spectral transformations of iir digital...

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

)ˆ(zGD

• Unit circles in z- and -planes defined by ,• Transformation from z-domain to

-domain given by

• Then

ω= jez ω= ˆˆ jez

)ˆ(zFz =

)}ˆ({)ˆ( zFGzG LD =

• From , thus , hence

• Recall that a stable allpass function A(z)satisfies the condition

)ˆ(zFz = )ˆ(zFz =

<<==>>

1if,11if,11if,1

)ˆ(zzz

• Therefore must be a stable allpassfunction whose general form is

<>==><

1if,11if,11if,1

)ˆ(/1 zF

ˆ1)ˆ(

α−±= ∏=

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

)ˆ(zGD

cωcω

α−α−==−

zFz ˆ

ˆ1)ˆ(

• On the unit circle we have

• From the above we get

• Taking the ratios of the above two expressions

ω−ω−

α−α−= ˆ

)2/ˆtan(11)2/tan( ω

α−α+=ω

ω−ω−

α−−⋅α±=

α−α−= ˆ

11)1(1

eee mm

• 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

)3917.06763.01)(2593.01()1(0662.0)( 211

−−−

+−−+=

zzzzzGL

π25.0

π35.0

• Here

• Hence, the desired lowpass transfer function is

1934.0)3.0sin()05.0sin( −=

ππ−=α

ˆ1934.011934.0ˆ)()ˆ(

−−

zzzLD zGzG

0 0.2 0.4 0.6 0.8 1-40

B GL(z) G

• The lowpass-to-lowpass transformation

can also be used as highpass-to-highpass,bandpass-to-bandpass and bandstop-to-bandstop transformations

α−α−==−

zFz ˆ

ˆ1)ˆ(

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

−−

α+α+−=

( )( )2/)ˆ(cos

2/)ˆ(coscc

ccω−ωω+ω

−=α

cωcω

• 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

−−−

+−−+=

zzzzzGL

π25.0π55.0

3468.0)15.0cos(/)4.0cos( −=ππ−=α

ˆ3468.013468.0ˆ

−−

−−−=

• The desired highpass filter is

ˆ3468.013468.0ˆ)()ˆ(

−−

−−−==

zzzD zGzG

0 0.2π 0.4π 0.6π 0.8π π

Normalized frequency

• 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

LowpassLowpass-to--to-BandpassBandpassSpectral TransformationSpectral Transformation

++βαβ−

+β−β

+β−β+

+βαβ−

−=−−

−−

• 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

( ) )2/tan(2/)ˆˆ(cot 12 ccc ωω−ω=β

( )( )2/)ˆˆ(cos

2/)ˆˆ(cos12

ccω−ωω+ω

cω1ˆ cω 2ˆ cω

• 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

ˆ1ˆˆ −

−−−

α−α−−=

LowpassLowpass-to--to-BandstopBandstopSpectral TransformationSpectral Transformation

1ˆ12ˆ

−+−

=−−

−−

βαβ

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

1ˆ cω 2ˆ cω

( )( )2/)ˆˆ(cos

2/)ˆˆ(cos12

ccω−ωω+ω

( ) )2/tan(2/)ˆˆ(tan 12 ccc ωω−ω=β

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

)( ωjd eH

)( ωjd eH ω

∫ ∞≤≤∞−ωπ

ωω ndeeHnh njjdd ,)(

∑=∞

−∞=

ω−ω

jd enheH ][)(

• 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

)( ωjt eH{ }][nht

{ }][nhd

• Commonly used approximation criterion -Minimize the integral-squared error

ω∫ −π

ωω deHeH jd

∑=−=

ω−ω M

jt enheH ][)(

• 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][][

∑ ∑++∑ −=−−

−∞=

+=−=

222 ][][][][M

n Mndd

Mndt nhnhnhnh

Φ][][ nhnh dt = MnM ≤≤−

• 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

][][ Mnhnh t −=

Impulse Responses of IdealImpulse Responses of IdealFiltersFilters

• Ideal lowpass filter -

• Ideal highpass filter -

∞≤≤∞−πω= nn

nnh cLP ,sin][

0 c – c

HLP(e j )

0 c – c

HHP(e j )

≠πω−

=πω−

=0,)sin(

• Ideal bandpass filter -

=πω−π

≠πω−π

0,)sin()sin(

– c1 c1 – c2 c2

HBP (e j )

• Ideal bandstop filter -

– c1 c1 – c2 c2

HBS(e j )

≠πω−π

=πω−ω−

=0,)sin()sin(

0,)(1][

• Ideal multiband filter -

1 2 3 4

HML(e j )

,)( kj

ML AeH =ω

,1 kk ω≤ω≤ω −

Lk ,,2,1 K=

+ πω⋅−=

ML nnAAnh

)sin()(][l

• Ideal discrete-time Hilbert transformer -

π<ω<−<ω<π−

oddforn,2/evenfor,0

• Ideal discrete-time differentiator -

π≤ω≤ω=ω 0,)( jeH jDIF

≠π=

= 0,cos0,0

][ nnn

nnhDIF

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

N = 20N = 60

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

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

ϕ−ωϕω deeHeH jjd

jt )()(

21)( )(

)( ωjt eH )( ωΨ je

][nht ][nw

GibbsGibbs Phenomenon Phenomenon• Thus is obtained by a periodic

continuous convolution of with)( ωj

)( ωΨ je)( ωj

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

)( ωjt eH

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

)( ωjt eH

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

deRectangular window

M = 10

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

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

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