52
ELECTRONIC FILTERS Celso José Faria de Araújo, M.Sc.
ELE
CT
RO
NIC
FIL
TE
RS
Cel
so Jo
sé F
aria
de
Ara
újo,
M.S
c.
1El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
�A
“Id
eal E
lect
roni
c Fi
lters
” al
low
dis
torti
onle
sstra
nsm
issi
on o
f a c
erta
in b
and
of fr
eque
ncie
s and
supp
ress
al
l the
rem
aini
ng fr
eque
ncie
s of t
he sp
ectru
m o
f the
inpu
t si
gnal
.
�Th
e fr
eque
ncy
spec
trum
is a
repr
esen
tatio
n of
am
plitu
de
vers
us fr
eque
ncie
s of t
his s
igna
l.
2El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
CLA
SS
IFIC
AT
ION
Elec
troni
cFi
lters
Cla
ssifi
catio
nin
term
sof
Tech
nolo
gyU
sed
Cla
ssifi
catio
nin
term
sof
func
tion
acco
mpl
ishe
d
Cla
ssifi
catio
nin
term
sof
Res
pons
e-Fu
nctio
n
Pass
ive
Filte
r
Activ
eFi
lter
Dig
ital
Filte
r
Low
-Pas
sFi
lter
Hig
h-Pa
ssFi
lter
Band
-Pas
sFi
lter
Band
-Rej
ect
Filte
r
Butte
rwor
thFi
lter
Che
bysh
evFi
lter
Elip
ticor
Cau
erFi
lter
3El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
FUN
CT
ION
AC
CO
MP
LIS
HE
D
�
(log
scal
e)�
c
K 2K
Idea
l
Rea
l )(�j
H
Hig
h-Pa
ss F
ilter
�
(log
scal
e)�
c
K 2K
Idea
l Rea
l
)(�j
H Lo
w-P
ass F
ilter
�
(log
scal
e)�
o
K 2K
Idea
l Rea
l
)(�j
H
Ban
d-Pa
ss F
ilter
�c2
�
c1
�
(log
scal
e)�
o
K 2K
Idea
l
Rea
l
)(�j
H
Ban
d-R
ejec
t Filt
er
�c1
�
c2
4El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
AC
TIV
ES
FIL
TE
RS
�A
dvan
tage
:�
Supp
ress
use
of i
nduc
tors
;�
Easy
Des
ign
of C
ompl
ex F
ilter
s by
casc
ade
of si
mpl
e st
age;
�A
con
side
rabl
e am
plifi
catio
n of
inpu
t sig
nal i
s pos
sibl
e;�
Des
ign
Flex
ibili
ty.
�D
isad
vant
ages
:�
Pow
er su
pply
is n
eces
sary
;�
The
freq
uenc
y re
spon
se i
s lim
ited
by a
ctiv
e de
vice
s (O
p-A
mps
, Tr
ansi
stor
s) fr
eque
ncy
resp
onse
;�
It’s n
ot o
ften
appl
y in
med
ium
and
hig
h po
wer
syst
em.
Des
pite
thes
e di
sadv
anta
ges i
t’s w
idel
y us
ed in
seve
ral a
pplic
atio
n, su
ch
as: t
elec
omm
unic
atio
n an
d in
dust
rial i
nstru
men
tatio
n.
5El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
ST
AN
DA
RD
RE
QU
IRE
ME
NT
S F
OR
D
ES
IGN
OF
AP
PR
OX
IMA
TIO
N
FUN
CT
ION
LOW
-PA
SS
FIL
TE
R
�
(log
scal
e)�
p �
s
Loss dB
Am
ax
Am
in
Pass
band
Stop
band
Tran
sitio
n Pa
ssba
nd
6El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
NO
N-F
LAT
ST
OP
BA
ND
R
EQ
UIR
EM
EN
TS
FO
R L
OW
-PA
SS
FI
LTE
R
�
(log
scal
e)�
p �
s1
Loss dB
Am
axAmin1
Pass
band
Amin2
�s2
7El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
ST
AN
DA
RD
RE
QU
IRE
ME
NT
S F
OR
D
ES
IGN
OF
AP
PR
OX
IMA
TIO
N
FUN
CT
ION
HIG
H-P
AS
S F
ILT
ER
Am
in
�
(log
scal
e)�
s �
p
Loss dB
Am
ax
Pass
band
Stop
band
Tran
sitio
n Pa
ssba
nd
8El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
NO
N-F
LAT
ST
OP
BA
ND
R
EQ
UIR
EM
EN
TS
FO
R H
IGH
-PA
SS
FI
LTE
R
Amin1
�
(log
scal
e)�
s1
�p
Loss dB
Am
ax
Pass
band
�s2
Amin1
Amin2
9El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
ST
AN
DA
RD
RE
QU
IRE
ME
NT
S F
OR
D
ES
IGN
OF
AP
PR
OX
IMA
TIO
N
FUN
CT
ION
BA
ND
-PA
SS
FIL
TE
R
�
(log
scal
e)
Loss dB
Pass
band
St
opba
nd
Tran
sitio
nPa
ssba
nd
Tran
sitio
nPa
ssba
nd
Stop
band
�3
�1
�2
�4
Am
ax
Am
in
21
2�
��
�o
12
��
��
�B
43
2�
��
�o
•B�
is th
epa
ssba
ndw
idth
of B
P fil
ter
•�o
is th
ece
nter
(geo
met
ricm
ean)
of t
hepa
ssba
ndof
BP
filte
r
10El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
NO
NS
YM
ME
TR
ICA
L R
EQ
UIR
EM
EN
TS
FO
R B
AN
D-P
AS
S F
ILT
ER
�
(log
scal
e)
Loss dB
Pass
band
�3
�1
�2
�4’
Am
ax
�4
Amin2
Amin1
21
2�
��
�o
12
��
��
�B
' 43
2�
��
�o
•B�
is th
epa
ssba
ndw
idth
of B
P fil
ter
•�o
is th
ece
nter
(geo
met
ricm
ean)
of t
hepa
ssba
ndof
BP
filte
r
11El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
ST
AN
DA
RD
RE
QU
IRE
ME
NT
S F
OR
D
ES
IGN
OF
AP
PR
OX
IMA
TIO
N
FUN
CT
ION
BA
ND
-RE
JEC
T F
ILT
ER
�
(log
scal
e)
Loss dB
Tran
sitio
nPa
ssba
nd
Tran
sitio
nPa
ssba
nd
�1
�3
�4
�2
Am
ax
Am
in
Stop
band
Pass
band
Pa
ssba
nd
21
2�
��
�o
12
��
��
�B
43
2�
��
�o
•B�
is th
epa
ssba
ndw
idth
of B
R fi
lter
•�o
is th
ece
nter
(geo
met
ricm
ean)
of t
hest
opba
ndof
BR
filte
r
12El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
NO
NS
YM
ME
TR
ICA
L R
EQ
UIR
EM
EN
TS
FO
R B
AN
D-R
EJE
CT
FIL
TE
R
�
(log
scal
e)
Loss dB
�1
�3
�4
�2
Amin �
3’
Amax2
Amax1 2
12
��
��
o1
2�
��
��
B4
'2
3�
��
�o
•B�
is th
epa
ssba
ndw
idth
of B
R fi
lter
•�o
is th
ece
nter
(geo
met
ricm
ean)
of t
hest
opba
ndof
BR
filte
r
13El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
A S
EC
ON
D-O
RD
ER
GA
IN F
UN
CT
ION
FO
R L
OW
-PA
SS
FIL
TE
R
22
2
ppp
p
INO
sQ
sVV
GAIN
�
�
�
��
��
�
(log
scal
e)�
p
Loss dB
Slop
e =
40dB
/dec
ade
0dB
�
j� G
ain
14El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
A S
EC
ON
D-O
RD
ER
GA
IN F
UN
CT
ION
FO
R H
IGH
-PA
SS
FIL
TE
R
22
2
ppp
INO
sQ
s
sVV
GAIN
�
�
��
��
�
(log
scal
e)�
p
Loss dB
Slop
e =
-40d
B/d
ecad
e
0dB
�
j�
Gai
n
15El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
A S
EC
ON
D-O
RD
ER
GA
IN F
UN
CT
ION
FO
R B
AN
D-P
AS
S F
ILT
ER
22
ppppp
INO
sQ
s
sQ
VVGAIN
�
��
��
��
�
(log
scal
e)�
p
Loss dB
Slop
e =
-40d
B/d
ecad
e
0dB
Slop
e =
40dB
/dec
ade
�
j�
Gai
n
16El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
A S
EC
ON
D-O
RD
ER
GA
IN F
UN
CT
ION
FO
R B
AN
D-R
EJE
CT
FIL
TE
R
22
22
ppp
p
INO
sQ
s
sVV
GAIN
�
�
�
��
�
��
�
(log
scal
e)�
p
Loss dB
0dB
�
j�
Gai
n
�p
17El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
A S
EC
ON
D-O
RD
ER
GA
IN F
UN
CT
ION
FO
R L
OW
-PA
SS
NO
TC
H F
ILT
ER
pz
ppp
z
INO
sQ
s
sVV
GAIN
��
�
�
�
��
��
��
�;
22
22
�
j�
Gai
n j�
z -j�z �
p
�
(log
scal
e)�
p
Loss dB
�z
18El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
A S
EC
ON
D-O
RD
ER
GA
IN F
UN
CT
ION
FO
R H
IGH
-PA
SS
NO
TC
H F
ILT
ER
pz
ppp
z
INO
sQ
s
sVV
GAIN
��
�
�
�
��
��
��
�;
22
22
�
j�
Gai
n
j�z
-j�z �
p
�
(log
scal
e)
Loss dB
�p
�z
19El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
GA
IN E
QU
ALI
ZE
RS
20El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
DE
LAY
EQ
UA
LIZ
ER
S
)(
)(
oin
oT
tv
tv
��
osT
ino
es
Vs
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()
(
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Vs
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)(
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ej
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)(
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H
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(��
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dddelay
21El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
DE
LAY
EQ
UA
LIZ
ER
SS
econ
d-or
der
(all-
pass
)
22
22
ppp
ppp
INO
sQ
s
sQ
s
VVGAIN
�
�
�
�
��
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)(
)(
0)
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22
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2
22
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ppp
p
pp
Q
Qdd
delay
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j�
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n
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22El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
EQ
UA
LIZ
AT
ION
OF
CA
BLE
DE
LAY
23El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
Filt
erpo
le-z
ero
patt
erns
24El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
Bod
e P
lot
App
roxi
mat
ion
Tec
hniq
ue
�
100
400
Loss dB
3dB
35dB
32
/40
0/
100
3535
3
�
��
��
�
order
oitava
srad
srad
dBdB
dB
��
�� ��
�� ���
��
�2
210
010
010
0)
(s
Qs
s
Ks
H
p
62
010
110
010
0)
(�
��
��
KK
sH
s ��
13
100
100
100
10lo
g20
100
22
6
��
��
�� ��
�� �
�
p
js
p
Qs
Qs
s
)10
010
0)(
100
(10
)(
22
6
��
�
�
ss
ss
H
Freq
uenc
y(r
ad/s
)
Loss(dB)
25El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
BU
TT
ER
WO
RT
H A
PP
OX
IMA
TIO
NR
equi
rem
ents
: �
p�
Am
ax�
s�
Am
in
2
1,0
1,0
log
110
110
log
max
min �� ��
�� ��
�� ���� ��
��
�
psAA
n
��
Ord
er:
�C
orre
ctio
n Fa
ctor
:1
101,0
��
máx
A
Roo
tsof
a th
ird-o
rder
norm
aliz
edfil
ters
�
j �
Gai
n
-1
1
�/3
The
pole
s are
just
the
root
s at l
eft h
alf p
lane
Roo
tsof
the
norm
aliz
edfil
ters
:
nk
onde
eS
nnk
j
k2
....,
,2,11
22
��
� ��� ��
� � ��
�
�
n1/
H(S
)1
S+
12
S2+
1,41
4S+
13
(S2
+S
+1)
(S+1
)4
(S2
+0,
7653
7S+
1)(S
2+
1,84
776S
+1)
5(S
2+
0,61
803S
+1)
(S2
+1,
6180
3S+
1)(S
+1)
�� ��
�� ���
pns
S��
1
Den
orm
aliz
atio
n:
26El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
Mag
nitu
de re
spon
se fo
r But
terw
orth
filte
rs o
f var
ious
ord
er w
ith�
= 1.
Not
e th
at a
s the
ord
er in
crea
ses,
the
resp
onse
app
roac
hest
he
idea
l bric
kwal
ltyp
e tra
nsm
issi
on.
27El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
Gra
phic
al c
onstr
uctio
n fo
r det
erm
inin
g th
e po
les o
f a B
utte
rwor
th fi
lter o
f ord
er N
. A
ll th
e po
les l
ie in
the
left
half
of th
e s-
plan
e on
a
circ
le o
f rad
ius �
0= �
p(1/�
)1/N, w
here
�is
the
pass
band
devi
atio
n pa
ram
eter
: (a
)the
gen
eral
cas
e, (
b)N
= 2,
(c)N
= 3,
(d)N
= 4.
�� 1
1010/
max
��
A�
28El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
BU
TT
ER
WO
RT
H A
PP
OX
IMA
TIO
NE
xam
ple
Req
uire
men
ts: �
p=
100
rad/
sA
max
= 0.
5 dB
�
s=
400
rad/
sA
min
= 12
dB
273.1
100
400
log
110
110
log
:2
5.01,0
121,0
��
�
� ��� ��
�� ���� ��
�xx
norder
35.01
10:
5.01,0
��
�x
factor
Correction
�
ss
Sation
Denormaliz
pn00
59.0
:1
��� ��
�� ���
��
141
4.11
)(
2�
�
�
SS
SH
Normalized
4.28
727
6.23
94.
2872
7)
(2
��
�
ss
sH
edDenormaliz
29El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
CH
EB
YS
HE
V A
PP
OX
IMA
TIO
NR
equi
rem
ents
: �
p�
Am
ax�
s�
Am
in
��
pss
sAA
where
nOrder
máx
�
��
�
�� ��
�� ��
��
�
1
1,0
1,01 co
sh
110
110
cosh
:
min
�
j �
Gai
n
The
pole
s are
just
the
root
s at l
eft h
alf p
lane
� ��� ��
�
�11
cosh
1sinh
n
� ��� ��
�
�11
1sinh
nsinh
110
1,0�
�máx
A�
Roo
tsof
a th
ird-o
rder
norm
aliz
edfil
ters
�
Roo
tsof
the
norm
aliz
edfil
ters
:
� ��� ��
� � ��
� ��� ��
�
�
� ��� ��
� � ��
� ��� ��
�
�
��
�
��
�
�
�
�
�
�
��
1se
nh1
cosh
21
2co
s
1se
nh1
senh
21
2se
n
12
,...,
2,1,0
11
nnk
nnk
nk
where
jS kk
kk
k
psS
ation
Denormaliz
�
�:
30El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
Sket
ches
of t
he tr
ansm
issi
on c
hara
cter
istic
s of a
repr
esen
tativ
eev
en-a
nd o
dd-o
rder
Che
bysh
evfil
ters
.
31El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
CH
EB
YS
HE
V A
PP
OX
IMA
TIO
N(O
rder
from
Plo
t)
Req
uire
men
ts: �
p�
Am
ax�
s�
Am
in1
��
��
��
��
pps
sp
��
��
Loss
of L
P C
heby
shev
appr
oxim
atio
nfo
r Am
ax=
0.50
dBLo
ssof
LP
Che
bysh
evap
prox
imat
ion
for A
max
= 1d
BLo
ssof
LP
Che
bysh
evap
prox
imat
ion
for A
max
= 0.
25dB
32El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
CH
EB
YS
HE
V A
PP
OX
IMA
TIO
N(P
olyn
omin
alfr
omT
able
)R
equi
rem
ents
: �
p�
Am
ax�
s�
Am
in
A=
0.25
dBN
umer
ator
of 1
/H(S
)D
enom
inat
or(k
)m
axn 1
S+
4.10
811
4.10
811
2S2
+1.
7966
8S+
2.11
403
2.05
405
3(S
2+
0.76
722S
+1.
3386
3)(S
+0.7
6722
)1.
0270
24
(S2
+0.
4250
4S+
1.16
195)
(S2
+1.
0261
3S+
0.45
485)
0.51
352
5(S
2+
0.27
005S
+1.
0954
3)(S
2+
0.70
700S
+0.
5364
2)(S
+0.
4369
5)0.
2567
6
Am
ax=
0.50
dBn
Num
erat
orof
1/H
S)D
enom
inat
or(k
)
1S
+2.
8627
82.
8627
82
S2+
1.42
562S
+1.
5162
01.
4313
83
(S2
+0.
6264
6S+
1.14
245)
(S+0
.626
46)
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570
4(S
2+
0.35
071S
+1.
0635
2)(S
2+
0.84
668S
+0.
3564
12)
0.35
785
5(S
2+
0.22
393S
+1.
0357
8)(S
2+
0.58
625S
+0.
4767
7)(S
+0.
3623
32)
0.17
892
Am
ax=
1dB
nN
umer
ator
of 1
/H(S
)D
enom
inat
or(k
)
1S
+1.
9652
31.
9652
32
S2+
1.09
773S
+1.
1025
10.
9826
13
(S2
+0.
4941
7S+
0.99
420)
(S+0
.494
17)
0.49
130
4(S
2+
0.27
907S
+0.
9865
0)(S
2+
0.67
374S
+0.
2794
0)0.
2456
5
5(S
2+
0.17
892S
+0.
9883
1)(S
2+
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841S
+0.
4293
0)(S
+0.
2894
9)0.
1228
3
psS
ation
Denormaliz
�
�:
33El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
CH
EB
YS
HE
V A
PP
OX
IMA
TIO
NE
xam
ple
Req
uire
men
ts: �
p=
200
rad/
sA
max
= 0.
5 dB
�
s=
600
rad/
sA
min
= 20
dB
33
200
600
��
��
�n
graph
from
s
200
:s
Sation
Denormaliz
�
)(14
245
.162
646
.0(
7157
0.0
)(
2�
�
�
SS
SS
H
)3.12
5)(
4569
83.
125
(57
2560
0)
(2
��
�
�
ss
ss
H
edDenormaliz
From
tabl
e H
(S) n
orm
aliz
ed is
obta
ined
)62
646
.0�
Ban
dpas
sDet
ails
34El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
ELL
IPT
IC (
CA
UE
R)
AP
PO
XIM
AT
ION
Req
uire
men
ts: �
p�
Am
ax�
s�
Am
in
pss
��
��
psS
ation
Denormaliz
�
�:
H(s
) = L
ossF
unct
ion
35El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
ELL
IPT
IC A
PP
OX
IMA
TIO
NE
xam
ple
Req
uire
men
ts: �
p=
200
rad/
sA
max
= 0.
5 dB
�
s=
600
rad/
sA
min
= 20
dB
23
200
600
��
��
�n
table
from
s
200
:s
Sation
Denormaliz
�
3571
5.1
()
4852
8.
17(
0839
74.0
)(
2
2
��
��
SS
SS
H
)8.62
212
4.27
1(
)69
9411
(08
3974
.0)
(2
2
��
��
ss
ss
H
edDenormaliz
From
tabl
e H
(S) n
orm
aliz
ed is
obta
ined
Ban
dpas
sDet
ails
)55
532
.1
36El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
Pha
sean
dde
lay
char
acte
rist
ics
Cha
ract
eris
tics o
f a fo
urth
-ord
er C
heby
shev
(Am
ax=
0.5d
B);
(a) L
oss,
(b) D
elay
, (c)
Ste
p in
put,
(d) S
tep
resp
onse
.C
hara
cter
istic
s of a
four
th-o
rder
But
terw
orth
(Am
ax=
3dB
);
(a) L
oss,
(b) D
elay
, (c)
Ste
p re
spon
se.
37El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
BE
SS
EL
AP
PR
OX
IMA
TIO
NH
(S) n
orm
aliz
ed is
)(
)0()
(S
BBS
Hnn
�
Whe
re B
n(S)
is th
e nt
h or
der p
olyn
omia
l whi
ch
is d
efin
ed b
y th
e fo
llow
ing
recu
rsiv
e eq
uatio
n
1)
(1
)(
10
���
SS
BS
B
)(
)(
)12(
)(
22
1S
BS
SB
nS
Bn
nn
��
��
�an
d
945
(S2 +
6.70
391S
+14.
2725
) (S2 +
4.64
934S
+18.
1563
1) (S
+3.6
4674
)5
105
(S2 +
5.79
242S
+9.1
4013
) (S2 +
4.20
758S
+11.
4878
)4
15(S
2 +3.
6778
2S+6
.459
44)(
S+2.
3221
9)3
3S2 +
3S+3
21
S+1
1N
umer
ator
Den
omin
ator
nB
esse
l App
roxi
mat
ion
Func
tion
in N
orm
aliz
ed a
nd F
acto
red
Form
Loss
of L
P B
esse
l App
roxi
mat
ions
oT�
��
pps
s�
��
��
ppo
oT
sTS
���
��
��� ��
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�
�� ��
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2(1
22
2
250
%10
0(%
)n
nn
pp
o
p
p
en
TTDelay
�
�
�
�
Den
orm
aliz
atio
n
Del
ay o
f LP
Bes
sel A
ppro
xim
atio
ns
38El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
1.T
ry a
four
th-o
rder
Filt
er.
1% �
0 …
�p=
1.9
(fro
m d
elay
B
esse
l app
roxi
mat
ions
plo
ts)
�s=
(6/2
)1.9
= 5
.7A
tten
uatio
n is
onl
y 22
dB
��
p= 5
.7 (
from
loss
Bes
sel
appr
oxim
atio
ns p
lots
)
Den
orm
aliz
atio
n
opp
osT
SK
T�
��
��
��
�
sec
1098
9.1
22
5.24
��
Solu
tion
2.T
ry a
fift
h-or
der
Filte
r. 1
% �
0 …
�p=
2.5
(fro
m d
elay
B
esse
l app
roxi
mat
ions
plo
ts)
�s=
(6/2
)2.5
= 7
.5A
tten
uatio
n is
29.
5 dB
��
p= 5
.7 (
from
loss
Bes
sel
appr
oxim
atio
ns p
lots
)
3.64
674)
(S 18
.156
31)
4.64
934S
(S 14
.272
5)6.
7039
1S(S
945
)(
22
��
��
�
�S
HFrom
tabl
e :
4
4
84
2
8
84
2
8
1083
35.1
1083
35.1
1058
94.4
s10
338
.2s
1058
94.4
10
608
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1037
0.3
s10
608
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(�
�
�
��
��
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��
��
��
ss
H%
52,0%
100
(%)
2
��
�
�kHz
foTT
Delay
BE
SS
EL
AP
PR
OX
IMA
TIO
NE
xam
ple
Req
uire
men
ts:
a)Th
e de
lay
mus
t be
flat w
ithin
1 p
erce
nt o
f the
DC
val
ue u
p to
2K
Hz.
b)Th
e at
tenu
atio
n at
6K
Hz
mus
t exc
eed
25dB
.
39El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
DE
LAY
EQ
UA
LIZ
ER
SFu
ncti
onA
ppro
xim
atio
n
bas
sb
ass
VVGAIN
INO
��
��
��
22 b�
��
� ��
�
��
��
N ii
i
ii
INO
bsa
sb
sa
sVV
GAIN
122
The
num
ber o
f del
ay se
ctio
ns N
and
thei
r def
inin
g pa
ram
eter
s (ai
, bi)
for a
ppro
xim
atin
g a
give
n de
lay
shap
e ar
e us
ually
obt
aine
d by
com
pute
r op
timiz
atio
n.
40El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
DE
LAY
EQ
UA
LIZ
ER
SE
xam
ple
Del
ay E
qual
izat
ion
of a
fout
h-or
der C
heby
shev
(Am
ax=0
.25d
B, p
assb
and
edge
= 1
rad/
sec)
41El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
FRE
QU
EN
CY
TR
AN
SFO
RM
AT
ION
S
HP,
BP
or B
Rre
quire
men
ts
HP
HB
P(s)
H
BR(s
) LP
requ
irem
ents
H
LP(s
) H
(s)
Blo
ck d
iagr
am o
f the
freq
uenc
y tra
nsfo
rmat
ion
proc
edur
e
42El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
HIG
H-P
AS
S F
ILT
ER
SR
equi
rem
ents
sps
s
pp
AA
AA
��
��
��
�
��
���
1m
axm
axm
inm
in
Nor
mal
ized
LPH
P
�H
(S) L
PN
orm
aliz
ed is
obt
aine
d fr
om L
P R
equi
rem
ents
Nor
mal
ized
�H
(S) L
Pno
rmal
ized
can
be
trans
form
ed
to a
hig
h-pa
ss fu
nctio
n by
the
freq
uenc
y tra
nsfo
rmat
ion
sS
p�
�
sS
LPHP
pS
Hs
H�
�
�)
()
(
Exam
ple:
2/
500
1/
1000
3m
ax3
max
15m
in15
min
Nor
mal
ized
LPH
P
��
��
�
��
��
��
�
��
�
sps
srad
s
ps
rad
p
dBA
dBA
dBA
dBA
��
�
�
)1)(1
(1
)(
2�
��
�
SS
SS
LP
)10
)(10
10(
)(
36
32
3
��
�
�
ss
ss
sHHP
sS
LPHP
SH
sH
1000
)(
)(
�
�
1K ra
d/s
-> -
3dB
500
rad/
s->
-18
dB
HB
utte
rwor
th
43El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
BA
ND
-PA
SS
FIL
TE
RS
Req
uire
men
ts
�
��
��
��
��
�
�
34
12
43
21
1m
axm
axm
inm
in
Nor
mal
ized
LPB
P
��
��
�
��
�
��
�
s
po
AA
AA
5553
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3571
5.1
4852
8.
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3947
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(2
2
��
��
SS
SS
HLP
Ellip
tic
1410
27
33
4
142
84
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1042.8
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1026.4
1089.3
1012.2
084
.0)
(�
��
��
��
�
��
��
�
ss
ss
ss
sHHP
ss
SLP
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Hs
H50
02
5000
004
22
)(
)(
�
��
��
�H
(S) B
PN
orm
aliz
ed is
obt
aine
d fr
om L
P R
equi
rem
ents
Nor
mal
ized
�H
(S) L
Pno
rmal
ized
can
be
trans
form
ed
to a
ban
d-pa
ss fu
nctio
n by
the
freq
uenc
y tra
nsfo
rmat
ion
ss
So
�
�2
2�
�
ss
SLP
BPo
SH
sH
�
�2
2)
()
(�
�
�
Exam
ple:
08.350
010
00
275
1818
/18
182
'
/27
52
1/
1000
2
/50
02
5.0m
ax5.0
max
20m
in20
min
Nor
mal
ized
LPB
P
4321
�
���
��
��
��
�
��
��
�
��
�
ss
rad
srad
ps
rad
srad
dBA
dBA
dBA
dBA
��
��
��
��
Am
ax=
0.5d
B
Am
in=
20 d
BPa
ssba
nd=
500
Hz
to 1
000
Hz
Stop
band
s= D
C to
275
Hz
and
2000
Hz
to �
44El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
BA
ND
-RE
JEC
T F
ILT
ER
S 1025
1.1
0977
3.1
9826
1.0
)(
2�
�
�
SS
SHLP
6000
000
450
002
22
)(
)(
�
�
�
�
�
ss
SLP
BPS
Hs
H
Che
bysh
evR
equi
rem
ents
34
12
43
21
1m
axm
axm
inm
in
Nor
mal
ized
LPB
R
��
��
�
�
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s
po
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AA
1612
23
4
162
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105.
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409
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610
347
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��
��
��
ss
ss
ss
sHHP
�H
(S) B
RN
orm
aliz
ed is
obt
aine
d fr
om
LP R
equi
rem
ents
Nor
mal
ized
�H
(S) L
Pno
rmal
ized
can
be
trans
form
ed
to a
ban
d-re
ject
func
tion
by th
e fr
eque
ncy
trans
form
atio
n
22
os
sS
�
� �
�
22
)(
)(
os
sS
LPBP
SH
sH
�
� �
�
�
Exam
ple:
50002
3000
000
100
06
/30
002
/20
002
'
1/
6000
2
/10
002
1m
ax1
max
20m
in20
min
Nor
mal
ized
LPB
P
4321
�
���
��
��
��
�
��
��
�
��
�
ss
rad
srad
ps
rad
srad
dBA
dBA
dBA
dBA
��
��
��
��
Am
ax=
1dB
A
min
= 20
dB
Pass
band
s= b
elow
100
0 H
z an
d ab
ove
6000
Hz
Stop
band
= 25
00 H
z to
300
0 H
z
45El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
Firs
t-ord
er fi
lters
.
46El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
Firs
t-ord
er a
ll-pa
ss fi
lter.
47El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
Seco
nd-o
rder
filte
ring
func
tions
.
48El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
Seco
nd-o
rder
filte
ring
func
tions
. (co
ntin
ued
1)
49El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.Se
cond
-ord
er fi
lterin
g fu
nctio
ns. (
cont
inue
d 2)
50El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
Rea
lizat
ion
of v
ario
us se
cond
-ord
er fi
lter f
unct
ions
usin
g th
e LC
R re
sona
tor:
(a)g
ener
al st
ruct
ure,
(b)L
P, (
c)H
P, (
d)B
P, (
e)no
tch
at �
0, (f
)gen
eral
not
ch,
(g)L
PN (�
n��
0), (
h)LP
N a
s s �
�,
(i) H
PN (�
n< �
0).
51El
ectro
nic
Filte
rsEl
ectri
cal C
ircui
ts -
Cel
so J
osé
Faria
de
Araú
jo, M
.Sc.
RE
FER
EN
CE
S�
SED
RA
, A
del
S. a
nd S
MIT
H,
Ken
neth
C.
Mic
roel
ectro
nic
Circ
uits.
Oxf
ord
Uni
vers
ity P
ress
.�
DA
RY
AN
AN
I, G
obin
d.
Prin
cipl
esof
Ac
tive
Net
work
Synt
hesis
and
Des
ign.
John
Wile
y&
Son
s.�
LATH
I, B
. P. S
igna
l Pro
cess
ing
& L
inea
r Sys
tem
s. B
erke
ley-
Cam
brid
ge P
ress
.�
RU
STO
N,
Hen
ry
and
BO
RD
OG
NA
, Jo
seph
. El
ectri
cN
etwo
rks:
func
tions
, filt
ers,
anal
ysis.
Mac
Gra
w-H
ill.
�N
OC
ETI
FILH
O,
Sidn
ei.
Filto
sSe
leto
res
de S
inai
s. Ed
itora
daU
FSC
.
