simple digital filters - catholic university of...
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1Copyright © 2001, S. K. Mitra
Simple Digital FiltersSimple Digital Filters
• Later in the course we shall review variousmethods of designing frequency-selectivefilters satisfying prescribed specifications
• We now describe several low-order FIR andIIR digital filters with reasonable selectivefrequency responses that often aresatisfactory in a number of applications
2Copyright © 2001, S. K. Mitra
Simple FIR Digital FiltersSimple FIR Digital Filters
• FIR digital filters considered here haveinteger-valued impulse response coefficients
• These filters are employed in a number ofpractical applications, primarily because oftheir simplicity, which makes them amenableto inexpensive hardware implementations
3Copyright © 2001, S. K. Mitra
Simple FIR Digital FiltersSimple FIR Digital FiltersLowpass FIR Digital Filters• The simplest lowpass FIR digital filter is the
2-point moving-average filter given by
• The above transfer function has a zero at and a pole at z = 0
• Note that here the pole vector has a unitymagnitude for all values of ω
zzzzH2
11 121
0+=+= − )()(
1−=z
4Copyright © 2001, S. K. Mitra
Simple FIR Digital FiltersSimple FIR Digital Filters
• On the other hand, as ω increases from 0 toπ, the magnitude of the zero vectordecreases from a value of 2, the diameter ofthe unit circle, to 0
• Hence, the magnitude response isa monotonically decreasing function of ωfrom ω = 0 to ω = π
|)(| 0ωjeH
5Copyright © 2001, S. K. Mitra
Simple FIR Digital FiltersSimple FIR Digital Filters
• The maximum value of the magnitudefunction is 1 at ω = 0, and the minimumvalue is 0 at ω = π, i.e.,
• The frequency response of the above filteris given by
01 00
0 == |)(|,|)(| πjj eHeH
)2/cos()( 2/0 ω= ω−ω jj eeH
6Copyright © 2001, S. K. Mitra
Simple FIR Digital FiltersSimple FIR Digital Filters
• The magnitude responsecan be seen to be a monotonicallydecreasing function of ω
)2/cos(|)(| 0 ω=ωjeH
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
ω/π
Mag
nitu
de
First-order FIR lowpass filter
7Copyright © 2001, S. K. Mitra
Simple FIR Digital FiltersSimple FIR Digital Filters• The frequency at which
is of practical interest since here the gainin dB is given by
since the dc gain
cω=ω
)(2
1)( 000
jj eHeH c =ω
)( cωG
dB32log20)(log20 100
10 −≅−= jeH)( cωG )(log20 10
cjeH ω=
0200 010
== )(log)( jeHG
8Copyright © 2001, S. K. Mitra
Simple FIR Digital FiltersSimple FIR Digital Filters
• Thus, the gain G(ω) at isapproximately 3 dB less than the gain at ω= 0
• As a result, is called the 3-dB cutofffrequency
• To determine the value of we set
which yields
cω=ω
cω
cω
2/π=ωc2122
0 )2/(cos|)(| =ω=ωc
j ceH
9Copyright © 2001, S. K. Mitra
Simple FIR Digital FiltersSimple FIR Digital Filters
• The 3-dB cutoff frequency can beconsidered as the passband edge frequency
• As a result, for the filter the passbandwidth is approximately π/2
• The stopband is from π/2 to π• Note: has a zero at or ω = π,
which is in the stopband of the filter
cω
)(zH0
)(zH0 1−=z
10Copyright © 2001, S. K. Mitra
Simple FIR Digital FiltersSimple FIR Digital Filters• A cascade of the simple FIR filter
results in an improved lowpass frequencyresponse as illustrated below for a cascadeof 3 sections
)()( 121
0 1 −+= zzH
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
ω/π
Mag
nitu
de
First-order FIR lowpass filter cascade
11Copyright © 2001, S. K. Mitra
Simple FIR Digital FiltersSimple FIR Digital Filters
• The 3-dB cutoff frequency of a cascade ofM sections is given by
• For M = 3, the above yields• Thus, the cascade of first-order sections
yields a sharper magnitude response but atthe expense of a decrease in the width of thepassband
)2(cos2 2/11 Mc
−−=ωπ=ω 302.0c
12Copyright © 2001, S. K. Mitra
Simple FIR Digital FiltersSimple FIR Digital Filters• A better approximation to the ideal lowpass
filter is given by a higher-order moving-average filter
• Signals with rapid fluctuations in samplevalues are generally associated with high-frequency components
• These high-frequency components areessentially removed by an moving-averagefilter resulting in a smoother outputwaveform
13Copyright © 2001, S. K. Mitra
Simple FIR Digital FiltersSimple FIR Digital Filters
Highpass FIR Digital Filters• The simplest highpass FIR filter is obtained
from the simplest lowpass FIR filter byreplacing z with
• This results in)()( 1
21
1 1 −−= zzH
z−
14Copyright © 2001, S. K. Mitra
Simple FIR Digital FiltersSimple FIR Digital Filters• Corresponding frequency response is given
by
whose magnitude response is plotted below)2/sin()( 2/
1 ω= ω−ω jj ejeH
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
ω/π
Mag
nitu
de
First-order FIR highpass filter
15Copyright © 2001, S. K. Mitra
Simple FIR Digital FiltersSimple FIR Digital Filters
• The monotonically increasing behavior ofthe magnitude function can again bedemonstrated by examining the pole-zeropattern of the transfer function
• The highpass transfer function has azero at z = 1 or ω = 0 which is in thestopband of the filter
)(zH1
)(zH1
16Copyright © 2001, S. K. Mitra
Simple FIR Digital FiltersSimple FIR Digital Filters
• Improved highpass magnitude response canagain be obtained by cascading severalsections of the first-order highpass filter
• Alternately, a higher-order highpass filter ofthe form
is obtained by replacing z with in thetransfer function of a moving average filter
z−
nMn
nM
zzH −−=∑ −= 1
01
1 1)()(
17Copyright © 2001, S. K. Mitra
Simple FIR Digital FiltersSimple FIR Digital Filters
• An application of the FIR highpass filters isin moving-target-indicator (MTI) radars
• In these radars, interfering signals, calledclutters, are generated from fixed objects inthe path of the radar beam
• The clutter, generated mainly from groundechoes and weather returns, has frequencycomponents near zero frequency (dc)
18Copyright © 2001, S. K. Mitra
Simple FIR Digital FiltersSimple FIR Digital Filters
• The clutter can be removed by filtering theradar return signal through a two-pulsecanceler, which is the first-order FIRhighpass filter
• For a more effective removal it may benecessary to use a three-pulse cancelerobtained by cascading two two-pulsecancelers
)()( 121
1 1 −−= zzH
19Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital FiltersLowpass IIR Digital Filters• A first-order causal lowpass IIR digital
filter has a transfer function given by
where |α| < 1 for stability• The above transfer function has a zero at
i.e., at ω = π which is in the stopband
α−+α−= −
−
1
1
11
21)(
zzzHLP
1−=z
20Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters• has a real pole at z = α• As ω increases from 0 to π, the magnitude
of the zero vector decreases from a value of2 to 0, whereas, for a positive value of α,the magnitude of the pole vector increasesfrom a value of to
• The maximum value of the magnitudefunction is 1 at ω = 0, and the minimumvalue is 0 at ω = π
α−1 α+1
)(zHLP
21Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters• i.e.,• Therefore, is a monotonically
decreasing function of ω from ω = 0 to ω = πas indicated below
0|)(|,1|)(| 0 == πjLP
jLP eHeH
|)(| ωjLP eH
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
ω/π
Mag
nitu
de
α = 0.8α = 0.7α = 0.5
10-2 10-1 100-20
-15
-10
-5
0
ω/π
Gai
n, d
B α = 0.8α = 0.7α = 0.5
22Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters
• The squared magnitude function is given by
• The derivative of with respectto ω is given by
)cos21(2)cos1()1(|)(| 2
22
ωα−α+ω+α−=ωj
LP eH
2|)(| ωjLP eH
22
222
)cos21(2sin)21()1(|)(|
α+ωα−ωα+α+α−−=
ω
ω
deHd j
LP
23Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters
in the rangeverifying again the monotonically decreasingbehavior of the magnitude function
• To determine the 3-dB cutoff frequencywe set
in the expression for the square magnitudefunction resulting in
0/|)(| 2 ≤ωω deHd jLP π≤ω≤0
21|)(| 2 =ωcj
LP eH
24Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters
or
which when solved yields
• The above quadratic equation can be solvedfor α yielding two solutions
21
)cos21(2)cos1()1(
2
2=
ωα−α+ω+α−
c
c
cc ωα−α+=ω+α− cos21)cos1()1( 22
212cos
α+α=ωc
25Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters• The solution resulting in a stable transfer
function is given by
• It follows from
that is a BR function for |α| < 1)cos21(2
)cos1()1(|)(| 2
22
ωα−α+ω+α−=ωj
LP eH
)(zHLP
)(zHLP
cc
ωω−=α cos
sin1
26Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters
Highpass IIR Digital Filters• A first-order causal highpass IIR digital filter
has a transfer function given by
where |α| < 1 for stability• The above transfer function has a zero at z = 1
i.e., at ω = 0 which is in the stopband
−−+= −
−
1
1
11
21
zzzHHP
αα
)(
27Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters• Its 3-dB cutoff frequency is given by
which is the same as that of• Magnitude and gain responses of
are shown below
cc ωω−=α cos/)sin1(cω
)(zHLP)(zHHP
10-2 10-1 100-20
-15
-10
-5
0
ω/π
Gai
n, d
B
α = 0.8α = 0.7α = 0.5
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
ω/π
Mag
nitu
de
α = 0.8α = 0.7α = 0.5
28Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters
• is a BR function for |α| < 1• Example - Design a first-order highpass
digital filter with a 3-dB cutoff frequency of0.8π
• Now, and
• Therefore
)(zHHP
587785.0)8.0sin()sin( =π=ωc80902.0)8.0cos( −=π
5095245.0cos/)sin1( −=ωω−=α cc
29Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters
• Therefore,
+−= −
−
1
1
50952450112452380
zz
..
−−+= −
−
1
1
11
21
zzzHHP
αα
)(
30Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital FiltersBandpass IIR Digital Filters• A 2nd-order bandpass digital transfer
function is given by
• Its squared magnitude function is
α+α+β−−α−= −−
−
21
2
)1(11
21)(
zzzzHBP
2)( ωj
BP eH
]2cos2cos)1(2)1(1[2)2cos1()1(
2222
2
ωα+ωα+β−α+α+β+ω−α−=
31Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters
• goes to zero at ω = 0 and ω = π• It assumes a maximum value of 1 at ,
called the center frequency of the bandpassfilter, where
• The frequencies and where becomes 1/2 are called the 3-dB cutofffrequencies
2|)(| ωjBP eH
2|)(| ωjBP eH
oω=ω
)(cos 1 β=ω −o
1cω 2cω
32Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters
• The difference between the two cutofffrequencies, assuming is calledthe 3-dB bandwidth and is given by
• The transfer function is a BRfunction if |α| < 1 and |β| < 1
12 cc ω>ω
)(zHBP
α+α
212
112 cos−=ω−ω= ccwB
α+α
212
33Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters
• Plots of are shown below|)(| ωjBP eH
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
ω/π
Mag
nitu
de
β = 0.34
α = 0.8α = 0.5α = 0.2
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
ω/π
Mag
nitu
de
α = 0.6
β = 0.8β = 0.5β = 0.2
34Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters• Example - Design a 2nd order bandpass
digital filter with center frequency at 0.4πand a 3-dB bandwidth of 0.1π
• Hereand
• The solution of the above equation yields:α = 1.376382 and α = 0.72654253
9510565.0)1.0cos()cos(1
22 =π==
α+α
wB
309017.0)4.0cos()cos( =π=ω=β o
35Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters
• The corresponding transfer functions are
and
• The poles of are at z = 0.3671712 and have a magnitude > 1
±114256361.j
21
2
3763817343424011188190 −−
−
+−−−=
zzzzHBP
...)('
21
2
726542530533531011136730 −−
−
+−−=
zzzzHBP
...)("
)(' zHBP
36Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters
• Thus, the poles of are outside theunit circle making the transfer functionunstable
• On the other hand, the poles of areat z = and have amagnitude of 0.8523746
• Hence is BIBO stable• Later we outline a simpler stability test
)(' zHBP
)(" zHBP8095546026676550 .. j±
)(" zHBP
37Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters
• Figures below show the plots of themagnitude function and the group delay of
)(" zHBP
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
ω/π
Mag
nitu
de
0 0.2 0.4 0.6 0.8 1-1
0
1
2
3
4
5
6
7
ω/π
Gro
up d
elay
, sam
ples
38Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters
Bandstop IIR Digital Filters• A 2nd-order bandstop digital filter has a
transfer function given by
• The transfer function is a BRfunction if |α| < 1 and |β| < 1
α+α+β−+β−α+= −−
−−
21
21
)1(121
21)(
zzzzzHBS
)(zHBS
39Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters
• Its magnitude response is plotted below
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
ω/π
Mag
nitu
de
α = 0.8α = 0.5α = 0.2
0 0.2 0.4 0.6 0.8 10
0.2
0.4
0.6
0.8
1
ω/π
Mag
nitu
de
β = 0.8β = 0.5β = 0.2
40Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters
• Here, the magnitude function takes themaximum value of 1 at ω = 0 and ω = π
• It goes to 0 at , where , called thenotch frequency, is given by
• The digital transfer function is morecommonly called a notch filter
oω=ω oω
)(cos 1 β=ω −o
)(zHBS
41Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters
• The frequencies and where becomes 1/2 are called the 3-dB cutofffrequencies
• The difference between the two cutofffrequencies, assuming is calledthe 3-dB notch bandwidth and is given by
2|)(| ωjBS eH1cω 2cω
12 cc ω>ω
112 cos−=ω−ω= ccwB
α+α
212
42Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters
Higher-Order IIR Digital Filters• By cascading the simple digital filters
discussed so far, we can implement digitalfilters with sharper magnitude responses
• Consider a cascade of K first-order lowpasssections characterized by the transferfunction
α−+α−= −
−
1
1
11
21)(
zzzHLP
43Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters
• The overall structure has a transfer functiongiven by
• The corresponding squared-magnitudefunction is given by
K
LP zzzG
α−+⋅α−= −
−
1
1
11
21)(
Kj
LP eG
ωα−α+ω+α−=ω
)cos21(2)cos1()1(|)(| 2
22
44Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters• To determine the relation between its 3-dB
cutoff frequency and the parameter α,we set
which when solved for α, yields for a stable :
cω
21
)cos21(2)cos1()1(
2
2=
ωα−α+ω+α−
K
c
c
)(zGLP
c
ccC
CCCω+−
−ω−ω−+=αcos1
2sincos)1(1 2
45Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters
where
• It should be noted that the expression for αgiven earlier reduces to
for K = 1
KKC /)( 12 −=
cc
ωω−=α cos
sin1
46Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters
• Example - Design a lowpass filter with a 3-dB cutoff frequency at using asingle first-order section and a cascade of 4first-order sections, and compare their gainresponses
• For the single first-order lowpass filter wehave
π=ω 4.0c
1584.0)4.0cos(
)4.0sin(1cos
sin1=
ππ+
=ω
ω+=α
c
c
47Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters
• For the cascade of 4 first-order sections, wesubstitute K = 4 and get
• Next we compute6818122 4141 ./)(/)( === −− KKC
c
ccC
CCCω+−
−ω−ω−+=αcos1
2sincos)1(1 2
)4.0cos(6818.11)6818.1()6818.1(2)4.0sin()4.0cos()6818.11(1 2
π+−−π−π−+=
2510.−=
48Copyright © 2001, S. K. Mitra
Simple IIR Digital FiltersSimple IIR Digital Filters• The gain responses of the two filters are
shown below• As can be seen, cascading has resulted in a
sharper roll-off in the gain response
10-2 10-1 100-20
-15
-10
-5
0
ω/π
Gai
n, d
B
K=4
K=1Passband details
10-2 10-1 100
-4
-2
0
ω/π
Gai
n, d
B K=1K=4