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PDS_Modulo5
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Digital Signal Processing
Module 5: Linear FiltersDigital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Module Overview:
Module 5.15.3: LTI systems and convolution; examples; stability
Module 5.4: Convolution theorem
Module 5.55.6: Ideal filters and their approximation
Module 5.75.8: Realizable filters; implementation
Module 5.95.10: Filter design
5 1
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Digital Signal Processing
Module 5.1: Linear FiltersDigital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Overview:
Linearity and time invariance
Convolution
5.1 2
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Overview:
Linearity and time invariance
Convolution
5.1 2
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
A generic signal processing device
x [n] H y [n]
y [n] = H{x [n]}
5.1 3
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Linearity
H{ x1[n] + x2[n]} = H{x1[n]}+ H{x2[n]}
5.1 4
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Linearity
5.1 5
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
(Non) Linearity
5.1 6
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Time invariance
y [n] = H{x [n]} H{x [n n0]} = y [n n0]
5.1 7
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Time invariance
5.1 8
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Time (in)variance
5.1 9
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Linear, time-invariant systems
x [n] H y [n]
additionscalarmultiplication
delays
5.1 10
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Linear, time-invariant systems
y [n] = H(x [n], x [n 1], x [n 2], . . . , y [n 1], y [n 2], . . .)
with H() a linear function of its arguments
5.1 11
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Impulse response
h[n] = H{[n]}
Fundamental result: impulse response fully characterizes the LTI system!
5.1 12
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Example
h[n]
b b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
3 0 3 6 9 12 150
1
h[n] = n u[n]
x [n]
b b
b
b
b
b b b
2 1 0 1 2 3 4 50
1
2
3
4
x [n] =
2 n = 0
3 n = 1
1 n = 2
0 otherwise
5.1 13
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Example
x [n] = 2[n] + 3[n 1] + [n 2]
we know the impulse response h[n] = H{[n]};
compute y [n] = H{x [n]} exploiting linearity and time-invariance
5.1 14
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Example
x [n] = 2[n] + 3[n 1] + [n 2]
we know the impulse response h[n] = H{[n]};
compute y [n] = H{x [n]} exploiting linearity and time-invariance
5.1 14
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Example
x [n] = 2[n] + 3[n 1] + [n 2]
we know the impulse response h[n] = H{[n]};
compute y [n] = H{x [n]} exploiting linearity and time-invariance
5.1 14
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Example
y [n] = H{2[n] + 3[n 1] + [n 2]}
= 2H{[n]} + 3H{[n 1]}+H{[n 2]}
= 2h[n] + 3h[n 1] + h[n 2]
5.1 15
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Example
y [n] = H{2[n] + 3[n 1] + [n 2]}
= 2H{[n]} + 3H{[n 1]}+H{[n 2]}
= 2h[n] + 3h[n 1] + h[n 2]
5.1 15
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Example
y [n] = H{2[n] + 3[n 1] + [n 2]}
= 2H{[n]} + 3H{[n 1]}+H{[n 2]}
= 2h[n] + 3h[n 1] + h[n 2]
5.1 15
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Example
b b b b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b b b b
2h[n]
5 0 5 10 15 20 250
1
2
3
4
5
5.1 16
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Example
b b b b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b b b b
b b b b b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b
3h[n 1]
5 0 5 10 15 20 250
1
2
3
4
5
5.1 16
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Example
b b b b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b b b b
b b b b b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b
b b b b b b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b b b b b
h[n 2]
5 0 5 10 15 20 250
1
2
3
4
5
5.1 16
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Example
b b b b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
y [n]
5 0 5 10 15 20 250
1
2
3
4
5
5.1 16
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution
We can always write (remember Module 3.2):
x [n] =
k=
x [k][n k]
by linearity and time invariance:
y [n] =
k=
x [k]h[n k]
= x [n] h[n]
5.1 17
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution
We can always write (remember Module 3.2):
x [n] =
k=
x [k][n k]
by linearity and time invariance:
y [n] =
k=
x [k]h[n k]
= x [n] h[n]
5.1 17
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Performing the convolution algorithmically
x [n] h[n] =
k=
x [k]h[n k]
Ingredients:
a sequence x [m]
a second sequence h[m]
The recipe:
time-reverse h[m]
at each step n (from to ):
center the time-reversed h[m] in n(i.e. shift by n)
compute the inner product
5.1 18
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Performing the convolution algorithmically
x [n] h[n] =
k=
x [k]h[n k]
Ingredients:
a sequence x [m]
a second sequence h[m]
The recipe:
time-reverse h[m]
at each step n (from to ):
center the time-reversed h[m] in n(i.e. shift by n)
compute the inner product
5.1 18
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Performing the convolution algorithmically
x [n] h[n] =
k=
x [k]h[n k]
Ingredients:
a sequence x [m]
a second sequence h[m]
The recipe:
time-reverse h[m]
at each step n (from to ):
center the time-reversed h[m] in n(i.e. shift by n)
compute the inner product
5.1 18
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Same example, different perspective
h[n]
b b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
3 0 3 6 9 12 150
1
h[n] = n u[n]
x [n]
b b
b
b
b
b b b
2 1 0 1 2 3 4 50
1
2
3
4
x [n] =
2 n = 0
3 n = 1
1 n = 2
0 otherwise
5.1 19
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b
b
b
b
b
b
b b b b b b b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b
b
b
b
b
b
b
b b b b b b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b
b
b
b
b
b
b
b
b b b b b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b
b
b
b
b
b
b
b
b
b b b b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b
b
b
b
b
b
b
b
b
b
b b b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b b
b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b
b
b
b
b
b
b
b
b
b
b
b b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b b
b
b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b
b
b
b
b
b
b
b
b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b b
b
b
b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b
b
b
b
b
b
b
b
b
b
b
b
b
b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b b
b
b
b
b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b b
b
b
b
b
b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b b
b
b
b
b
b
b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b b
b
b
b
b
b
b
b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b b
b
b
b
b
b
b
b
b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b b b b b b b b b b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b b
b
b
b
b
b
b
b
b
b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b b b b b b b b b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b b
b
b
b
b
b
b
b
b
b
b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b b b b b b b b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b b
b
b
b
b
b
b
b
b
b
b
b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b b b b b b b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b b
b
b
b
b
b
b
b
b
b
b
b
b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b b
b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b b b b b b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b b
b
b
b
b
b
b
b
b
b
b
b
b
b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b b b
b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b b b b b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b b b
b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b b b b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b b b
b b
b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b b b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b b b b
b b
b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b b b b
b b b
b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b b b b b
b b b
b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b b b b b b
b b b
b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution example
b b b b b b b b b
b
b
b
b b b b b b b b b b b b b b b b b b
8 4 0 4 8 12 16 200
2
4x[k]
b b b b b b
b b b
b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
8 4 0 4 8 12 16 200
1
h[n
k]
b b b b b b b b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b
8 4 0 4 8 12 16 200246
y[n]
5.1 20
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution properties
linearity and time invariance (by definition)
commutativity: (x h)[n] = (h x)[n]
associativity for absolutely- and square-summable sequences:((x h) w)[n] = (x (h w))[n]
5.1 21
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution properties
linearity and time invariance (by definition)
commutativity: (x h)[n] = (h x)[n]
associativity for absolutely- and square-summable sequences:((x h) w)[n] = (x (h w))[n]
5.1 21
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Convolution properties
linearity and time invariance (by definition)
commutativity: (x h)[n] = (h x)[n]
associativity for absolutely- and square-summable sequences:((x h) w)[n] = (x (h w))[n]
x [n] h[n] w [n] y [n]
x [n] (h w)[n] y [n]
5.1 21
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
END OF MODULE 5.1
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Digital Signal Processing
Module 5.2: Filtering by exampleDigital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Overview:
Moving average filter
Leaky integrator
5.2 22
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Overview:
Moving average filter
Leaky integrator
5.2 22
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Typical filtering scenario: denoising
0 100 200 300 400 500
6420246
5.2 23
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Typical filtering scenario: denoising
0 100 200 300 400 500
6420246
0 100 200 300 400 500
6
0
6
5.2 23
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Typical filtering scenario: denoising
0 100 200 300 400 500
6420246
0 100 200 300 400 500
6
0
6
0 100 200 300 400 500
6420246
5.2 23
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Denoising by Moving Average
idea: replace each sample by the local average
for instance: y [n] = (x [n] + x [n 1])/2)
more generally:
y [n] =1
M
M1k=0
x [n k]
5.2 24
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Denoising by Moving Average
idea: replace each sample by the local average
for instance: y [n] = (x [n] + x [n 1])/2)
more generally:
y [n] =1
M
M1k=0
x [n k]
5.2 24
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Denoising by Moving Average
idea: replace each sample by the local average
for instance: y [n] = (x [n] + x [n 1])/2)
more generally:
y [n] =1
M
M1k=0
x [n k]
5.2 24
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Denoising by Moving Average
M = 2
0 100 200 300 400 500
6
4
2
0
2
4
6
5.2 25
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Denoising by Moving Average
M = 4
0 100 200 300 400 500
6
4
2
0
2
4
6
5.2 25
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Denoising by Moving Average
M = 12
0 100 200 300 400 500
6
4
2
0
2
4
6
5.2 25
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Denoising by Moving Average
M = 100
0 100 200 300 400 500
6
4
2
0
2
4
6
5.2 25
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
MA: impulse response
h[n] =1
M
M1k=0
[n k]
=
{1/M for 0 n < M
0 otherwise
5.2 26
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
MA: impulse response
h[n] =1
M
M1k=0
[n k]
=
{1/M for 0 n < M
0 otherwise
5.2 26
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
MA: impulse response
b b b b b b b b b b b b b b b b b b b b
b b b b b b b b b
b b b b b b b b b b b b
0 M 10
1/M
5.2 27
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
MA: analysis
smoothing effect proportional to M
number of operations and storage also proportional to M
5.2 28
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
From the MA to a first-order recursion
yM [n] =1
M(x [n] + x [n 1] + x [n 2] + . . .+ x [n M + 1])
moving average over M points
5.2 29
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
From the MA to a first-order recursion
yM [n] =1
Mx [n] +
1
M(x [n 1] + x [n 2] + . . .+ x [n M + 1])
5.2 30
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
From the MA to a first-order recursion
yM [n] =1
Mx [n] +
1
M(x [n 1] + x [n 2] + . . .+ x [n M + 1])
almostyM1[n 1]
i.e., moving average over M 1 points, delayed by one
5.2 30
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
From the MA to a first-order recursion
Formally:
yM [n] =1
M
M1k=0
x [n k]
yM [n 1] =1
M
M1k=0
x [(n 1) k]
5.2 31
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
From the MA to a first-order recursion
Formally:
yM [n] =1
M
M1k=0
x [n k]
yM [n 1] =1
M
M1k=0
x [(n 1) k]
5.2 31
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
From the MA to a first-order recursion
Formally:
yM [n] =1
M
M1k=0
x [n k]
yM [n 1] =1
M
M1k=0
x [n (k + 1)]
5.2 31
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
From the MA to a first-order recursion
Formally:
yM [n] =1
M
M1k=0
x [n k]
yM [n 1] =1
M
M1+1k=0+1
x [n k]
5.2 31
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
From the MA to a first-order recursion
Formally:
yM [n] =1
M
M1k=0
x [n k]
yM [n 1] =1
M
Mk=1
x [n k]
5.2 31
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
From the MA to a first-order recursion
Formally:
yM [n] =1
M
M1k=0
x [n k]
yM [n 1] =1
M
Mk=1
x [n k]
yM1[n] =1
M 1
M2k=0
x [n k]
5.2 31
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
From the MA to a first-order recursion
Formally:
yM [n] =1
M
M1k=0
x [n k]
yM [n 1] =1
M
Mk=1
x [n k]
yM1[n 1] =1
M 1
M1k=1
x [n k]
5.2 31
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
From the MA to a first-order recursion
yM [n] =1
M
M1k=0
x [n k] yM1[n 1] =1
M 1
M1k=1
x [n k]
M1k=0
x [n k] = x [n] +
M1k=1
x [n k]
MyM [n] = x [n] + (M 1)yM1[n 1]
5.2 32
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
From the MA to a first-order recursion
yM [n] =1
M
M1k=0
x [n k] yM1[n 1] =1
M 1
M1k=1
x [n k]
M1k=0
x [n k] = x [n] +
M1k=1
x [n k]
MyM [n] = x [n] + (M 1)yM1[n 1]
5.2 32
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
From the MA to a first-order recursion
yM [n] =1
M
M1k=0
x [n k] yM1[n 1] =1
M 1
M1k=1
x [n k]
M1k=0
x [n k] = x [n] +
M1k=1
x [n k]
MyM [n] = x [n] + (M 1)yM1[n 1]
5.2 32
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
From the MA to a first-order recursion
yM [n] =1
M
M1k=0
x [n k] yM1[n 1] =1
M 1
M1k=1
x [n k]
M1k=0
x [n k] = x [n] +
M1k=1
x [n k]
MyM [n] = x [n] + (M 1)yM1[n 1]
5.2 32
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
From the MA to a first-order recursion
yM [n] =M 1
MyM1[n 1] +
1
Mx [n]
yM [n] = yM1[n 1] + (1 )x [n], =M 1
M
5.2 33
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
From the MA to a first-order recursion
yM [n] =M 1
MyM1[n 1] +
1
Mx [n]
yM [n] = yM1[n 1] + (1 )x [n], =M 1
M
5.2 33
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
The Leaky Integrator
when M is large, yM1[n] yM [n] (and 1)
try the filtery [n] = y [n 1] + (1 )x [n]
filter is now recursive, since it uses its previous output value
5.2 34
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
The Leaky Integrator
when M is large, yM1[n] yM [n] (and 1)
try the filtery [n] = y [n 1] + (1 )x [n]
filter is now recursive, since it uses its previous output value
5.2 34
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
The Leaky Integrator
when M is large, yM1[n] yM [n] (and 1)
try the filtery [n] = y [n 1] + (1 )x [n]
filter is now recursive, since it uses its previous output value
5.2 34
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Denoising recursively with the Leaky Integrator
= 0.2
0 100 200 300 400 500
6
4
2
0
2
4
6
5.2 35
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Denoising recursively with the Leaky Integrator
= 0.5
0 100 200 300 400 500
6
4
2
0
2
4
6
5.2 35
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Denoising recursively with the Leaky Integrator
= 0.8
0 100 200 300 400 500
6
4
2
0
2
4
6
5.2 35
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Denoising recursively with the Leaky Integrator
= 0.98
0 100 200 300 400 500
6
4
2
0
2
4
6
5.2 35
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
What about the impulse response?
y [n] = y [n 1] + (1 )[n]
y [n] = 0 for all n < 0
y [0] = y [1] + (1 )[0] = (1 )
y [1] = y [0] + (1 )[1] = (1 )
y [2] = y [1] + (1 )[2] = 2(1 )
y [3] = y [2] + (1 )[3] = 3(1 )
. . .
5.2 36
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
What about the impulse response?
y [n] = y [n 1] + (1 )[n]
y [n] = 0 for all n < 0
y [0] = y [1] + (1 )[0] = (1 )
y [1] = y [0] + (1 )[1] = (1 )
y [2] = y [1] + (1 )[2] = 2(1 )
y [3] = y [2] + (1 )[3] = 3(1 )
. . .
5.2 36
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
What about the impulse response?
y [n] = y [n 1] + (1 )[n]
y [n] = 0 for all n < 0
y [0] = y [1] + (1 )[0] = (1 )
y [1] = y [0] + (1 )[1] = (1 )
y [2] = y [1] + (1 )[2] = 2(1 )
y [3] = y [2] + (1 )[3] = 3(1 )
. . .
5.2 36
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
What about the impulse response?
y [n] = y [n 1] + (1 )[n]
y [n] = 0 for all n < 0
y [0] = y [1] + (1 )[0] = (1 )
y [1] = y [0] + (1 )[1] = (1 )
y [2] = y [1] + (1 )[2] = 2(1 )
y [3] = y [2] + (1 )[3] = 3(1 )
. . .
5.2 36
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
What about the impulse response?
y [n] = y [n 1] + (1 )[n]
y [n] = 0 for all n < 0
y [0] = y [1] + (1 )[0] = (1 )
y [1] = y [0] + (1 )[1] = (1 )
y [2] = y [1] + (1 )[2] = 2(1 )
y [3] = y [2] + (1 )[3] = 3(1 )
. . .
5.2 36
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
What about the impulse response?
y [n] = y [n 1] + (1 )[n]
y [n] = 0 for all n < 0
y [0] = y [1] + (1 )[0] = (1 )
y [1] = y [0] + (1 )[1] = (1 )
y [2] = y [1] + (1 )[2] = 2(1 )
y [3] = y [2] + (1 )[3] = 3(1 )
. . .
5.2 36
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Impulse response
h[n] = (1 )n u[n]
b b b b b b b b b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
1
0 15 300
5.2 37
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Leaky Integrator: why the name
Discrete-time integrator is a boundless accumulator:
y [n] =
nk=
x [k]
We can rewrite the integrator as
y [n] = y [n 1] + x [n]
5.2 38
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Leaky Integrator: why the name
To prevent explosion pick < 1
y [n] = y [n 1] + (1 )x [n]
keep only a fraction ofthe accumulated valueso far and forget(leak) a fraction 1
add only a fraction 1 of the current value tothe accumulator
5.2 39
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
END OF MODULE 5.2
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Digital Signal Processing
Module 5.3: Filter StabilityDigital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Overview:
Filter classification in the time domain
Filter stability
5.3 40
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Overview:
Filter classification in the time domain
Filter stability
5.3 40
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Filter types according to impulse response
Finite Impulse Response (FIR)
Infinite Impulse Response (IIR)
causal
noncausal
5.3 41
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Filter types according to impulse response
Finite Impulse Response (FIR)
Infinite Impulse Response (IIR)
causal
noncausal
5.3 41
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Filter types according to impulse response
Finite Impulse Response (FIR)
Infinite Impulse Response (IIR)
causal
noncausal
5.3 41
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Filter types according to impulse response
Finite Impulse Response (FIR)
Infinite Impulse Response (IIR)
causal
noncausal
5.3 41
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
FIR
impulse response has finite support
only a finite number of samples are involved in the computation of each output sample
5.3 42
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
FIR (example)
Moving Average filter
b b b b b b b b b b b b b b b b b b b b
b b b b b b b b b
b b b b b b b b b b b b
M 1
1/M
15 0 150
5.3 43
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
IIR
impulse response has infinite support
a potentially infinite number of samples are involved in the computation of each outputsample
surprisingly, in many cases the computation can still be performed in a finite amount ofsteps
5.3 44
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
IIR (example)
Leaky Integrator
b b b b b b b b b b b b b b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b b
15 0 150
5.3 45
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Causal vs Noncausal
causal:
impulse response is zero for n < 0
only past samples (with respect to the present) are involved in the computation of eachoutput sample
causal filters can work on line since they only need the past
noncausal:
impulse response is nonzero for some (or all) n < 0
can still be implemented in a oine fashion (when all input data is available on storage, e.g.in Image Processing)
5.3 46
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Causal example
Moving Average filter
b b b b b b b b b b b b b b b b b b b b
b b b b b b
b b b b b b b b b b b b b b b
18 12 6 0 6 12 180
5.3 47
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Noncausal example
Zero-centered Moving Average filter
b b b b b b b b b b b b b b b b b b
b b b b b
b b b b b b b b b b b b b b b b b b
18 12 6 0 6 12 180
5.3 48
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Stability
key concept: avoid explosions if the input is nice
a nice signal is a bounded signal: |x [n]| < M for all n
Bounded-Input Bounded-Output (BIBO) stability: if the input is nice the output shouldbe nice
5.3 49
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Stability
key concept: avoid explosions if the input is nice
a nice signal is a bounded signal: |x [n]| < M for all n
Bounded-Input Bounded-Output (BIBO) stability: if the input is nice the output shouldbe nice
5.3 49
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Stability
key concept: avoid explosions if the input is nice
a nice signal is a bounded signal: |x [n]| < M for all n
Bounded-Input Bounded-Output (BIBO) stability: if the input is nice the output shouldbe nice
5.3 49
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Fundamental Stability Theorem
A filter is BIBO stable if and only if its impulse response is absolutely summable
5.3 50
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Proof ()
Hypotheses:
|x [n]| < M
n |h[n]| = L
Proof ()
Hypotheses:
|x [n]| < M
n |h[n]| = L
Proof ()
Hypotheses:
|x [n]| < M
n |h[n]| = L
Proof ()
Hypotheses:
|x [n]| < M
n |h[n]| = L
Proof ()
Hypotheses:
|x [n]| < M
n |h[n]| = L
Proof ()
Hypotheses:
|x [n]| < M
|y [n]| < P
Thesis:
h[n] absolutelysummable
Proof (by contradiction):
assume
n |h[n]| =
build x [n] =
{+1 if h[n] 0
1 if h[n] < 0
clearly, x [n] is bounded
however
y [0] = (xh)[0] =
k=
h[k]x [k] =
k=
|h[k]| =
5.3 52
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Proof ()
Hypotheses:
|x [n]| < M
|y [n]| < P
Thesis:
h[n] absolutelysummable
Proof (by contradiction):
assume
n |h[n]| =
build x [n] =
{+1 if h[n] 0
1 if h[n] < 0
clearly, x [n] is bounded
however
y [0] = (xh)[0] =
k=
h[k]x [k] =
k=
|h[k]| =
5.3 52
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Proof ()
Hypotheses:
|x [n]| < M
|y [n]| < P
Thesis:
h[n] absolutelysummable
Proof (by contradiction):
assume
n |h[n]| =
build x [n] =
{+1 if h[n] 0
1 if h[n] < 0
clearly, x [n] is bounded
however
y [0] = (xh)[0] =
k=
h[k]x [k] =
k=
|h[k]| =
5.3 52
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Proof ()
Hypotheses:
|x [n]| < M
|y [n]| < P
Thesis:
h[n] absolutelysummable
Proof (by contradiction):
assume
n |h[n]| =
build x [n] =
{+1 if h[n] 0
1 if h[n] < 0
clearly, x [n] is bounded
however
y [0] = (xh)[0] =
k=
h[k]x [k] =
k=
|h[k]| =
5.3 52
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Proof ()
Hypotheses:
|x [n]| < M
|y [n]| < P
Thesis:
h[n] absolutelysummable
Proof (by contradiction):
assume
n |h[n]| =
build x [n] =
{+1 if h[n] 0
1 if h[n] < 0
clearly, x [n] is bounded
however
y [0] = (xh)[0] =
k=
h[k]x [k] =
k=
|h[k]| =
5.3 52
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
The good news
FIR filters are always stable
5.3 53
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Checking the stability of IIRs
Lets check the Leaky Integrator:
n=
|h[n]| = |1 |
n=0
||n
= limn
|1 |1 ||n+1
1 ||
Checking the stability of IIRs
Lets check the Leaky Integrator:
n=
|h[n]| = |1 |
n=0
||n
= limn
|1 |1 ||n+1
1 ||
Checking the stability of IIRs
Lets check the Leaky Integrator:
n=
|h[n]| = |1 |
n=0
||n
= limn
|1 |1 ||n+1
1 ||
Checking the stability of IIRs
Lets check the Leaky Integrator:
n=
|h[n]| = |1 |
n=0
||n
= limn
|1 |1 ||n+1
1 ||
Checking the stability of IIRs
We will study indirect methods for filter stability later in this Module.
5.3 55
Digital Sig
nal Proces
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Paolo Pran
doni and M
artin Vett
erli
2013
END OF MODULE 5.3
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Digital Signal Processing
Module 5.4: Frequency ResponseDigital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Overview:
Eigensequences
Convolution theorem
Frequency and phase response
5.4 56
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Overview:
Eigensequences
Convolution theorem
Frequency and phase response
5.4 56
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Overview:
Eigensequences
Convolution theorem
Frequency and phase response
5.4 56
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
A remarkable result
e j0n H ?
5.4 57
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
A remarkable result
y [n] = e j0n h[n]
= h[n] e j0n
=
k=
h[k]e j0(nk)
= e j0n
k=
h[k]ej0k
= H(e j0) e j0n
5.4 58
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
A remarkable result
y [n] = e j0n h[n]
= h[n] e j0n
=
k=
h[k]e j0(nk)
= e j0n
k=
h[k]ej0k
= H(e j0) e j0n
5.4 58
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
A remarkable result
y [n] = e j0n h[n]
= h[n] e j0n
=
k=
h[k]e j0(nk)
= e j0n
k=
h[k]ej0k
= H(e j0) e j0n
5.4 58
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
A remarkable result
y [n] = e j0n h[n]
= h[n] e j0n
=
k=
h[k]e j0(nk)
= e j0n
k=
h[k]ej0k
= H(e j0) e j0n
5.4 58
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
A remarkable result
y [n] = e j0n h[n]
= h[n] e j0n
=
k=
h[k]e j0(nk)
= e j0n
k=
h[k]ej0k
= H(e j0) e j0n
5.4 58
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
A remarkable result
e j0n H H(e j0) e j0n
complex exponentials are eigensequences of LTI systems, i.e., linear filters cannot changethe frequency of sinusoids
DTFT of impulse response determines the frequency characteristic of a filter
5.4 59
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
A remarkable result
e j0n H H(e j0) e j0n
complex exponentials are eigensequences of LTI systems, i.e., linear filters cannot changethe frequency of sinusoids
DTFT of impulse response determines the frequency characteristic of a filter
5.4 59
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Magnitude and phase
If H(e j0) = Ae j, then
H{e j0n} = Ae j(0n+)
amplitude:amplification (A > 1)or attenuation (0 A < 1)
phase shift:delay ( < 0)or advancement ( > 0)
5.4 60
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
The convolution theorem
In general:
DTFT {x [n] h[n]} =?
Intuition: the DTFT reconstruction formula tells us how to build x [n]from a set of complex exponential basis functions. By linearity...
5.4 61
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
The convolution theorem
In general:
DTFT {x [n] h[n]} =?
Intuition: the DTFT reconstruction formula tells us how to build x [n]from a set of complex exponential basis functions. By linearity...
5.4 61
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
The convolution theorem
DTFT {x [n] h[n]} =
n=
(x h)[n]ejn
=
n=
k=
x [k]h[n k]ejn
=
n=
k=
x [k]h[n k]ej(nk)ejk
=
k=
x [k]ejk
n=
h[n k]ej(nk)
= H(e j)X (e j)
5.4 62
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
The convolution theorem
DTFT {x [n] h[n]} =
n=
(x h)[n]ejn
=
n=
k=
x [k]h[n k]ejn
=
n=
k=
x [k]h[n k]ej(nk)ejk
=
k=
x [k]ejk
n=
h[n k]ej(nk)
= H(e j)X (e j)
5.4 62
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
The convolution theorem
DTFT {x [n] h[n]} =
n=
(x h)[n]ejn
=
n=
k=
x [k]h[n k]ejn
=
n=
k=
x [k]h[n k]ej(nk)ejk
=
k=
x [k]ejk
n=
h[n k]ej(nk)
= H(e j)X (e j)
5.4 62
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
The convolution theorem
DTFT {x [n] h[n]} =
n=
(x h)[n]ejn
=
n=
k=
x [k]h[n k]ejn
=
n=
k=
x [k]h[n k]ej(nk)ejk
=
k=
x [k]ejk
n=
h[n k]ej(nk)
= H(e j)X (e j)
5.4 62
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
The convolution theorem
DTFT {x [n] h[n]} =
n=
(x h)[n]ejn
=
n=
k=
x [k]h[n k]ejn
=
n=
k=
x [k]h[n k]ej(nk)ejk
=
k=
x [k]ejk
n=
h[n k]ej(nk)
= H(e j)X (e j)
5.4 62
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Frequency response
H(e j) = DTFT {h[n]}
Two effects:
magnitude: amplification (|H(e j)| > 1) or attenuation (|H(e j)| < 1) of inputfrequencies
phase: overall delay and shape changes
5.4 63
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Frequency response
H(e j) = DTFT {h[n]}
Two effects:
magnitude: amplification (|H(e j)| > 1) or attenuation (|H(e j)| < 1) of inputfrequencies
phase: overall delay and shape changes
5.4 63
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Moving Average revisited
h[n] = (u[n] u[nM])/M
b b b b b b b b b b b b b b b b b b b b
b b b b b b b b b b
b b b b b b b b b b b
0 M 10
1/M
5.4 64
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Moving Average, magnitude response
|H(e j)| = 1M
sin(2 M)sin(2 ) (see Module 4.7)
M = 9
pi pi/2 0 pi/2 pi0
1
|H(e
j)|
5.4 65
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Moving Average, magnitude response
|H(e j)| = 1M
sin(2 M)sin(2 ) (see Module 4.7)
M = 20
pi pi/2 0 pi/2 pi0
1
|H(e
j)|
5.4 65
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Moving Average, magnitude response
|H(e j)| = 1M
sin(2 M)sin(2 ) (see Module 4.7)
M = 100
pi pi/2 0 pi/2 pi0
1
|H(e
j)|
5.4 65
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Denoising revisited
0 100 200 300 400 500
6420246
0 100 200 300 400 500
6
0
6
5.4 66
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Denoising revisited
0 100 200 300 400 500
6420246
0 100 200 300 400 500
6
0
6
pi pi/2 0 pi/2 pi0
1
pi pi/2 0 pi/2 pi0
1
5.4 66
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Denoising revisited
pi pi/2 0 pi/2 pi0
1
5.4 67
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Denoising revisited
pi pi/2 0 pi/2 pi0
1
5.4 67
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Denoising revisited
pi pi/2 0 pi/2 pi0
1
5.4 67
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Denoising revisited
pi pi/2 0 pi/2 pi0
1
5.4 67
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Denoising revisited
pi pi/2 0 pi/2 pi0
1
5.4 67
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Denoising revisited
pi pi/2 0 pi/2 pi0
1
5.4 67
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
By the way, remember the time-domain analysis...
0 100 200 300 400 500
6
4
2
0
2
4
6
5.4 68
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
By the way, remember the time-domain analysis...
M = 12
0 100 200 300 400 500
6
4
2
0
2
4
6
5.4 68
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
What about the phase?
Assume |H(e j)| = 1
zero phase: H(e j) = 0
linear phase: H(e j) = d
nonlinear phase
5.4 69
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
What about the phase?
Assume |H(e j)| = 1
zero phase: H(e j) = 0
linear phase: H(e j) = d
nonlinear phase
5.4 69
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
What about the phase?
Assume |H(e j)| = 1
zero phase: H(e j) = 0
linear phase: H(e j) = d
nonlinear phase
5.4 69
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Phase and signal shape
x [n] =1
2sin(0n) + cos(20n) 0 =
2
40
0 28 56 84 112 140 168
1
0
1
5.4 70
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Phase and signal shape: linear phase
x [n] =1
2sin(0n + 0) + cos(20n+ 20) 0 =
8
5
0 28 56 84 112 140 168
1
0
1
5.4 71
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Phase and signal shape: nonlinear phase
x [n] =1
2sin(0n) + cos(20n + 20)
0 28 56 84 112 140 168
1
0
1
5.4 72
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Linear phase
x [n] D x [n d ]
y [n] = x [n d ]
Y (e j) = ejd X (e j)
H(e j) = ejd
linear phase term
5.4 73
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Linear phase
x [n] D x [n d ]
y [n] = x [n d ]
Y (e j) = ejd X (e j)
H(e j) = ejd
linear phase term
5.4 73
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Linear phase
x [n] D x [n d ]
y [n] = x [n d ]
Y (e j) = ejd X (e j)
H(e j) = ejd
linear phase term
5.4 73
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Linear phase
x [n] D x [n d ]
y [n] = x [n d ]
Y (e j) = ejd X (e j)
H(e j) = ejd
linear phase term
5.4 73
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Linear phase
In general, if H(e j) = A(e j)ejd , with A(e j) R
x [n] A(e j) delay by d y [n]
5.4 74
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Moving Average is linear phase
H(e j) =1
M
sin(2M
)sin(2
) ej M12
5.4 75
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Leaky integrator revisited
h[n] = (1 )n u[n]
b b b b b b b b b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
1
0 15 300
5.4 76
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Leaky integrator revisited
H(e j) =1
1 ej(Module 4.4)
Finding magnitude and phase require a little algebra...
5.4 77
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Leaky integrator revisited
Recall from complex algebra:1
a + jb=
a jb
a2 + b2
so that if x = 1/(a + jb),
|x |2 =1
a2 + b2
x = tan1[b
a
]
5.4 78
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Leaky integrator revisited
H(e j) =1
(1 cos) j sin
so that:
|H(e j)|2 =(1 )2
1 2 cos + 2
H(e j) = tan1[
sin
1 cos
]
5.4 79
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Leaky integrator, magnitude response
= 0.9
pi pi/2 0 pi/2 pi0
1
|H(e
j)|
5.4 80
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Leaky integrator, magnitude response
= 0.95
pi pi/2 0 pi/2 pi0
1
|H(e
j)|
5.4 80
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Leaky integrator, magnitude response
= 0.98
pi pi/2 0 pi/2 pi0
1
|H(e
j)|
5.4 80
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Leaky integrator, phase response
= 0.9
pi/2
pi/2
pi pi/2 0 pi/2 pi
H(e
j)
5.4 81
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Leaky integrator, phase response
= 0.95
pi/2
pi/2
pi pi/2 0 pi/2 pi
H(e
j)
5.4 81
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Leaky integrator, phase response
= 0.98
pi/2
pi/2
pi pi/2 0 pi/2 pi
H(e
j)
5.4 81
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Phase is sufficiently linear where it matters
pi pi/2 0 pi/2 pi0
1|H
(ej)|
pi/2
pi/2
pi pi/2 0 pi/2 pi
H(e
j)
5.4 82
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Phase is sufficiently linear where it matters
pi pi/2 0 pi/2 pi0
1|H
(ej)|
pi/2
pi/2
pi pi/2 0 pi/2 pi
H(e
j)
5.4 82
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Karplus-Strong revisited, again!
x [n] + b y [n]
zM
y [n] = y [n M] + x [n]
5.4 83
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
KS revisited
y [n] = n/Mx [n mod M] u[n]
0 166 332 498 664 830 996
1
0
1
5.4 84
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Karplus-Strong revisited
y [n] = x [0], x [1], . . . , x [M 1], x [0], x [1], . . . , x [M 1], 2x [0], 2x [1], . . .
5.4 85
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Karplus-Strong revisited
y [n] = x [0], x [1], . . . , x [M 1], x [0], x [1], . . . , x [M 1], 2x [0], 2x [1], . . .
1st period 2nd period . . .
5.4 85
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
DTFT of KS signal, using the convolution theorem
key observation:
y [n] = x [n] w [n], w [n] =
{k for n = kM
0 otherwise
Y (e j) = X (e j)W (e j)
5.4 86
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
DTFT of KS signal, using the convolution theorem
key observation:
y [n] = x [n] w [n], w [n] =
{k for n = kM
0 otherwise
Y (e j) = X (e j)W (e j)
5.4 86
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
DTFT of KS signal, using the convolution theorem
X (e j) = ej(M + 1
M 1
)1 ej(M1)
(1 ej)2
1 ej(M+1)
(1 ej)2
W (e j) =1
1 ejM
5.4 87
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
DTFT of KS
|X (e j)|
pi pi/2 0 pi/2 pi
5.4 88
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
DTFT of KS
|W (e j)|
pi pi/2 0 pi/2 pi
5.4 88
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
DTFT of KS
|Y (e j)|
pi pi/2 0 pi/2 pi
5.4 88
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
END OF MODULE 5.4
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Digital Signal Processing
Module 5.5: Ideal FiltersDigital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Overview:
Filter classification in the frequency domain
Ideal filters
Demodulation revisited
5.5 89
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Overview:
Filter classification in the frequency domain
Ideal filters
Demodulation revisited
5.5 89
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Overview:
Filter classification in the frequency domain
Ideal filters
Demodulation revisited
5.5 89
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Filter types according to magnitude response
Lowpass
Highpass
Bandpass
Allpass
Moving Average and Leaky Integrator are lowpass filters
5.5 90
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Filter types according to magnitude response
Lowpass
Highpass
Bandpass
Allpass
Moving Average and Leaky Integrator are lowpass filters
5.5 90
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Filter types according to magnitude response
Lowpass
Highpass
Bandpass
Allpass
Moving Average and Leaky Integrator are lowpass filters
5.5 90
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Filter types according to magnitude response
Lowpass
Highpass
Bandpass
Allpass
Moving Average and Leaky Integrator are lowpass filters
5.5 90
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Filter types according to magnitude response
Lowpass
Highpass
Bandpass
Allpass
Moving Average and Leaky Integrator are lowpass filters
5.5 90
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Filter types according to phase response
Linear phase
Nonlinear phase
5.5 91
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Filter types according to phase response
Linear phase
Nonlinear phase
5.5 91
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
What is the best lowpass we can think of?
ccpi 0 pi0
1
H(e
j)
5.5 92
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
What is the best lowpass we can think of?
cc
b = 2c
pi 0 pi0
1
H(e
j)
5.5 92
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Ideal lowpass filter
H(e j) =
{1 for || c
0 otherwise(2-periodicity implicit)
perfectly flat passband
infinite attenuation in stopband
zero-phase (no delay)
5.5 93
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Ideal lowpass filter
H(e j) =
{1 for || c
0 otherwise(2-periodicity implicit)
perfectly flat passband
infinite attenuation in stopband
zero-phase (no delay)
5.5 93
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Ideal lowpass filter
H(e j) =
{1 for || c
0 otherwise(2-periodicity implicit)
perfectly flat passband
infinite attenuation in stopband
zero-phase (no delay)
5.5 93
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Ideal lowpass filter: impulse response
h[n] = IDTFT{H(e j)
}=
1
2
pipi
H(e j)e jnd
=1
2
cc
e jnd
=1
n
e jcn ejcn
2j
=sincn
n
5.5 94
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Ideal lowpass filter: impulse response
h[n] = IDTFT{H(e j)
}=
1
2
pipi
H(e j)e jnd
=1
2
cc
e jnd
=1
n
e jcn ejcn
2j
=sincn
n
5.5 94
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Ideal lowpass filter: impulse response
h[n] = IDTFT{H(e j)
}=
1
2
pipi
H(e j)e jnd
=1
2
cc
e jnd
=1
n
e jcn ejcn
2j
=sincn
n
5.5 94
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Ideal lowpass filter: impulse response
h[n] = IDTFT{H(e j)
}=
1
2
pipi
H(e j)e jnd
=1
2
cc
e jnd
=1
n
e jcn ejcn
2j
=sincn
n
5.5 94
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Ideal lowpass filter: impulse response
h[n] = IDTFT{H(e j)
}=
1
2
pipi
H(e j)e jnd
=1
2
cc
e jnd
=1
n
e jcn ejcn
2j
=sincn
n
5.5 94
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Ideal lowpass filter: impulse response
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
0
c/pi
20 10 0 10 20
5.5 95
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Ideal lowpass filter: impulse response
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
0
c/pi
20 10 0 10 20
5.5 95
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
The bad news
impulse response is infinite support, two-sided= cannot compute the output in a finite amount of time= thats why its called ideal
impulse response decays slowly in time= we need a lot of samples for a good approximation
5.5 96
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
The bad news
impulse response is infinite support, two-sided= cannot compute the output in a finite amount of time= thats why its called ideal
impulse response decays slowly in time= we need a lot of samples for a good approximation
5.5 96
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Nevertheless...
The sinc-rect pair:
rect(x) =
{1 |x | 1/20 |x | > 1/2
sinc(x) =
sin(x)
xx 6= 0
1 x = 0
(note that sinc(x) = 0 when x is a nonzero integer)
5.5 97
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
The ideal lowpass in canonical form
rect
(
2c
)DTFT
cpi
sinc(cpin
)
5.5 98
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Example
c = /3: H(ej) = rect(3/2), h[n] = (1/3)sinc(n/3)
pi 2pi/3 pi/3 0 pi/3 2pi/3 pi0
1
H(e
j)
b
b b
b
b b
b
b b
b
b b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b b
b
b b
b
b b
b
b b
b
1/3
30 20 10 0 10 20 30
0
h[n]
5.5 99
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Little-known fact
the sinc is not absolutely summable
the ideal lowpass is not BIBO stable!
example for c = /3: h[n] = (1/3) sinc(n/3)
take x [n] = sign{sinc(n/3)} and
y [0] = (x h)[0] =1
3
k=
| sinc(k/3)| =
5.5 100
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Little-known fact
the sinc is not absolutely summable
the ideal lowpass is not BIBO stable!
example for c = /3: h[n] = (1/3) sinc(n/3)
take x [n] = sign{sinc(n/3)} and
y [0] = (x h)[0] =1
3
k=
| sinc(k/3)| =
5.5 100
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Little-known fact
the sinc is not absolutely summable
the ideal lowpass is not BIBO stable!
example for c = /3: h[n] = (1/3) sinc(n/3)
take x [n] = sign{sinc(n/3)} and
y [0] = (x h)[0] =1
3
k=
| sinc(k/3)| =
5.5 100
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Little-known fact
the sinc is not absolutely summable
the ideal lowpass is not BIBO stable!
example for c = /3: h[n] = (1/3) sinc(n/3)
take x [n] = sign{sinc(n/3)} and
y [0] = (x h)[0] =1
3
k=
| sinc(k/3)| =
5.5 100
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Divergence is however very slow...
s(n) = (1/3)n
k=n
| sinc(k/3)|
1 5000 100000
1
2
3
4
5.5 101
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Ideal highpass filter
ccpi 0 pi0
1
H(e
j)
5.5 102
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Ideal highpass filter
Hhp(ej) =
{1 for || c
0 otherwise(2-periodicity implicit)
Hhp(ej) = 1 Hlp(e
j)
hhp[n] = [n]csinc(
cn)
5.5 103
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Ideal highpass filter
Hhp(ej) =
{1 for || c
0 otherwise(2-periodicity implicit)
Hhp(ej) = 1 Hlp(e
j)
hhp[n] = [n]csinc(
cn)
5.5 103
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Ideal highpass filter
Hhp(ej) =
{1 for || c
0 otherwise(2-periodicity implicit)
Hhp(ej) = 1 Hlp(e
j)
hhp[n] = [n]csinc(
cn)
5.5 103
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Ideal bandpass filter
0 c 0 0 + c0pi 0 pi0
1
H(e
j)
5.5 104
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Ideal bandpass filter
ccpi 0 pi0
1
5.5 105
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Ideal bandpass filter
0pi 0 pi0
1
5.5 105
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Ideal bandpass filter
00pi 0 pi0
1
5.5 105
Digital Sig
nal Proces
sing
Paolo Pran
doni and M
artin Vett
erli
2013
Ideal bandpass filter
Hbp(ej) =
{1 for | 0| c
0 otherwise(2-periodicity implicit)
hbp[n] = 2 cos(0n)csinc(
cn)
5.5 106
Digital Sig
nal Proces
sing