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

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Digital Sign al Pro cessing: I IR Filter Design F orm ula
Sheet for GA TE
I IR Filter Basics
• Definition : Infinite Impulse Resp onse (I IR) filters ha v e an impulse resp onse of infinite
duration due to feedbac k.
• T ransfer F unction : General form of an I IR filter in the z-domain.
H(z) =
?
M
m=0
b
m
z
-m
?
N
k=0
a
k
z
-k
where b
m
are feedforw ard co e?icien ts, a
k
are feedbac k co e?icien ts, M and N are the
orders of n umerator and denominator.
• Difference Equation :
N
?
k=0
a
k
y[n-k] =
M
?
m=0
b
m
x[n-m]
I IR Filter Design Metho ds
• Analog Protot yp e Filters : I IR filters are designed b y transforming analog filters
to digital domain.
– Butterw orth Filter : Flat passband, monotonic resp onse.
|H
a
(s)|
2
=
1
1+
(
s
j?c
)
2N
where N is t he filter order, ?
c
is the cutoff frequency .
– Cheb yshev T yp e I : Equiripple in passband, monotonic in stopband.
|H
a
(s)|
2
=
1
1+?
2
T
2
N
(?/?
c
)
where ? con trols ripple, T
N
is the N -th order Cheb yshev p olynomial.
– Cheb yshev T yp e I I : Monotonic in passband, equiripple in stopband.
– Elliptic Filter : Equiripple in b oth passband and stopband, sharp est transition.
• Filter Order Calculation : F or Butterw orth, order N is determined b y:
N =
log
(v
10
0.1as
-1
10
0.1ap
-1
)
log
(
?s
?p
)
where a
p
is passband atten uation (dB), a
s
is stopband atten uation (dB), ?
p
is pass-
band frequency , ?
s
is stopband frequency .
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Digital Sign al Pro cessing: I IR Filter Design F orm ula
Sheet for GA TE
I IR Filter Basics
• Definition : Infinite Impulse Resp onse (I IR) filters ha v e an impulse resp onse of infinite
duration due to feedbac k.
• T ransfer F unction : General form of an I IR filter in the z-domain.
H(z) =
?
M
m=0
b
m
z
-m
?
N
k=0
a
k
z
-k
where b
m
are feedforw ard co e?icien ts, a
k
are feedbac k co e?icien ts, M and N are the
orders of n umerator and denominator.
• Difference Equation :
N
?
k=0
a
k
y[n-k] =
M
?
m=0
b
m
x[n-m]
I IR Filter Design Metho ds
• Analog Protot yp e Filters : I IR filters are designed b y transforming analog filters
to digital domain.
– Butterw orth Filter : Flat passband, monotonic resp onse.
|H
a
(s)|
2
=
1
1+
(
s
j?c
)
2N
where N is t he filter order, ?
c
is the cutoff frequency .
– Cheb yshev T yp e I : Equiripple in passband, monotonic in stopband.
|H
a
(s)|
2
=
1
1+?
2
T
2
N
(?/?
c
)
where ? con trols ripple, T
N
is the N -th order Cheb yshev p olynomial.
– Cheb yshev T yp e I I : Monotonic in passband, equiripple in stopband.
– Elliptic Filter : Equiripple in b oth passband and stopband, sharp est transition.
• Filter Order Calculation : F or Butterw orth, order N is determined b y:
N =
log
(v
10
0.1as
-1
10
0.1ap
-1
)
log
(
?s
?p
)
where a
p
is passband atten uation (dB), a
s
is stopband atten uation (dB), ?
p
is pass-
band frequency , ?
s
is stopband frequency .
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Analog-to-Digital T ransformation
• Impulse In v ariance Metho d : Maps analog impulse resp onse to digital.
h[n] =Th
a
(nT), H(z) =
N
?
k=1
A
k
1-e
s
k
T
z
-1
where h
a
(t) is the analog impulse resp onse, T is the sampling p erio d, s
k
are p oles of
the analog filter, A
k
are residues.
• Bilinear T ransformation : Maps s-plane to z-plane, a v oiding aliasing.
s =
2
T
1-z
-1
1+z
-1
H(z) =H
a
(
2
T
1-z
-1
1+z
-1
)
F requency W arping :
?
d
=
2
T
tan
-1
(
?
a
T
2
)
where ?
a
is analog frequency , ?
d
is digital frequency .
• Pre-w arping : A djust analog frequency to comp ensate for bilinear transformation.
?
a
=
2
T
tan
(
?
d
T
2
)
Stabilit y
• Stabilit y Condition : An I IR filter is stable if all p oles of H(z) lie inside the unit
circle (|z|< 1 ).
• Routh-Hurwitz Criterion : Applied to analog protot yp e to ensure stabilit y b efore
transformation.
Key Notes
• Common GA TE F o cus : Bilinear transformation and Butterw orth filter design are
frequen tly tested.
• Units : F requencies in radians/sample (digital) or radians/second (analog); ensure
consistency .
• Filter T yp es : Lo w-pass, high-pass, band-pass, band-stop; transform using frequency
mappings.
• Design Steps : Sp ecify passband/stopband frequencies, calculate order, apply trans-
formation, v erify stabilit y .
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