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Digital Signal Processing
H. Introduction to FIR filters design
Athanassios C. Iossifides
February 2013
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.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
. Introduction to FIR digital filter design
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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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H.1 Discrete time systems and filters
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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H.1 Discrete time systems and filters
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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H.1 Discrete time systems and filters
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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H.1 Discrete time systems and filters
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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H.1 Discrete time systems and filters
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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H.1 Discrete time systems and filters
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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H.1 Discrete time systems and filters
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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H.1 Discrete time systems and filters
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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H.1 Discrete time systems and filters
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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H.1 Discrete time systems and filters
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
. Introduction to FIR digital filter design
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. 2 FIR filter design with windows
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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. 2 FIR filter design with windows
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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. 2 FIR filter design with windows
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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. 2 FIR filter design with windows
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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. 2 FIR filter design with windows
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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. 2 FIR filter design with windows
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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. 2 FIR filter design with windows
( ) ( ) ( )d h n h n w n
Windows comparison
Rectangular
Hanning
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. 2 FIR filter design with windows
Windows comparison
Hamming
Blackman
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. 2 FIR filter design with windows
Windows comparison
Kaiser = 2
Kaiser = 8
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. 2 FIR filter design with windows
Example (LPF with cutoff frequency 0.2 )Rectangular Hanning
Hamming Blackman
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. 2 FIR filter design with windows
Example (LPF with cutoff frequency 0.2 )Rectangular Hanning
Hamming Blackman
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. 2 FIR filter design with windows
Example (LPF with cutoff frequency 0.2 )Hamming, = 31 Hamming, = 63
Hamming, = 127 Hamming, = 255
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. 2 FIR filter design with windows
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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. 2 FIR filter design with windows
Example of FIR filter design with windows
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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. 2 FIR filter design with windows
Example of FIR filter design with windows
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 FIR filter design with windows
Example of FIR filter design with windows
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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. 2 FIR filter design with windows
Example of FIR filter design with windows
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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. 2 FIR filter design with windows
Example of FIR filter design with windows
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 FIR filter design with windows
Example of FIR filter design with windows
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