Formula sheet: Digital Filter Structure | Digital Signal Processing - Electronics and Communication Engineering (ECE) PDF Download

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Digital Signal Pro cessing: Digital Filter Structures
F orm ula Sheet for GA TE
General Filter Concepts
• T ransfer F unction : F or a linear time-in v arian t (L TI) digital filter.
H(z)=
Y(z)
X(z)
=
?
M
k=0
b
k
z
-k
1+
?
N
k=1
a
k
z
-k
where b
k
are feedforw ard co e?icien ts, a
k
are feedbac k co e?icien ts, M is the n um b er of
zeros, N is the n um b er of p oles.
• Difference Equation : Relates input x(n) and output y(n) .
y(n)=
M
?
k=0
b
k
x(n-k)-
N
?
k=1
a
k
y(n-k)
Direct F orm Structures
• Direct F orm I : Non-canonical structure with separate dela y lines for input and out-
put.
H(z)=
?
M
k=0
b
k
z
-k
1+
?
N
k=1
a
k
z
-k
Requires M +N dela y eleme n ts.
• Direct F orm I I : Canonical structure with shared dela y line.
H(z)=
?
M
k=0
b
k
z
-k
1+
?
N
k=1
a
k
z
-k
Requires max(M,N) dela y elemen ts.
w(n)=x(n)-
N
?
k=1
a
k
w(n-k), y(n)=
M
?
k=0
b
k
w(n-k)
where w(n) is the in termediate signa l at the dela y line.
Cascade F orm
• Cascade Structure : Filter imple men ted as a series of first- or second-order sections.
H(z)=
K
?
i=1
H
i
(z), H
i
(z)=
b
i0
+b
i1
z
-1
+b
i2
z
-2
1+a
i1
z
-1
+a
i2
z
-2
where K is the n um b er of sections, t ypically second-order for stabilit y .
• A dv an tage : Reduces co e?icien t sensitivit y and round-off errors.
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Digital Signal Pro cessing: Digital Filter Structures
F orm ula Sheet for GA TE
General Filter Concepts
• T ransfer F unction : F or a linear time-in v arian t (L TI) digital filter.
H(z)=
Y(z)
X(z)
=
?
M
k=0
b
k
z
-k
1+
?
N
k=1
a
k
z
-k
where b
k
are feedforw ard co e?icien ts, a
k
are feedbac k co e?icien ts, M is the n um b er of
zeros, N is the n um b er of p oles.
• Difference Equation : Relates input x(n) and output y(n) .
y(n)=
M
?
k=0
b
k
x(n-k)-
N
?
k=1
a
k
y(n-k)
Direct F orm Structures
• Direct F orm I : Non-canonical structure with separate dela y lines for input and out-
put.
H(z)=
?
M
k=0
b
k
z
-k
1+
?
N
k=1
a
k
z
-k
Requires M +N dela y eleme n ts.
• Direct F orm I I : Canonical structure with shared dela y line.
H(z)=
?
M
k=0
b
k
z
-k
1+
?
N
k=1
a
k
z
-k
Requires max(M,N) dela y elemen ts.
w(n)=x(n)-
N
?
k=1
a
k
w(n-k), y(n)=
M
?
k=0
b
k
w(n-k)
where w(n) is the in termediate signa l at the dela y line.
Cascade F orm
• Cascade Structure : Filter imple men ted as a series of first- or second-order sections.
H(z)=
K
?
i=1
H
i
(z), H
i
(z)=
b
i0
+b
i1
z
-1
+b
i2
z
-2
1+a
i1
z
-1
+a
i2
z
-2
where K is the n um b er of sections, t ypically second-order for stabilit y .
• A dv an tage : Reduces co e?icien t sensitivit y and round-off errors.
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P arallel F orm
• P arallel Structure : Filter implemen ted as a sum of first- or second-order sections.
H(z)=C +
K
?
i=1
b
i0
+b
i1
z
-1
1+a
i1
z
-1
+a
i2
z
-2
where C is a constan t (for systems with M >N ), K is the n um b er of sections.
• A dv an tage : Suitable for partial fraction expansion of transfer function.
Lattice Structure
• Lattice Structure (FIR) : F or finite i mpulse resp onse filters.
H(z)=
M
?
k=0
h(k)z
-k
Uses reflection co e?icien ts k
i
for implemen tation.
• Lattice Structure (I IR) : F or infinite impulse resp onse filters.
f
m
(n)=f
m-1
(n)+k
m
g
m-1
(n-1), g
m
(n)=k
m
f
m-1
(n)+g
m-1
(n-1)
where k
m
are lattice co e?icien ts, f
m
(n) and g
m
(n) are forw ard and bac kw ard signals.
• A dv an tage : Robust to co e?icien t quan tization, used in adaptiv e filtering.
FIR and I IR Filter Structures
• FIR Filters : Non-recursiv e, inheren tly stable.
H(z)=
M
?
k=0
b
k
z
-k
Structures: Direct form, cascade, linear phase.
• I IR Filters : Recursiv e, feedbac k included.
H(z)=
?
M
k=0
b
k
z
-k
1+
?
N
k=1
a
k
z
-k
Structures: Direct form I/I I, cascade, parallel, lattice.
Key N otes
• Direct F orm I I is preferred for I IR filters due to few er dela y elemen ts.
• Cascade and parallel forms are deriv ed from p ole-zero factorization or partial fraction
expansion.
• Lattice structures are useful for adaptiv e filters and sp eec h pro cessing.
• Ensure prop er co e?icien t pairing in cascade/parallel forms for real-v alued systems.
• F or GA TE, fo cus on blo c k diagram in terpretation and transfer function deriv ation.
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