1Copyright © S. K. Mitra
Spectral Transformations ofSpectral Transformations ofIIR Digital FiltersIIR Digital Filters
• Objective - Transform a given lowpass digitaltransfer function to another digitaltransfer function that could be alowpass, highpass, bandpass or bandstop filter
• has been used to denote the unit delay inthe prototype lowpass filter andto denote the unit delay in the transformedfilter to avoid confusion
)(zGL)ˆ(zGD
1ˆ−z1−z
)(zGL
)ˆ(zGD
2Copyright © S. K. Mitra
Spectral Transformations ofSpectral Transformations ofIIR Digital FiltersIIR Digital Filters
• Unit circles in z- and -planes defined by ,• Transformation from z-domain to
-domain given by
• Then
z
z
ω= jez ω= ˆˆ jez
)ˆ(zFz =
)}ˆ({)ˆ( zFGzG LD =
3Copyright © S. K. Mitra
Spectral Transformations ofSpectral Transformations ofIIR Digital FiltersIIR Digital Filters
• From , thus , hence
• Recall that a stable allpass function A(z)satisfies the condition
)ˆ(zFz = )ˆ(zFz =
<<==>>
1if,11if,11if,1
)ˆ(zzz
zF
4Copyright © S. K. Mitra
Spectral Transformations ofSpectral Transformations ofIIR Digital FiltersIIR Digital Filters
• Therefore must be a stable allpassfunction whose general form is
<>==><
1if,11if,11if,1
)(zzz
zA
)ˆ(/1 zF
1,ˆ
ˆ1)ˆ(
11
*<α
α−
α−±= ∏=
ll l
lL
zz
zF
5Copyright © S. K. Mitra
LowpassLowpass-to--to-LowpassLowpassSpectral TransformationSpectral Transformation
• To transform a lowpass filter with acutoff frequency to another lowpass filter
with a cutoff frequency , thetransformation is
where α is a function of the two specifiedcutoff frequencies
)(zGL
)ˆ(zGD
cωcω
α−α−==−
zz
zFz ˆ
ˆ1)ˆ(
11
6Copyright © S. K. Mitra
LowpassLowpass-to--to-LowpassLowpassSpectral TransformationSpectral Transformation
• On the unit circle we have
• From the above we get
• Taking the ratios of the above two expressions
ω−
ω−ω−
α−α−= ˆ
ˆ
1 j
jj
eee
)2/ˆtan(11)2/tan( ω
α−α+=ω
ω−
ω−
ω−
ω−ω−
α−−⋅α±=
α−α−= ˆ
ˆ
ˆ
ˆ
11)1(1
11 j
j
j
jj
ee
eee mm
7Copyright © S. K. Mitra
LowpassLowpass-to--to-LowpassLowpassSpectral TransformationSpectral Transformation
• Solving we get
• Example - Consider the lowpass digital filter
which has a passband from dc to witha 0.5 dB ripple
• Redesign the above filter to move thepassband edge to
( )( )2/)ˆ(sin
2/)ˆ(sincc
ccω+ωω−ω
=α
)3917.06763.01)(2593.01()1(0662.0)( 211
31
−−−
−
+−−+=
zzzzzGL
π25.0
π35.0
8Copyright © S. K. Mitra
LowpassLowpass-to--to-LowpassLowpassSpectral TransformationSpectral Transformation
• Here
• Hence, the desired lowpass transfer function is
1934.0)3.0sin()05.0sin( −=
ππ−=α
1
11
ˆ1934.011934.0ˆ)()ˆ(
−
−−
++==
zzzLD zGzG
0 0.2 0.4 0.6 0.8 1-40
-30
-20
-10
0
ω/π
Gai
n, d
B GL(z) G
D(z)
9Copyright © S. K. Mitra
LowpassLowpass-to--to-LowpassLowpassSpectral TransformationSpectral Transformation
• The lowpass-to-lowpass transformation
can also be used as highpass-to-highpass,bandpass-to-bandpass and bandstop-to-bandstop transformations
α−α−==−
zz
zFz ˆ
ˆ1)ˆ(
11
10Copyright © S. K. Mitra
LowpassLowpass-to--to-HighpassHighpassSpectral TransformationSpectral Transformation
• Desired transformation
• The transformation parameter is given by
where is the cutoff frequency of thelowpass filter and is the cutoff frequencyof the desired highpass filter
1
11
ˆ1ˆ
−
−−
α+α+−=
zzz
( )( )2/)ˆ(cos
2/)ˆ(coscc
ccω−ωω+ω
−=α
α
cωcω
11Copyright © S. K. Mitra
LowpassLowpass-to--to-HighpassHighpassSpectral TransformationSpectral Transformation
• Example - Transform the lowpass filter
• with a passband edge at to a highpassfilter with a passband edge at
• Here• The desired transformation is
)3917.06763.01)(2593.01()1(0662.0)( 211
31
−−−
−
+−−+=
zzzzzGL
π25.0π55.0
3468.0)15.0cos(/)4.0cos( −=ππ−=α
1
11
ˆ3468.013468.0ˆ
−
−−
−−−=
zzz
12Copyright © S. K. Mitra
LowpassLowpass-to--to-HighpassHighpassSpectral TransformationSpectral Transformation
• The desired highpass filter is
1
11
ˆ3468.013468.0ˆ)()ˆ(
−
−−
−−−==
zzzD zGzG
0 0.2π 0.4π 0.6π 0.8π π
−80
−60
−40
−20
0
Normalized frequency
Gai
n, d
B
13Copyright © S. K. Mitra
LowpassLowpass-to--to-HighpassHighpassSpectral TransformationSpectral Transformation
• The lowpass-to-highpass transformation canalso be used to transform a highpass filterwith a cutoff at to a lowpass filter witha cutoff atand transform a bandpass filter with a centerfrequency at to a bandstop filter with acenter frequency at
cωcω
oωoω
14Copyright © S. K. Mitra
LowpassLowpass-to--to-BandpassBandpassSpectral TransformationSpectral Transformation
• Desired transformation
1ˆ1
2ˆ11
11ˆ
12ˆ
12
12
1
++βαβ−
+β−β
+β−β+
+βαβ−
−=−−
−−
−
zz
zzz
15Copyright © S. K. Mitra
LowpassLowpass-to--to-BandpassBandpassSpectral TransformationSpectral Transformation
• The parameters and are given by
where is the cutoff frequency of thelowpass filter, and and are thedesired upper and lower cutoff frequenciesof the bandpass filter
α β
( ) )2/tan(2/)ˆˆ(cot 12 ccc ωω−ω=β
( )( )2/)ˆˆ(cos
2/)ˆˆ(cos12
12
cc
ccω−ωω+ω
=α
cω1ˆ cω 2ˆ cω
16Copyright © S. K. Mitra
LowpassLowpass-to--to-BandpassBandpassSpectral TransformationSpectral Transformation
• Special Case - The transformation can besimplified if
• Then the transformation reduces to
where with denoting thedesired center frequency of the bandpassfilter
12 ˆˆ ccc ω−ω=ω
oω=α ˆcos oω
1
111
ˆ1ˆˆ −
−−−
α−α−−=
zzzz
17Copyright © S. K. Mitra
LowpassLowpass-to--to-BandstopBandstopSpectral TransformationSpectral Transformation
• Desired transformation
1ˆ12ˆ
11
11ˆ
12ˆ
12
12
1
++
−+−
+−+
+−
=−−
−−
−
zz
zzz
βαβ
ββ
ββ
βαβ
18Copyright © S. K. Mitra
LowpassLowpass-to--to-BandstopBandstopSpectral TransformationSpectral Transformation
• The parameters and are given by
where is the cutoff frequency of thelowpass filter, and and are thedesired upper and lower cutoff frequenciesof the bandstop filter
cω
α β
1ˆ cω 2ˆ cω
( )( )2/)ˆˆ(cos
2/)ˆˆ(cos12
12
cc
ccω−ωω+ω
=α
( ) )2/tan(2/)ˆˆ(tan 12 ccc ωω−ω=β
19Copyright © S. K. Mitra
Least Integral-Squared ErrorLeast Integral-Squared ErrorDesign of FIR FiltersDesign of FIR Filters
• Let denote the desired frequencyresponse
• Since is a periodic function ofwith a period , it can be expressed as aFourier series
where
)( ωjd eH
)( ωjd eH ω
π2
∫ ∞≤≤∞−ωπ
=π
π−
ωω ndeeHnh njjdd ,)(
21][
∑=∞
−∞=
ω−ω
n
njd
jd enheH ][)(
20Copyright © S. K. Mitra
Least Integral-Squared ErrorLeast Integral-Squared ErrorDesign of FIR FiltersDesign of FIR Filters
• In general, is piecewise constantwith sharp transitions between bands
• In which case, is of infinite lengthand noncausal
• Objective - Find a finite-durationof length 2M+1 whose DTFTapproximates the desired DTFT insome sense
)( ωjd eH
)( ωjd eH
)( ωjt eH{ }][nht
{ }][nhd
21Copyright © S. K. Mitra
Least Integral-Squared ErrorLeast Integral-Squared ErrorDesign of FIR FiltersDesign of FIR Filters
• Commonly used approximation criterion -Minimize the integral-squared error
where
ω∫ −π
=Φπ
π−
ωω deHeH jd
jt
2)()(
21
∑=−=
ω−ω M
Mn
njt
jt enheH ][)(
22Copyright © S. K. Mitra
Least Integral-Squared ErrorLeast Integral-Squared ErrorDesign of FIR FiltersDesign of FIR Filters
• Using Parseval’s relation we can write
• It follows from the above that isminimum when for
• Best finite-length approximation to idealinfinite-length impulse response in themean-square sense is obtained by truncation
∑ −=Φ∞
−∞=ndt nhnh 2][][
∑ ∑++∑ −=−−
−∞=
∞
+=−=
1
1
222 ][][][][M
n Mndd
M
Mndt nhnhnhnh
Φ][][ nhnh dt = MnM ≤≤−
⇒
23Copyright © S. K. Mitra
Least Integral-Squared ErrorLeast Integral-Squared ErrorDesign of FIR FiltersDesign of FIR Filters
• A causal FIR filter with an impulse responseh[n] can be derived from by delaying:
• The causal FIR filter h[n] has the samemagnitude response as and its phaseresponse has a linear phase shift ofradians with respect to that of
][nht
][nht
][nht
][][ Mnhnh t −=
Mω
24Copyright © S. K. Mitra
Impulse Responses of IdealImpulse Responses of IdealFiltersFilters
• Ideal lowpass filter -
• Ideal highpass filter -
∞≤≤∞−πω= nn
nnh cLP ,sin][
1
0 c – c
HLP(e j )
0 c – c
1
HHP(e j )
≠πω−
=πω−
=0,)sin(
0,1][
nnn
nnh
c
c
HP
25Copyright © S. K. Mitra
Impulse Responses of IdealImpulse Responses of IdealFiltersFilters
• Ideal bandpass filter -
=πω−π
ω
≠πω−π
ω
=0,
0,)sin()sin(
][12
12
n
nnn
nn
nhcc
cc
BP
11–
– c1 c1 – c2 c2
HBP (e j )
26Copyright © S. K. Mitra
Impulse Responses of IdealImpulse Responses of IdealFiltersFilters
• Ideal bandstop filter -
1
– c1 c1 – c2 c2
HBS(e j )
≠πω−π
ω
=πω−ω−
=0,)sin()sin(
0,)(1][
21
12
nnn
nn
nnh
cc
cc
BS
27Copyright © S. K. Mitra
Impulse Responses of IdealImpulse Responses of IdealFiltersFilters
• Ideal multiband filter -
0
1 2 3 4
HML(e j )
A5
A 4
A3
A2
A1
,)( kj
ML AeH =ω
,1 kk ω≤ω≤ω −
Lk ,,2,1 K=
∑=
+ πω⋅−=
LL
ML nnAAnh
11
)sin()(][l
ll
28Copyright © S. K. Mitra
Impulse Responses of IdealImpulse Responses of IdealFiltersFilters
• Ideal discrete-time Hilbert transformer -
π<ω<−<ω<π−
=ω
0,0,
)(j
jeH j
HT
π=
oddforn,2/evenfor,0
][nn
nhHT
29Copyright © S. K. Mitra
Impulse Responses of IdealImpulse Responses of IdealFiltersFilters
• Ideal discrete-time differentiator -
π≤ω≤ω=ω 0,)( jeH jDIF
≠π=
= 0,cos0,0
][ nnn
nnhDIF
30Copyright © S. K. Mitra
Gibbs Gibbs PhenomenonPhenomenon• Gibbs phenomenon - Oscillatory behavior in
the magnitude responses of causal FIR filtersobtained by truncating the impulse responsecoefficients of ideal filters
0 0.2 0.4 0.6 0.8 10
0.5
1
1.5
ω/π
Mag
nitu
de
N = 20N = 60
31Copyright © S. K. Mitra
GibbsGibbs Phenomenon Phenomenon• As can be seen, as the length of the lowpass
filter is increased, the number of ripples inboth passband and stopband increases, witha corresponding decrease in the ripplewidths
• Height of the largest ripples remain thesame independent of length
• Similar oscillatory behavior observed in themagnitude responses of the truncatedversions of other types of ideal filters
32Copyright © S. K. Mitra
GibbsGibbs Phenomenon Phenomenon• Gibbs phenomenon can be explained by
treating the truncation operation as anwindowing operation:
• In the frequency domain
• where and are the DTFTsof and , respectively
][][][ nwnhnh dt ⋅=
∫ ϕΨπ
=π
π−
ϕ−ωϕω deeHeH jjd
jt )()(
21)( )(
)( ωjt eH )( ωΨ je
][nht ][nw
33Copyright © S. K. Mitra
GibbsGibbs Phenomenon Phenomenon• Thus is obtained by a periodic
continuous convolution of with)( ωj
t eH
)( ωΨ je)( ωj
d eH
34Copyright © S. K. Mitra
GibbsGibbs Phenomenon Phenomenon
• If is a very narrow pulse centered at (ideally a delta function) compared to
variations in , then willapproximate very closely
• Length 2M+1 of w[n] should be very large• On the other hand, length 2M+1 of
should be as small as possible to reducecomputational complexity
)( ωΨ je
)( ωjd eH
)( ωjd eH
)( ωjt eH
][nht
0=ω
35Copyright © S. K. Mitra
GibbsGibbs Phenomenon Phenomenon• A rectangular window is used to achieve
simple truncation:
• Presence of oscillatory behavior inis basically due to:– 1) is infinitely long and not absolutely
summable, and hence filter is unstable– 2) Rectangular window has an abrupt transition
to zero
≤≤
=otherwise,00,1
][Mn
nwR
)( ωjt eH
][nhd
36Copyright © S. K. Mitra
GibbsGibbs Phenomenon Phenomenon• Oscillatory behavior can be explained by
examining the DTFT of :
• has a main lobe centered at• Other ripples are called sidelobes
][nwR)( ωΨ jR e
)( ωΨ jR e 0=ω
-1 -0.5 0 0.5 1-10
0
10
20
30
ω/π
Am
plitu
deRectangular window
M = 4
M = 10
37Copyright © S. K. Mitra
GibbsGibbs Phenomenon Phenomenon• Main lobe of characterized by its
width defined by first zerocrossings on both sides of
• As M increases, width of main lobedecreases as desired
• Area under each lobe remains constantwhile width of each lobe decreases with anincrease in M
• Ripples in around the point ofdiscontinuity occur more closely but withno decrease in amplitude as M increases
)( ωΨ jR e
0=ω)12/(4 +π M
)( ωjt eH
38Copyright © S. K. Mitra
GibbsGibbs Phenomenon Phenomenon• Rectangular window has an abrupt transition
to zero outside the range , whichresults in Gibbs phenomenon in
• Gibbs phenomenon can be reduced either:(1) Using a window that tapers smoothly tozero at each end, or(2) Providing a smooth transition frompassband to stopband in the magnitudespecifications
MnM ≤≤−)( ωj
t eH