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9/3/2004 I. Discrete-Time Signals and Systems 1
MM2: Synthesis of IIR DT filtersMM2: Synthesis of IIR DT filters
Reading material: p.160-163, 439 – 458 and 824-829
1. Explanation of last exercise2. Continuous time filters3. Impulse-invariance method4. Bilinear transformation method
9/3/2004 I. Discrete-Time Signals and Systems 2
Explanation of Exercise One
Analog filter Discrete filter
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Effect of Effect of FilterFiltering ing
System frequency response: H(ejωωωω) = |H(ejωωωω)| eH(ejωωωω)
Input and output relationship|Y(ejωωωω)| = |H(ejωωωω)| |X(ejωωωω)|
Y(ejωωωω) = H(ejωωωω) + X(ejωωωω)
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9/3/2004 I. Discrete-Time Signals and Systems 4
Requirements for Requirements for Filter DesignFilter Design
Frequency-selective filters
Lowpass, highpass, bandpass, bandstop filters…
Linear Phase filters
Causal filters
Stable filters
9/3/2004 I. Discrete-Time Signals and Systems 5
FrequencyFrequency--Selective FiltersSelective Filters Ideal lowpass filter
Ideal highpass filter
Ideal bandstop filter
Ideal bandpass filter
0 ωc-ωc π-π
0 ωc-ωc π-π
0 ωa-ωa π-π
0 ωa-ωa π-π
≤
=others
ceH j
lp
,0||,1
|)(|ωωω
≥
=others
ceH j
hp
,0||,1
|)(|ωωω
≤≤≥≤
=aa
baajbs b
andeH
||,0
||||,1|)(|
ωωωωωωω
≤≤≥≤
=aa
baajbp b
andeH
||,1
||||,0|)(|
ωωωωωωω
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9/3/2004 I. Discrete-Time Signals and Systems 7
Synthesis of Discrete-Time IIR Filtersfrom Continuous-Time Filters
C/D DT Filter D/C
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DT specificationsDT specifications CT specificationsCT specifications
CT FilterHc(s), hc(t)
(Butterworth, Chebyshev,
Elliptic)
CT FilterHc(s), hc(t)
(Butterworth, Chebyshev,
Elliptic)
CT Design
DT FilterH(z), h[n]
(Butterworth, Chebyshev,
Elliptic)
DT FilterH(z), h[n]
(Butterworth, Chebyshev,
Elliptic)
transform
inv. trans.
??
?
9/3/2004 I. Discrete-Time Signals and Systems 9
Synthesis of Continuous-Time Filters
Analog FilterSee page See page 824824--829829
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9/3/2004 I. Discrete-Time Signals and Systems 10
ButterworthButterworth LLowpassowpass FFiltersilters (I)(I) Characteristics
The magnitude response is maximally flat in the passband
The magnitude response is monotonic in the passband and stopband
Magnitude-squared function
Nc
c jjjH 2
2
)/(11
|)(|ΩΩ+
=Ω
Nc
cc jjjHjH 2)/(1
1)()(
ΩΩ+=Ω−Ω
9/3/2004 I. Discrete-Time Signals and Systems 11
Butterworth Butterworth LLowpassowpass FFiltersilters (II)(II) Filter Construction
System function
Roots of denominator polynomial see Fig.8.3 (p.826)
Stable and causal system Hc(s)
Select the poles on the left-half-plane of the s-plane
Nc
cc jjsHsH 2)/(1
1)()(
ΩΩ+=−
9/3/2004 I. Discrete-Time Signals and Systems 12
Nc
cc jjsHsH 2)/(1
1)()(
ΩΩ+=−
3dB
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9/3/2004 I. Discrete-Time Signals and Systems 13
ChebyshevChebyshev FiltersFilters (I)(I) Motivation: to distribute the accuracy of the approximation
over the passband or the stopband, leading to a lower order filter Characteristics
Type I Chebyshev filter: the magnitude response is equiripple in the passband and monotonic in the stopband
Type II Chebyshev filter: the magnitude response is monotonic in thepassband and equiripple in the stopband
Magnitude-squared function of type I
Where VN(x) is the Nth-order Chebyshev polynomial, which can be recurrently calculated by
)coscos()()/(1
1|)(| 1
222 xNxV
VjH N
cNc
−=ΩΩ+
=Ωε
)()(2)( 11 xVxxVxV NNN −+ −=continue..
9/3/2004 I. Discrete-Time Signals and Systems 14
ChebyshevChebyshev FiltersFilters (II)(II) Design parameters
εεεε can be specified by the allowable passband ripple ΩΩΩΩc can be specified by the desired cutoff frequency N can be chosen that the stopband specification are met
Location of poles: on an ellipse Length of the minor axis – 2aΩΩΩΩc
Length of the major axis – 2bΩΩΩΩc
Type II chebyshev filters21
)(21
)(21
1
/1/1
/1/1
−++=
+=
−=
−
−
−
εεα
αα
αα
NN
NN
b
a
1222
)]/([11
|)(| −ΩΩ+=Ω
cNc V
jHε
9/3/2004 I. Discrete-Time Signals and Systems 15
Elliptic FiltersElliptic Filters Characteristics: equiripple in the passband and stopband Elliptic filters is the best that can be achieved for a given filter
order N, in the sense that for a given Ωp, δ1,δ2, the transition band (Ωs- Ωp) is as small as possible
Magnitude-squared function
UN(Ω) is a Jacobian elliptic function)(1
1|)(| 222
Ω+=Ω
Nc U
jHε
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9/3/2004 I. Discrete-Time Signals and Systems 16
DT specificationsDT specifications CT specificationsCT specifications
CT FilterHc(s), hc(t)
(Butterworth, Chebyshev,
Elliptic)
CT FilterHc(s), hc(t)
(Butterworth, Chebyshev,
Elliptic)
CT Design
DT FilterH(z), h[n]
(Butterworth, Chebyshev,
Elliptic)
DT FilterH(z), h[n]
(Butterworth, Chebyshev,
Elliptic)
transform
inv. trans.
√√√√?
?
9/3/2004 I. Discrete-Time Signals and Systems 17
Synthesis of Discrete-Time IIR Filtersusing Impulse Invariance Method
C/D DT Filter D/C
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Impulse Invariance Impulse Invariance MethodMethod Fundamental formula:
Frequency property:
DT specification CT specification: Ω=ω/Td
Pole relationship: (p.445) (stable CT filter stable DT filter) dkTs
k ez =
),(][ dcd nThTnh =
πωωω ≤= ||)()(d
cj
TjHeH
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9/3/2004 I. Discrete-Time Signals and Systems 19
Practical ProcedurePractical Procedure Step1: Transform DT specification to CT specification
using Ω=ω/Td
Step2: Design a CT filter based on CT specification
Step3: Transform Hc(s) to be H(z)
=
−= −
=−
=N
kTs
kdN
k k
kc ze
ATzH
ssA
sHdk
11
1 1)()(
9/3/2004 I. Discrete-Time Signals and Systems 20
DT specificationsDT specifications CT specificationsCT specifications
CT Filter(Butterworth, Chebyshev,
Elliptic)
CT Filter(Butterworth, Chebyshev,
Elliptic)
CT Design
DT Filter(Butterworth, Chebyshev,
Elliptic)
DT Filter(Butterworth, Chebyshev,
Elliptic)
DT IIR Filter by Impulse InvarianceDT IIR Filter by Impulse Invariance
Ω=ω/Td
=
−= −
=−
=N
kTs
kdN
k k
kc ze
ATzH
ssA
sHdk
11
1 1)()(
πωωω ≤= ||)()(d
cj
TjHeH
9/3/2004 I. Discrete-Time Signals and Systems 21
ExampleExample: : Impulse Invariance DesignImpulse Invariance Design (p.446)(p.446)
Desire a lowpass DT filter with
Select Td=1, CT specification
CT butterwoth filter
Sixth-order Hc(s) Matlab auto-design …
πωππω
ω
ω
≤≤≤≤≤≤≤
||3.0,17783.0|)(|
2.0||0,1|)(|89125.0j
j
eH
eH
πππ
≤Ω≤≤Ω≤Ω≤≤Ω≤
||3.0,17783.0|)(|
2.0||0,1|)(|89125.0
jH
jH c
Nc
c
c
jH
jHjH
22
)/(11
|)(|
17783.0|)3.0(||,)2.0(|89125.0
ΩΩ+=Ω
≤≤ ππ
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9/3/2004 I. Discrete-Time Signals and Systems 22
Remarks about Impulse Invariance Remarks about Impulse Invariance Remarks about Impulse Invariance The basis for this method is to choose a DT impulse
response that is similar to the CT impulse response
The CT and DT frequencies have linear relationship, except for aliasing, the shape is preserved
This method is appropriate only for bandlimited filters
),(][ dcd nThTnh =
πωωω ≤= ||)()(d
cj
TjHeH
Hc(jΩ)=0, π/ Td ≤ | Ω|,
9/3/2004 I. Discrete-Time Signals and Systems 23
Synthesis of Discrete-Time IIR Filtersusing Bilinear Method
C/D DT Filter D/C
9/3/2004 I. Discrete-Time Signals and Systems 24
Bilinear TransformBilinear Transform
Fundemantal formula
))11
(2
()(),11
(2
1
1
1
1
−
−
−
−
+−=
+−=
zz
THzH
zz
Ts
dc
d
)))2/(1)2/(1
()(,)2/(1)2/(1
sTsT
HsHsTsT
zd
dc
d
d
−+=
−+=
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9/3/2004 I. Discrete-Time Signals and Systems 25
Characteristics of Characteristics of Bilinear TransformBilinear Transform
An algebraic transformation between s-plane and z-plane, i.e., mapping the jΩ-axis in s-plane to one revolution of the unit circle in the z-plane
the CT and DT frequencies have nonlinear relationship
)2
arctan(2)2
tan(2 d
d
TT
Ω==Ω ωω
9/3/2004 I. Discrete-Time Signals and Systems 26
)11
(2
1
1
−
−
+−=
zz
Ts
d
sTsT
zd
d
)2/(1)2/(1
−+=
9/3/2004 I. Discrete-Time Signals and Systems 27
DT IIR Filter Design by Bilinear TransformDT IIR Filter Design by Bilinear Transform
DT specificationsDT specifications CT specificationsCT specifications
)2
arctan(2 dTΩ=ω
)2
tan(2 ω
dT=Ω
CT Filter(Butterworth, Chebyshev,
Elliptic)
CT Filter(Butterworth, Chebyshev,
Elliptic)
CT Design
DT Filter(Butterworth, Chebyshev,
Elliptic)
DT Filter(Butterworth, Chebyshev,
Elliptic)
))11
(2
()(),11
(2
1
1
1
1
−
−
−
−
+−=
+−=
zz
THzH
zz
Ts
dc
d
10
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Remarks about Remarks about Bilinear Bilinear MethodMethod
Sampling period Td will not affect the design, therefore, in specific problems it can be chosen as any convenient value
Stable CT filter stable DT filter
Avoid the aliasing problem
the CT and DT frequencies have nonlinear relationship
9/3/2004 I. Discrete-Time Signals and Systems 30
Ω=
=Ω
)2
arctan(2
)2
tan(2
d
d
T
T
ω
ω
)2
tan(2 ω
α dTj
j ee−
Ω− =
Linear phase property will lose !!
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9/3/2004 I. Discrete-Time Signals and Systems 31
Examples of bilinear transform design (see p.454-465)