Transfer Filters

Dorf, R.C., Wan, Z. Transfer Functions of Filters
The Electrical Engineering Handbook
Ed. Richard C. Dorf
Boca Raton: CRC Press LLC, 2000
© 2000 by CRC Press LLC
10
Transler !uncfIons
ol !IIfers
10.1 Intioduction
10.2 Ideal Filteis
10.3 The Ideal Lineai-Phase Low-Pass Filtei
10.4 Ideal Lineai-Phase Bandpass Filteis
10.5 Causal Filteis
10.6 Butteiwoith Filteis
10.7 Chebyshev Filteis
10.1 Intruductiun
Filteis aie widely used to pass signals at selected fiequencies and ieject signals at othei fiequencies. An e|etìríta|
f|ìer is a ciicuit that is designed to intioduce gain oi loss ovei a piesciibed iange of fiequencies. In this section,
we will desciibe ideal flteis and then a selected set of piactical flteis.
10.2 Idea! Fi!ters
An ideal nlter is a system that completely iejects sinusoidal inputs of the foim x(ì) · À cos uì, -~ < ì < ~,
foi u in ceitain fiequency ianges and does not attenuate sinusoidal inputs whose fiequencies aie outside these
ianges. Theie aie foui basic types of ideal flteis: low-pass, high-pass, bandpass, and bandstop. The magnitude
functions of these foui types of flteis aie displayed in Fig. 10.1. Mathematical expiessions foi these magnitude
functions aie as follows:
(10.1)
(10.2)
(10.3)
(10.4)
Ideal low-pass:

H
B B
B
( )
,
,
u
u
u

÷ s s
>
¹
,
¹
¹
¹
1
0
Ideal high-pass:

H
B B
B
( )
,
,
u
u
u

÷ < <
>
¹
,
¹
¹
¹
0
1
Ideal bandpass:
all othei


H
B B
( )
,
,
u
u
u

s s
¹
,
¹
¹
¹
1
0
1 2
Ideal bandstop:
all othei


H
B B
( )
,
,
u
u
u

s s
¹
,
¹
¹
¹
0
1
1 2
RIchard C. Iorl
Iníver·íry of Co|ífornío, Doví·
Zhen Wan
Iníver·íry of Co|ífornío, Doví·
© 2000 by CRC Press LLC
The stopband of an ideal fltei is defned to be
the set of all fiequencies u foi which the fltei
completely stops the sinusoidal input x(ì) · À cos
uì, -~ < ì < ~. The passband of the filtei is the
set of all fiequencies u foi which the input x(ì) is
passed without attenuation.
Moie complicated examples of ideal flteis can
be constiucted by cascading ideal low-pass, high-
pass, bandpass, and bandstop flteis. Foi instance,
by cascading bandstop flteis with diffeient values
of B
1
and B
2
, we can constiuct an ideal comb fltei,
whose magnitude function is illustiated in Fig. 10.2.
10.3 The Idea! Linear-Phase Luv-Pass Fi!ter
Considei the ideal low-pass fltei with the fiequency function
(10.5)
wheie ì
J
is a positive ieal numbei. Equation (10.5) is the polai-foim iepiesentation of H(u). Fiom Eq. (10.5)
we have
and
FIGURE 10.1 Magnitude functions of ideal flteis:(a) low-pass; (b) high-pass; (c) bandpass; (d) bandstop.
| |
1
0
÷
(a)
1
1
0
÷
1
(c)
2
÷
2
1
0
(d)
1
0
(b)
u
| |
| | | |
÷
u u
u
1
÷
1 2
÷
2
FIGURE 10.2 Magnitude function of an ideal comb fltei.
| |
1
0
÷
4
÷
3
÷
2
÷
1 1 2 3 4
H
e B B
B B
, ì
J
( )
,
, ,
u
u
u u
u

÷ s s
< ÷ >
¹
,
¹
¹
¹
÷
0
H
B B
B B
( )
,
, ,
u
u
u u

÷ s s
< ÷ >
¹
,
¹
¹
¹
1
0
/
( )
,
, ,
H
ì B B
B B
J
u
u u
u u

÷ ÷ s s
< ÷ >
¹
,
¹
¹
¹0
© 2000 by CRC Press LLC
The phase function
/
H(u) of the fltei is plotted in Fig. 10.3. Note that ovei the fiequency iange 0 to B, the
phase function of the system is lineai with slope equal to -ì
J
.
The impulse iesponse of the low-pass fltei defned by Eq. (10.5) can be computed by taking the inveise
Fouiiei tiansfoim of the fiequency function H(u). The impulse iesponse of the ideal lowpass fltei is
(10.6)
wheie Sa(x) · (sin x)/x. The impulse iesponse |(ì) of the ideal low-pass fltei is not zeio foi ì < 0. Thus, the
fltei has a iesponse befoie the impulse at ì · 0 and is said to be noncausal. As a iesult, it is not possible to
build an ideal low-pass fltei.
10.4 Idea! Linear-Phase Bandpass Fi!ters
One can extend the analysis to ideal lineai-phase bandpass flteis. The fiequency function of an ideal lineai-
phase bandpass fltei is given by
wheie ì
J
, B
1
, and B
2
aie positive ieal numbeis. The magnitude function is plotted in Fig. 10.1(c) and the phase
function is plotted in Fig. 10.4. The passband of the fltei is fiom B
1
to B
2
. The fltei will pass the signal within
the band with no distoition, although theie will be a time delay of ì
J
seconds.
FIGURE 10.3 Phase function of ideal low-pass fltei defned by Eq. (10.5).
FIGURE 10.4 Phase function of ideal lineai-phase bandpass fltei.
(u)
0
÷
Slope = ÷
u
(u)
2
0
u
Slope = ÷
1
÷
2
÷
1 2 1
| ì
B
Sa B ì ì ì
J
( ) ¦ ( )], ÷ ÷ ~ < < ~
r
H
e B B
, ì
J
( )
,
,
u
u
u
u

s s
¹
,
¹
¹
¹
÷
1 2
0

all othei
© 2000 by CRC Press LLC
10.5 Causa! Fi!ters
As obseived in the pieceding section, ideal flteis cannot be utilized in ieal-time flteiing applications, since
they aie noncausal. In such applications, one must use causal nlters, which aie necessaiily nonideal; that is,
the tiansition fiom the passband to the stopband (and vice veisa) is giadual. In paiticulai, the magnitude
functions of causal veisions of low-pass, high-pass, bandpass, and bandstop flteis have giadual tiansitions
fiom the passband to the stopband. Examples of magnitude functions foi the basic fltei types aie shown in
Fig. 10.5.
Foi a causal fltei with fiequency function H(u), the passband is defned as the set of all fiequencies u foi
which
(10.7)
wheie u
p
is the value of u foi which H(u) is maximum. Note that Eq. (10.7) is equivalent to the condition
that H(u)
dB
is less than 3 dB down fiom the peak value H(u
p
)
dB
. Foi low-pass oi bandpass flteis, the width
of the passband is called the 3-dB bandwidth.
A stopband in a causal fltei is a set of fiequencies u foi which H(u)
dB
is down some desiied amount (e.g., 40
oi 50 dB) fiom the peak value H(u
p
)
dB
. The iange of fiequencies between a passband and a stopband is called a
transition region. In causal fltei design, a key objective is to have the tiansition iegions be suitably small in extent.
10.6 Buttervurth Fi!ters
The tiansfei function of the two-pole Butteiwoith fltei is
Factoiing the denominatoi of H(s), we see that the poles aie located at
FIGURE 10.5 Causal fltei magnitude functions: (a) low-pass; (b) high-pass; (c) bandpass; (d) bandstop.
0
u
u
p
÷u
p
1 0.707
(a)
0
u
1
(b)
0
u
1
(c)
0
u
1
(d)
H H H
¡ ¡
( ) ( ) . ( ) u u u >
1
2
0 707
H s
s s
n
n n
( )


+ +
u
u u
2
2 2
2
s ,
n n
÷ ±
u u
2 2
© 2000 by CRC Press LLC
Note that the magnitude of each of the poles is equal to u
n
.
Setting s · ,u in H(s), we have that the magnitude function of the two-pole Butteiwoith fltei is
(10.8)
Fiom Eq. (10.8) we see that the 3-dB bandwidth of the Butteiwoith fltei is equal to u
n
. Foi the case u
n
· 2
iad/s, the fiequency iesponse cuives of the Butteiwoith fltei aie plotted in Fig. 10.6. Also displayed aie the
fiequency iesponse cuives foi the one-pole low-pass fltei with tiansfei function H(s) · 2/(s - 2), and the two-
pole low-pass fltei with ¸ · 1 and with 3-dB bandwidth equal to 2 iad/s. Note that the Butteiwoith fltei has
the shaipest cutoff of all thiee flteis.
10.7 Chebyshev Fi!ters
The magnitude function of the n-pole Butteiwoith fltei has a monotone chaiacteiistic in both the passband
and stopband of the fltei. Heie monoìone means that the magnitude cuive is giadually decieasing ovei the
passband and stopband. In contiast to the Butteiwoith fltei, the magnitude function of a type 1 Chebyshev
fltei has iipple in the passband and is monotone decieasing in the stopband (a type 2 Chebyshev fltei has the
opposite chaiacteiistic). By allowing iipple in the passband oi stopband, we aie able to achieve a shaipei
tiansition between the passband and stopband in compaiison with the Butteiwoith fltei.
The n-pole type 1 Chebyshev fltei is given by the fiequency function
(10.9)
wheie T
n
(u/u
1
) is the nth-oidei Chebyshev polynomial. Note that r is a numeiical paiametei ielated to the
level of iipple in the passband. The Chebyshev polynomials can be geneiated fiom the iecuision
T
n
(x) · 2xT
n - 1
(x) - T
n - 2
(x)
wheie T
0
(x) · 1 and T
1
(x) · x. The polynomials foi n · 2, 3, 4, 5 aie
T
2
(x) · 2x(x) - 1 · 2x
2
- 1
T
3
(x) · 2x(2x
2
- 1) - x · 4x
3
- 3x
T
4
(x) · 2x(4x
3
- 3x) - (2x
2
- 1) · 8x
4
- 8x
2
- 1
T
5
(x) · 2x(8x
4
- 8x
2
- 1) - (4x
3
- 3x) · 16x
5
- 20x
3
- 5x (10.10)
FIGURE 10.6 Magnitude cuives of one- and two-pole low-pass flteis.
2
+2
u
Two-pole Butterworth filter
Two-pole filter with ¸ = 1
One-pole filter ( ) =
1 2 3 4 5 6 7 8 9 0 10
0.707
1
0.8
0.6
0.4
0.2
0
Passband
| (u)|
H
n
( )
( / )
u
u u

+
1
1
4
H
T
n
( )
( )
u
u u

+
1
1
2 2
1
/
© 2000 by CRC Press LLC
Using Eq. (10.10), the two-pole type 1 Chebyshev fltei has the following fiequency function
Foi the case of a 3-dB iipple ( · 1), the tiansfei functions of the two-pole and thiee-pole type 1 Chebyshev
flteis aie
wheie u
t
· 3-dB bandwidth. The fiequency cuives foi these two flteis aie plotted in Fig. 10.7 foi the case u
t
· 2.5 iad.
The magnitude iesponse functions of the thiee-pole Butteiwoith fltei and the thiee-pole type 1 Chebyshev
fltei aie compaied in Fig. 10.8 with the 3-dB bandwidth of both flteis equal to 2 iad. Note that the tiansition
fiom passband to stopband is shaipei in the Chebyshev fltei; howevei, the Chebyshev fltei does have the 3-
dB iipple ovei the passband.
FIGURE 10.7 Fiequency cuives of two- and thiee-pole Chebyshev flteis with u
t
· 2.5 iad/s: (a) magnitude cuives; (b)
phase cuives.
u
Three-pole filter
Two-pole filter
1 2 3 4 5 6 7 8 9 0 10
0.707
1
0.8
0.6
0.4
0.2
0
Passband
| (u)|
(a)
u
Three-pole filter
Two-pole filter
1 2 3 4 5 6 7 8 9
0
10
÷50°
÷100°
÷150°
÷200°
÷250°
÷300°
0
| (u)|
(b)
H( )
¦ ( / ) ]
u
u u

+ ÷
1
1 2 1
2
1
2 2

H s
s s
H s
s s s
t
t t
t
t t t
( )
.
. .
( )
.
. . .

+ +

+ + +
0 50
0 645 0 708
0 251
0 597 0 928 0 251
2
2 2
3
3 2 2 3
u
u u
u
u u u
© 2000 by CRC Press LLC
Dehning Terms
Causal nlter: A fltei of which the tiansition fiom the passband to the stopband is giadual, not ideal. This
fltei is iealizable.
3-dB bandwidth: Foi a causal low-pass oi bandpass fltei with a fiequency function H(,u): the fiequency at
which H(u)
dB
is less than 3 dB down fiom the peak value H(u
¡
)
dB
.
Ideal nlter: An ideal fltei is a system that completely iejects sinusoidal inputs of the foim x(ì) · À cos uì,
-~ < ì < ~, foi u within a ceitain fiequency iange, and does not attenuate sinusoidal inputs whose
fiequencies aie outside this iange. Theie aie foui basic types of ideal flteis:low-pass, high-pass, bandpass,
and bandstop.
Passband: Range of fiequencies u foi which the input is passed without attenuation.
Stopband: Range of fiequencies u foi which the fltei completely stops the input signal.
Transition region: The iange of fiequencies of a fltei between a passband and a stopband.
Re!ated Tupics
4.2 Low-Pass Filtei Functions · 4.3 Low Pass Filteis · 11.1 Intioduction · 29.1 Synthesis of Low-Pass Foims
Relerences
R.C. Doif, InìroJutìíon ìo E|etìríta| Círtuíìs, 3id ed., New Yoik: Wiley, 1996.
E.W. Kamen, InìroJutìíon ìo Sígna|s anJ Sysìems, 2nd ed., New Yoik: Macmillan, 1990.
G.R. Coopei and C.D. McGillem, MoJern Communítaìíons anJ S¡reaJ S¡etìrum, New Yoik: McGiaw-Hill, 1986.
Further Inlurmatiun
IEEE Transatìíons on Círtuíìs anJ Sysìems, Pait I: Fundamental Theoiy and Applications.
IEEE Transatìíons on Círtuíìs anJ Sysìems, Pait II: Analog and Digital Signal Piocessing.
Available fiom IEEE.
FIGURE 10.8 Magnitude cuives of thiee-pole Butteiwoith and thiee-pole Chebyshev flteis with 3-dB bandwidth equal
to 2.5 iad/s.
u
Three-pole Chebyshev
1 2 3 4 5 6 7 8 9 0 10
0.707
1
0.8
0.6
0.4
0.2
0
Passband
| (u)|
Three-pole Butterworth