ECE 2006
Theory and Design of Analog
Filters
Dr. Shweta B. Thomas
School of Electronics Engineering
Dr. Shweta B. Thomas, VIT Vellore
Frequency Selective Filters
 A filter is one, which rejects unwanted frequencies from the input signal &
allow the desired frequencies to obtain the required shape of the output
signal.
 The range of frequencies of the signal that are passed through the filter is
called passband & those frequencies that are blocked is called stopband.
 Types of filters are:
 Lowpass filter
 Highpass filter
 Bandpass filter - Anti-notch Filter
 Bandreject filter - Notch Filter
Dr. Shweta B. Thomas, VIT Vellore
Magnitude response of the filters
Low and high pass filter
The magnitude response of an ideal low pass filter allows low frequencies in the
pass-band 0 < Ω < Ωc to pass, whereas the higher frequencies in the stop band Ω >
Ωc are blocked. Opposite to that, the high pass filter allows high frequencies above
Ω > Ωc and rejects the frequencies between Ω = 0 and Ω = Ωc. The frequency Ωc
between the two bands is the cutoff frequency, where the magnitude 𝐻(𝑗ω) =
1/ 2.
Dr. Shweta B. Thomas, VIT Vellore
Dr. Shweta B. Thomas, VIT Vellore
Band pass and band reject filter
 It allows only a band of frequencies Ω1 to Ω2 to pass and stops all other
frequencies.
 Opposite to band pass, band reject filter rejects all the frequencies between Ω1
and Ω2 and allows remaining frequencies.
Dr. Shweta B. Thomas, VIT Vellore
Dr. Shweta B. Thomas, VIT Vellore
Dr. Shweta B. Thomas, VIT Vellore
Specifications for the magnitude response of analog LPF
Dr. Shweta B. Thomas, VIT Vellore
Specifications for the magnitude response of digital LPF
Dr. Shweta B. Thomas, VIT Vellore
Alternative specifications for the magnitude response of LPF
Dr. Shweta B. Thomas, VIT Vellore
Analog LPF Design
Dr. Shweta B. Thomas, VIT Vellore
Analog LPF Butterworth Filter
 Increasing the filter order N, the magnitude response of the Butterworth filter
approximates the ideal response. At Ω = Ωc, the curve pass through 0.707, which
corresponds to -3 dB point. However, the phase response becomes more non-linear with
increasing N.
Dr. Shweta B. Thomas, VIT Vellore
Analog LPF Butterworth Filter
Dr. Shweta B. Thomas, VIT Vellore
Analog LPF Butterworth Filter
 For N odd, 𝑠 2𝑁 =1= 𝑒 𝑖2π𝑘 .
Now roots can be found as , 𝑠𝑘 = 𝑒 𝑖π𝑘/𝑁 k = 1,2,…….2N.
 For N even, 𝑠 2𝑁 = -1= 𝑒 𝑖(2𝑘−1)π .
Now roots can be found as , 𝑠𝑘 = 𝑒 𝑖(2𝑘−1)π/2𝑁 k = 1,2,…….2N.
 For N = 3, k = 1,2,…….6 (k = 1,2,…….2N).
π
π
 𝑠1 = 𝑒 𝑗π/3 = cos 3 +jsin 3 = 0.5 +j0.866
 𝑠2 = 𝑒 𝑗2π/3 = cos
2π
3
+jsin
2π
=
3
-0.5 +j0.866
 𝑠3 = 𝑒 𝑗π = cos π +jsin π = -1
 𝑠4 = 𝑒 𝑗4π/3 = cos
4π
3
 𝑠5 = 𝑒 𝑗5π/3 = cos
5π
3
+j sin
+jsin
4π
3
= −0.5 −j0.866
5π
=
3
0.5 -j0.866
 𝑠6 = 𝑒 𝑗2π =cos 2π +jsin 2π = 1
Dr. Shweta B. Thomas, VIT Vellore
Pole location for magnitude square function of B. F. (N=3)
 All the above poles are located in the s-plane and separation between the poles
are given by 3600/2N, which in this case is equal to 600.
 To ensure stability, consider the poles that lie on the left half of the s-plane.
Dr. Shweta B. Thomas, VIT Vellore
Poles at stable condition
Dr. Shweta B. Thomas, VIT Vellore
Butterworth approximations for magnitude response
Dr. Shweta B. Thomas, VIT Vellore
Order of Butterworth Filter
Dr. Shweta B. Thomas, VIT Vellore
Dr. Shweta B. Thomas, VIT Vellore
Steps to design analog low pass B. F.
 From the given specifications find the order of the filter N.
 Round off it to the next higher integer.
 Find the transfer function H(s) for Ωc = 1 rad/sec for the value of N.
 Calculate the value of cutoff frequency Ωc.
 Find the transfer function Ha(s) for the above value of Ωc by substituting s = s/
Ωc in H(s).
Dr. Shweta B. Thomas, VIT Vellore
List of Butterworth Polynomials
Dr. Shweta B. Thomas, VIT Vellore
Problem: Design a low pass Butterworth filter that has a -2dB pass band
attenuation at a frequency of 20 rad/sec and at least -10 dB stop band attenuation
at 30 rad/sec.
Dr. Shweta B. Thomas, VIT Vellore
Dr. Shweta B. Thomas, VIT Vellore
Dr. Shweta B. Thomas, VIT Vellore
Dr. Shweta B. Thomas, VIT Vellore
Dr. Shweta B. Thomas, VIT Vellore
Problem: For the given specifications, design a low pass Butterworth filter
Dr. Shweta B. Thomas, VIT Vellore
Dr. Shweta B. Thomas, VIT Vellore
Dr. Shweta B. Thomas, VIT Vellore
Difference between Butterworth and Chebyshev Filter
Dr. Shweta B. Thomas, VIT Vellore
Magnitude response of Chebyshev Low Pass Filter
Dr. Shweta B. Thomas, VIT Vellore
Magnitude response of Chebyshev Low Pass Type I Filter
Dr. Shweta B. Thomas, VIT Vellore
Analog Low Pass Chebyshev Filter (Type I)
Dr. Shweta B. Thomas, VIT Vellore
Order of Low Pass Chebyshev Filter (Type I)
Dr. Shweta B. Thomas, VIT Vellore
Analog Low Pass Chebyshev Filter (Type II)
Dr. Shweta B. Thomas, VIT Vellore
Steps to Design Analog Low Pass Chebyshev Filter
 From the given specifications find the order of the filter N.
 Round off it to the next higher integer.
 Using following expression find the values of a and b, which are minor and major
axis of the ellipse respectively.
 Calculate the poles of the Chebyshev filter which lies an ellipse by using the
formula
Dr. Shweta B. Thomas, VIT Vellore
 Find the denominator of the transfer function using above poles as:
(s – s1)(s – s2)…….
 The numerator of the polynomial depends on the value of N.
 For N=odd, substitute s = 0 in the denominator polynomial and find the value.
This value is equal to the numerator of the Transfer function. (Since for N odd,
the mag. Response starts from unity).
 For N even, substitute s = 0 in the denominator polynomial and devide the result
by
.. This value is equal to the numerator.
Dr. Shweta B. Thomas, VIT Vellore
Dr. Shweta B. Thomas, VIT Vellore
Dr. Shweta B. Thomas, VIT Vellore
Dr. Shweta B. Thomas, VIT Vellore
Dr. Shweta B. Thomas, VIT Vellore
Dr. Shweta B. Thomas, VIT Vellore
Dr. Shweta B. Thomas, VIT Vellore
Dr. Shweta B. Thomas, VIT Vellore
Design steps of Chebyshev Type II (Inverse Chebyshev) Filter
 From the given specifications find the order of the filter N.

cosh 1  
1
cosh
A



N


 cosh 1 1
1  s
cosh 

k

p



  10
0.1 p

1
0.5
and   10
0.1 s
 1
0.5
 Round off it to the next higher integer.
 Find the values of a and b, which are minor and major axis of the ellipse
respectively.
    1 
Dr. Shweta B. Thomas, VIT Vellore
2
 The zeros are located on the imaginary axis at the points
s
s k j
k  1,2,
sin k
,N
 The poles are located at the points (xk, yk), where
 s k
xk  2
k  1,2,
2
 k  k
,N
s k
yk  2
k  1,2,
2
 k  k
,N
 k  a cos k and  k  b sin k
Dr. Shweta B. Thomas, VIT Vellore
 The transfer function can be found as
pk  s  zk 
H  s   
k 1 zk  s  pk 
N
Problem: Find the transfer function of the analog Chebyshev type II filter for the
following digital filter specifications. Use bilinear transformation.
1  H  e j 
H  e j 
dB
dB
0
0    0.2
 20
Dr. Shweta B. Thomas, VIT Vellore
  0.3
Dr. Shweta B. Thomas, VIT Vellore