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Tutorial 6 - Analog Filters

This document provides information on digital signal processing and includes examples of designing various analog filters. It discusses determining the order of a Chebyshev lowpass filter based on passband and stopband ripple specifications. It also demonstrates using frequency transformation to derive highpass and lowpass filters from a given transfer function. Examples are provided on designing Butterworth highpass and lowpass filters based on given specifications.

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Anon Poet
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46 views22 pages

Tutorial 6 - Analog Filters

This document provides information on digital signal processing and includes examples of designing various analog filters. It discusses determining the order of a Chebyshev lowpass filter based on passband and stopband ripple specifications. It also demonstrates using frequency transformation to derive highpass and lowpass filters from a given transfer function. Examples are provided on designing Butterworth highpass and lowpass filters based on given specifications.

Uploaded by

Anon Poet
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PPT, PDF, TXT or read online on Scribd
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Digital Signal Processing

TUTORIAL 6

Department of EEE
BITS-Pilani, Hyderabad campus

November 23, 2023 Digital Signal Processing 1


Problem-1

Determine the lowest order (N) of a lowpass


Chebyshev filter with αp = 2 dB, αs = 30 dB
at frequencies 20 rad/s and 40 rad/s
respectively.

(1) Find the 3-dB cut-off frequency


(2) Find the transfer function of the filter
(3) Find the location of the poles

November 23, 2023 Digital Signal Processing 2


ε2 = 0.584893
A2 = 1000 N = 3.3589 ≈ 4

Formulas
1 1  1  
 3dB   p cosh  cosh  
N   

November 23, 2023 Digital Signal Processing 3


November 23, 2023 Digital Signal Processing 4
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November 23, 2023 Digital Signal Processing 10
Using Matlab
2.615e04
-----------------------------------------------
s^4 + 14.32 s^3 + 502.6 s^2 + 4134 s + 3.292e04

November 23, 2023 Digital Signal Processing 11


Problem-2
Let H(s) = 1/ (s2 + s + 1) represent the transfer
function of a lowpass filter with a passband of 1
rad/s.
Use frequency transformation to find the
following transfer functions
(1) A lowpass filter with a passband of 10rad/s
(2) A highpass filter with a cut-off frequency of 1
rad/s
(3) A highpass filter with a cut-off frequency of 10
rad/s
November 23, 2023 Digital Signal Processing 12
Answers

(1) H(s) = 100 / s2 +10s +100

(2) H(s) = s2 / s2 + s +1

(3) H(s) = s2 / s2 + 10s +100

November 23, 2023 Digital Signal Processing 13


Problem-3

(i) Design a high-pass Butterworth filter with


αp = 3 dB, αs = 15 dB at frequencies
1000 rad/s and 500 rad/s respectively.
(ii) Find its transfer function and
(iii) locate its poles in the s-plane.

Ωs = 1000/500 = 2

November 23, 2023 Digital Signal Processing 14


Analog HighPass Filter Design
ε2 = 0.9953 A2 = 31.623
N =2.471 ≈ 3

Ωc = 1000 rad/s

November 23, 2023 Digital Signal Processing 15


November 23, 2023 Digital Signal Processing 16
HHP(s) = s → ?

HHP(s) = s → 1000 / S

HHP(s) = S3/(S3 + 2S2 + 2S + 1)

November 23, 2023 Digital Signal Processing 17


Problem-4
The transfer function of a third order analog
Butterworth lowpass filter with a passband
edge at 0.24 Hz and a passband ripple of 0.5 dB
is given by

HLP (s) = 10 / [s3 + 4.309 s2 + 9.2835 s + 10]

Determine the transfer function HHP(s) of analog


highpass filter with a passband edge at 3 Hz and a
passband ripple of 0.5 dB.
November 23, 2023 Digital Signal Processing 18
Solution

November 23, 2023 Digital Signal Processing 19


Problem - 5
A Butterworth analog highpass filter is to be
designed with the following specifications :
ˆˆ  66..55kHz
FF , ˆ
F kHz,, pp  00..55dB
kHz , Fˆss 11..55kHz dB,, ss  40
40dB
dB
pp

(i)What are the band edges and the order of the


corresponding analog lowpass filter ?
(ii) What is the order of the highpass filter ?

November 23, 2023 Digital Signal Processing 20


Solution

November 23, 2023 Digital Signal Processing 21


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