆ω = 0.1
Ripple: δ = ǫ = 0.02
20 log10 ǫ = −34 dB
20 log10 (1 + δ) = 0.17 dB
1
0.5
0
0
1
ω
2
3
Order:
M≈
−5.6−4.3 log10 (δǫ)
ω2 −ω1
= 92
Ideal Impulse Response:
Difference of two lowpass filters
h[n] =
sin ω2 n
πn
sin ω1 n
πn
−
Kaiser Window: β = 2.5
M=92
0
DSP and Digital Filters (2016-8872)
Windows: 6 – 9 / 12
Example Design
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Specifications:
Bandpass: ω1 = 0.5, ω2 = 1
Transition bandwidth: ∆ω = 0.1
Ripple: δ = ǫ = 0.02
20 log10 ǫ = −34 dB
20 log10 (1 + δ) = 0.17 dB
1
0.5
0
0
1
ω
2
1
Order:
M≈
−5.6−4.3 log10 (δǫ)
ω2 −ω1
= 92
Ideal Impulse Response:
Difference of two lowpass filters
h[n] =
sin ω2 n
πn
sin ω1 n
πn
−
3
M=92
β = 2.5
0.5
0
0
1
ω
2
3
Kaiser Window: β = 2.5
M=92
0
DSP and Digital Filters (2016-8872)
Windows: 6 – 9 / 12
Example Design
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Specifications:
Bandpass: ω1 = 0.5, ω2 = 1
Transition bandwidth: ∆ω = 0.1
Ripple: δ = ǫ = 0.02
20 log10 ǫ = −34 dB
20 log10 (1 + δ) = 0.17 dB
1
0.5
0
0
1
ω
2
1
Order:
M≈
−5.6−4.3 log10 (δǫ)
ω2 −ω1
= 92
Ideal Impulse Response:
Difference of two lowpass filters
h[n] =
sin ω2 n
πn
sin ω1 n
πn
−
3
M=92
β = 2.5
0.5
0
0
1
ω
2
0
3
M=92
β = 2.5
Kaiser Window: β = 2.5
-20
M=92
-40
0
-60
0
DSP and Digital Filters (2016-8872)
1
ω
2
3
Windows: 6 – 9 / 12
Frequency sampling
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Take M + 1 uniform samples of H(ejω )
DSP and Digital Filters (2016-8872)
Windows: 6 – 10 / 12
Frequency sampling
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Take M + 1 uniform samples of H(ejω )
1
M+1=93
0.5
0
DSP and Digital Filters (2016-8872)
-2
0
ω
2
Windows: 6 – 10 / 12
Frequency sampling
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Take M + 1 uniform samples of H(ejω ); take IDFT to obtain h[n]
1
M+1=93
0.5
0
DSP and Digital Filters (2016-8872)
-2
0
ω
2
Windows: 6 – 10 / 12
Frequency sampling
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Take M + 1 uniform samples of H(ejω ); take IDFT to obtain h[n]
1
M+1=93
1
0.5
0
DSP and Digital Filters (2016-8872)
M+1=93
0.5
-2
0
ω
2
0
0
1
ω
2
3
Windows: 6 – 10 / 12
Frequency sampling
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Take M + 1 uniform samples of H(ejω ); take IDFT to obtain h[n]
Advantage:
exact match at sample points
1
M+1=93
1
0.5
0
DSP and Digital Filters (2016-8872)
M+1=93
0.5
-2
0
ω
2
0
0
1
ω
2
3
Windows: 6 – 10 / 12
Frequency sampling
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Take M + 1 uniform samples of H(ejω ); take IDFT to obtain h[n]
Advantage:
exact match at sample points
Disadvantage:
poor intermediate approximation if spectrum is varying rapidly
1
M+1=93
1
0.5
0
DSP and Digital Filters (2016-8872)
M+1=93
0.5
-2
0
ω
2
0
0
1
ω
2
3
Windows: 6 – 10 / 12
Frequency sampling
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Take M + 1 uniform samples of H(ejω ); take IDFT to obtain h[n]
Advantage:
exact match at sample points
Disadvantage:
poor intermediate approximation if spectrum is varying rapidly
Solutions:
(1) make the filter transitions smooth over ∆ω width
1
M+1=93
1
0.5
0
DSP and Digital Filters (2016-8872)
M+1=93
0.5
-2
0
ω
2
0
0
1
ω
2
3
Windows: 6 – 10 / 12
Frequency sampling
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Take M + 1 uniform samples of H(ejω ); take IDFT to obtain h[n]
Advantage:
exact match at sample points
Disadvantage:
poor intermediate approximation if spectrum is varying rapidly
Solutions:
(1) make the filter transitions smooth over ∆ω width
(2) oversample and do least squares fit (can’t use IDFT)
1
M+1=93
1
0.5
0
DSP and Digital Filters (2016-8872)
M+1=93
0.5
-2
0
ω
2
0
0
1
ω
2
3
Windows: 6 – 10 / 12
Frequency sampling
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
Take M + 1 uniform samples of H(ejω ); take IDFT to obtain h[n]
Advantage:
exact match at sample points
Disadvantage:
poor intermediate approximation if spectrum is varying rapidly
Solutions:
(1) make the filter transitions smooth over ∆ω width
(2) oversample and do least squares fit (can’t use IDFT)
(3) use non-uniform points with more near transition (can’t use IDFT)
1
M+1=93
1
0.5
0
DSP and Digital Filters (2016-8872)
M+1=93
0.5
-2
0
ω
2
0
0
1
ω
2
3
Windows: 6 – 10 / 12
Summary
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
• Make an FIR filter by windowing the IDTFT of the ideal response
DSP and Digital Filters (2016-8872)
Windows: 6 – 11 / 12
Summary
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
• Make an FIR filter by windowing the IDTFT of the ideal response
ω0 n
◦ Ideal lowpass has h[n] = sinπn
DSP and Digital Filters (2016-8872)
Windows: 6 – 11 / 12
Summary
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
• Make an FIR filter by windowing the IDTFT of the ideal response
ω0 n
◦ Ideal lowpass has h[n] = sinπn
◦ Add/subtract lowpass filters to make any piecewise constant
DSP and Digital Filters (2016-8872)
response
Windows: 6 – 11 / 12
Summary
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
• Make an FIR filter by windowing the IDTFT of the ideal response
ω0 n
◦ Ideal lowpass has h[n] = sinπn
◦ Add/subtract lowpass filters to make any piecewise constant
response
• Ideal filter response is ⊛ with the DTFT of the window
DSP and Digital Filters (2016-8872)
Windows: 6 – 11 / 12
Summary
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
• Make an FIR filter by windowing the IDTFT of the ideal response
ω0 n
◦ Ideal lowpass has h[n] = sinπn
◦ Add/subtract lowpass filters to make any piecewise constant
response
• Ideal filter response is ⊛ with the DTFT of the window
◦ Rectangular window (W (z) = Dirichlet kernel) has −13 dB
DSP and Digital Filters (2016-8872)
sidelobes and is always a bad idea
Windows: 6 – 11 / 12
Summary
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
• Make an FIR filter by windowing the IDTFT of the ideal response
ω0 n
◦ Ideal lowpass has h[n] = sinπn
◦ Add/subtract lowpass filters to make any piecewise constant
response
• Ideal filter response is ⊛ with the DTFT of the window
◦ Rectangular window (W (z) = Dirichlet kernel) has −13 dB
sidelobes and is always a bad idea
◦ Hamming, Blackman-Harris are good
DSP and Digital Filters (2016-8872)
Windows: 6 – 11 / 12
Summary
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
• Make an FIR filter by windowing the IDTFT of the ideal response
ω0 n
◦ Ideal lowpass has h[n] = sinπn
◦ Add/subtract lowpass filters to make any piecewise constant
response
• Ideal filter response is ⊛ with the DTFT of the window
◦ Rectangular window (W (z) = Dirichlet kernel) has −13 dB
sidelobes and is always a bad idea
◦ Hamming, Blackman-Harris are good
◦ Kaiser good with β trading off main lobe width v. sidelobes
DSP and Digital Filters (2016-8872)
Windows: 6 – 11 / 12
Summary
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
• Make an FIR filter by windowing the IDTFT of the ideal response
ω0 n
◦ Ideal lowpass has h[n] = sinπn
◦ Add/subtract lowpass filters to make any piecewise constant
response
• Ideal filter response is ⊛ with the DTFT of the window
◦ Rectangular window (W (z) = Dirichlet kernel) has −13 dB
sidelobes and is always a bad idea
◦ Hamming, Blackman-Harris are good
◦ Kaiser good with β trading off main lobe width v. sidelobes
• Uncertainty principle: cannot be concentrated in both time and
frequency
DSP and Digital Filters (2016-8872)
Windows: 6 – 11 / 12
Summary
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
• Make an FIR filter by windowing the IDTFT of the ideal response
ω0 n
◦ Ideal lowpass has h[n] = sinπn
◦ Add/subtract lowpass filters to make any piecewise constant
response
• Ideal filter response is ⊛ with the DTFT of the window
◦ Rectangular window (W (z) = Dirichlet kernel) has −13 dB
sidelobes and is always a bad idea
◦ Hamming, Blackman-Harris are good
◦ Kaiser good with β trading off main lobe width v. sidelobes
• Uncertainty principle: cannot be concentrated in both time and
frequency
• Frequency sampling: IDFT of uniform frequency samples: not so great
DSP and Digital Filters (2016-8872)
Windows: 6 – 11 / 12
Summary
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
• Make an FIR filter by windowing the IDTFT of the ideal response
ω0 n
◦ Ideal lowpass has h[n] = sinπn
◦ Add/subtract lowpass filters to make any piecewise constant
response
• Ideal filter response is ⊛ with the DTFT of the window
◦ Rectangular window (W (z) = Dirichlet kernel) has −13 dB
sidelobes and is always a bad idea
◦ Hamming, Blackman-Harris are good
◦ Kaiser good with β trading off main lobe width v. sidelobes
• Uncertainty principle: cannot be concentrated in both time and
frequency
• Frequency sampling: IDFT of uniform frequency samples: not so great
For further details see Mitra: 7, 10.
DSP and Digital Filters (2016-8872)
Windows: 6 – 11 / 12
MATLAB routines
6: Window Filter Design
• Inverse DTFT
• Rectangular window
• Dirichlet Kernel
• Window relationships
• Common Windows
• Uncertainty principle
• Order Estimation
• Example Design
• Frequency sampling
• Summary
• MATLAB routines
diric(x,n)
hanning
hamming
kaiser
kaiserord
DSP and Digital Filters (2016-8872)
0.5nx
Dirichlet kernel: sin
sin 0.5x
Window functions
(Note ’periodic’ option)
Estimate required filter order and β
Windows: 6 – 12 / 12