10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
10: Digital Filter Structures
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 1 / 19
Direct Forms
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
B(z)
Filter: H(z) = A(z) with input x[n] and output y[n]
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 2 / 19
Direct Forms
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
B(z)
Filter: H(z) = A(z) with input x[n] and output y[n]
y[n] =
PM
k=0 b[k]x[n
− k] −
PN
k=1 a[k]y[n
− k]
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 2 / 19
Direct Forms
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
B(z)
Filter: H(z) = A(z) with input x[n] and output y[n]
y[n] =
PM
k=0 b[k]x[n
− k] −
PN
k=1 a[k]y[n
− k]
Direct forms use coefficients a[k] and b[k] directly
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 2 / 19
Direct Forms
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
B(z)
Filter: H(z) = A(z) with input x[n] and output y[n]
y[n] =
PM
k=0 b[k]x[n
− k] −
PN
k=1 a[k]y[n
− k]
Direct forms use coefficients a[k] and b[k] directly
Direct Form 1:
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 2 / 19
Direct Forms
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
B(z)
Filter: H(z) = A(z) with input x[n] and output y[n]
y[n] =
PM
k=0 b[k]x[n
− k] −
PN
k=1 a[k]y[n
− k]
Direct forms use coefficients a[k] and b[k] directly
Direct Form 1:
• Direct implementation of difference equation
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 2 / 19
Direct Forms
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
B(z)
Filter: H(z) = A(z) with input x[n] and output y[n]
y[n] =
PM
k=0 b[k]x[n
− k] −
PN
k=1 a[k]y[n
− k]
Direct forms use coefficients a[k] and b[k] directly
Direct Form 1:
• Direct implementation of difference equation
1
• Can view as B(z) followed by A(z)
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 2 / 19
Direct Forms
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
B(z)
Filter: H(z) = A(z) with input x[n] and output y[n]
y[n] =
PM
k=0 b[k]x[n
− k] −
PN
k=1 a[k]y[n
− k]
Direct forms use coefficients a[k] and b[k] directly
Direct Form 1:
• Direct implementation of difference equation
1
• Can view as B(z) followed by A(z)
Direct Form II:
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 2 / 19
Direct Forms
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
B(z)
Filter: H(z) = A(z) with input x[n] and output y[n]
y[n] =
PM
k=0 b[k]x[n
− k] −
PN
k=1 a[k]y[n
− k]
Direct forms use coefficients a[k] and b[k] directly
Direct Form 1:
• Direct implementation of difference equation
1
• Can view as B(z) followed by A(z)
Direct Form II:
1
• Implements A(z)
followed by B(z)
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 2 / 19
Direct Forms
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Numerator
• Summary
• MATLAB routines
B(z)
Filter: H(z) = A(z) with input x[n] and output y[n]
y[n] =
PM
k=0 b[k]x[n
− k] −
PN
k=1 a[k]y[n
− k]
Direct forms use coefficients a[k] and b[k] directly
Direct Form 1:
• Direct implementation of difference equation
1
• Can view as B(z) followed by A(z)
Direct Form II:
1
• Implements A(z)
followed by B(z)
• Saves on delays (= storage)
DSP and Digital Filters (2017-10122)
Structures: 10 – 2 / 19
Transposition
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Can convert any block diagram into an equivalent transposed form:
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 3 / 19
Transposition
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Can convert any block diagram into an equivalent transposed form:
• Reverse direction of each interconnection
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 3 / 19
Transposition
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Can convert any block diagram into an equivalent transposed form:
• Reverse direction of each interconnection
• Reverse direction of each multiplier
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 3 / 19
Transposition
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Can convert any block diagram into an equivalent transposed form:
• Reverse direction of each interconnection
• Reverse direction of each multiplier
• Change junctions to adders and vice-versa
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 3 / 19
Transposition
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Can convert any block diagram into an equivalent transposed form:
• Reverse direction of each interconnection
• Reverse direction of each multiplier
• Change junctions to adders and vice-versa
• Interchange the input and output signals
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 3 / 19
Transposition
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Can convert any block diagram into an equivalent transposed form:
• Reverse direction of each interconnection
• Reverse direction of each multiplier
• Change junctions to adders and vice-versa
• Interchange the input and output signals
Example:
Direct form II → Direct Form IIt
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 3 / 19
Transposition
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Can convert any block diagram into an equivalent transposed form:
• Reverse direction of each interconnection
• Reverse direction of each multiplier
• Change junctions to adders and vice-versa
• Interchange the input and output signals
Example:
Direct form II → Direct Form IIt
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 3 / 19
Transposition
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Can convert any block diagram into an equivalent transposed form:
• Reverse direction of each interconnection
• Reverse direction of each multiplier
• Change junctions to adders and vice-versa
• Interchange the input and output signals
Example:
Direct form II → Direct Form IIt
Would normally be drawn with input on the left
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 3 / 19
Transposition
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Numerator
Can convert any block diagram into an equivalent transposed form:
• Reverse direction of each interconnection
• Reverse direction of each multiplier
• Change junctions to adders and vice-versa
• Interchange the input and output signals
Example:
Direct form II → Direct Form IIt
Would normally be drawn with input on the left
Note: A valid block diagram must never have any feedback loops that don’t
go through a delay (z −1 block).
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 3 / 19
State Space
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
+
v[n] is a vector of delay element outputs
Can write: v[n + 1] = Pv[n] + qx[n]
y[n] = rT v[n] + sx[n]
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 4 / 19
State Space
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
+
v[n] is a vector of delay element outputs
Can write: v[n + 1] = Pv[n] + qx[n]
y[n] = rT v[n] + sx[n]
Numerator
• Summary
• MATLAB routines
Example: Direct Form IIt
P=
DSP and Digital Filters (2017-10122)
−a[1] 1
−a[2] 0
Structures: 10 – 4 / 19
State Space
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
+
v[n] is a vector of delay element outputs
Can write: v[n + 1] = Pv[n] + qx[n]
y[n] = rT v[n] + sx[n]
Numerator
• Summary
• MATLAB routines
Example: Direct Form IIt
P=
DSP and Digital Filters (2017-10122)
−a[1] 1
−a[2] 0
q=
b[1] − b[0]a[1]
b[2] − b[0]a[2]
Structures: 10 – 4 / 19
State Space
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
+
v[n] is a vector of delay element outputs
Can write: v[n + 1] = Pv[n] + qx[n]
y[n] = rT v[n] + sx[n]
Numerator
• Summary
• MATLAB routines
Example: Direct Form IIt
P=
rT =
DSP and Digital Filters (2017-10122)
−a[1] 1
−a[2] 0
1 0
q=
b[1] − b[0]a[1]
b[2] − b[0]a[2]
Structures: 10 – 4 / 19
State Space
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
+
v[n] is a vector of delay element outputs
Can write: v[n + 1] = Pv[n] + qx[n]
y[n] = rT v[n] + sx[n]
Numerator
• Summary
• MATLAB routines
Example: Direct Form IIt
P=
rT =
DSP and Digital Filters (2017-10122)
−a[1] 1
−a[2] 0
1 0
q=
b[1] − b[0]a[1]
b[2] − b[0]a[2]
s = b[0]
Structures: 10 – 4 / 19
State Space
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
+
v[n] is a vector of delay element outputs
Can write: v[n + 1] = Pv[n] + qx[n]
y[n] = rT v[n] + sx[n]
P, q, rT , s is the state-space
representation of the filter structure.
Numerator
• Summary
• MATLAB routines
Example: Direct Form IIt
P=
rT =
DSP and Digital Filters (2017-10122)
−a[1] 1
−a[2] 0
1 0
q=
b[1] − b[0]a[1]
b[2] − b[0]a[2]
s = b[0]
Structures: 10 – 4 / 19
State Space
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
+
v[n] is a vector of delay element outputs
Can write: v[n + 1] = Pv[n] + qx[n]
y[n] = rT v[n] + sx[n]
P, q, rT , s is the state-space
representation of the filter structure.
The transfer function is given by:
H(z) =
B(z)
A(z)
=
det(zI−P+qrT )
det(zI−P)
+s−1
Numerator
• Summary
• MATLAB routines
Example: Direct Form IIt
P=
rT =
DSP and Digital Filters (2017-10122)
−a[1] 1
−a[2] 0
1 0
q=
b[1] − b[0]a[1]
b[2] − b[0]a[2]
s = b[0]
Structures: 10 – 4 / 19
State Space
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
+
v[n] is a vector of delay element outputs
Can write: v[n + 1] = Pv[n] + qx[n]
y[n] = rT v[n] + sx[n]
P, q, rT , s is the state-space
representation of the filter structure.
The transfer function is given by:
H(z) =
B(z)
A(z)
=
det(zI−P+qrT )
det(zI−P)
+s−1
Numerator
• Summary
• MATLAB routines
Example: Direct Form IIt
P=
rT =
−a[1] 1
−a[2] 0
1 0
From which H(z) =
DSP and Digital Filters (2017-10122)
q=
b[1] − b[0]a[1]
b[2] − b[0]a[2]
s = b[0]
b[0]z 2 +b[1]z+b[2]
z 2 +a[1]z+a[2]
Structures: 10 – 4 / 19
State Space
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
+
v[n] is a vector of delay element outputs
Can write: v[n + 1] = Pv[n] + qx[n]
y[n] = rT v[n] + sx[n]
P, q, rT , s is the state-space
representation of the filter structure.
The transfer function is given by:
H(z) =
B(z)
A(z)
=
det(zI−P+qrT )
det(zI−P)
T
The transposed form has P → P
+s−1
and q ↔ r
⇒
same H(z)
Numerator
• Summary
• MATLAB routines
Example: Direct Form IIt
P=
rT =
−a[1] 1
−a[2] 0
1 0
From which H(z) =
DSP and Digital Filters (2017-10122)
q=
b[1] − b[0]a[1]
b[2] − b[0]a[2]
s = b[0]
b[0]z 2 +b[1]z+b[2]
z 2 +a[1]z+a[2]
Structures: 10 – 4 / 19
Precision Issues
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
If all computations were exact, it would not make any difference which of
the equivalent structures was used.
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 5 / 19
Precision Issues
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
If all computations were exact, it would not make any difference which of
the equivalent structures was used. However ...
• Coefficient precision
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 5 / 19
Precision Issues
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
If all computations were exact, it would not make any difference which of
the equivalent structures was used. However ...
• Coefficient precision
Coefficients are stored to finite precision and so are not exact.
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 5 / 19
Precision Issues
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
If all computations were exact, it would not make any difference which of
the equivalent structures was used. However ...
• Coefficient precision
Coefficients are stored to finite precision and so are not exact.
The filter actually implemented is therefore incorrect.
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 5 / 19
Precision Issues
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
If all computations were exact, it would not make any difference which of
the equivalent structures was used. However ...
• Coefficient precision
Coefficients are stored to finite precision and so are not exact.
The filter actually implemented is therefore incorrect.
• Arithmetic precision
Arithmetic calculations are not exact.
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 5 / 19
Precision Issues
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
If all computations were exact, it would not make any difference which of
the equivalent structures was used. However ...
• Coefficient precision
Coefficients are stored to finite precision and so are not exact.
The filter actually implemented is therefore incorrect.
• Arithmetic precision
Arithmetic calculations are not exact.
◦ Worst case for arithmetic errors is when calculating the
difference between two similar values:
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 5 / 19
Precision Issues
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Numerator
If all computations were exact, it would not make any difference which of
the equivalent structures was used. However ...
• Coefficient precision
Coefficients are stored to finite precision and so are not exact.
The filter actually implemented is therefore incorrect.
• Arithmetic precision
Arithmetic calculations are not exact.
◦ Worst case for arithmetic errors is when calculating the
difference between two similar values:
1.23456789 − 1.23455678 = 0.00001111: 9 s.f. → 4 s.f.
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 5 / 19
Precision Issues
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Numerator
• Summary
• MATLAB routines
If all computations were exact, it would not make any difference which of
the equivalent structures was used. However ...
• Coefficient precision
Coefficients are stored to finite precision and so are not exact.
The filter actually implemented is therefore incorrect.
• Arithmetic precision
Arithmetic calculations are not exact.
◦ Worst case for arithmetic errors is when calculating the
difference between two similar values:
1.23456789 − 1.23455678 = 0.00001111: 9 s.f. → 4 s.f.
Arithmetic errors introduce noise that is then filtered by the transfer
function between the point of noise creation and the output.
DSP and Digital Filters (2017-10122)
Structures: 10 – 5 / 19
Coefficient Sensitivity
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
The roots of high order polynomials can be very sensitive to small changes
in coefficient values.
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 6 / 19
Coefficient Sensitivity
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
The roots of high order polynomials can be very sensitive to small changes
in coefficient values.
Wilkinson’s polynomial: (famous example)
f (x) =
Q20
20
19
18
(x
−
n)
=
x
−
210x
+
20615x
− ...
n=1
has roots well separated on the real axis.
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 6 / 19
Coefficient Sensitivity
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
The roots of high order polynomials can be very sensitive to small changes
in coefficient values.
Wilkinson’s polynomial: (famous example)
f (x) =
Q20
20
19
18
(x
−
n)
=
x
−
210x
+
20615x
− ...
n=1
has roots well separated on the real axis.
Numerator
• Summary
• MATLAB routines
5
0
-5
0
DSP and Digital Filters (2017-10122)
5
10
15
20
25
Structures: 10 – 6 / 19
Coefficient Sensitivity
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
The roots of high order polynomials can be very sensitive to small changes
in coefficient values.
Wilkinson’s polynomial: (famous example)
f (x) =
Q20
20
19
18
(x
−
n)
=
x
−
210x
+
20615x
− ...
n=1
has roots well separated on the real axis.
Multiplying the coefficient of x19 by 1.000001 moves the roots a lot.
Numerator
• Summary
• MATLAB routines
5
0
-5
0
DSP and Digital Filters (2017-10122)
5
10
15
20
25
Structures: 10 – 6 / 19
Coefficient Sensitivity
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
The roots of high order polynomials can be very sensitive to small changes
in coefficient values.
Wilkinson’s polynomial: (famous example)
f (x) =
Q20
20
19
18
(x
−
n)
=
x
−
210x
+
20615x
− ...
n=1
has roots well separated on the real axis.
Multiplying the coefficient of x19 by 1.000001 moves the roots a lot.
Numerator
• Summary
• MATLAB routines
5
5
0
0
-5
-5
0
DSP and Digital Filters (2017-10122)
5
10
15
20
25
0
5
10
15
20
25
Structures: 10 – 6 / 19
Coefficient Sensitivity
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
The roots of high order polynomials can be very sensitive to small changes
in coefficient values.
Wilkinson’s polynomial: (famous example)
f (x) =
Q20
20
19
18
(x
−
n)
=
x
−
210x
+
20615x
− ...
n=1
has roots well separated on the real axis.
Multiplying the coefficient of x19 by 1.000001 moves the roots a lot.
“Speaking for myself I regard it as the most traumatic experience in
my career as a numerical analyst”, James Wilkinson 1984
Numerator
• Summary
• MATLAB routines
5
5
0
0
-5
-5
0
DSP and Digital Filters (2017-10122)
5
10
15
20
25
0
5
10
15
20
25
Structures: 10 – 6 / 19
Coefficient Sensitivity
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
The roots of high order polynomials can be very sensitive to small changes
in coefficient values.
Wilkinson’s polynomial: (famous example)
f (x) =
Q20
20
19
18
(x
−
n)
=
x
−
210x
+
20615x
− ...
n=1
has roots well separated on the real axis.
Multiplying the coefficient of x19 by 1.000001 moves the roots a lot.
“Speaking for myself I regard it as the most traumatic experience in
my career as a numerical analyst”, James Wilkinson 1984
Numerator
• Summary
• MATLAB routines
5
5
0
0
-5
-5
0
5
10
15
20
25
0
5
10
15
20
25
Moral: Avoid using direct form for filters orders over about 10.
DSP and Digital Filters (2017-10122)
Structures: 10 – 6 / 19
Cascaded Biquads
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Avoid high order polynomials by factorizing into quadratic terms:
B(z)
A(z)
=
Q
(1+bk,1 z−1 +bk,2 z−2 )
Q
g (1+ak,1 z−1 +ak,2 z−2 )
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 7 / 19
Cascaded Biquads
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Avoid high order polynomials by factorizing into quadratic terms:
B(z)
A(z)
=
Q
(1+bk,1 z−1 +bk,2 z−2 )
Q
g (1+ak,1 z−1 +ak,2 z−2 ) =
g
1+bk,1 z −1 +bk,2 z −2
k=1 1+ak,1 z −1 +ak,2 z −2
QK
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 7 / 19
Cascaded Biquads
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Avoid high order polynomials by factorizing into quadratic terms:
B(z)
A(z)
=
Q
(1+bk,1 z−1 +bk,2 z−2 )
Q
g (1+ak,1 z−1 +ak,2 z−2 ) =
g
M N where K = max
.
2 ,
2
1+bk,1 z −1 +bk,2 z −2
k=1 1+ak,1 z −1 +ak,2 z −2
QK
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 7 / 19
Cascaded Biquads
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Avoid high order polynomials by factorizing into quadratic terms:
B(z)
A(z)
=
Q
(1+bk,1 z−1 +bk,2 z−2 )
Q
g (1+ak,1 z−1 +ak,2 z−2 ) =
g
M N where K = max
.
2 ,
2
The term
1+bk,1 z −1 +bk,2 z −2
1+ak,1 z −1 +ak,2 z −2
1+bk,1 z −1 +bk,2 z −2
k=1 1+ak,1 z −1 +ak,2 z −2
QK
is a biquad (bi-quadratic section).
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 7 / 19
Cascaded Biquads
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Avoid high order polynomials by factorizing into quadratic terms:
B(z)
A(z)
=
Q
(1+bk,1 z−1 +bk,2 z−2 )
Q
g (1+ak,1 z−1 +ak,2 z−2 ) =
g
M N where K = max
.
2 ,
2
The term
1+bk,1 z −1 +bk,2 z −2
1+ak,1 z −1 +ak,2 z −2
1+bk,1 z −1 +bk,2 z −2
k=1 1+ak,1 z −1 +ak,2 z −2
QK
is a biquad (bi-quadratic section).
Numerator
• Summary
• MATLAB routines
Direct Form II
Transposed
DSP and Digital Filters (2017-10122)
Structures: 10 – 7 / 19
Cascaded Biquads
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Avoid high order polynomials by factorizing into quadratic terms:
B(z)
A(z)
=
Q
(1+bk,1 z−1 +bk,2 z−2 )
Q
g (1+ak,1 z−1 +ak,2 z−2 ) =
g
M N where K = max
.
2 ,
2
The term
1+bk,1 z −1 +bk,2 z −2
1+ak,1 z −1 +ak,2 z −2
1+bk,1 z −1 +bk,2 z −2
k=1 1+ak,1 z −1 +ak,2 z −2
QK
is a biquad (bi-quadratic section).
We need to choose:
(a) which poles to pair with which zeros in each biquad
Numerator
• Summary
• MATLAB routines
Direct Form II
Transposed
DSP and Digital Filters (2017-10122)
Structures: 10 – 7 / 19
Cascaded Biquads
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Numerator
Avoid high order polynomials by factorizing into quadratic terms:
B(z)
A(z)
=
Q
(1+bk,1 z−1 +bk,2 z−2 )
Q
g (1+ak,1 z−1 +ak,2 z−2 ) =
g
M N where K = max
.
2 ,
2
The term
1+bk,1 z −1 +bk,2 z −2
1+ak,1 z −1 +ak,2 z −2
1+bk,1 z −1 +bk,2 z −2
k=1 1+ak,1 z −1 +ak,2 z −2
QK
is a biquad (bi-quadratic section).
We need to choose:
(a) which poles to pair with which zeros in each biquad
(b) how to order the biquads
• Summary
• MATLAB routines
Direct Form II
Transposed
DSP and Digital Filters (2017-10122)
Structures: 10 – 7 / 19
Pole-zero Pairing/Ordering
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Example: Elliptic lowpass filter
1
0.5
0
-0.5
-1
-1
0
z
1
Numerator
• Summary
• MATLAB routines
Gain (dB)
20
0
-20
-40
0
DSP and Digital Filters (2017-10122)
0.5
1
1.5
ω
2
2.5
3
Structures: 10 – 8 / 19
Pole-zero Pairing/Ordering
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Example: Elliptic lowpass filter
2 pole pairs and 2 zero pairs
1
0.5
0
-0.5
-1
-1
0
z
1
Numerator
• Summary
• MATLAB routines
Gain (dB)
20
0
-20
-40
0
DSP and Digital Filters (2017-10122)
0.5
1
1.5
ω
2
2.5
3
Structures: 10 – 8 / 19
Pole-zero Pairing/Ordering
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Example: Elliptic lowpass filter
2 pole pairs and 2 zero pairs
need 2 biquads
1
0.5
0
-0.5
-1
-1
0
z
1
Numerator
• Summary
• MATLAB routines
Gain (dB)
20
0
-20
-40
0
DSP and Digital Filters (2017-10122)
0.5
1
1.5
ω
2
2.5
3
Structures: 10 – 8 / 19
Pole-zero Pairing/Ordering
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Example: Elliptic lowpass filter
2 pole pairs and 2 zero pairs
need 2 biquads
Noise introduced in one biquad is amplified
by all the subsequent ones:
1
0.5
0
-0.5
-1
-1
0
z
1
Numerator
• Summary
• MATLAB routines
Gain (dB)
20
0
-20
-40
0
DSP and Digital Filters (2017-10122)
0.5
1
1.5
ω
2
2.5
3
Structures: 10 – 8 / 19
Pole-zero Pairing/Ordering
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Example: Elliptic lowpass filter
2 pole pairs and 2 zero pairs
need 2 biquads
Noise introduced in one biquad is amplified
by all the subsequent ones:
1
0.5
0
-0.5
-1
-1
0
z
1
• Make the peak gain of each biquad as small as possible
Numerator
• Summary
• MATLAB routines
Gain (dB)
20
0
-20
-40
0
DSP and Digital Filters (2017-10122)
0.5
1
1.5
ω
2
2.5
3
Structures: 10 – 8 / 19
Pole-zero Pairing/Ordering
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Example: Elliptic lowpass filter
2 pole pairs and 2 zero pairs
need 2 biquads
Noise introduced in one biquad is amplified
by all the subsequent ones:
1
0.5
0
-0.5
-1
-1
0
z
1
• Make the peak gain of each biquad as small as possible
◦ Pair poles with nearest zeros to get lowest peak gain
Numerator
• Summary
• MATLAB routines
Nearest
Gain (dB)
20
0
-20
-40
0
DSP and Digital Filters (2017-10122)
0.5
1
1.5
ω
2
2.5
3
Structures: 10 – 8 / 19
Pole-zero Pairing/Ordering
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Example: Elliptic lowpass filter
2 pole pairs and 2 zero pairs
need 2 biquads
Noise introduced in one biquad is amplified
by all the subsequent ones:
1
0.5
0
-0.5
-1
-1
0
z
1
• Make the peak gain of each biquad as small as possible
◦ Pair poles with nearest zeros to get lowest peak gain
begin with the pole nearest the unit circle
Numerator
• Summary
• MATLAB routines
Nearest
Gain (dB)
20
0
-20
-40
0
DSP and Digital Filters (2017-10122)
0.5
1
1.5
ω
2
2.5
3
Structures: 10 – 8 / 19
Pole-zero Pairing/Ordering
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Example: Elliptic lowpass filter
1
0.5
2 pole pairs and 2 zero pairs
need 2 biquads
0
-0.5
-1
Noise introduced in one biquad is amplified
by all the subsequent ones:
-1
0
z
1
• Make the peak gain of each biquad as small as possible
◦ Pair poles with nearest zeros to get lowest peak gain
◦
begin with the pole nearest the unit circle
Pairing with farthest zeros gives higher peak biquad gain
Numerator
• Summary
• MATLAB routines
Nearest
0
-20
-40
0
DSP and Digital Filters (2017-10122)
0.5
1
1.5
ω
2
2.5
3
Farthest
20
Gain (dB)
Gain (dB)
20
0
-20
-40
0
0.5
1
1.5
ω
2
2.5
3
Structures: 10 – 8 / 19
Pole-zero Pairing/Ordering
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Example: Elliptic lowpass filter
1
0.5
2 pole pairs and 2 zero pairs
need 2 biquads
0
-0.5
-1
Noise introduced in one biquad is amplified
by all the subsequent ones:
-1
0
z
1
• Make the peak gain of each biquad as small as possible
◦ Pair poles with nearest zeros to get lowest peak gain
◦
begin with the pole nearest the unit circle
Pairing with farthest zeros gives higher peak biquad gain
Numerator
• Poles near the unit circle have the highest peaks and introduce most
• Summary
• MATLAB routines
noise so place them last in the chain
Nearest
0
-20
-40
0
DSP and Digital Filters (2017-10122)
0.5
1
1.5
ω
2
2.5
3
Farthest
20
Gain (dB)
Gain (dB)
20
0
-20
-40
0
0.5
1
1.5
ω
2
2.5
3
Structures: 10 – 8 / 19
Linear Phase
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Implementation can take advantage of any symmetry in the coefficients.
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 9 / 19
Linear Phase
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Implementation can take advantage of any symmetry in the coefficients.
Linear phase filters are always FIR and have symmetric (or, more rarely,
antisymmetric) coefficients.
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 9 / 19
Linear Phase
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Implementation can take advantage of any symmetry in the coefficients.
Linear phase filters are always FIR and have symmetric (or, more rarely,
antisymmetric) coefficients.
H(z) =
PM
−m
h[m]z
m=0
h[M − m] = h[m]
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 9 / 19
Linear Phase
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Implementation can take advantage of any symmetry in the coefficients.
Linear phase filters are always FIR and have symmetric (or, more rarely,
antisymmetric) coefficients.
PM
−m
h[M − m] = h[m]
h[m]z
m=0
M − M P M2 −1
−m
m−M
= h 2 z 2 + m=0 h[m] z
+z
H(z) =
[m even]
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 9 / 19
Linear Phase
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Implementation can take advantage of any symmetry in the coefficients.
Linear phase filters are always FIR and have symmetric (or, more rarely,
antisymmetric) coefficients.
PM
−m
h[M − m] = h[m]
h[m]z
m=0
M − M P M2 −1
−m
m−M
= h 2 z 2 + m=0 h[m] z
+z
H(z) =
[m even]
For M even, we only need M
2 + 1 multiplies instead of M + 1.
We still need M additions and M delays.
Numerator
• Summary
• MATLAB routines
M = 6:
DSP and Digital Filters (2017-10122)
Structures: 10 – 9 / 19
Linear Phase
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Implementation can take advantage of any symmetry in the coefficients.
Linear phase filters are always FIR and have symmetric (or, more rarely,
antisymmetric) coefficients.
PM
−m
h[M − m] = h[m]
h[m]z
m=0
M − M P M2 −1
−m
m−M
= h 2 z 2 + m=0 h[m] z
+z
H(z) =
[m even]
For M even, we only need M
2 + 1 multiplies instead of M + 1.
We still need M additions and M delays.
Numerator
• Summary
• MATLAB routines
M = 6:
For M odd (no central coefficient), we only need M2+1 multiplies.
DSP and Digital Filters (2017-10122)
Structures: 10 – 9 / 19
Hardware Implementation
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Software Implementation:
All that matters is the total number of multiplies and adds
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 10 / 19
Hardware Implementation
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Software Implementation:
All that matters is the total number of multiplies and adds
Hardware Implementation:
Delay elements (z −1 ) represent storage registers
The maximum clock speed is limited by the number of sequential
operations between registers
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 10 / 19
Hardware Implementation
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Software Implementation:
All that matters is the total number of multiplies and adds
Hardware Implementation:
Delay elements (z −1 ) represent storage registers
The maximum clock speed is limited by the number of sequential
operations between registers
Example: Symmetric Linear Phase Filter
Direct form: Maximum sequential delay = 4a + m
Numerator
a and m are the delays of adder and multiplier respectively
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 10 / 19
Hardware Implementation
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Numerator
Software Implementation:
All that matters is the total number of multiplies and adds
Hardware Implementation:
Delay elements (z −1 ) represent storage registers
The maximum clock speed is limited by the number of sequential
operations between registers
Example: Symmetric Linear Phase Filter
Direct form: Maximum sequential delay = 4a + m
Transpose form: Maximum sequential delay = a + m ,
a and m are the delays of adder and multiplier respectively
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 10 / 19
Allpass Filters
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Allpass filters have mirror image numerator and denominator coefficients:
b[n] = a[N − n]
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 11 / 19
Allpass Filters
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Allpass filters have mirror image numerator and denominator coefficients:
b[n] = a[N − n]
⇔
B(z) = z −N A(z −1 )
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 11 / 19
Allpass Filters
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Allpass filters have mirror image numerator and denominator coefficients:
b[n] = a[N − n]
⇔
B(z) = z −N A(z −1 )
⇒ H(ejω ) ≡ 1∀ω
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 11 / 19
Allpass Filters
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Allpass filters have mirror image numerator and denominator coefficients:
b[n] = a[N − n]
⇔
B(z) = z −N A(z −1 )
⇒ H(ejω ) ≡ 1∀ω
There are several efficient structures, e.g.
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 11 / 19
Allpass Filters
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Allpass filters have mirror image numerator and denominator coefficients:
b[n] = a[N − n]
⇔
B(z) = z −N A(z −1 )
⇒ H(ejω ) ≡ 1∀ω
There are several efficient structures, e.g.
• First Order: H(z) =
a[1]+z −1
1+a[1]z −1
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 11 / 19
Allpass Filters
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Allpass filters have mirror image numerator and denominator coefficients:
b[n] = a[N − n]
⇔
B(z) = z −N A(z −1 )
⇒ H(ejω ) ≡ 1∀ω
There are several efficient structures, e.g.
• First Order: H(z) =
a[1]+z −1
1+a[1]z −1
• Second Order: H(z) =
a[2]+a[1]z −1 +z −2
1+a[1]z −1 +a[2]z −2
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 11 / 19
Allpass Filters
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Allpass filters have mirror image numerator and denominator coefficients:
b[n] = a[N − n]
⇔
B(z) = z −N A(z −1 )
⇒ H(ejω ) ≡ 1∀ω
There are several efficient structures, e.g.
• First Order: H(z) =
a[1]+z −1
1+a[1]z −1
• Second Order: H(z) =
a[2]+a[1]z −1 +z −2
1+a[1]z −1 +a[2]z −2
Numerator
• Summary
• MATLAB routines
Allpass filters have a gain magnitude of 1 even with coefficient errors.
DSP and Digital Filters (2017-10122)
Structures: 10 – 11 / 19
Lattice Stage
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Suppose G is allpass: G(z) =
+
z −N A(z −1 )
A(z)
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 12 / 19
Lattice Stage
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Suppose G is allpass: G(z) =
+
z −N A(z −1 )
A(z)
V (z) = X(z) − kGz −1 V (z)
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 12 / 19
Lattice Stage
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
+
Suppose G is allpass: G(z) =
z −N A(z −1 )
A(z)
V (z) = X(z) − kGz −1 V (z)
⇒ V (z) =
1
1+kGz −1 X(z)
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 12 / 19
Lattice Stage
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
+
Suppose G is allpass: G(z) =
z −N A(z −1 )
A(z)
V (z) = X(z) − kGz −1 V (z)
⇒ V (z) =
1
1+kGz −1 X(z)
Y (z) = kV (z) + Gz −1 V (z)
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 12 / 19
Lattice Stage
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
+
Suppose G is allpass: G(z) =
z −N A(z −1 )
A(z)
V (z) = X(z) − kGz −1 V (z)
⇒ V (z) =
1
1+kGz −1 X(z)
Y (z) = kV (z) + Gz
−1
V (z) =
k+z −1 G
1+kGz −1 X(z)
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 12 / 19
Lattice Stage
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
+
Suppose G is allpass: G(z) =
z −N A(z −1 )
A(z)
V (z) = X(z) − kGz −1 V (z)
⇒ V (z) =
1
1+kGz −1 X(z)
Y (z) = kV (z) + Gz
Y (z)
X(z)
=
−1
V (z) =
k+z −1 G
1+kGz −1 X(z)
kA(z)+z −N −1 A(z −1 )
A(z)+kz −N −1 A(z −1 )
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 12 / 19
Lattice Stage
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
+
Suppose G is allpass: G(z) =
z −N A(z −1 )
A(z)
V (z) = X(z) − kGz −1 V (z)
⇒ V (z) =
1
1+kGz −1 X(z)
Y (z) = kV (z) + Gz
Y (z)
X(z)
=
−1
V (z) =
kA(z)+z −N −1 A(z −1 )
A(z)+kz −N −1 A(z −1 )
,
k+z −1 G
1+kGz −1 X(z)
z −(N +1) D(z −1 )
D(z)
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 12 / 19
Lattice Stage
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Numerator
• Summary
• MATLAB routines
+
Suppose G is allpass: G(z) =
z −N A(z −1 )
A(z)
V (z) = X(z) − kGz −1 V (z)
⇒ V (z) =
1
1+kGz −1 X(z)
Y (z) = kV (z) + Gz
Y (z)
X(z)
=
−1
V (z) =
kA(z)+z −N −1 A(z −1 )
A(z)+kz −N −1 A(z −1 )
,
k+z −1 G
1+kGz −1 X(z)
z −(N +1) D(z −1 )
D(z)
Obtaining {d[n]}
 from {a[n]}:

n=0
1
d[n] = a[n] + ka[N + 1 − n] 1 ≤ n ≤ N


k
n=N +1
DSP and Digital Filters (2017-10122)
Structures: 10 – 12 / 19
Lattice Stage
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Numerator
• Summary
• MATLAB routines
+
Suppose G is allpass: G(z) =
z −N A(z −1 )
A(z)
V (z) = X(z) − kGz −1 V (z)
⇒ V (z) =
1
1+kGz −1 X(z)
Y (z) = kV (z) + Gz
Y (z)
X(z)
=
−1
V (z) =
kA(z)+z −N −1 A(z −1 )
A(z)+kz −N −1 A(z −1 )
,
k+z −1 G
1+kGz −1 X(z)
z −(N +1) D(z −1 )
D(z)
Obtaining {d[n]}
 from {a[n]}:

n=0
1
d[n] = a[n] + ka[N + 1 − n] 1 ≤ n ≤ N


k
n=N +1
Obtaining {a[n]} from {d[n]}:
k = d[N + 1]
DSP and Digital Filters (2017-10122)
a[n] =
d[n]−kd[N +1−n]
1−k2
Structures: 10 – 12 / 19
Lattice Stage
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Numerator
• Summary
• MATLAB routines
+
Suppose G is allpass: G(z) =
z −N A(z −1 )
A(z)
V (z) = X(z) − kGz −1 V (z)
⇒ V (z) =
1
1+kGz −1 X(z)
Y (z) = kV (z) + Gz
Y (z)
X(z)
=
−1
V (z) =
kA(z)+z −N −1 A(z −1 )
A(z)+kz −N −1 A(z −1 )
,
k+z −1 G
1+kGz −1 X(z)
z −(N +1) D(z −1 )
D(z)
Obtaining {d[n]}
 from {a[n]}:

n=0
1
d[n] = a[n] + ka[N + 1 − n] 1 ≤ n ≤ N


k
n=N +1
Obtaining {a[n]} from {d[n]}:
k = d[N + 1]
a[n] =
d[n]−kd[N +1−n]
1−k2
Y (z)
If G(z) is stable then X(z) is stable if and only if |k| < 1 (see note)
DSP and Digital Filters (2017-10122)
Structures: 10 – 12 / 19
Example A(z) ↔ D(z)
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Suppose N = 3, k = 0.5 and
A(z) = 1 + 4z −1 − 6z −2 + 10z −3
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 13 / 19
Example A(z) ↔ D(z)
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Suppose N = 3, k = 0.5 and
A(z) = 1 + 4z −1 − 6z −2 + 10z −3
A(z) → D(z)
A(z)
z −4 A(z −1 )
D(z) = A(z) + kz −4 A(z −1 )
z0
1
1
z −1
4
10
9
z −2
−6
−6
−9
z −3
10
4
12
z −4
1
0.5
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 13 / 19
Example A(z) ↔ D(z)
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Numerator
• Summary
• MATLAB routines
Suppose N = 3, k = 0.5 and
A(z) = 1 + 4z −1 − 6z −2 + 10z −3
A(z) → D(z)
z0
1
A(z)
z −4 A(z −1 )
D(z) = A(z) + kz −4 A(z −1 )
1
z −1
4
10
9
z −2
−6
−6
−9
z −3
10
4
12
z −4
1
0.5
D(z) → A(z)
D(z)
k = d[N + 1]
z −4 D(z −1 )
D(z) − kz −4 D(z −1 )
A(z) =
DSP and Digital Filters (2017-10122)
D(z)−kz −4 D(z −1 )
1−k2
z0
1
z −1
9
z −2
−9
z −3
12
0.5
0.75
12
3
−9
−4.5
9
7.5
z −4
0.5
0.5
1
0
1
4
−6
10
0
Structures: 10 – 13 / 19
Allpass Lattice
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
We can implement any allpass filter H(z) =
z −M A(z −1 )
A(z)
as a lattice filter
with M stages:
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 14 / 19
Allpass Lattice
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
We can implement any allpass filter H(z) =
z −M A(z −1 )
A(z)
as a lattice filter
with M stages:
• Initialize AM (z) = A(z)
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 14 / 19
Allpass Lattice
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
We can implement any allpass filter H(z) =
z −M A(z −1 )
A(z)
as a lattice filter
with M stages:
• Initialize AM (z) = A(z)
• Repeat for m = M : −1 : 1
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 14 / 19
Allpass Lattice
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
We can implement any allpass filter H(z) =
z −M A(z −1 )
A(z)
as a lattice filter
with M stages:
• Initialize AM (z) = A(z)
• Repeat for m = M : −1 : 1
◦ k[m] = am [m]
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 14 / 19
Allpass Lattice
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
We can implement any allpass filter H(z) =
z −M A(z −1 )
A(z)
as a lattice filter
with M stages:
• Initialize AM (z) = A(z)
• Repeat for m = M : −1 : 1
◦ k[m] = am [m]
◦ am−1 [n] =
am [n]−k[m]am [m−n]
1−k2 [m]
for 0 ≤ n ≤ m − 1
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 14 / 19
Allpass Lattice
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
We can implement any allpass filter H(z) =
z −M A(z −1 )
A(z)
as a lattice filter
with M stages:
• Initialize AM (z) = A(z)
• Repeat for m = M : −1 : 1
◦ k[m] = am [m]
◦ am−1 [n] =
am [n]−k[m]am [m−n]
1−k2 [m]
equivalently Am−1 (z) =
for 0 ≤ n ≤ m − 1
Am (z)−k[m]z −m Am (z −1 )
1−k2 [m]
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 14 / 19
Allpass Lattice
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
We can implement any allpass filter H(z) =
z −M A(z −1 )
A(z)
as a lattice filter
with M stages:
• Initialize AM (z) = A(z)
• Repeat for m = M : −1 : 1
◦ k[m] = am [m]
◦ am−1 [n] =
am [n]−k[m]am [m−n]
1−k2 [m]
equivalently Am−1 (z) =
for 0 ≤ n ≤ m − 1
Am (z)−k[m]z −m Am (z −1 )
1−k2 [m]
Numerator
• Summary
• MATLAB routines
A(z) is stable iff |k[m]| < 1 for all m (good stability test)
DSP and Digital Filters (2017-10122)
Structures: 10 – 14 / 19
Lattice Filter
Label outputs um [n] and vm [n] and define Hm (z) =
DSP and Digital Filters (2017-10122)
Vm (z)
Um (z)
=
z −m Am (z −1 )
Am (z)
Structures: 10 – 15 / 19
Lattice Filter
Label outputs um [n] and vm [n] and define Hm (z) =
Vm (z)
Um (z)
=
z −m Am (z −1 )
Am (z)
From earlier slide (slide 12):
Um−1 (z)
Um (z)
=
1
1+k[m]z −1 Hm−1 (z)
DSP and Digital Filters (2017-10122)
Structures: 10 – 15 / 19
Lattice Filter
Label outputs um [n] and vm [n] and define Hm (z) =
Vm (z)
Um (z)
=
z −m Am (z −1 )
Am (z)
From earlier slide (slide 12):
Um−1 (z)
Um (z)
=
1
1+k[m]z −1 H
DSP and Digital Filters (2017-10122)
m−1 (z)
=
Am−1 (z)
Am−1 (z)+k[m]z −m Am−1 (z −1 )
Structures: 10 – 15 / 19
Lattice Filter
Label outputs um [n] and vm [n] and define Hm (z) =
Vm (z)
Um (z)
=
z −m Am (z −1 )
Am (z)
From earlier slide (slide 12):
Um−1 (z)
Um (z)
=
1
1+k[m]z −1 H
DSP and Digital Filters (2017-10122)
m−1 (z)
=
Am−1 (z)
Am−1 (z)+k[m]z −m Am−1 (z −1 )
=
Am−1 (z)
Am (z)
Structures: 10 – 15 / 19
Lattice Filter
Label outputs um [n] and vm [n] and define Hm (z) =
Vm (z)
Um (z)
=
z −m Am (z −1 )
Am (z)
From earlier slide (slide 12):
Um−1 (z)
Um (z)
=
1
1+k[m]z −1 H
m−1 (z)
=
Am−1 (z)
Am−1 (z)+k[m]z −m Am−1 (z −1 )
=
Am−1 (z)
Am (z)
Hence:
Um (z)
X(z)
=
Am (z)
A(z)
DSP and Digital Filters (2017-10122)
Structures: 10 – 15 / 19
Lattice Filter
Label outputs um [n] and vm [n] and define Hm (z) =
Vm (z)
Um (z)
=
z −m Am (z −1 )
Am (z)
From earlier slide (slide 12):
Um−1 (z)
Um (z)
=
1
1+k[m]z −1 H
m−1 (z)
=
Am−1 (z)
Am−1 (z)+k[m]z −m Am−1 (z −1 )
=
Am−1 (z)
Am (z)
Hence:
Um (z)
X(z)
=
Am (z)
A(z)
DSP and Digital Filters (2017-10122)
and
Vm (z)
X(z)
=
Um (z)
X(z)
×
Vm (z)
Um (z)
Structures: 10 – 15 / 19
Lattice Filter
Label outputs um [n] and vm [n] and define Hm (z) =
Vm (z)
Um (z)
=
z −m Am (z −1 )
Am (z)
From earlier slide (slide 12):
Um−1 (z)
Um (z)
=
1
1+k[m]z −1 H
m−1 (z)
=
Am−1 (z)
Am−1 (z)+k[m]z −m Am−1 (z −1 )
=
Am−1 (z)
Am (z)
Hence:
Um (z)
X(z)
=
Am (z)
A(z)
DSP and Digital Filters (2017-10122)
and
Vm (z)
X(z)
=
Um (z)
X(z)
×
Vm (z)
Um (z)
=
z −m Am (z −1 )
A(z)
Structures: 10 – 15 / 19
Lattice Filter
Label outputs um [n] and vm [n] and define Hm (z) =
Vm (z)
Um (z)
=
z −m Am (z −1 )
Am (z)
From earlier slide (slide 12):
Um−1 (z)
Um (z)
=
1
1+k[m]z −1 H
m−1 (z)
=
Am−1 (z)
Am−1 (z)+k[m]z −m Am−1 (z −1 )
=
Am−1 (z)
Am (z)
Hence:
Um (z)
X(z)
=
Am (z)
A(z)
and
Vm (z)
X(z)
=
Um (z)
X(z)
×
Vm (z)
Um (z)
=
z −m Am (z −1 )
A(z)
V (z)
m
is of order m so you can create any numerator of order M by
The numerator of X(z)
summing appropriate multiples of Vm (z):
w[n] =
PM
m=0 cm vm [n]
DSP and Digital Filters (2017-10122)
Structures: 10 – 15 / 19
Lattice Filter
Label outputs um [n] and vm [n] and define Hm (z) =
Vm (z)
Um (z)
=
z −m Am (z −1 )
Am (z)
From earlier slide (slide 12):
Um−1 (z)
Um (z)
=
1
1+k[m]z −1 H
m−1 (z)
=
Am−1 (z)
Am−1 (z)+k[m]z −m Am−1 (z −1 )
=
Am−1 (z)
Am (z)
Hence:
Um (z)
X(z)
=
Am (z)
A(z)
and
Vm (z)
X(z)
=
Um (z)
X(z)
×
Vm (z)
Um (z)
=
z −m Am (z −1 )
A(z)
V (z)
m
is of order m so you can create any numerator of order M by
The numerator of X(z)
summing appropriate multiples of Vm (z):
w[n] =
PM
m=0 cm vm [n]
DSP and Digital Filters (2017-10122)
⇒ W (z) =
PM
m=0
cm z −m Am (z −1 )
A(z)
Structures: 10 – 15 / 19
Lattice Example
A(z) = A3 (z) = 1 + 0.2z −1 − 0.23z −2 + 0.2z −3
DSP and Digital Filters (2017-10122)
Structures: 10 – 16 / 19
Lattice Example
A(z) = A3 (z) = 1 + 0.2z −1 − 0.23z −2 + 0.2z −3
• k[3] = 0.2 ⇒a2 [ ] =
DSP and Digital Filters (2017-10122)
[1, 0.2, −0.23]−0.2[0.2, −0.23, 0.2]
1−0.22
= [1, 0.256, −0.281]
Structures: 10 – 16 / 19
Lattice Example
A(z) = A3 (z) = 1 + 0.2z −1 − 0.23z −2 + 0.2z −3
• k[3] = 0.2 ⇒a2 [ ] =
[1, 0.2, −0.23]−0.2[0.2, −0.23, 0.2]
1−0.22
• k[2] = −0.281 ⇒a1 [ ] =
DSP and Digital Filters (2017-10122)
= [1, 0.256, −0.281]
[1, 0.256]+0.281[−0.281, 0.256]
1−0.2812
= [1, 0.357]
Structures: 10 – 16 / 19
Lattice Example
A(z) = A3 (z) = 1 + 0.2z −1 − 0.23z −2 + 0.2z −3
• k[3] = 0.2 ⇒a2 [ ] =
[1, 0.2, −0.23]−0.2[0.2, −0.23, 0.2]
1−0.22
• k[2] = −0.281 ⇒a1 [ ] =
= [1, 0.256, −0.281]
[1, 0.256]+0.281[−0.281, 0.256]
1−0.2812
= [1, 0.357]
• k[1] = 0.357 ⇒a0 [ ] = 1
DSP and Digital Filters (2017-10122)
Structures: 10 – 16 / 19
Lattice Example
A(z) = A3 (z) = 1 + 0.2z −1 − 0.23z −2 + 0.2z −3
• k[3] = 0.2 ⇒a2 [ ] =
[1, 0.2, −0.23]−0.2[0.2, −0.23, 0.2]
1−0.22
• k[2] = −0.281 ⇒a1 [ ] =
= [1, 0.256, −0.281]
[1, 0.256]+0.281[−0.281, 0.256]
1−0.2812
= [1, 0.357]
• k[1] = 0.357 ⇒a0 [ ] = 1
V0 (z)
X(z)
=
1
1+0.2z −1 −0.23z −2 +0.2z −3
DSP and Digital Filters (2017-10122)
Structures: 10 – 16 / 19
Lattice Example
A(z) = A3 (z) = 1 + 0.2z −1 − 0.23z −2 + 0.2z −3
• k[3] = 0.2 ⇒a2 [ ] =
[1, 0.2, −0.23]−0.2[0.2, −0.23, 0.2]
1−0.22
• k[2] = −0.281 ⇒a1 [ ] =
= [1, 0.256, −0.281]
[1, 0.256]+0.281[−0.281, 0.256]
1−0.2812
= [1, 0.357]
• k[1] = 0.357 ⇒a0 [ ] = 1
V0 (z)
X(z)
=
1
−1
1+0.2z −0.23z −2 +0.2z −3
DSP and Digital Filters (2017-10122)
V1 (z)
X(z)
=
0.357+z −1
1+0.2z −1 −0.23z −2 +0.2z −3
Structures: 10 – 16 / 19
Lattice Example
A(z) = A3 (z) = 1 + 0.2z −1 − 0.23z −2 + 0.2z −3
• k[3] = 0.2 ⇒a2 [ ] =
[1, 0.2, −0.23]−0.2[0.2, −0.23, 0.2]
1−0.22
• k[2] = −0.281 ⇒a1 [ ] =
= [1, 0.256, −0.281]
[1, 0.256]+0.281[−0.281, 0.256]
1−0.2812
= [1, 0.357]
• k[1] = 0.357 ⇒a0 [ ] = 1
V0 (z)
X(z)
=
1
−1
1+0.2z −0.23z −2 +0.2z −3
V2 (z)
X(z)
=
−0.281+0.256z −1 +z −2
1+0.2z −1 −0.23z −2 +0.2z −3
DSP and Digital Filters (2017-10122)
V1 (z)
X(z)
=
0.357+z −1
1+0.2z −1 −0.23z −2 +0.2z −3
Structures: 10 – 16 / 19
Lattice Example
A(z) = A3 (z) = 1 + 0.2z −1 − 0.23z −2 + 0.2z −3
• k[3] = 0.2 ⇒a2 [ ] =
[1, 0.2, −0.23]−0.2[0.2, −0.23, 0.2]
1−0.22
• k[2] = −0.281 ⇒a1 [ ] =
= [1, 0.256, −0.281]
[1, 0.256]+0.281[−0.281, 0.256]
1−0.2812
= [1, 0.357]
• k[1] = 0.357 ⇒a0 [ ] = 1
V0 (z)
X(z)
V2 (z)
X(z)
=
1
−1
1+0.2z −0.23z −2 +0.2z −3
V1 (z)
X(z)
=
0.357+z −1
1+0.2z −1 −0.23z −2 +0.2z −3
=
−0.281+0.256z −1 +z −2
1+0.2z −1 −0.23z −2 +0.2z −3
V3 (z)
X(z)
=
0.2−0.23z −1 +0.2z −2 +z −3
1+0.2z −1 −0.23z −2 +0.2z −3
DSP and Digital Filters (2017-10122)
Structures: 10 – 16 / 19
Lattice Example
A(z) = A3 (z) = 1 + 0.2z −1 − 0.23z −2 + 0.2z −3
• k[3] = 0.2 ⇒a2 [ ] =
[1, 0.2, −0.23]−0.2[0.2, −0.23, 0.2]
1−0.22
• k[2] = −0.281 ⇒a1 [ ] =
= [1, 0.256, −0.281]
[1, 0.256]+0.281[−0.281, 0.256]
1−0.2812
= [1, 0.357]
• k[1] = 0.357 ⇒a0 [ ] = 1
V0 (z)
X(z)
V2 (z)
X(z)
=
1
−1
1+0.2z −0.23z −2 +0.2z −3
V1 (z)
X(z)
=
0.357+z −1
1+0.2z −1 −0.23z −2 +0.2z −3
=
−0.281+0.256z −1 +z −2
1+0.2z −1 −0.23z −2 +0.2z −3
V3 (z)
X(z)
=
0.2−0.23z −1 +0.2z −2 +z −3
1+0.2z −1 −0.23z −2 +0.2z −3
V (z)
B(z)
m
to create an arbitrary 1+0.2z −1 −0.23z −2 +0.2z −3
Add together multiples of X(z)
DSP and Digital Filters (2017-10122)
Structures: 10 – 16 / 19
Lattice Example Numerator
Form a new output signal as w[n] =
DSP and Digital Filters (2017-10122)
PM
m=0 cm vm [n]
Structures: 10 – 17 / 19
Lattice Example Numerator
Form a new output signal as w[n] =
W (z) =
PM
m=0 cm Vm (z) =
B(z)
−1
1+0.2z −0.23z −2 +0.2z −3 X(z)
DSP and Digital Filters (2017-10122)
PM
m=0 cm vm [n]
Structures: 10 – 17 / 19
Lattice Example Numerator
Form a new output signal as w[n] =
W (z) =
PM
m=0 cm Vm (z) =
B(z)
−1
1+0.2z −0.23z −2 +0.2z −3 X(z)
V0 (z)
X(z)
V2 (z)
X(z)
PM
m=0 cm vm [n]
=
1
1+0.2z −1 −0.23z −2 +0.2z −3
V1 (z)
X(z)
=
−0.281+0.256z −1 +z −2
1+0.2z −1 −0.23z −2 +0.2z −3
V3 (z)
X(z)
DSP and Digital Filters (2017-10122)
=
0.357+z −1
1+0.2z −1 −0.23z −2 +0.2z −3
=
0.2−0.23z −1 +0.2z −2 +z −3
1+0.2z −1 −0.23z −2 +0.2z −3
Structures: 10 – 17 / 19
Lattice Example Numerator
Form a new output signal as w[n] =
W (z) =
PM
m=0 cm Vm (z) =
B(z)
−1
1+0.2z −0.23z −2 +0.2z −3 X(z)
V0 (z)
X(z)
V2 (z)
X(z)
=
PM
m=0 cm vm [n]
1
1+0.2z −1 −0.23z −2 +0.2z −3
V1 (z)
X(z)
−0.281+0.256z −1 +z −2
1+0.2z −1 −0.23z −2 +0.2z −3
V3 (z)
X(z)
=

b[0]
 b[1]
We have
 b[2]
b[3]


=
0.357+z −1
1+0.2z −1 −0.23z −2 +0.2z −3
=
0.2−0.23z −1 +0.2z −2 +z −3
1+0.2z −1 −0.23z −2 +0.2z −3


c0
1 0.357 −0.281
0.2


  0
1
0.256 −0.23 
  c1 
=
  0
0
1
0.2   c2 
c3
0
0
0
1
DSP and Digital Filters (2017-10122)
Structures: 10 – 17 / 19
Lattice Example Numerator
Form a new output signal as w[n] =
W (z) =
PM
m=0 cm Vm (z) =
B(z)
−1
1+0.2z −0.23z −2 +0.2z −3 X(z)
V0 (z)
X(z)
V2 (z)
X(z)
=
PM
m=0 cm vm [n]
1
1+0.2z −1 −0.23z −2 +0.2z −3
V1 (z)
X(z)
−0.281+0.256z −1 +z −2
1+0.2z −1 −0.23z −2 +0.2z −3
V3 (z)
X(z)
=

b[0]
 b[1]
We have
 b[2]
b[3]


1 0.357
  0
1
=
  0
0
0
0
 

c0
 c1  
 
Hence choose cm as 
 c2  = 
c3
DSP and Digital Filters (2017-10122)
=
0.357+z −1
1+0.2z −1 −0.23z −2 +0.2z −3
=
0.2−0.23z −1 +0.2z −2 +z −3
1+0.2z −1 −0.23z −2 +0.2z −3


c0
−0.281
0.2


0.256 −0.23 
  c1 
1
0.2   c2 
c3
0
1
−1 
1 0.357 −0.281
0.2

0
1
0.256 −0.23 
 
0
0
1
0.2  
0
0
0
1

b[0]
b[1] 

b[2] 
b[3]
Structures: 10 – 17 / 19
Summary
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
• Filter block diagrams
◦ Direct forms
◦ Transposition
◦ State space representation
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 18 / 19
Summary
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
• Filter block diagrams
◦ Direct forms
◦ Transposition
◦ State space representation
• Precision issues: coefficient error, arithmetic error
◦ cascaded biquads
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 18 / 19
Summary
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
• Filter block diagrams
◦ Direct forms
◦ Transposition
◦ State space representation
• Precision issues: coefficient error, arithmetic error
◦ cascaded biquads
• Allpass filters
◦ first and second order sections
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 18 / 19
Summary
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Numerator
• Summary
• MATLAB routines
• Filter block diagrams
◦ Direct forms
◦ Transposition
◦ State space representation
• Precision issues: coefficient error, arithmetic error
◦ cascaded biquads
• Allpass filters
◦ first and second order sections
• Lattice filters
◦ Arbitrary allpass response
◦ Arbitrary IIR response by summing intermediate outputs
DSP and Digital Filters (2017-10122)
Structures: 10 – 18 / 19
Summary
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
Numerator
• Summary
• MATLAB routines
• Filter block diagrams
◦ Direct forms
◦ Transposition
◦ State space representation
• Precision issues: coefficient error, arithmetic error
◦ cascaded biquads
• Allpass filters
◦ first and second order sections
• Lattice filters
◦ Arbitrary allpass response
◦ Arbitrary IIR response by summing intermediate outputs
For further details see Mitra: 8.
DSP and Digital Filters (2017-10122)
Structures: 10 – 18 / 19
MATLAB routines
10: Digital Filter Structures
• Direct Forms
• Transposition
• State Space
+
• Precision Issues
• Coefficient Sensitivity
• Cascaded Biquads
• Pole-zero Pairing/Ordering
• Linear Phase
• Hardware Implementation
• Allpass Filters
• Lattice Stage
+
• Example
A(z) ↔ D(z)
• Allpass Lattice
• Lattice Filter
• Lattice Example
• Lattice Example
residuez
tf2sos,sos2tf
zp2sos,sos2zp
zp2ss,ss2zp
tf2ss,ss2tf
poly
b(z −1)
a(z −1 )
b(z −1)
a(z −1 )
↔
→ k 1−prkkz−1
Q b0,l +b1,l z−1 +b2,l z−2
{zm , pk , g} ↔
P
l 1+a1,l z −1 +a2,l z −2
Q b0,l +b1,l z−1 +b2,l z−2
l 1+a∈1,l z −1 +a2,l z −2
(
′
x = Ax + Bu
{zm , pk , g} ↔
y = Cx + Du
(
x′ = Ax + Bu
b(z −1)
a(z −1 ) ↔
y = Cx + Du
poly(A) = det (zI−A)
Numerator
• Summary
• MATLAB routines
DSP and Digital Filters (2017-10122)
Structures: 10 – 19 / 19