1. By
Imtiyaz Mohiuddin
10361A0434
In charge
Dr.M.Narayana
Electronics and Communications Engineering

2. SECTION OUTLINE:
 Introduction to Digital filter design
 Introduction to FIR Filter
 Design of FIR Filter using WINDOW Techniques
 Introduction to IIR Filter
 Design of IIR Filter using Approximation Method
 MATLAB Code of Designed Filters

3. Introduction:
 A digital filter is a system that performs mathematical
operations on a sampled, discrete-time signal to reduce or
enhance certain aspects of that signal.
 In digital signal processing, there are two important types
of systems:
 Digital filters: perform signal filtering in the time
domain
 Spectrum analyzers: provide signal representation in the
frequency domain

4. Digital Filter:
xn yn
Digital Filter
Sampling
frequency
fS
A
D
C
D
A
C
x(t) y(t)
Analog
anti-
aliasing
filter
Analog
smoothing
filter

5. Preliminaries:
 The design of a digital filter is carried out in three steps:
 Specifications: they are determined by the applications
 Approximations: once the specification are defined, we use various
concepts and mathematics that we studied so far to come up with a
filter description that approximates the given set of specifications.
(in detail)
 Implementation: The product of the above step is a filter
description in the form of either a difference equation, or a system
function H(z), or an impulse response h(n). From this description
we implement the filter in hardware or through software on a
computer.

6. Classification:
 Digital filters are classified into one of two basic forms,
according to how they respond to a unit impulse:
 Finite impulse response
 Infinite impulse response

7. Finite Impulse Response:
 In signal processing, a finite impulse response
(FIR) filter is a filter whose impulse response is
of finite duration, because it settles to zero in finite
time.
 FIR digital filters use only current and past input
samples, and none of the filter's previous output
samples, to obtain a current output sample value

8. The transfer function is given by
 The length of Impulse Response is N
 All poles are at Z=0. .
 Zeros can be placed anywhere on the z-plane




1
0
).()(
N
n
n
znhzH

9. Filter Design by Windowing
 Simplest way of designing FIR filters
 Start with ideal frequency response
 Choose ideal frequency response as desired response
 Most ideal impulse responses are of infinite length
   




n
nj
d
j
d enheH     

 



deeH
2
1
nh njj
dd
   


 

else0
Mn0nh
nh d

10. Rectangular:
Bartlett:
Hamming:
Blackman:
Kaiser:
2
1

N
nN
n2
1






N
n2
cos1






N
n2
cos46.054.0











N
n
N
n  4
cos08.0
2
cos5.042.0
)(
1
2
1 0
2
0  J
N
n
J
















Commonly used windows

11. Kaiser window
 Kaiser window
β Transition
width (Hz)
Min. stop attn
dB
2.12 1.5/N 30
4.54 2.9/N 50
6.76 4.3/N 70
8.96 5.7/N 90

12. Rectangular Window
 


 

else0
Mn01
nw
 Narrowest main lob
– 4/(M+1)
– Sharpest transitions at
discontinuities in frequency
 Large side lobs
– Large oscillation around
discontinuities
– -13 dB
Simplest window possible

13. Hamming Window
 









 


else0
Mn0
M
n2
cos46.054.0
nw
 Medium main lob
– 8/M
 Good side lobs
– -41 dB
– Simpler than
Blackman

14. Kaiser Window
 Parameterized equation forming a set
of windows
 Parameter to change main-lob width
and side-lob area trade-off
 I0(.) represents zeroth-order
modified Bessel function of 1st kind
 
 






















 


else0
Mn0
I
2/M
2/Mn
1I
nw
0
2
0

15. MATLAB CODE:
 %Design of LPF&HPF using rectangular,hamming and kaiser windows
 clc;clear all;close all;
 rp=input('enter attenuation in pass band');
 rs=input('enter attenuation in stop band');
 fp=input('enter pass band frequency');
 fs=input('enter stop band frequency');
 Fs=input('enter sampling frequency');
 wp=2*pi*fp/Fs;
 ws=2*pi*fs/Fs;
 %formula for FIR filter
 num=-20*log10(sqrt(rp*rs))-13;
 den=14.6*(fs-fp)/Fs;
 n=ceil(num/den);
 disp('order of filter is n');
 disp(n);
 disp('press any key to continue');
 pause;
 n1=n+1;
 %For even order
 if(rem(n,2)~=0)
 n1=n;
 end
 %LPF

16.  %LPF
 s1=input('enter the value for window 0-rectangularLPF 1-kaiserLPF 2-hammingLPF 3-
rectangularHPF 4-kaiserHPF 5-hammingHPF');
 switch(s1);
 case 0
 y=rectwin(n1);
 [b,a]=fir1(n,wp,'low',y);
 freqz(b,a,512);
 case 1
 y=kaiser(n1);
 [b,a]=fir1(n,wp,'low',y);
 freqz(b,a,512);
 case 2
 y=hamming(n1);
 [b,a]=fir1(n,wp,'low',y);
 freqz(b,a,512);
 case 3
 y=rectwin(n1);
 [b,a]=fir1(n,wp,'high',y);
 freqz(b,a,512);
 case 4
 y=kaiser(n1);
 [b,a]=fir1(n,wp,'high',y);
 freqz(b,a,512);
 case 5
 y=hamming(n1);
 [b,a]=fir1(n,wp,'high',y);
 freqz(b,a,512);
 end

17. Pros & Cons:
FIR filters have the following
advantages:
 Exactly linear phase is possible
 Always stable, even when quantized
 Design methods are generally linear
 Efficient hardware realizations
 Startup transients have finite
duration
FIR filters have the following
disadvantages:
• Higher filter order than IIR
filters
• Corresponding greater
delays

18. Infinite Impulse Response Filter:
 IIR systems have an impulse response function that is
non-zero over an infinite length of time. This is in
contrast to finite impulse response (FIR) filters, which
have fixed-duration impulse responses
 IIR filters may be implemented as
either analog or digital filters

19. Cont..
 While designing a digital IIR filter , an analog filter
(e.g. Chebyshev filter, Butterworth filter) is first
designed and then is converted to a digital filter by
applying discretization techniques such as Bilinear
transform or Impulse invariance.

20. Discretization techniques

21. Chebyshev Filter:
 Chebyshev filters are analog or digital filters having a
steeper roll-off and more passband ripple (type I) or
stopband ripple (type II)
 Chebyshev filters have the property that they
minimize the error between the idealized and the
actual filter characteristic over the range of the filter,
but with ripples in the passband

22. Cont..
 Type-1 Chebyshev Filter
 Type-2 Chebyshev Filter:

23. Butterworth filter
 The Butterworth filter is a type of signal processing
filter designed to have as flat a frequency response as
possible in the pass band. It is also referred to as a
maximally flat magnitude filter

24. MATLAB Prototype Filter Design Commands
 [B,A] = BUTTER(N,Wn)
 [B,A] = CHEBY1(N,R,Wn)
 [B,A] = CHEBY2(N,R,Wn)
 [B,A] = ELLIP(N,Rp,Rs,Wn)
– N = filter order
– R = pass band ripple (cheby1) or stop-band ripple (cheby2) in
dB. (Rp and Rs respectively for the elliptic filter)
– Wn = cut-off frequency (radians/sec for analog filters or
normalized digital frequencies for digital filters)
– [B,A] = filter coefficients, s-domain (analog filter) or z-domain
(digital filter)

25. Design Example
 Filter Specifications:
 Butterworth response
 Pass-band edges = 400 Hz and 600 Hz
 Stop-band edges = 300 Hz and 700 Hz
 Pass-band ripple = 1 dB
 Stop-band attenuation = -20 dB
 Sampling Frequency = 2000 Hz

26. Design Example Results
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Frequency (kHz)
Magnitude
Magnitude Response
Band Edges
(-1dB and -20 dB)

27. Design Example
Chebyshev II High-Pass Filter
 Filter specifications:
 Chebyshev II response (stop-band ripple)
 Pass-band edge = 1000 Hz
 Stop-band edge = 900 Hz
 Pass-band ripple = 1 dB
 Stop-band attenuation = -40 dB
 Sampling frequency = 8 kHz

28. MATLAB Code for Design Example
>> fs=8000;
>> Wp=[2*1000/fs]; % Pass-band edge normalized digital frequency
>> Ws=[2*900/fs]; % Stop-band edge normalized digital frequency
>> [N,Wn]=cheb2ord(Wp,Ws,1,40); % The “order” command
>> [B,A]=cheby2(N,40,Wn,'high');
% cheby2 is the “filter” command. In this command
% the syntax requires the stop-band attenuation
% as the second parameter
>> fvtool(B,A)

29. Design Example Results
0 0.5 1 1.5 2 2.5 3 3.5
-100
-80
-60
-40
-20
0
20
Frequency (kHz)
Magnitude(dB)
Magnitude Response (dB)

30. MATLAB CODE:
 %Design of IIR filters
 fp1=input('enter pass band frequency');
 fs1=input('enter stop band frequency');
 Fs1=input('enter sampling frequency');
 wp1=fp1/Fs1;
 ws1=fs1/Fs1;
 [n1,wn1]=buttord(wp1,ws1,2,60);
 [x,y]=butter(n1,wn1,'low');
 figure;
 freqz(x,y,512);
 [n1,wn1]=buttord(wp1,ws1,2,60);
 [x,y]=butter(n1,wn1,'high');
 figure;
 freqz(x,y,512);
 [n1,wn1]=cheb1ord(wp1,ws1,2,60);
 [x,y]=cheby1(n1,3,wn1,'low');
 figure;
 freqz(x,y,512);
 [n1,wn1]=cheb1ord(wp1,ws1,2,60);
 [x,y]=cheby1(n1,3,wn1,'high');
 figure;
 freqz(x,y,512);

31. Summary of IIR Filter:
 IIR filters can be design by pole-zero location
– Digital oscillators: poles on the unit circle
– Notch filters: zeros on the unit circle with nearby poles to
control notch width
 Classic analog filters can be designed using the
bilinear transformation
 IIR filters have the advantage of smaller filter order for
a given frequency response.
 IIR filters have the disadvantages of possible instability
due to coefficient quantization effects and non-linear
phase response.

32. References:
 “Design of IIR Filter” by K.S Chandra, M.Tech, IIT-Bombay, Jan-2006
 “Digital Filter Design” by Prof. A.G. Constantinides, University of
Auckland, 2006
 “FIR Filter Design”, Gao Xinbo,School of E.E., Xidian Univ.
xbgao@ieee.org
 “FIR Filter by Windowing”- The lab Book Pages.com
 “Digital Signal Processing”, Prof.Ramesh Babu, Pondicherry Govt.
College, TataMcgraw-Hill publication.
 Wikipedia.org