ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Continuous-Time Signal Processing (ET 2004)
Chapter 6 : Introduction to Analog Filters
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Irma Zakia
School of Electrical Engineering and Informatics, Institut Teknologi Bandung
*Untuk kalangan terbatas
Butterworth
Filter
Chebyshev Fil.
March 28, 2019
Frequency
Transformation
Equalization
1 / 89
Presentation Outline
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
1 Introduction
2 Distortionless Transmission
3 Nonideal Filters
4 Filter Design
5 Butterworth Filter
6 Chebyshev Fil.
7 Frequency Transformation
Chebyshev Fil.
Frequency
Transformation
8 Equalization
Equalization
2 / 89
Presentation Outline
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
1 Introduction
2 Distortionless Transmission
3 Nonideal Filters
4 Filter Design
5 Butterworth Filter
6 Chebyshev Fil.
7 Frequency Transformation
Chebyshev Fil.
Frequency
Transformation
8 Equalization
Equalization
3 / 89
Filtering
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
The convolution property of LTI system is the whole basis
of filtering, for which the input-output of LTI system is
related by the filter H(jω)
The system performs filtering on the input signal by
presenting a different response to components of the input
that are at different frequencies
This also means that by filtering, the frequency
components of the input are affected in terms of
magnitude and phase
Chebyshev Fil.
Frequency
Transformation
Equalization
4 / 89
Typical Parameters in Filtering
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
Passband : range of frequencies that are passed
Stopband : range of frequencies that are attenuated
Transition band : range of frequencies for which the transition from
passband to stopband occurs, and vice versa
The magnitude squared response |H(jω)|2 is commonly described in
units of dB as
|H(jω)|dB = 20 log10 |H(jω)| dB
Cut-off frequency ωc
frequency for which the magnitude squared response
|H(jω)|2 decreases to 21 its maximum value, or |H(jω)|
decreases to √12 of its maximum value
Cut off frequency is also called the −3 dB point
At cut off frequency, the filter passes half of the input
power
Majority of filtering involves real impulse response h(t), which yields
|H(jω)| to be an even function. Thus plotting |H(jω)| for ω ≥ 0
suffices
5 / 89
Presentation Outline
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
1 Introduction
2 Distortionless Transmission
3 Nonideal Filters
4 Filter Design
5 Butterworth Filter
6 Chebyshev Fil.
7 Frequency Transformation
Chebyshev Fil.
Frequency
Transformation
8 Equalization
Equalization
6 / 89
Definition
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Block diagram of a continuous time LTI system
F
x(t) −
→ X (jω)
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
F
y (t) −
→ Y (jω)
Distortionless transmission : output is an exact replica of
the input, except, for a possible modifications
1
Introduction
F
h(t) −
→ H(jω)
2
scaling of amplitude C
time delay t0
In time-domain, distortionless transmission yields
y (t) = C x(t − to )
The system impulse response for distortionless
transmission becomes
h(t) = C δ(t − to )
whereas the frequency response
H(jω) = C e −jωt0
7 / 89
Frequency Response for Distortionless Transmission
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Two requirements for distortionless transmission
For the range frequencies of interest:
1
Magnitude response must be constant |H(jω)| = C
C
Irma Zakia
Introduction
ω
Distortionless
Transmission
Nonideal
Filters
Filter Design
2
0
Phase response must be linear and intercept zero
arg(H(jω)) = −ωt0
arg(H(jω))
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
ω
0
slope
= −t0
8 / 89
Frequency Response of Ideal LPF
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
The frequency response of an ideal LPF
arg(H(jω))
|H(jω)|
1
ωc
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
−ωc
0
ω
ωc
−ωc
(a) Magnitude Response
H(jω) =
e −jωt0 ,
0,
0
ω
slope
= −t0
(b) Phase Response
|ω| ≤ ωc
|ω| > ωc
where C = 1
Distortionless transmission is achieved by ideal filters
It requires abrupt transition from passband to stopband
In this chapter, we put emphasis on the frequency characterization of
system (frequency response), although we know that the step
response is also significant in the design of some applications, like
automobile suspension.
9 / 89
Presentation Outline
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
1 Introduction
2 Distortionless Transmission
3 Nonideal Filters
4 Filter Design
5 Butterworth Filter
6 Chebyshev Fil.
7 Frequency Transformation
Chebyshev Fil.
Frequency
Transformation
8 Equalization
Equalization
10 / 89
Practical Filters
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Ideal filters is impractical, since the discontinuous
transition band is not realizable. It is actually noncausal as
well.
On the other hand, nonideal filters with gradual transition
from passband to stopband is generally preferable.
Distortionless
Transmission
Nonideal
Filters
X (jω)
X2 (jω)
X1 (jω)
Filter Design
Butterworth
Filter
Chebyshev Fil.
ω
Frequency
Transformation
Equalization
11 / 89
Characteristic of RC LPF
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
First order RC LPF circuit
R
x(t) +
−
C
20
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
−
y (t)
3 dB
apprx.
|H(jω)|dB
Nonideal
Filters
+
1
The transfer function H(s) = 1+sRC
Plot of the magnitude response in Bode diagram
0 dB
Distortionless
Transmission
−
+
−20
−40
0.1
RC
1
RC
10
RC
100
RC
ω
1
The cutoff frequency is RC
1
The pole is real at s = − RC
1
The roll-off (asymptotic slope ω >> RC
) is −20 dB/decade. This
roll-off is related to the transition band
Higher roll-off can be achieved by cascading these first-order RC
LPFs, e.g. cascade of N filters yields roll-off −20N dB/decade
12 / 89
Fourth Order Cascaded RC LPF
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Cascade interconnection of N = 4 first-order RC LPF
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
Op-amp is used to avoid loading effect, thus, separate the
individual filter stages
The transfer function becomes
H(s) =
1
(1 + sRC )4
1
Yet, real poles (4 poles) at s = − RC
Cutoff of each individual filter is
cascaded interconnection
1
RC
times higher than the
13 / 89
Frequency Response of Fourth Order Cascaded RC
LPF
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Roll-off becomes −80 dB/decade
Distortionless
Transmission
|H(jω)|dB
Introduction
arg(H(jω))
Irma Zakia
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
normalized frequency ω̃
(a)
magnitude response
normalized frequency ω̃
(b)
phase response
Reference: Texas Instruments, Chapter 16: Active Filter Design Techniques, Literature Number
SLOA088
Equalization
14 / 89
Cascaded First-order Filters
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Compare to ideal fourth order filter, the cascaded RC has :
Not flat passband gain
Wider transition band (the 80 dB roll-off is shifted 1.5
octaves above cut-off)
Nonlinear phase response
Consider transfer function of first-order H(s) =
Cascading two first-order filters yields
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
1
s+ω0
H(s) =
1
1
1
= 2
s + ω0 s + ω0
s + 2ω0 s + ω02
This results in quality factor Q = 0.5 which means that:
gradual transition from passband to stopband
significant attenuation in the passband
Equalization
15 / 89
Optimizing the Frequency Response
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
using real poles are not enough, so use complex conjugate
poles
complex conjugate poles allow designer to optimize one of
the following filter criteria :
Nonideal
Filters
1
2
Filter Design
3
Maximum flatness in passband
Smaller transition band
Linear phase response (at least in the passband)
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
16 / 89
Optimizing the Frequency Response continues
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
the transfer function becomes
H(s) = b̃
1
(s 2 + c1 s + d1 )(s 2 + c2 s + d2 ) · · · (s 2 + cn s + dn )
the denominator is cascade of second-order LPF, where the
filter coefficients ck and dk , k = 1, 2, · · · n are real
the filter coefficients determine the complex pole locations
the filter coefficients are determined based on specific criteria
the Butterworth filter optimize maximum flatness in
passband
the Chebyshev filter sharpening the transition band
the Bessel filter linearize the phase response (in passband)
Equalization
17 / 89
Presentation Outline
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
1 Introduction
2 Distortionless Transmission
3 Nonideal Filters
4 Filter Design
5 Butterworth Filter
6 Chebyshev Fil.
7 Frequency Transformation
Chebyshev Fil.
Frequency
Transformation
8 Equalization
Equalization
18 / 89
Background
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Although signal processing is done mainly in digital domain, the
physical signal is analog.
In digital processing of analog signals, LPFs are used.
Front-end: restrict bandwidth for sampling
Far-end: smoothing / remove higher frequency components
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Analog filters are used as prototype for the design of the so-called
Infinite Impulse Response (IIR) digital filters by using transformation
from the s−plane to the z−plane (bilinear transformation)
Focus is on the design of LPF filters, since other types of filters are
obtained simply by frequency transformation.
Equalization
19 / 89
Filter Design Specifications
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Nonideal/practical filters involve an acceptable values of ’distortion’
if compared to ideal filters
Those values are the filter characteristics in frequency domain, which
translates to filter specifications from a design perspective
|H(jω)|
1
(1 − δp )
δp
Distortionless
Transmission
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
stopband
passband
Nonideal
Filters
transition
band
δs
δs
0
ωp
ωs
ω
Filter specifications:
tolerance
tolerance
passband
stopband
in passband : δp
in stopband : δs
edge : ωp
edge : ωs
20 / 89
Design Rule of Thumb
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
The design can’t always meet the specifications exactly.
From financial perspective, the design has to be met with
minimum cost. If with a filter order N, the specifications
are fulfilled, don’t increase the order to N + 1.
The passband specifications are a priority
If the passband specifications are satisfied, then the
stopband will be oversatisfied in terms of a lower ωs value
or narrower transition width
The design delivers the filter frequency response H(jω) or
alternatively the transfer function H(s)
After obtaining the transfer function, the filter is
implemented in a physical system
Equalization
21 / 89
Analog Filter Types
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Analog filter prototypes: Bessel filter, Butterworth, Chebyshev,
elliptic
Bessel
Introduction
Distortionless
Transmission
Nonideal
Filters
|H(jω)|dB
Irma Zakia
Butterworth
Chebyshev I
Chebyshev II
elliptic
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
normalized frequency ω̃
Figure: Fifth order filter
Equalization
Reference: inst.eecs.berkeley.edu/ ee247/fa04/fa04/lectures/L2 f04.pdf
22 / 89
Group Delay Comparison
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
23 / 89
Comparison of Chebyshev Type 1 and Type 2
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
(a)
s-plane Type 1
Nonideal
Filters
Filter Design
Butterworth
Filter
Figure: magnitude and phase response
Chebyshev Fil.
Frequency
Transformation
Equalization
(b)
Reference:2.161 Signal Processing: Continuous and Discrete Fall
2008, MIT OpenCourseWare, ocw.mit.edu
s-plane Type 2
24 / 89
Which filter type?
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Trade-off between transition width and phase response
Filters with high signal attenuation per pole has poor phase
response
For a given signal attenuation requirement of preserving
constant group delay, we require higher order filter, but more
cost and area
In cases where analog filter is followed by digital processing, it is
possible to digitally correct for phase non-linearities incurred by
the analog circuitry by using phase equalizers
Elliptic : optimum but requires numerical computation.
Optimum in the sense that no other filter of the same order has
narrower transition width for a given passband and stopband
tolerance
This chapter considers the design of Butterworth and Chebyshev
Equalization
25 / 89
Presentation Outline
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
1 Introduction
2 Distortionless Transmission
3 Nonideal Filters
4 Filter Design
5 Butterworth Filter
6 Chebyshev Fil.
7 Frequency Transformation
Chebyshev Fil.
Frequency
Transformation
8 Equalization
Equalization
26 / 89
Description of a Butterworth Filter
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Butterworth filter of order N is described by the
magnitude squared of its frequency response
1
|H(jω)|2 =
2N
1 + ωωc
where ωc is the cutoff frequency
Distortionless
Transmission
N=∞
1
Nonideal
Filters
Butterworth
Filter
N = 25
|H(jω)|2
Filter Design
N=2
N=1
0.5
Chebyshev Fil.
Frequency
Transformation
0
ωc
ω
Equalization
27 / 89
Properties of Butterworth Filter
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
For any order N, |H(jω)|2
|H(jω)|
=1
ω=0
For any order N
2
|H(jω)|
ω=ωc
ω=ωc
1
=
2
=
, −3.0103 dB
0.707
2
|H(jω)| is a monotonic function (no maxima or minima)
As the order N increases, |H(jω)|2 approaches an ideal LPF
|H(jω)| is maximally flat at the origin
By using Binomial series, the magnitude response
|H(jω)| =
1+
ω
ωc
2N !− 1
2
=1−
1
2
ω
ωc
2N
+ higher order of
ω
ωc
2N
Taking the derivative
d k |H(jω)|
dω k
ω=0
, k = 1, 2, .., 2N − 1
Butterworth filter has 2N − 1 (a maximum possible number)
derivatives which vanish at ω = 0 (flat curve at ω = 0)
28 / 89
Asymptotic Property of All-Pole Filters
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Asymptotic property of |H(jω)| for ω → ∞
This property is valid for any all-pole filters, not only
Butterworth
−N
ω
|H(jω)|ω→∞ ≈
ωc
ω
|H(jω)|dB ≈ −20N log
ωc
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Roll-off/asymptotic slope
−20N dB/decade
Equalization
29 / 89
Example of Butterworth LPF
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
First order RC circuit
+ Vr (t) −
R
+
Vs (t) −
+
Vc (t)
−
C
Introduction
Distortionless
Transmission
Nonideal
Filters
RC circuit acts as LPF for the capacitor voltage Vc (t)
The frequency response
H(jω)
=
1
1 + jωRC
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
The magnitude squared response yields
|H(jω)|2
=
1+
1
ω
1
RC
2
which shows the response of a first order Butterworth filter with
1
ωc = RC
30 / 89
Design the Butterworth Filter
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Need to define H(s)
What we get from the filter specification is |H(jω)|2
Assume h(t) real, thus H(jω)∗ = H(−jω)
|H(jω)|2
=
H(s)H(−s)
=
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
1+
1+
Poles for H(s)H(−s) are derived from
−s 2
ωc2
N
1
ω
ωc
1
2N
−s 2
ωc2
N
=
−1
!N
=
e j(2k−1)π , k = 1, 2, · · · , N
Frequency
Transformation
sk2
=
−ωc2 e j
Equalization
sk
=
±jωc e j
Filter Design
Butterworth
Filter
Chebyshev Fil.
−sk2
ωc2
(2k−1)π
N
(2k−1)π
2N
= ωc e
j
(2k−1)π
±π
2N
2
31 / 89
Poles Characteristic
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Poles of H(s)H(−s) are at sk = ωc e
ℑ{s}
N=1
j
(2k−1)π
±π
2N
2
, k = 1, 2, · · · , N
ℑ{s}
N=2
π
2
ωc ℜ{s}
Irma Zakia
ωc
ℜ{s}
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
ℑ{s}
N=3
N=6
π
3
Chebyshev Fil.
Frequency
Transformation
ℑ{s}
ωc
π
6
ℜ{s}
ωc
ℜ{s}
Equalization
32 / 89
Poles of Butterworth Filter
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Focusing on poles at left-half of s-plane (LHP) → take the +
sign
sk
=
=
j
(2k−1)π
+π
2N
2 , k = 1, 2, · · · , N
ωc e
(2k − 1)π
(2k − 1)π
ωc − sin
+ j cos
2N
2N
Poles are at radius ωc and equally spaced in a circle, with
2π
spacing 2N
A pole never lies on the ℑ{s} axis, and occur at ℜ{s} axis for
N odd
Further work on the transfer function considers the normalized
Butterworth filter, such that ωc = 1 rad/s.
If not normalized, the parameter s is replaced by
transfer function
s
ωc
in the
Equalization
33 / 89
Determining the Transfer Function
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Order N = 1, pole at s = −1, thus
1
H(s) =
s +1
Order N = 2, poles at s =
H(s) =
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
−1
√
2
s2
± j √12 , thus
1
√
+ 2s + 1
Order N = 3, poles at s = −1, s =
H(s) =
1
−1
2
±j
√
3
2 ,
thus
(s + 1)(s 2 + s + 1)
If order N odd, there exists a single real pole at s = −1, besides
N−1
pairs of complex poles
2
If order N even, all poles are complex conjugate
Denominator of H(s) is the Butterworth polynomial
34 / 89
Determining the Transfer Function continues..
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
For N > 1, complex conjugate poles are present
Rewrite sk (assume normalized filter ωc = 1 rad/s)
sk = − sin
Irma Zakia
(2k − 1)π
2N
+ j cos
(2k − 1)π
2N
Each complex pole results in the denominator of H(s) as
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
=
s + sin
2
s + 2 sin
|
(2k − 1)π
2N
(2k − 1)π
2N
{z
bk
Equalization
(2k − 1)π
2N
Define a constant bk = 2 sin
k=
s + sin
(2k − 1)π
2N
− j cos
(2k − 1)π
2N
s+1
}
Chebyshev Fil.
Frequency
Transformation
+ j cos

 1, 2, · · · ,

1, 2, · · · ,
(2k−1)π
2N
, where
N
2
for N even
N−1
2
for N odd
35 / 89
Determining the Transfer Function continues..
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
General transfer function

1
,
N even

N


2

 Q (s 2 +bk s+1)
k=1
H(s) =
1
, N odd

N−1


2
Q

2
 (s+1)
(s +bk s+1)
k=1
If not normalized (ωc 6= 1), replace s → ωsc

ωcN

,
N even

N


2
Q

2
2

(s +bk ωc s+ωc )
k=1
H(s) =
ωcN

, N odd

N−1


2
Q

2
2
 (s+ωc )
(s +bk ωc s+ωc )
k=1
Butterworth polynomial only requires to determine bk , no
need to know pole locations
36 / 89
Transfer Function for N = 4
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
The constants
π
8
=
3π
b2 = 2 sin
8
=
b1 = 2 sin
avoid rounding numbers
The transfer function
Distortionless
Transmission
H(s) =
1.84776
| {z }
avoid rounding numbers
Irma Zakia
Introduction
0.76537
| {z }
1
(s 2 + 0.76537s + 1)(s 2 + 1.84776s + 1)
Nonideal
Filters
Filter Design
Butterworth
Filter
If filter is not normalized (ωc 6= 1)
H(s)
=
Chebyshev Fil.
Frequency
Transformation
=
s
ωc
2
+ 0.76537
s
ωc
1
2
s
s
+1
+
1.84776
+
1
ω
ω
c
c
ωc4
(s 2 + 0.76537ωc s + ωc2 )(s 2 + 1.84776ωc s + ωc2 )
Equalization
37 / 89
Transfer Function for N = 5
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
The constants
π
10
3π
b2 = 2 sin
10
b1 = 2 sin
Irma Zakia
Introduction
Filter Design
Butterworth
Filter
H(s) =
=
1.61803
1
(s + 1)(s 2 + 0.61803s + 1)(s 2 + 1.61803s + 1)
If filter is not normalized (ωc 6= 1)
H(s)
=
Chebyshev Fil.
Frequency
Transformation
0.61803
The transfer function
Distortionless
Transmission
Nonideal
Filters
=
=
s
ωc
+1
s
ωc
2
+ 0.61803
ωc5
s
ωc
1
+1
s
ωc
2
+ 1.61803
s
ωc
+1
(s + ωc )(s 2 + 0.61803ωc s + ωc2 )(s 2 + 1.61803ωc s + ωc2 )
Equalization
38 / 89
Determining Filter Order
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
From specifications of δp , δs , ωp , ωs → filter order N, and next
ωc
|H(jω)|
1
(1 − δp )
Introduction
δs
Distortionless
Transmission
ω
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
ωp
0
Nonideal
Filters
Recall : |H(jω)|2 = 1/ 1+( ωωc )2N
Inserting the specifications
1
2N
ω
1+( ωpc )
ωp
ωc
2N
ωs
1
ωs
1+( ω
c
= (1 − δp )2 ,
=
1
(1−δp )2
− 1,
ωs
ωc
)
2N
2N
=
= δs2
1
δs2
−1
39 / 89
Determining Filter Order continues
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
ωs
ωp
2N
=
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
1
−1
δs2
1
−
(1−δp )2
log10
N ≥
s
log10
1
1
−1
δs2
1
−1
(1−δp )2
ωs
ωp
Chebyshev Fil.
Frequency
Transformation
Equalization
40 / 89
After Filter Order... what next?
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Passband requirement must be fulfilled exactly
Two alternatives in determining ωc
Take the one which satisfies the passband requirement
exactly
ωc = ωp
1
(1−δp )2
Determine ωs from equation
1
2N
−1
1+
1
ωs
ωc
2N
= δs2
Frequency for which |H(jω)| = δs is to the left of ωs .
Let’s call this frequency ωs ′ .
Do we perform better or worse?
Equalization
41 / 89
Butterworth LPF Filter Design Example
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Design an analog Butterworth filter that has passband
attenuation 2 dB at 20 rad/s and at least 35 dB attenuation at
40 rad/s
(1 − δp )2 = 10−2/10 = 0.63096, ωp = 20 rad/s
δs2 = 10−35/10 = 0.00031622, ωs = 40 rad/s
r
filter order
1
log10
ωc =
Butterworth
Filter
Frequency
Transformation
Equalization
≥
N
=
log10
7
Cut-off frequency
Filter Design
Chebyshev Fil.
N
20
1
0.63096
−1
−1
0.00031622
1
−1
0.63096
40
20
= 6.2
1/14 = 20.78106 rad/s
The frequency which corresponds
to δs
1/14
′
ωs = 20.78106
1
−1
0.00031622
= 36.95393 rad/s < ωs
42 / 89
LPF Butterworth Design Example continues
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Transfer function for N = 7
H(s) = h
ωc7
The constants
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Equalization
b1
=
b2
=
b3
=
π
= 0.44504
14
3π
= 1.24698
2 sin
14
5π
2 sin
= 1.80194
14
2 sin
Inserting those values, the transfer function yields
Chebyshev Fil.
Frequency
Transformation
i
(s+ωc )(s 2 +b1 ωc s+ωc2 )
(s 2 +b2 ωc s+ωc2 )(s 2 +b3 ωc s+ωc2 )
H(s) = h
1.67369 109
i
(s+20.78106)(s 2 +9.24840s+431.85245)
(s 2 +25.91357s+431.85245)(s 2 +37.44622s+431.85245)
43 / 89
LPF Butterworth Design Example continues
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Dari hasil fungsi transfer H(s)
H(s) = h
1.67369 109
(s+20.78106)(s 2 +9.24840s+431.85245)
(s 2 +25.91357s+431.85245)(s 2 +37.44622s+431.85245)
(a)
(b)
i
Berapa nilai magnituda |H(jω) saat ω = ωp = 20 rad/s?
Berapa nilai magnituda |H(jω) saat ω = ωs = 40 rad/s?
Misal Anda mengambil orde filter N ≤ 6.2 = 6,
(a)
(b)
Berapa nilai magnituda |H(jω) saat ω = ωp = 20 rad/s?
Berapa nilai magnituda |H(jω) saat ω = ωs = 40 rad/s?
Mengapa demikian?
Frequency
Transformation
Equalization
44 / 89
Presentation Outline
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
1 Introduction
2 Distortionless Transmission
3 Nonideal Filters
4 Filter Design
5 Butterworth Filter
6 Chebyshev Fil.
7 Frequency Transformation
Chebyshev Fil.
Frequency
Transformation
8 Equalization
Equalization
45 / 89
Characteristics
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Performance of filter is increased by allowing ripple → equiripple
For a given δp , δs , ωp , N, transition width of Chebyshev is narrower
than Butterworth
For a given δp , ωp , ωs , N, stopband attenuation of Chebyshev is
larger than Butterworth
Two types of Chebyshev : Type I and Type II
Nonideal
Filters
|H(jω)|
|H(jω)|
Distortionless
Transmission
√1
1
1
√1
1+ǫ2
1+ǫ2
Filter Design
δs
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
δs
ω
ω
0
ωp ωs
(a) Type I
0
ωp ωs
(b) Type II
Preferred: Type I (Chebyshev Type II has complex zeros, poor roll-off)
Parameter ǫ is chosen such that the peak-to-peak passband ripple 1 − √ 1
1+ǫ2
is acceptable
Equalization
46 / 89
Description of Chebyshev Filter
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Magnitude frequency response of order N
|H(jω)| = q
A
1 + ǫ2 CN2 ( ωωp )
A need not be 1, and
CN (x) ≡
cos(Ncos−1 x),
cosh(Ncosh−1 x),
|x| < 1
|x| > 1
|x| < 1 translates to |ω| < ωp , since cosine is oscillatory →
equiripple in passband
|x| > 1 translates to |ω| > ωp , since cosh is monotonic →
monotonic in stopband
Without loss of generality, assume A = 1
Equalization
47 / 89
Chebyshev Polynomials
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
A recursive formula in determining the polynomial
CN (x) = 2xCN−1 (x) − CN−2 (x), N ≥ 2
Irma Zakia
N
CN (x)
0
1
2
3
4
5
6
7
8
9
10
1
x
2x 2 − 1
4x 3 − 3x
8x 4 − 8x 2 + 1
16x 5 − 20x 3 + 5x
32x 6 − 48x 4 + 18x 2 − 1
64x 7 − 112x 5 + 56x 3 − 7x
128x 8 − 256x 6 + 160x 4 − 32x 2 + 1
256x 9 − 576x 7 + 432x 5 − 120x 3 + 9x
512x 10 − 1280x 8 + 1120x 6 − 400x 4 + 50x 4 − 1
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
48 / 89
Polynomial Values at Origin and Passband
Frequency
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Take a look at |H(jω)| =
A
q
1+ǫ2 CN2 ( ωωp )
What is |H(jω)| at ω = 0 and ω = ωp ?
N = 0,
CN ( ωωp )
ω=0
N = 1,
CN ( ωωp )
ω=0
N = 2,
CN ( ωωp )
ω=0
N = 3,
CN ( ωωp )
ω=0
N = 4,
CN ( ωωp )
ω=0
= 1,
CN ( ωωp )
= 0,
CN ( ωωp )
= −1,
ω=ωp
ω=ωp
CN ( ωωp )
= 0,
CN ( ωωp )
= 1,
CN ( ωωp )
=1
=1
ω=ωp
ω=ωp
ω=ωp
=1
=1
=1
and so on
Equalization
49 / 89
Ripple Characteristic
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Recall : |H(jω)| =
q
1
1+ǫ2 CN2 ( ωωp )
Filter magnitude at the origin
|H(jω)|
ω=0
=
(
1,
√1
1+ǫ
,
2
N odd
N even
Filter magnitude at passband frequency |H(jω)|
ω=ωp
=
√ 1
1+ǫ2
Total number of maximum and minimum in passband is N
|H(jω)|
|H(jω)|
1
1
√ 1
1+ǫ2
√ 1
1+ǫ2
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
0
ωp
(a) Even (N = 4)
ω
0
ωp
ω
(b) Odd (N = 5)
50 / 89
Cut-off Frequency
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Cut-off frequency is determined from
1
1+
ǫ2 CN2 ( ωωp )
ω=ωc
=
ωc
CN
=
ωp
−1 ωc
=
cosh Ncosh
ωp
ωc
1
2
1
>1
ǫ
1
ǫ
= ωp cosh
1
cosh−1
N
1
ǫ
Chebyshev Fil.
Frequency
Transformation
Cut-off frequency in Chebyshev is not as significant as in
Butterworth from a design perspective. Why?
Equalization
51 / 89
Poles of Chebyshev
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Need to define H(s)
No need to know poles location
H(s)H(−s)
1
1 + ǫ2 CN2 ( jωs )
=
p
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Poles for H(s)H(−s) are
derived from
ǫ2 CN2
s
jωp
=
−1
Take the LHP poles
Poles are located in an ellipse
(compare to Butterworth!)
Figure: Fourth order Chebyshev
filter
Equalization
52 / 89
Transfer Function
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
Denominator of H(s) contains poles in the form
(s + ωp c0 ) and(s 2 + bk ωp s + ck ωp2 )
But bk is not the same as Butterworth
Transfer function

N
2

Q

1
N
√

ck
ω
p

1+ǫ2

k=1

,
N even

N


2
Q


(s 2 +bk ωp s+ck ωp2 )
k=1
H(s) =
N−1

2
Q


ωpN c0
ck


k=1

, N odd

N−1


2

2
2
 (s+ωp c0 ) Q
(s +bk ωp s+ck ωp )
k=1
53 / 89
Constants in the Transfer Function
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Let’s define
 "r
#1/N "r
#−1/N 


1
1
1
1
1
1+ 2 +
1+ 2 +
−
dN =

2
ǫ
ǫ
ǫ
ǫ
The constants
Distortionless
Transmission
c 0 = dN
Nonideal
Filters
ck
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
bk
(2k − 1)π
2N
(2k − 1)π
= 2dN sin
2N
= dN2 + cos2
Compare to Butterworth : c0 = 1, ck = 1 and dN = 1
Compare coefficient bk to Butterworth. For Chebyshev
dN 6= 1, hence poles location of Chebyshev traces an ellipse
54 / 89
What to do given filter specifications?
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
replacements
|H(jω)|
1
(1 − δp )
δs
ω
0
ωp
ωs
Determine ǫ and N
Frequency
Transformation
Equalization
55 / 89
Determining ǫ and Filter Order
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Evaluating |H(jω)| at ω = ωp
1
√
= (1 − δp ) → ǫ =
1 + ǫ2
Nonideal
Filters
Filter Design
Butterworth
Filter
q
1
1 + ǫ2 CN2 ( ωωps )
cosh Ncosh−1
≤ δs
ωs
ωp
→
≥
Chebyshev Fil.
Frequency
Transformation
Equalization
1
−1
(1 − δp )2
Evaluating |H(jω)| at ω = ωs , and inserting ǫ, remember
that we perform better at stopband
Introduction
Distortionless
Transmission
s
v
u
u δ12 − 1
ωs
CN
≥ t 1s
ωp
−1
(1−δp )2
| {z }
v
u
u
t
>1
1
−1
δs2
1
−
(1−δp )2
cosh−1
N
≥
s
cosh−1
1
1 −1
δs2
1
−1
(1−δp )2
ωs
ωp
56 / 89
Alternative Way in Determining Filter Order
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Defining N of Chebyshev requires cosh−1 , which may not be
available
Alternative way:
1 trial and error: increment N until the inequality is fulfilled,
drawback?
Irma Zakia
q
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
2
Equalization
1 + ǫ2 CN2 ( ωωps )
ωs
CN
ωp
≤
δs
≥
1
ǫ
r
1
−1
δs2
p
Use the relation cosh−1 y = ln y + y 2 − 1 , which
results in
log10
Chebyshev Fil.
Frequency
Transformation
1
N≥
s
1 −1
δs2
1
−1
(1−δp )2
log10
ωs
ωp
+
+
s
r
ωs
ωp
1 −1
δs2
1
−1
(1−δp )2
2
−1
!
−1
!
57 / 89
Filter Order Formulas of Butterworth and
Chebyshev
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
N Butterworth N Chebyshev
v
u
u
u
log10 t
1 −1
δs2
1
−1
(1−δp )2
ω
log10 ω s
p
v
u
u
u
−1
cosh t
1 −1
δs2
1
−1
(1−δp )2
cosh−1 ωωps
Frequency
Transformation
Equalization
58 / 89
LPF Chebyshev Design Example
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
Design analog Chebyshev LPF with acceptable passband ripple
of 2 dB at 20 rad/s, stopband attenuation of 35 dB or more at
40 rad/s
(1 − δp )2 = 10−2/10 , ωp = 20 rad/s
δs2 = 10−35/10 , ωs = 40 rad/s
Find ǫ
1
= (1 − δp )2 , ǫ = 0.76478
1 + ǫ2
Find N with trial and error
CN
ωs
ωp
≥
Initial guess, try N = 3,
ωs
ωp
C3 (2)
=
C4 (2)
=
1
ǫ
r
1
− 1 = 73.51819
δs2
=2=x
4(2)3 − 3(2) = 26
8(2)4 − 8(2)2 + 1 = 97
So N = 4 → compare to N = 7 Butterworth
59 / 89
LPF Chebyshev Design Example continues
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Transfer function for N = 4
H(s) =
(s 2 + b1 ωp s
√ 1
ω4 c c
1+ǫ2 p 1 2
+ c1 ωp2 )(s 2 + b2 ωp s
+ c2 ωp2 )
The constants
dN =
1
2
( "r
1+
1
1
+
ǫ2
ǫ
#1/N
π
= 0.92867,
8
π
b1 = 2dN sin = 0.20977,
8
c1 = dN2 + cos2
−
"r
1+
1
1
+
ǫ2
ǫ
#−1/N )
= 0.27408
3π
= 0.22157
8
3π
b2 = 2dN sin
= 0.50643
8
c2 = dN2 + cos2
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Inserting those values, the transfer function yields
H(s) =
2.61513
(s 2 + 4.19540s + 371.46800)(s 2 + 10.12860s + 88.62800)
Equalization
60 / 89
Exercise
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
1
Design Chebyshev LPF that has passband from 0 to 200
Hz with acceptable ripple of 1 dB, and a monotonic
stopband that is down at least 40 dB at 250 Hz
2
Repeat for Butterworth
3
Sketch both filters
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
61 / 89
Kuis 5 Sem I 2016/2017
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
1
Seorang user menghendaki filter analog band-pass (BPF)
dengan spesifikasi:
Melewatkan sinyal dengan redaman passband 2 dB pada
frekuensi antara 3 KHz dan 5 KHz
Menghalangi sinyal di bawah 2 KHz dan di atas 6 KHz
dengan redaman setidaknya 35 dB
Tidak mengizinkan adanya ripple
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
(a)
(b)
Tentukan fungsi transfer H(s)
Apa filter yang dihasilkan memenuhi spesifikasi user?
Jelaskan!
Chebyshev Fil.
Frequency
Transformation
Equalization
62 / 89
Presentation Outline
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
1 Introduction
2 Distortionless Transmission
3 Nonideal Filters
4 Filter Design
5 Butterworth Filter
6 Chebyshev Fil.
7 Frequency Transformation
Chebyshev Fil.
Frequency
Transformation
8 Equalization
Equalization
63 / 89
What is Frequency Transformation?
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Any analog filters, i.e. LPF, HPF, BPF, BSF is designed by applying
freq.transformation from its corresponding Normalized LPF (NLPF) form
Filter specs, LPF/HPF: δp , δs , ωp , ωs
BPF/BSF: δp , δs , ωp1 , ωp2 , ωs1 , ωs2
Conversion to
NLPF specifications
NLPF specifications
Design of NLPF
NLPF transfer function H(S)
Frequency transformation
Chebyshev Fil.
Frequency
Transformation
Equalization
Desired filter transfer function H(s)
Note : symbol S is used to distinguish NLPF with the desired analog filter
64 / 89
What is Normalized LPF
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
NLPF is LPF filter with normalized passband edge
frequency, i.e. ωp of the NLPF is 1 rad/s
|H(jω)|
1
(1 − δp )
δs
ω
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
0
1
ωs
Finding ωs of the NLPF are simple for LPF and HPF, but
tedious for BPF and BSF
Equalization
65 / 89
Frequency Transformation for LPF
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Replace
S→
Here, ωp belongs to LPF
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
s
ωp
S
s
0
1
ωs
∞
0
ωp
ωs ωp
∞
|H(jω)|
|H(jω)|
1
(1 − δp )
1
(1 − δp )
δs
δs
ω
0
1
ωs
(a) NLPF
0
ω p ωp ω s
ω
(b) LPF
66 / 89
Frequency Transformation for HPF
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Replace
S→
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
ωp
s
S
s
0
∞
j1
jωs
∞
0
jωp
ω
j ωp
s
Remember : |H(jω)| is even
|H(jω)|
|H(jω)|
1
(1 − δp )
1
(1 − δp )
δs
δs
ω
0
1
ωs
(a) NLPF
0
ωp
ωs
ωp
ω
(b) HPF
67 / 89
Determining ωs of the NLPF Prototype in
Designing LPF and HPF
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
LPF design S =
inserting s = jωs
S=
S=
Thus in LPF design
ωs of NLPF =
HPF design S =
ωp
s
inserting s = jωs
jωs
ωp
Filter Design
Butterworth
Filter
s
ωp
ωs
ωp
ωp
jωs
Thus in HPF design
ωs of NLPF =
ωp
ωs
Chebyshev Fil.
Frequency
Transformation
Equalization
68 / 89
Frequency Transformation for BPF
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Passband edges (ωp1 , ωp2 ) and
stopband edges (e
ωs1 , ω
es2 ) are
geometric symmetric w.r.t. ω0
(compare to arithmetic symmetric!)
Replace
S→
s 2 + ω02
Ws
|H(jω)|
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
S
s
0
∞
j1
jωs
jω0
0, ∞
jωp1 , jωp2
jω
es1 , j ω
es2
1
(1−δp )
ω0 : BPF center freq.
ω02 = ωp1 ωp2 = ω
es1 ω
es2
W = ωp2 − ωp1
δs
0
ω
es1 ωp1 ω0 ωp2
ω
ω
es2
Any two frequency with identical value
of |H(jω)|, satisfy the geometric
symmetry property
Equalization
69 / 89
Frequency Transformation for BSF
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Passband edges and stopband edges
are also geometric symmetric
Replace
S→
Ws
s 2 + ω02
|H(jω)|
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
S
s
0
∞
j1
jωs
0, ∞
jω0
jωp1 , jωp2
jω
es1 , j ω
es2
1
(1−δp )
ω0 : BSF center freq.
ω02 = ωp1 ωp2 = ω
es1 ω
es2
W = ωp2 − ωp1
δs
0
ωp1 ω
es1 ω0 ω
es2
ωp2
ω
Any two frequency with identical value
of |H(jω)|, satisfy the geometric
symmetry property
Equalization
70 / 89
Stopband Edges Adjustment in Designing BPF and
BSF
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
User specifications: δp , δs , ωp1 , ωp2 , ωs1 , ωs2
In the design process : the passband and stopband edges must be
geometric symmetric
Need to make adjustment in stopband
Since passband is fix ω02 = ωp1 ωp2 6= ωs1 ωs2
There are two cases
1
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
2
ωp1 ωp2 < ωs1 ωs2
BPF : ω
es2 < ωs2 , ω
es1 = ωs1
BSF : ω
es1 < ωs1 , ω
es2 = ωs2
ωp1 ωp2 > ωs1 ωs2
BPF : ω
es1 > ωs1 , ω
es2 = ωs2
BSF : ω
es2 > ωs2 , ω
es1 = ωs1
After adjustment ωp1 ωp2 = ω
es1 ω
es2 = ω02
71 / 89
Determining ωs of the NLPF Prototype in
Designing BPF and BSF
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
BPF design S =
inserting s = j ω
es1
jωs =
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Frequency
Transformation
Equalization
2 + ω2
−e
ωs1
0
Wj ω
es1
inserting s = j ω
es2
jωs =
Butterworth
Filter
Chebyshev Fil.
s 2 +ω02
Ws
2 + ω2
−e
ωs2
0
Wj ω
es2
BSF design S =
(1)
ω
es2 − ω
es1
ωp2 − ωp1
inserting s = j ω
es1
jωs =
Wj ω
es1
2
−e
ωs1 + ω02
(3)
inserting s = j ω̃s2
(2)
Either (1) or (2) leads to
ωs =
Ws
s 2 +ω02
jωs =
Wj ω
es2
2
−e
ωs2 + ω02
(4)
Either (3) or (4) leads to
ωs =
ωp2 − ωp1
ω
es2 − ω
es1
72 / 89
Example on HPF Design
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Design a Butterworth analog HPF that will pass all signals
greater than 27 kHz with 6 dB attenuation, and have a
stopband attenuation at least 20 dB for all frequencies
below 18 kHz
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
73 / 89
Example on BSF Design
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Design a Chebyshev analog BSF that will reject signals
between 4.6 kHz and 7.4 kHz with at least 10 dB
attenuation, and have a acceptable passband ripple of 3
dB for frequencies below 2 kHz and above 10 kHz
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
74 / 89
Presentation Outline
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
1 Introduction
2 Distortionless Transmission
3 Nonideal Filters
4 Filter Design
5 Butterworth Filter
6 Chebyshev Fil.
7 Frequency Transformation
Chebyshev Fil.
Frequency
Transformation
8 Equalization
Equalization
75 / 89
Why equalize?
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Recall : Distortionless transmission requires that the
system has constant magnitude and linear phase response
If not: use equalizer
Heq (jω) =
1
1
=
e −jarg(H(jω))
H(jω)
|H(jω)|
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
76 / 89
Delay equalizer
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
If Htot = H(jω)Heq (jω), then we wish to have Htot = 1 ?
It is impractical to construct an equalizer with inverse
phase response of the channel (Htot = 1)
Instead, make the total phase response to be linear or
phase proportional to ω, i.e. Htot ≈ e −jωt0
Delay-equalizer can be achieved by series interconnection
with all-pass networks
When designing Butterworth/Chebyshev analog filters :
are their phases linear?
Chebyshev Fil.
Frequency
Transformation
Equalization
77 / 89
Phase Response of Butterworth and Chebyshev
Filters
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
(a) 5th order Butterworth
(b) 5th order Chebyshev
Reference: inst.eecs.berkeley.edu/ ee247/fa04/fa04/lectures/L2 f04.pdf
78 / 89
Further Phase Response of Chebyshev Filters
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
(a) LPF
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
(b) BPF
Reference: Y. Su, Group Delay Variations in Microwave Filters and Equalization Methodologies,
Master ’s Thesis in Microtechnology and Nanoscience, 2012
79 / 89
Frequency Response of Several Filters
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
(a) magnitude response
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
(b) group delay
80 / 89
Intersymbol Interference (ISI) due to Channel
Impairments
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
(a) linear phase response channel
(b) nonlinear phase response
channel
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Reference: Y. Su, Group Delay Variations in Microwave Filters and Equalization Methodologies,
Master ’s Thesis in Microtechnology and Nanoscience, 2012
Equalization
81 / 89
ISI Effect on Data Transmission
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
82 / 89
First-order All-pass Systems
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
s − α0
s + α0
|H(jω)| = 1
H(s) =
argH(jω) = −2tan−1
ω
α0
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
83 / 89
Second-order All-pass Systems
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
H(s) =
|H(jω)| = 1
argH(jω) = −2tan
Distortionless
Transmission
Nonideal
Filters
Filter Design
s2 −
(s − α1 )2 + β12
=
s2 +
(s + α1 )2 + β12
where ωr =
−1
ωr ω
Q
ωr2 − ω 2
q
α12 + β12 , and Q =
ωr
Qs
ωr
Qs
+ ωr2
+ ωr2
ωr
2α1
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
84 / 89
Before and After Equalization
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
(a) magnitude response
(b) group delay
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Reference: Y. Su, Group Delay Variations in Microwave Filters and Equalization Methodologies,
Master ’s Thesis in Microtechnology and Nanoscience, 2012
Equalization
85 / 89
After Delay Equalization
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Figure: Group delay of (a)BPF, (b)second-order all-pass network,
(c)overall circuit
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Reference: Y. Su, Group Delay Variations in Microwave Filters and Equalization Methodologies,
Master ’s Thesis in Microtechnology and Nanoscience, 2012
Equalization
86 / 89
Chebyshev with Allpass
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
(a) magnitude response
(b) phase response
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
(c) group delay
Reference: inst.eecs.berkeley.edu/ ee247/fa04/fa04/lectures/L2 f04.pdf
87 / 89
UAS Sem 2 2015/2016
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Chebyshev Fil.
Frequency
Transformation
Equalization
Seorang user menghendaki filter analog low-pass (LPF) dengan
spesifikasi:
Mengizinkan adanya ripple di passband sebesar 3 dB pada
frekuensi di bawah 2 kHz
Respons monotonic di stopband dengan redaman
sedikitnya 10 dB untuk frekuensi di atas 3 kHz
(a) Tentukan fungsi transfer H(s) untuk memenuhi spesifikasi tersebut.
(b) Apakah filter rancangan Anda memenuhi spesifikasi user ? Jelaskan.
(c) Sketsakan respons magnitude (skala linear) LPF dengan
memperhatikan bentuk/nilai ripple yang tepat serta titik-titik
signifikan pada passband dan stopband.
(d) Sketsakan respons fasa (skala linear) dari LPF tersebut.
(e) Perkirakan gambar group delay LPF dengan menggunakan hasil
sketsa respons fasa serta formulasi
group delay = −
d arg(H(jω))
dω
(f) Jika LPF diinterkoneksi secara seri/cascade dengan filter all-pass,
perkirakan gambar group delay filter all-pass yang dibutuhkan
sehingga respons fasa sistem setelah diinterkoneksi adalah linear.
88 / 89
Reference
ContinuousTime Signal
Processing
(ET 2004)
Chapter 6 :
Introduction
to Analog
Filters
Alan V. Oppenheim, Alan S. Willsky, with S. Hamid, Signals
and Systems, 2nd edition, Prentice-Hall, 1996.
Irma Zakia
Introduction
Distortionless
Transmission
Nonideal
Filters
Filter Design
Butterworth
Filter
Simon Haykin, Barry Van Veen, Signals and Systems, 2nd
edition, John Wiley & Sons, Inc., 2004
inst.eecs.berkeley.edu/ ee247/fa04/fa04/lectures/L2 f04.pdf
Y. Su, Group Delay Variations in Microwave Filters and
Equalization Methodologies, Master ’s Thesis in
Microtechnology and Nanoscience, 2012
Chebyshev Fil.
Frequency
Transformation
Equalization
89 / 89