FIR Filter Design_new

7/31/2019 FIR Filter Design_new

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op cs:

1. Linear Phase FIR Digital Filter.

Introduction

2. Linear-Phase FIR Digital Filter Design:n ow n ow ng e o

1

op c:

.

advanta es and disadvanta es of linear hase FIR di ital

filters,

linear phase conditions for FIR filters,

four groups/kinds of linear phase FIR digital filters.

2

op c:

-Window (Windowing) Method

basic principles and algorithms,

method description in time- and frequency-domain,

Example A.: FIR filter design-rectangular window application,

Gibbs phenomenon and different windowing applications,

Example B.: FIR filter design at different window

applications.

3

Special operations

Differentiation:( )

( )dx t

y tdt

= ( ) ( )Y j j X j =

Integration:

( ) ( )y t x d

=

( ) ( ) (0) ( )Y j X j X j

= +

4


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Digital Filter Design

Objective - Determination of a realizabletrans er unct on z approx mat ng a g venfrequency response specification.

Digital filter design is the process of derivingthe transfer function G(z).

Two possibilities: IIR or FIR.

,

stable real rational function

5

Digital Filter Specifications

The magnitude and/or the phase (delay)response is specified for the design of a

di ital filter for most a lications

In most practical applications, the problem

v z

approximation to a given magnituderesponse specification

6


approximation problem

magnitude responses as shown belowj

1

LP e

1

HP e

0 cc 0 cc H j

11

BP (e )

1

7 c1 c1c2 c2 c1 c1c2 c2


As the im ulse res onse corres ondin to

each of these ideal filters is noncausal and

,

realizable

In practice, the magnitu e response

specifications of a digital filter in the

passband and in the stopband are given with

In addition, a transition band is specified

8

etween t e pass an an stop an


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For exam le, the ma nitude res onse| |j

G e

of adigital lowpass filter may be given as

9


As indicated in the figure, in thepassband,

defined by , we require that

with an error i.e.1jeG

p0

jeG + ,1)(1

require that with an error ,0)( jeG

s

s

. .,

ssj

eG ,)(

10


-

- stopband edge frequency

p

s

- peak ripple value in thepassbandp-

is a periodic function of and thes

)( j

eG

magnitude response of a real-coefficient

di ital filter is an even function of

As a result, filter specifications are given

11

on y or e requency range

.

response of ideal filters is linear:

0( ) t =

B. Comments on group delay function: Group delayfunction of ideal filters is constant:

( )d d = = = =0 0 .

d d

C. Note: It will be proved for linear phase FIR filters:

12

02

t =


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- -

response is identically a positive constant ( )( ) .jH e const =

. -

is not restricted and is allowed to vary arbitrarily as a.

In general, a rational filter is all-pass if only if it has the

same number of poles and zeros (including multiplicities),

and each zero is the conjugate inverse of a corresponding

pole:zk=1/pk.

Example:1

0.8( )

1 0.8

zH z

z

=

1 0.8p =1 1/0.8z =

131 11/ 0.8 1/ z p= =

Linear Phase FIR Digital Filter.Introduction

14

g a er as a n e num er o non-zero

coefficients of its impulse response:

: ( ) 0M N h n for n M = >

Mathematical model of a causal FIR digital filter:

1

0

( ) ( ) ( )M

k

y n h k x n k

=

=

Digital FIR filters cannot be derived from analogue

filters, since causal analogue filters cannot have a finiteimpulse response. In many digital signal processing

15applications, FIR filters are preferred over their IIR

counterparts.

FIR filters with exactly linear phase can be easilydesigned. This simplifies the approximation problem,

in many cases, when one is only interested in designing

of a filter that approximates an arbitrary magnituderesponse. Linear phase filters are important for

applications where frequency dispersion due to

nonlinear phase is harmful (e.g. speech processing and

data transmission).

implementing FIR filters. These include both non-

16

.


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FIR filters realized non-recursively are inherentlystable and free of limit cycle oscillations when

implemented on a finite-word length digital system.

The output noise due to multiplication round off

sensitivity to variations in the filter coefficients is

.

Excellent design methods are available for variouskinds of FIR filters with arbitrary specifications.

17

The relative com utational com lexit of FIR filter ishigher than that of IIR filters. This situation can be

met es eciall in a lications demandin narrowtransition bands or if it is required to approximate sharp

cut off fre uenc . The cost of im lementation of an FIR

filter can be reduced e.g. by using multiplier-efficient

realizations, fast convolution al orithms and multirate

filtering.

e group e ay unc on o near p ase ers

need not always be an integer number of samples.

18

Frequency Response of Linear Phase FIR Digital

Filters

FIR filter of length M:

1Mk

1M

0k

e e=

=0k

y n x n=

=

19

The linear phase condition is obtained by imposing

symmetry conditions on the impulse response of the

filter. In particular, we consider two different symmetry

conditions for h(k):

( ) ( 1 ) 0,1,2, , 1h k h M k for k M = = K

.

B. Antisymmetrical impulse response:

( ) ( 1 ) 0,1,2, , 1h k h M k for k M = = K

The length of the impulse response of the FIR filter(M) can be even or odd. Then, the four cases of linear

20phase FIR filters can be obtained.


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Symmetrical Impulse Response, M: Even

=7 = 8

=

=

=

21n

Example: M=4 (even), symmetrical impulse response

1 4 1 3 0,1,2,3M k = = =

(0) (3) (1) (2)h h h h= =

1 1 1 22 2 2 2

= = = =

, , , , K

(0) ( 1), (1) ( 2), (2) ( 3), ,h h M h h M h h M = = = K

1M M

h h

=

22

1 4 1 3

0 0 0

( ) ( ) ( ) ( )M

j j k j k j k

k k k

H e h k e h k e h k e

= = =

= = =

0 1 2 30 1 2 3j j j j jH e h e h e h e h e = + + + =

0 3 1 2j j j j

1

0

( ) j k

k

h k e e

=

= + =

( )

12

1

M

j M kj k

= =23

.0k=

.

11 2

1

M

MM kk k

0 0k k

e e e e= =

= = + =

11 2 22

22 ( )

M j k j kM

j e ee h k

+

=

1M

0k=

2

0

1( ) 2 ( )cos

2

jj

k

H e e h k k

=

=

Here, the real-valued frequency response is given by

12 1

( ) 2 ( )cos

M

MH h k k

=

240 2k=


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1

2( ) ( )M

jj

H e e H

=1

2( ) ( ) 0M

j

H e for H

1

2 0

Mj

H e or H

+

=

g(0)=f(-7)

=-

63n

Example: Magnitude Response

=

xamp e: ase esponse

( )

0

=

64


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Gibbs Phenomenon and Different Windowing

Direct truncation of impulse response leads to well known

.

It manifests itself as a fixed percentage overshoot and ripple

e ore an a er scon nu y n e requency response.

E.g. standard filters, the largest ripple in the frequency

response is about 18% of the size of discontinuity and its

amplitude does not decrease with increasing impulse response

. .

series does not decrease the amplitude of the largest ripple.

ns ea , e overs oo s con ne o a sma er an sma er

frequency range as is increased.

65

Example: Gibbs phenomenon illustration

-

FIR low-pass digital filters with normalized cut off

=, , , , .confirm the above given statements concerning the

.

66

Low-Pass FIR Filter: Rectangular Window Applicationj5N= =

j

/ 2 / 2

jG e

jG e

100N=50N=

67 / 2 / 2

The major effect is that discontinuities of became( )jH e

transition bands between values on the either side of the

discontinuity. Since the final frequency response of the

filter is the circular convolution of the ideal frequencyresponse with the windows frequency response

*j j j

on the width of the main (central) lobe of .( )jW e

68


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from the side lobes produces a ripple in the resulting

.

a) Small width of the main lobe of the frequency

response of the window containing as much the total

energy as possible.

b) Side lobes of the frequency response that decrease in

69

.

Some Commonly Used Windows

1

2

MN

=

( )w n R for N n N ( ) 0w n for n N = >

n

1N

+ec angu ar: =

Hann: ( ) 1 cos2 2 1

w nN

= +

Hamming: ( ) 0.54 0.46cos2 1

nw n

N

=

+

70Blackmann:2 4

( ) 0.42 0.5cos 0.08cos2 1 2 1

n nw n

N N

= +

+ +

Properties of Commonly Used Windows

Rectangular Bart let t ( t r iangular)

=

otherwise

nnw

,0

,,][

= MnMMnnw 2/,/22

,

][

otherwise,0Hanning Hamming

71

=

otherwisenw

,0

,..][

=

otherwise

MnMnnw

,0

0),/2cos(46.054.0][

Windows Magnitude of Frequency

Response

Window Based Design


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Summary of Windows

Characteristics

We see clearly that a wider transition region (wider main-

lobe) is compensated by much lower side-lobes and thus less

ripples.

Window Based DesignGiven specifications: p,, 21 an s

We employ the following procedure

1. Compute

= ),min( 21

sp =

Choose M, the filter order, to meet transition

10og

width

are given by[ ] [ ] [ ], 0

dh n h n w n n M =

=74

,c p s

Window type Window length

= .

Hanning M= 3.1/ f

Hamming M= 3.3/ f

Blackman M=5.5/ f

Kaiser Window

1 21

2nI

2/,0,)(

][0

MMnI

nw ==

is zeroth order modified Bessel function of theFirst Kind

(.)0I

2

)5(.)(

=m

xxI

controls sidelobe level (Stopband Attenuation)

=m

The filter or er M controls the Mainlo e wi th

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Desi n Method: define

ps =

),min(,log20 2110 == whereA

> 50,)7.8(1102. AA

+= 5021),21(07886.)21(5842.4.0

AAA

8

,

A

77=

285.2M

78

Normalized frequency ( )

79

Exam le:

By the windowing method, design a low-pass filter of

or er = w pass- an cu o requency .

Frequency sampling is .0 z=

4Sf kHz=

.

80

Example:FIR Filter Design by Windowing Method

Example:FIR Filter Design by Windowing Method


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* j

=10

[dB]

Bartlett Window

Rectangular Window

Hamming Window

81

* j

10

Kaiser Window: alfa=3

Kaiser Window: alfa=10

a ser n ow: a a=

=

82

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FIR Filter Design_new