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Digital Signal Processing, VIII, Zheng-Hua Tan, 20061
Digital Signal Processing, Fall 2006
Zheng-Hua Tan
Department of Electronic Systems Aalborg University, Denmark
Lecture 8: FIR Filter Design
Digital Signal Processing, VIII, Zheng-Hua Tan, 20062
Course at a glance
Discrete-time signals and systems
Fourier-domain representation
DFT/FFT
Systemanalysis
Filter design
z-transform
MM1
MM2
MM9, MM10MM3
MM6
MM4
MM7, MM8
Sampling andreconstruction MM5
Systemstructure
System
2
Digital Signal Processing, VIII, Zheng-Hua Tan, 20063
FIR filter design
Design problem: the FIR system function
Start from impulse response directly
Find the degree M andthe filter coefficients h[k]
to approximate a desired frequency response
⎩⎨⎧ ≤≤
=
= ∑=
−
otherwise ,00 ,
][
)(0
Mnbnh
zbzH
n
M
k
kk
MzMhzhhzH −− +++= ][...]1[]0[)( 1
Digital Signal Processing, VIII, Zheng-Hua Tan, 20064
Part I: FIR filter design
FIR filter designCommonly used windowsGeneralized linear-phase FIR filterThe Kaiser window filter design method
3
Digital Signal Processing, VIII, Zheng-Hua Tan, 20065
Lowpass filter – as an example
Ideal lowpass filter
IIR filter: based on transformations of continuous-time IIR system into discrete-time ones.
FIR filter: how? is non-causal, infinite! ][nh
∞<<∞−=
⎩⎨⎧
≤<<
=
nn
nnh
eH
c
c
cjlp
,sin][
,0|| ,1
)(
πω
πωωωωω
Nc
c jH 22
)/(11|)(|ΩΩ+
=Ω
poles
Digital Signal Processing, VIII, Zheng-Hua Tan, 20066
Design by windowing
Desired frequency responses are often piecewise-constant with discontinuities at the boundaries between bands, resulting in non-causal and infinteimpulse response extending from to , but
So, the most straightforward method is to truncate the ideal response by windowing and do time-shifting:
0][ , →±∞→ nhn d
∞∞−
][][otherwise ,0
|| ],[][
Mngnh
Mnnhng d
−=⎩⎨⎧ ≤
=
4
Digital Signal Processing, VIII, Zheng-Hua Tan, 20067
Design by windowing
After Champagne & Labeau, DSP Class Notes 2002.
Digital Signal Processing, VIII, Zheng-Hua Tan, 20068
Design by rectangular window
In general,
For simple truncation, the window is the rectangular window
∫−−=
π
π
θωθω θπ
deHeHeH jjd
j )()(21)( )(
⎩⎨⎧ ≤≤
=
=
otherwise ,00 ,1
][
][][][
Mnnw
nwnhnh d
5
Digital Signal Processing, VIII, Zheng-Hua Tan, 20069
Convolution process by truncation
Fig. 7.19
?t then wha, ,1][ ∞<<∞−= nnw )2(2)( reWr
j πωπδω ∑∞
−∞=
+=
band narrow be should )( ωjeW
Digital Signal Processing, VIII, Zheng-Hua Tan, 200610
Requirements on the window
Requirementsapproximates an impulse to faithfully
reproduce the desired frequency responsew[n] as short as possible in duration (the order of the filter) to minimize computation in the implementation of the filterConflicting
Take the rectangular window as an example.
, ,1][ ∞<<∞−= nnw )2(2)( reWr
j πωπδω ∑∞
−∞=
+=
band narrow be should )( ωjeW
)( ωjeW
6
Digital Signal Processing, VIII, Zheng-Hua Tan, 200611
Rectangular window
M+1
M=7
π4constant
Digital Signal Processing, VIII, Zheng-Hua Tan, 200612
Part II: Commonly used windows
FIR filter designCommonly used windowsGeneralized linear-phase FIR filterThe Kaiser window filter design method
7
Digital Signal Processing, VIII, Zheng-Hua Tan, 200613
Standard windows – time domain
Rectangular
Triangular
Hanning
Hamming
Blackman
⎩⎨⎧ ≤≤−
=otherwise ,00 ),/2cos(5.05.0
][MnMn
nwπ
⎪⎩
⎪⎨
⎧≤≤−
≤≤=
otherwise ,02/ ,/22
2/0 ,/2][ MnMMn
MnMnnw
⎩⎨⎧ ≤≤
=otherwise ,00 ,1
][Mn
nw
⎩⎨⎧ ≤≤−
=otherwise ,0
0 ),/2cos(46.054.0][
MnMnnw
π
⎩⎨⎧ ≤≤+−
=otherwise ,00 ),/4cos(08.0)/2cos(5.042.0
][MnMnMn
nwππ
Digital Signal Processing, VIII, Zheng-Hua Tan, 200614
Standard windows – figure
Fig. 7.21
Plotted for convenience. In fact, the window is defined only at integer values of n.
8
Digital Signal Processing, VIII, Zheng-Hua Tan, 200615
Standard windows – magnitude
Digital Signal Processing, VIII, Zheng-Hua Tan, 200616
Standard windows – comparison
Magnitude of side lobes vs width of main lobe
Independent of M!
9
Digital Signal Processing, VIII, Zheng-Hua Tan, 200617
Part III: Linear-phase FIR filter
FIR filter designCommonly used windowsGeneralized linear-phase FIR filterThe Kaiser window filter design method
Digital Signal Processing, VIII, Zheng-Hua Tan, 200618
Linear-phase FIR systems
Generalized linear-phase system
Causal FIR systems have generalized linear-phase if h[n] satisfies the symmetry condition
MnnhnMh
MnnhnMh
,...,1,0 ],[][ or
,...,1,0 ],[][
=−=−
==−
constants real are and , offunction real a is )(
)()(
βαωω
βωαωω
j
jjjj
eA
eeAeH +−=
10
Digital Signal Processing, VIII, Zheng-Hua Tan, 200619
Types of GLP FIR filters
1or 1,...,1,0 ],[][
−===−
εε MnnhnMh
Type IVType III
Type IIType I
M oddM even
1=ε
1−=ε
Digital Signal Processing, VIII, Zheng-Hua Tan, 200620
Generalized linear phase FIR filter
Often aim at designing causal systems with a generalized linear phase (stability is not a problem)
If the impulse response of the desired filter is symmetric about M/2, Choose windows being symmetric about the point M/2
is a real, even function of wthe resulting frequency response will have a generalized linear phase
2/)()( Mjje
j eeWeW ωωω −=
⎩⎨⎧ ≤≤−
=otherwise ,0
0 ],[][
MnnMwnw
)( ωje eW
][][ nhnMh dd =−
2/)()( Mjje
j eeAeH ωωω −= even and real is )( ωje eA
11
Digital Signal Processing, VIII, Zheng-Hua Tan, 200621
Linear-phase lowpass filter – an example
Desired frequency response
so
apply a symmetric window a linear-phase system
∞<<∞−−−
=
⎩⎨⎧
≤<<
=−
nMn
Mnnh
eeH
clp
c
cMj
jlp
,)2/(
)]2/(sin[][
,0|| ,
)(2/
πω
πωωωωω
ω
][][ nhnMh lplp =−
][)2/(
)]2/(sin[][][][ nw
MnMn
nwnhnh clp −
−==
πω
Digital Signal Processing, VIII, Zheng-Hua Tan, 200622
Window method approximations
even and real is )(
)()(
even and real is )(
)()(
2/
2/
ω
ωωω
ω
ωωω
je
Mjje
j
je
Mjje
jd
eW
eeWeW
eH
eeHeH
−
−
=
=
∫−−
−
=
=π
πθωθω
ωωω
θπ
deWeHeA
eeAeH
je
je
je
Mjje
j
)()(21)(
)()(
)(
2/
12
Digital Signal Processing, VIII, Zheng-Hua Tan, 200623
Key parameters
To meet the requirement of FIR filter, chooseShape of the windowDuration of the window
Trail and error is not a satisfactory method to design filters a simple formalization of the window method by Kaiser
Digital Signal Processing, VIII, Zheng-Hua Tan, 200624
Part IV: Kaiser window filter design
FIR filter designCommonly used windowsGeneralized linear-phase FIR filterThe Kaiser window filter design method
13
Digital Signal Processing, VIII, Zheng-Hua Tan, 200625
The Kaiser window filter design method
An easy way to find the trade-off between the main-lobe width and side-lobe areaThe Kaiser window
is the zeroth-order modified Bessel function of the first kind. is an adjustable design parameter.The length (M+1) and the shape parameter can be adjusted to trade side-lobe amplitude for main-lobe width (not possible for preceding windows!)
2/ otherwise ,0
0 ,)(
])]/)[(1([][ 0
2/120
M
MnInI
nw
=
⎪⎩
⎪⎨
⎧≤≤
−−=
α
βααβ
)(0 ⋅I0≥β
β
Digital Signal Processing, VIII, Zheng-Hua Tan, 200626
The Kaiser window
14
Digital Signal Processing, VIII, Zheng-Hua Tan, 200627
Design FIR filter by the Kaiser window
Calculate M and to meet the filter specificationThe peak approximation error is determined by
Define then
Passband cutoff frequency is determined by:
Stopband cutoff frequency by:Transition width M must satisfy
δω −≥1|)(| jeH
βδ
β
δω ≤|)(| jeH
⎪⎩
⎪⎨
⎧
<≤≤−+
>−
=21 ,0.0
5021 ),21(07886.021)-0.5842(A
50 ),7.8(1102.00.4
AAA
AA
β
ps ωωω −=Δ
pω
δ10log20−=A
sω
ωΔ−
=285.2
8AM
(Peak error is fixed for other windows)
(Rectangular)
Digital Signal Processing, VIII, Zheng-Hua Tan, 200628
A lowpass filter Specifications
Specifications for a discrete-time lowpass filter
πωω
πωωω
ω
6.0 ,001.0|)(|
4.00 ,01.01|)(|01.01
=≥≤
=≤≤+≤≤−
sj
pj
eH
eH
001.001.0
2
1
==
δδ
15
Digital Signal Processing, VIII, Zheng-Hua Tan, 200629
Design the lowpass filter by Kaiser window
Designing by window method indicating , we must setTransition width
The two parameters:Cutoff frequency of the ideal lowpass filter
Impulse response
60log20 10 =−= δA
001.0=δπωωω 2.0=−=Δ ps
21 δδ =
37 ,653.5 == Mβ
⎪⎩
⎪⎨
⎧≤≤
−−⋅
−−
=
otherwise ,0
0 ,)(
])]/)[(1([)(
)(sin][ 0
2/120 Mn
InI
nn
nhc
βααβ
απαω
πωωω 5.02/)( =+= psc
Digital Signal Processing, VIII, Zheng-Hua Tan, 200630
Design the lowpass filter
What is the group delay?
M/2=18.5
16
Digital Signal Processing, VIII, Zheng-Hua Tan, 200631
Summary
FIR filter designCommonly used windowsGeneralized linear-phase FIR filterThe Kaiser window filter design method
Digital Signal Processing, VIII, Zheng-Hua Tan, 200632
Course at a glance
Discrete-time signals and systems
Fourier-domain representation
DFT/FFT
Systemanalysis
Filter design
z-transform
MM1
MM2
MM9, MM10MM3
MM6
MM4
MM7, MM8
Sampling andreconstruction MM5
Systemstructure
System