Stability Condition in Terms of the Pole Locationssip.cua.edu/res/docs/courses/ee515/chapter04/ch4-3.pdf · terms of the pole locations of the transfer ... • The function fftfilt

1Copyright © 2001, S. K. Mitra

Stability Condition in Terms ofStability Condition in Terms ofthe Pole Locationsthe Pole Locations

• A causal LTI digital filter is BIBO stable ifand only if its impulse response h[n] isabsolutely summable, i.e.,

• We now develop a stability condition interms of the pole locations of the transferfunction H(z)

∞<= ∑∞

−∞=nnh ][S



• The ROC of the z-transform H(z) of theimpulse response sequence h[n] is definedby values of |z| = r for which isabsolutely summable

• Thus, if the ROC includes the unit circle |z|= 1, then the digital filter is stable, and viceversa

nrnh −][



• In addition, for a stable and causal digitalfilter for which h[n] is a right-sidedsequence, the ROC will include the unitcircle and entire z-plane including the point

• An FIR digital filter with bounded impulseresponse is always stable

∞=z



• On the other hand, an IIR filter may beunstable if not designed properly

• In addition, an originally stable IIR filtercharacterized by infinite precisioncoefficients may become unstable whencoefficients get quantized due toimplementation



• Example - Consider the causal IIR transferfunction

• The plot of the impulse response coefficientsis shown on the next slide

21 8505860845111

−− +−=

zzzH

..)(



• As can be seen from the above plot, theimpulse response coefficient h[n] decaysrapidly to zero value as n increases

0 10 20 30 40 50 60 700

2

4

6

Time index n

Am

plitu

de

h[n]



• The absolute summability condition of h[n]is satisfied

• Hence, H(z) is a stable transfer function• Now, consider the case when the transfer

function coefficients are rounded to valueswith 2 digits after the decimal point:

21 85085111

−− +−=

zzzH

..)(

^



• A plot of the impulse response of isshown below

][nh^

0 10 20 30 40 50 60 700

2

4

6

Time index n

Am

plitu

de

][nh



• In this case, the impulse response coefficient increases rapidly to a constant value as

n increases• Hence, the absolute summability condition of

is violated• Thus, is an unstable transfer function

][nh^

)(zH^



• The stability testing of a IIR transferfunction is therefore an important problem

• In most cases it is difficult to compute theinfinite sum

• For a causal IIR transfer function, the sum Scan be computed approximately as

∞<=∑∞−∞=n nh ][S

∑ −== 1

0Kn nh ][SK



• The partial sum is computed for increasingvalues of K until the difference between aseries of consecutive values of issmaller than some arbitrarily chosen smallnumber, which is typically

• For a transfer function of very high orderthis approach may not be satisfactory

• An alternate, easy-to-test, stability conditionis developed next

SK

610−



• Consider the causal IIR digital filter with arational transfer function H(z) given by

• Its impulse response {h[n]} is a right-sidedsequence

• The ROC of H(z) is exterior to a circlegoing through the pole furthest from z = 0

∑∑

=−

=−

= Nk

kk

Mk

kk

zdzp

zH0

0)(



• But stability requires that {h[n]} beabsolutely summable

• This in turn implies that the DTFTof {h[n]} exists

• Now, if the ROC of the z-transform H(z)includes the unit circle, then

ω=ω = jez

j zHeH )()(

)( ωjeH



• Conclusion: All poles of a causal stabletransfer function H(z) must be strictly insidethe unit circle

• The stability region (shown shaded) in thez-plane is shown below

1 Re z

j Im z

1−

j−

j

unit circle

stability region



• Example - The factored form of

is

which has a real pole at z = 0.902 and a realpole at z = 0.943

• Since both poles are inside the unit circle,H(z) is BIBO stable

21 850586.0845.011

)( −− +−=

zzzH

)943.01)(902.01(1

)( 11 −− −−=

zzzH



• Example - The factored form of

is

which has a real pole on the unit circle at z= 1 and the other pole inside the unit circle

• Since both poles are not inside the unitcircle, H(z) is unstable

21 85.085.111)( −− +−

=zz

zH

)85.01)(1(1

)( 11 −− −−=

zzzH


Types of Transfer FunctionsTypes of Transfer Functions

• The time-domain classification of an LTIdigital transfer function sequence is basedon the length of its impulse response:- Finite impulse response (FIR) transferfunction- Infinite impulse response (IIR) transferfunction


Types of Transfer FunctionsTypes of Transfer Functions

• Several other classifications are also used• In the case of digital transfer functions with

frequency-selective frequency responses,one classification is based on the shape ofthe magnitude function or the formof the phase function θ(ω)

• Based on this four types of ideal filters areusually defined

|)(| ωjeH


Ideal FiltersIdeal Filters

• A digital filter designed to pass signalcomponents of certain frequencies withoutdistortion should have a frequency responseequal to one at these frequencies, andshould have a frequency response equal tozero at all other frequencies



• The range of frequencies where thefrequency response takes the value of one iscalled the passband

• The range of frequencies where thefrequency response takes the value of zerois called the stopband


Ideal FiltersIdeal Filters• Frequency responses of the four popular types

of ideal digital filters with real impulseresponse coefficients are shown below:

Lowpass Highpass

Bandpass Bandstop


Ideal FiltersIdeal Filters• Lowpass filter: Passband - Stopband -• Highpass filter: Passband - Stopband -• Bandpass filter: Passband - Stopband -• Bandstop filter: Stopband - Passband -

cω≤ω≤0π≤ω<ωc

π≤ω≤ωc

cω<ω≤0

21 cc ω≤ω≤ω

10 cω<ω≤ π≤ω<ω 2cand

21 cc ω<ω<ω

10 cω≤ω≤ π≤ω≤ω 2cand



• The frequencies , , and are calledthe cutoff frequencies

• An ideal filter has a magnitude responseequal to one in the passband and zero in thestopband, and has a zero phase everywhere

cω 1cω 2cω


Ideal FiltersIdeal Filters• Earlier in the course we derived the inverse

DTFT of the frequency responseof the ideal lowpass filter:

• We have also shown that the above impulseresponse is not absolutely summable, andhence, the corresponding transfer functionis not BIBO stable

)( ωjLP eH

∞<<∞−πω= nn

nnh cLP ,sin][


Ideal FiltersIdeal Filters• Also, is not causal and is of doubly

infinite length• The remaining three ideal filters are also

characterized by doubly infinite, noncausalimpulse responses and are not absolutelysummable

• Thus, the ideal filters with the ideal “brickwall” frequency responses cannot berealized with finite dimensional LTI filter

][nhLP


Ideal FiltersIdeal Filters• To develop stable and realizable transfer

functions, the ideal frequency responsespecifications are relaxed by including atransition band between the passband andthe stopband

• This permits the magnitude response todecay slowly from its maximum value inthe passband to the zero value in thestopband


Ideal FiltersIdeal Filters• Moreover, the magnitude response is

allowed to vary by a small amount both inthe passband and the stopband

• Typical magnitude response specificationsof a lowpass filter are shown below


Zero-Phase and Linear-PhaseZero-Phase and Linear-PhaseTransfer FunctionsTransfer Functions

• A second classification of a transferfunction is with respect to its phasecharacteristics

• In many applications, it is necessary that thedigital filter designed does not distort thephase of the input signal components withfrequencies in the passband



• One way to avoid any phase distortion is tomake the frequency response of the filterreal and nonnegative, i.e., to design thefilter with a zero phase characteristic

• However, it is possible to design a causaldigital filter with a zero phase



• For non-real-time processing of real-valuedinput signals of finite length, zero-phasefiltering can be very simply implemented byrelaxing the causality requirement

• One zero-phase filtering scheme is sketchedbelow

x[n] v[n] u[n] w[n]H(z) H(z)

][][],[][ nwnynvnu −=−=



• It is easy to verify the above scheme in thefrequency domain

• Let , , , , and denote the DTFTs of x[n], v[n],

u[n], w[n], and y[n], respectively• From the figure shown earlier and making

use of the symmetry relations we arrive atthe relations between various DTFTs asgiven on the next slide

)( ωjeX )( ωjeV )( ωjeU )( ωjeW)( ωjeY



• Combining the above equations we get

x[n] v[n] u[n] w[n]H(z) H(z)

][][],[][ nwnynvnu −=−=

),()()( ωωω = jjj eXeHeV )()()( ωωω = jjj eUeHeW,)(*)( ωω = jj eVeU )(*)( ωω = jj eWeY

)(*)(*)(*)( ωωωω == jjjj eUeHeWeY)()()(*)()(* ωωωωω == jjjjj eXeHeHeVeH

)()(2 ωω= jj eXeH



• The function fftfilt implements theabove zero-phase filtering scheme

• In the case of a causal transfer function witha nonzero phase response, the phasedistortion can be avoided by ensuring thatthe transfer function has a unity magnitudeand a linear-phase characteristic in thefrequency band of interest



• The most general type of a filter with alinear phase has a frequency response givenby

which has a linear phase from ω = 0 to ω =2π

• Note also

Djj eeH ω−ω =)(

1)( =ωjeHD=ωτ )(



• The output y[n] of this filter to an input is then given by

• If and represent the continuous-time signals whose sampled versions,sampled at t = nT, are x[n] and y[n] givenabove, then the delay between andis precisely the group delay of amount D

njAenx ω=][)(][ DnjnjDj AeeAeny −ωωω− ==

)(txa

)(txa

)(tya

)(tya



• If D is an integer, then y[n] is identical tox[n], but delayed by D samples

• If D is not an integer, y[n], being delayed bya fractional part, is not identical to x[n]

• In the latter case, the waveform of theunderlying continuous-time output isidentical to the waveform of the underlyingcontinuous-time input and delayed D unitsof time



• If it is desired to pass input signalcomponents in a certain frequency rangeundistorted in both magnitude and phase,then the transfer function should exhibit aunity magnitude response and a linear-phaseresponse in the band of interest



• Figure below shows the frequency responseif a lowpass filter with a linear-phasecharacteristic in the passband



• Since the signal components in the stopbandare blocked, the phase response in thestopband can be of any shape

• Example - Determine the impulse responseof an ideal lowpass filter with a linear phaseresponse:

=ω)( j

LP eHπ≤ω≤ω

ω<ω<ω−

cc

nj oe,0

0,



• Applying the frequency-shifting property ofthe DTFT to the impulse response of anideal zero-phase lowpass filter we arrive at

• As before, the above filter is noncausal andof doubly infinite length, and hence,unrealizable

∞<<∞−−π−ω

= nnn

nnnho

ocLP ,

)()(sin][



• By truncating the impulse response to afinite number of terms, a realizable FIRapproximation to the ideal lowpass filtercan be developed

• The truncated approximation may or maynot exhibit linear phase, depending on thevalue of chosenon



• If we choose = N/2 with N a positiveinteger, the truncated and shiftedapproximation

will be a length N+1 causal linear-phaseFIR filter

on

NnNn

Nnnh cLP ≤≤

−π−ω= 0,

)2/()2/(sin][^



• Figure below shows the filter coefficientsobtained using the function sinc for twodifferent values of N

0 2 4 6 8 10 12-0.2

0

0.2

0.4

0.6

Time index n

Am

plitu

de

N = 12

0 2 4 6 8 10 12-0.2

0

0.2

0.4

0.6

Time index n

Am

plitu

de

N = 13



• Because of the symmetry of the impulseresponse coefficients as indicated in the twofigures, the frequency response of thetruncated approximation can be expressed as:

where , called the zero-phaseresponse or amplitude response, is a realfunction of ω

)(][)( 2/

0ω== ω−

=

ω−ω ∑ LPNj

N

n

njLP

jLP HeenheH ^^ ∼

)(ωLPH∼

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Stability Condition in Terms of the Pole Locationssip.cua.edu/res/docs/courses/ee515/chapter04/ch4-3.pdf · terms of the pole locations of the transfer ... • The function fftfilt