DSP4-FIR 1

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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)

[email protected]

Office: FR-404 (100 Campus )

Course Support

Official: ZABdesk

Wednesday, 11 March, 2015 HR Spring 15 DSP

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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)


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Introduction to

Digital Filters


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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

=

=


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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)


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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)


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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


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Digital FIR Filter Design:

Window Method


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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


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FIR -Windows

Ideal Low Pass FIR Filter Design:

-c-pi -c pi

D()

1, 0,


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FIR -Windows



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FIR -Windows


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


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FIR -Windows

To convert above into an ideal digital filter:

Truncation . . . . . . . . . Finite . . . . .

Digitize . . . . . . . . . . Digital . . . .

Shifted . . . . . . . . . . . . Causal

0 2or0 1


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FIR -Windows


-M M


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FIR -Windows



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FIR -Windows


Shift

. 0 1


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FIR -Windows


. 0 1


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FIR -Windows

Rectangular Window:

. 0 1

0 1

1


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FIR -Windows

Other Filters:

2. High pass

-c-pi c pi

D()

" # # sin " # #

0 1; 12Wednesday, 11 March, 2015 HR Spring 15 DSP

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FIR -Windows

Other Filters:

3. Band pass

-a-pi pi

D()

-b ba

' ( )' ( ) 0 1


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FIR -Windows

Other Filters:

4. Band stop

-a-pi pi

D()

-b ba

'* # ( )

'* # . ( ) 0 1Wednesday, 11 March, 2015 HR Spring 15 DSP

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FIR -Windows

Other Filters:


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FIR -Windows

Ideal digital FIR filter Specifications:

Stop band

Pass band

Cutoff

pi


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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


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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


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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:


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Example O (p.538):

FIR -Windows


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Example O (p.538):

FIR -Windows


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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:


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