11-Mar-15
1
FIR
CH 4
ME-4722
Digital Signal Processing
Chapter 4
Design of Digital FIR Filters(Elective)
Spring 2015, SZABIST, Karachi
FIR
CH 4
Instructor:
Engr. Humera Rafique
Assistant Professor (Mechatronics)
Office: FR-404 (100 Campus )
Course Support
Official: ZABdesk
Wednesday, 11 March, 2015 HR Spring 15 DSP
2
11-Mar-15
2
FIR
CH 4
Chapter Contents
Introduction to Digital Filters
Brief Comparison of FIR/IIR Filters
Design of Digital FIR Filters
Window Method
Rectangular
Hamming
Han
Kaiser
Other windows: Performance overview
Frequency Sampling Method*
Text: Introduction to Signal Processing (S.J. Orfanidis)
Wednesday, 11 March, 2015 HR Spring 15 DSP
3
FIR
CH 4
Introduction to
Digital Filters
Wednesday, 11 March, 2015 HR Spring 15 DSP
4
11-Mar-15
3
FIR
CH 4
Digital Filters
Discrete time LTI systems can be classified as:
1. FIR (Finite Impulse Response): h = [h0,h1,h2,. . . ,hN-1]
2. IIR (Infinite Impulse Response): h = [h0,h1,h2,. . . . .]
h(n)
nN
h(n)
n
0( ) ( ) ( )IIR ky n h k x n k
=
= 1
0( ) ( ) ( )NFIR ky n h k x n k
=
=
Wednesday, 11 March, 2015 HR Spring 15 DSP
5
FIR
CH 4
Digital Filters
FIR Filters
Finite length Impulse response
Linear phase property
Guaranteed stabilities
High computational cost
IIR Filters
Infinite length impulse response
Linear phase can not be achieved exactly
over entire Nyquist interval
Unstable (poles outside the unit circle)
Low computational cost and efficient
implementation (specially in sos cascade
configuration)
Wednesday, 11 March, 2015 HR Spring 15 DSP
6
11-Mar-15
4
FIR
CH 4
Digital Filters
Mathematical Description of Digital Filter:
1. Impulse response
2. Hardware realization
3. i/o difference equation
4. Pole zero constellation
5. Transfer function
6. (Magnitude/Phase responses)
Wednesday, 11 March, 2015 HR Spring 15 DSP
7
FIR
CH 4
Digital Filters
Digital Filter Design:
The process of constructing impulse response/ transfer function of a digital filter that meets the
prescribed requirements described in frequency response specification is called Digital Filter Design.
Filter design classes:
FIR Filter design:
Window method
Frequency Sampling method*
IIR Filter design:
Bilinear transformation
Wednesday, 11 March, 2015 HR Spring 15 DSP
8
11-Mar-15
5
FIR
CH 4
Digital FIR Filter Design:
Window Method
Wednesday, 11 March, 2015 HR Spring 15 DSP
9
FIR
CH 4
FIR -Windows Low pass filter design
Use of window
Rectangular window with Low pass filter
Advantages and limitations of rectangular window
Hamming, Hann and other windows:
Mathematical Models & Frequency responses
HPF, BPF & BSF design
Kaiser window:
Low pass filter design
Band pass filter design
Crossover design
Wednesday, 11 March, 2015 HR Spring 15 DSP
10
11-Mar-15
6
FIR
CH 4
FIR -Windows
Ideal Low Pass FIR Filter Design:
-c-pi -c pi
D()
1, 0,
Wednesday, 11 March, 2015 HR Spring 15 DSP
11
FIR
CH 4
FIR -Windows
Ideal Low Pass FIR Filter Design:
Wednesday, 11 March, 2015 HR Spring 15 DSP
12
11-Mar-15
7
FIR
CH 4
FIR -Windows
Ideal Low Pass FIR Filter Design:
Impulse Response of a digital filter
Impulse Response of an analog filter
d(k): Impulse Response of an analog filter:
Infinite
Continuous
Non-causalh(n): Impulse Response of a digital filter:
Finite
Discrete
Causal
Wednesday, 11 March, 2015 HR Spring 15 DSP
13
FIR
CH 4
FIR -Windows
To convert above into an ideal digital filter:
Truncation . . . . . . . . . Finite . . . . .
Digitize . . . . . . . . . . Digital . . . .
Shifted . . . . . . . . . . . . Causal
0 2or0 1
Wednesday, 11 March, 2015 HR Spring 15 DSP
14
11-Mar-15
8
FIR
CH 4
FIR -Windows
Ideal Low Pass FIR Filter Design:
-M M
Wednesday, 11 March, 2015 HR Spring 15 DSP
15
FIR
CH 4
FIR -Windows
Ideal Low Pass FIR Filter Design:
Wednesday, 11 March, 2015 HR Spring 15 DSP
16
11-Mar-15
9
FIR
CH 4
FIR -Windows
Ideal Low Pass FIR Filter Design:
Shift
. 0 1
Wednesday, 11 March, 2015 HR Spring 15 DSP
17
FIR
CH 4
FIR -Windows
Ideal Low Pass FIR Filter Design:
. 0 1
Wednesday, 11 March, 2015 HR Spring 15 DSP
18
11-Mar-15
10
FIR
CH 4
FIR -Windows
Rectangular Window:
. 0 1
0 1
1
Wednesday, 11 March, 2015 HR Spring 15 DSP
19
FIR
CH 4
FIR -Windows
Other Filters:
2. High pass
-c-pi c pi
D()
" # # sin " # #
0 1; 12Wednesday, 11 March, 2015 HR Spring 15 DSP
20
11-Mar-15
11
FIR
CH 4
FIR -Windows
Other Filters:
3. Band pass
-a-pi pi
D()
-b ba
' ( )' ( ) 0 1
Wednesday, 11 March, 2015 HR Spring 15 DSP
21
FIR
CH 4
FIR -Windows
Other Filters:
4. Band stop
-a-pi pi
D()
-b ba
'* # ( )
'* # . ( ) 0 1Wednesday, 11 March, 2015 HR Spring 15 DSP
22
11-Mar-15
12
FIR
CH 4
FIR -Windows
Other Filters:
Wednesday, 11 March, 2015 HR Spring 15 DSP
23
FIR
CH 4
FIR -Windows
Ideal digital FIR filter Specifications:
Stop band
Pass band
Cutoff
pi
Wednesday, 11 March, 2015 HR Spring 15 DSP
24
11-Mar-15
13
FIR
CH 4
Example O (10.1.1):
Determine the length-11, rectangularly windowed impulse response that approximates an ideal
lowpass filter of cutoff frequency pi/4 rad.s-1.
{11, pi/4}
1 2 3 4 5 6 7 8 9 10 11-0.05
0
0.05
0.1
0.15
0.2
0.25
0.3Impulse Response
Ampli
tude
Samples
FIR -Windows
+0.045, 0, 0.075, 0.1592, 0.2251, 0.25,0.2251, 0.1592, 0.075, 0, 0.0450
0 1N = 11,
M = (N-1)/2 = 5
c = pi/4 rad.s-1
Wednesday, 11 March, 2015 HR Spring 15 DSP
25
FIR
CH 4
Example O (10.1.2):
Repeat example 1 with length 5 and cutoff
frequency = 0.3 pi rad/sec. Pole-zero constellation is
also required.
Repeat example 1 with length 7.
h(n) = {0.075, 0.1592, 0.2251, 0.25, 0.2251, 0.1592, 0.0750}
Example O (3):
h(n) = {0.1514, 0.2575, 0.3, 0.2575, 0.1514}
-1 -0.5 0 0.5 1
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
4
Real Part
Imag
inar
y Pa
rt
FIR -Windows
Wednesday, 11 March, 2015 HR Spring 15 DSP
26
11-Mar-15
14
FIR
CH 4
Example O (p.538):
FIR -Windows
Design an ideal lowpass filter of cutoff frequency c = 0.3, approximated by a rectangularly windowed
response of length N = 41 and then by another one of length N = 121:
Wednesday, 11 March, 2015 HR Spring 15 DSP
27
FIR
CH 4
Example O (p.538):
FIR -Windows
Wednesday, 11 March, 2015 HR Spring 15 DSP
28
11-Mar-15
15
FIR
CH 4
Example O (p.538):
FIR -Windows
Wednesday, 11 March, 2015 HR Spring 15 DSP
29
FIR
CH 4
Rectangular Window Properties:
Simple mathematical model
Simple implementation
Ripples:
Passband
Stopband
Pass-Stop band
FIR -Windows
* 0 (min)* 100% (max)* 0
ApassAstop
Criterion of good filter design:
Wednesday, 11 March, 2015 HR Spring 15 DSP
30