Introduction to FIR Filter Design

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Digital Signal Processing

H. Introduction to FIR filters design

Athanassios C. Iossifides

February 2013


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2

.1 Discrete time systems and filters.2 FIR filter design with windows

. Introduction to FIR digital filter design


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.1 Discrete time systems and filters



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H.1 Discrete time systems and filters

LSI system as frequency filters

The frequency response H(e j ) of an LSI system, leads to the modificationof the input (e j ) in the frequency domain, and creates the output (e j )according to the formula

Therefore, every LSI system can be considered as a frequency filter, inthe sense that it passes, amplifies, attenuates or cuts frequencies of theinput . The terms LSI system and filter are used interchangeably.

The characteristics of a filter in the frequency domain depend on The positions of the zeros and the poles of the transfer (system)

function

The positions of zeros and poles are determined by the coefficients of the difference equation that describes the

system

( ) ( ) ( ) j j jY e H e X e


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Distortionless response

The response y (n) of an LSI system to an input x (n) has no distortion (ingeneral) when it is of the form

so that the input signal is only uniformly attenuated with a constant C anddelayed by n0 samples. Using the properties of DTFT, we have

so thatTherefore, a system does not introduce distortion when

The magnitude of the frequency response is constant The phase is a linear function of frequency

The group delay is defined as

and provides the delay that undergoes each frequency passing throughthe system. In the case of distortionless response systems, the groupdelay is constant.

0( ) ( )y n Cx n n

00

( ) ( ) j n j x n n e X eF

0 0( ) ( ) or ( ) j jn j j jnY e Ce X e H e Ce

( )( ) , ( ) ( ) j g

d H e

d


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Ideal filters

The frequency response of an ideal filter has unit amplitude (gain) at thepassband, zero amplitude at the stopband and linear phase.

0| ( )| j H e

cc

B

0| ( )| j H e

cc

0| ( )| j H e

00

B

1 2

0| ( )| j H e

00

0| ( )| j H e

Lowpass filter (LPF) Highpass filter (HPF)

Bandpass filter (BPF) Bandstop filter (BSF)

Allpass filter


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deal filters

he ideal Lowpass Filter (LPF) may be expressed by the frequencyresponse

The impulse response is calculated by the inverse DTFT as follows

This filter is not causal and therefore it is not realizable.

1, | |( )

0, | |c j

c

H e

1 1 1( ) ( ) ( )

2 2 2

sin( ),

cc c

c

j jn jn j n j n

c

h n X e e d e d e e jn

nn

n


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Transfer functions of real (non-ideal) filters

The principle of zeros and poles positioning for the production of thedesirable frequency response is to put

the poles near to the points of the unit circle that correspond tothe frequencies that we want to amplify (or not attenuate)

the zeros close (or exactly on) the frequencies that we want toattenuate (make zero)

Additionally, the following restrictions should apply: All the poles must be in the unto circle so that the filter is stable. All the complex zeros and poles must be complex conjugate pairs

so that the coefficients of the filter in the diefference equation arereal.1

0 10

1

1 1

(1 )( )

1 (1 )

MMk

k k k k

N Nk

k k k k

z zb z

H z b

a z p z


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Examples of real (non-ideal) filters

Lowpass

Highpass

Bandpass

1

1

1( )

1z

H zaz

1

1

1( )

1

zH z

az

2

2

1( )

1z

H zaz


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FIR filter realization

An FIR filter is described by the difference equation

and a transfer function (including only zeros) of the form

This can be realized in a direct form as drawn below

This requires + 1 multiplications, additions and memory places.

0( )

M

k k k

H z b z

0 0( ) ( ) ( ) ( )

M M

k k k

y n b x n k h k x n k

1z

0b 1b

1z 1z 1z

2b 3b 1Mb Mb

( ) x n

( )y n


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FIR direct form realization

Simplified form

1z

0b 1b

1z 1z 1z

2b 3b 1Mb Mb

( ) x n

( )y n

1z 1z 1z1z

0b 1b 2b 1Mb Mb

( )y n

( ) x n


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FIR filters of ljnear phase

A filter has a linear phase when

where = 0 or /2 and constant. For an FIR filter in the interval [0 , ] , the linear phase condition leads to the following symmetry conditions ofthe impulse response:

( ) , j H e

( ) ( ), / 2, 0

( ) ( ), / 2, / 2

h n h M n N

h n h M n N

V


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IIR filter realization

An IR filter is described by the difference equation

and a transfer function of the form

0

1

( )

1

Mk

k k

Nk

k k

b z

H z

a z

1 0( ) ( ) ( )

N M

k k k k

y n a y n k b x n k


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IIR filter realization

Direct Form of type I Direct Form of type II

Number of mult.: + + 1 Number of mult.: + + 1

Number of add.: + Number of add.: +

Memory places: + Memory places: max{ ,}

1z

1z

1z

1z

0b

1b

1Mb

Mb

( )y n( ) x n

1z

1z

1a

1Na

Na

1z

1z

0b

1b

1Mb

Mb

( )y n( ) x n

1z

1a

1Na

Na


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. 2 FIR filter design with windows



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The basic idea

The basic idea of the window method (windowing) is based on: The selection of a non-causal ideal filter of infinite impulse

response duration. The modification of the impulse response with a proper function

(window) so that to produce a causal and linear phase filter.

The modification of the impulse response h(n) is applied withmultiplication with a proper function w (n) which is called window.

An ideal Lowpass Filter (LPF) with cuttof frequency c has a frequencyresponse of the form

where a is a delay that does not affect the phase linearity of the filter andis mandatory in order to convert the non-causal filter to a causal one.

1 , | |( ) 0, | |

ja

c j d c

e H e


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Impulse response of ideal filter

The impulse response of an ideal LPF may be found with an IDFT of theprevious formula, which leads to

and is symmetrical with respect to delay a .

sin[ ( )]( )

( )c

d n a

h n n a


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Conversion to an FIR filter

In order to produce a FIR filter, causal and with linear phase, we restrictthe impulse response with a proper window function w (n), that issymmetrical with respect to a , so that to conserve the symmetry of thefinal impulse response.

The greater the delay a is, the greater the impulse response duration isand the better the ideal filter is approximated (this results in smallertransition band)

( ) ( ) ( )d h n h n w n


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Filter characteristics

The characteristics of the filters that are of interest are: The order (length) of the filter. The transition band width. The attenuation in the stopband.

With the window method it is not possible to control independently the

passband and the stopband and the ripples are not uniform (equal).

10

10

120log

1

20log1

p p

p

ss

p

R

A


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Classic windows ( h(n) = 0, n [0, N-1]):

21

21

2 41 1

Rectangular : ( ) 1, 0 1

Hanning : ( ) 0.5 0.5cos , 0 1

Hamming : ( ) 0.54 0.46cos , 0 1

Blackman : ( ) 0.42 0.5cos 0.08cos , 0 1

nN

nN

n nN N

w n n N

w n n N

w n n N

w n n N

windowTransition

bandwidth f Minimum stopband

attenuation

Rectangular 0.9/ 21dB

Hanning 3.1/ 44dB

Hamming 3.3/ 53dB

Blackman 5.5/ 74dB


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Kaiser parametric window

The parametric window Kaiser is defined as

Given p, s, R p, and As, the length of the filter and the parameter arecalculated as:

0.4

Transition bandwidth : 2

7.95Order (length) : 114.36

0.1102( 8.7), 50Parameter

0.5842( 21) 0.07886( 21), 21 50

s p

s

s s

s s s

f

AN f

A A

A A A

220 10

1 1, 0 1( ) ( )

0,

nNI n Nw n I


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( ) ( ) ( )d h n h n w n

Windows comparison

Rectangular

Hanning


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Windows comparison

Hamming

Blackman


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Windows comparison

Kaiser = 2

Kaiser = 8


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Example (LPF with cutoff frequency 0.2 )Rectangular Hanning

Hamming Blackman


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Example (LPF with cutoff frequency 0.2 )Rectangular Hanning

Hamming Blackman


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Example (LPF with cutoff frequency 0.2 )Hamming, = 31 Hamming, = 63

Hamming, = 127 Hamming, = 255


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Example of FIR filter design with windows

A. Design a filter with the following characteristics

------------------------------------------------

Attenuation more than 50 dB is given by the Hamming window. Wecalculate the transition bandwidth

The cutoff frequency is given by

The order of the filter is

The ripple of the passband is 0.039 dB (calculated only with a computer).

0.2 , 0.25 dB

0.3 , 50 dB p p

s s

R

A

0.3 0.2 0.05

2 2s p f

3.3 1 67

N N f

0.252

s pc


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1

2331

2

sin[ ( )] sin[0.25 ( 33)]( ) ( ) ( ) ( ) 0.54 0.46cos( 33)( )

Nc n

d N n nh n h n w n w n

n n


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B. Design the following filter with the Kaiser window

------------------------------------------------

We calculate the transition bandwidth

The cutoff frequency is

The ripple in the passband is 0.044 dB (calculated with computer).

0.2 , 0.25 dB

0.3 , 50 dB p p

s s

R

A

0.3 0.2 0.05

2 2s p f

0.252

s pc

7.95Order (length) : 1 1 6114.36 Parameter 0.1102( 8.7) 4.5512

s

s

AN f

A


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2

1 0 3021

02

4.5512 1 1sin[ ( )] sin[0.25 ( 30)]( ) ( ) ( ) ( )( 30) (4.5512)( )

nNcd N

I n nh n h n w n w n n I n


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C. Design the following bandpass filter

------------------------------------------------

Attenuation over 60 dB can be achieved by the Blackman or the Kaiserwindow. We will design the Blackman.

The impulse response of the bandpass filter has the form

The cutoff frequencies are

1 1

2 2

0.4 , 60 dB, 0.5 , 1 dB

0.7 , 1 dB, 0.8 , 60 dBs s p p

p p s s

A R

R A

1 1 2 21 20.45 , 0.752 2

s p s pc c

1 12 12 2

1 12 2

( ) ( ) ( )

sin[ ( )] sin[ ( )]

( ) ( ) ( )

d

N Nc c

d N N

h n h n w n

n n

h n n n


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The transition bandwidth is

The order of the filter is calculated as

The ripple in the passband is equal to R p 0.0033 dB (calculated by acomputer).

5.51 111N N f

1 1 2 2 0.052 2

p s s p f


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2 4110 110sin[0.75 ( 55)] sin[0.45 ( 55)]( ) 0.42 0.5cos 0.08cos( 55) ( 55)n n n nh n

n n

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Introduction to FIR Filter Design