Linear-Phase FIR Transfer Functionssip.cua.edu/res/docs/courses/ee515/chapter04/ch4-4.pdf ·...

1Copyright © 2001, S. K. Mitra

Linear-Phase FIR TransferLinear-Phase FIR TransferFunctionsFunctions

• It is nearly impossible to design a linear-phase IIR transfer function

• It is always possible to design an FIRtransfer function with an exact linear-phaseresponse

• Consider a causal FIR transfer function H(z)of length N+1, i.e., of order N:

∑ =−= N

nnznhzH 0 ][)(



• The above transfer function has a linearphase, if its impulse response h[n] is eithersymmetric, i.e.,

or is antisymmetric, i.e.,NnnNhnh ≤≤−= 0],[][

NnnNhnh ≤≤−−= 0],[][



• Since the length of the impulse response canbe either even or odd, we can define fourtypes of linear-phase FIR transfer functions

• For an antisymmetric FIR filter of oddlength, i.e., N even

h[N/2] = 0• We examine next the each of the 4 cases



Type 1: N = 8 Type 2: N = 7

Type 3: N = 8 Type 4: N = 7



Type 1: Symmetric Impulse Response withOdd Length

• In this case, the degree N is even• Assume N = 8 for simplicity• The transfer function H(z) is given by

321 3210 −−− +++= zhzhzhhzH ][][][][)(87654 87654 −−−−− +++++ zhzhzhzhzh ][][][][][



• Because of symmetry, we have h[0] = h[8],h[1] = h[7], h[2] = h[6], and h[3] = h[5]

• Thus, we can write)]([)]([)( 718 110 −−− +++= zzhzhzH

45362 432 −−−−− +++++ zhzzhzzh ][)]([)]([

)]([)]([{ 33444 10 −−− +++= zzhzzhz]}[)]([)]([ 432 122 hzzhzzh +++++ −−



• The corresponding frequency response isthen given by

• The quantity inside the braces is a realfunction of ω, and can assume positive ornegative values in the range π≤ω≤0

)3cos(]1[2)4cos(]0[2{)( 4 ω+ω= ω−ω hheeH jj

]}4[)cos(]3[2)2cos(]2[2 hhh +ω+ω+



• The phase function here is given by

where β is either 0 or π, and hence, it is alinear function of ω in the generalized sense

• The group delay is given by

indicating a constant group delay of 4 samples

β+ω−=ωθ 4)(

4)( )( =−=ωτωωθ

dd



• In the general case for Type 1 FIR filters,the frequency response is of the form

where the amplitude response , alsocalled the zero-phase response, is of theform

)()( 2/ ω= ω−ω HeeH jNj ~

)(ωH~

)(ωH~∑ ω−+==

2/

1 22 )cos(][2][N

n

NN nnhh



• Example - Consider

which is seen to be a slightly modifiedversion of a length-7 moving-average FIRfilter

• The above transfer function has asymmetric impulse response and therefore alinear phase response

][)( 62154321

21

61

0−−−−−− ++++++= zzzzzzzH



• A plot of the magnitude response ofalong with that of the 7-point moving-average filter is shown below

)(zH0

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

ω/π

Mag

nitu

de

modified filtermoving-average



• Note the improved magnitude responseobtained by simply changing the first and thelast impulse response coefficients of amoving-average (MA) filter

• It can be shown that we an express

which is seen to be a cascade of a 2-point MAfilter with a 6-point MA filter

• Thus, has a double zero at , i.e.,(ω = π)

)()()( 54321611

21

0 11 −−−−−− +++++⋅+= zzzzzzzH

1−=z)(zH0



Type 2: Symmetric Impulse Response withEven Length

• In this case, the degree N is odd• Assume N = 7 for simplicity• The transfer function is of the form

321 3210 −−− +++= zhzhzhhzH ][][][][)(7654 7654 −−−− ++++ zhzhzhzh ][][][][



• Making use of the symmetry of the impulseresponse coefficients, the transfer functioncan be written as

)]([)]([)( 617 110 −−− +++= zzhzhzH)]([)]([ 4352 32 −−−− ++++ zzhzzh

)]([)]([{ ///// 2525272727 10 −−− +++= zzhzzhz)}]([)]([ //// 21212323 32 −− ++++ zzhzzh



• The corresponding frequency response isgiven by

• As before, the quantity inside the braces is areal function of ω, and can assume positiveor negative values in the range

)cos(]1[2)cos(]0[2{)( 25

272/7 ωωω−ω += hheeH jj

)}cos(]3[2)cos(]2[2 223 ωω ++ hh

π≤ω≤0



• Here the phase function is given by

where again β is either 0 or π• As a result, the phase is also a linear

function of ω in the generalized sense• The corresponding group delay is

indicating a group delay of samples

β+ω−=ωθ 27)(

27

27)( =ωτ



• The expression for the frequency responsein the general case for Type 2 FIR filters isof the form

where the amplitude response is given by

)(ωH~ ∑ −ω−=+

=

+2/)1(

1 21

21 ))(cos(][2

N

n

N nnh

)()( 2/ ω= ω−ω HeeH jNj ~



Type 3: Antiymmetric Impulse Responsewith Odd Length

• In this case, the degree N is even• Assume N = 8 for simplicity• Applying the symmetry condition we get

)]([)]([{)( 33444 10 −−− −+−= zzhzzhzzH)}]([)]([ 122 32 −− −+−+ zzhzzh




• It also exhibits a generalized phase responsegiven by

where β is either 0 or π

)3sin(]1[2)4sin(]0[2{)( 2/4 ω+ω= π−ω−ω hheeeH jjj

)}sin(]3[2)2sin(]2[2 ω+ω+ hh

β++ω−=ωθ π24)(



• The group delay here is

indicating a constant group delay of 4 samples• In the general case

where the amplitude response is of the form

4)( =ωτ

)()( 2/ ω= ω−ω HjeeH jNj ~

)(ωH~ ∑ ω−==

2/

1 2 )sin(][2N

n

N nnh



Type 4: Antiymmetric Impulse Responsewith Even Length

• In this case, the degree N is even• Assume N = 7 for simplicity• Applying the symmetry condition we get

)]([)]([{)( ///// 2525272727 10 −−− −+−= zzhzzhzzH)}]([)]([ //// 21212323 32 −− −+−+ zzhzzh




• It again exhibits a generalized phaseresponse given by

where β is either 0 or π

)sin(]1[2)sin(]0[2{)( 25

272/2/7 ωωπ−ω−ω += hheeeH jjj

)}sin(]3[2)sin(]2[2 223 ωω ++ hh

β++ω−=ωθ π22

7)(



• The group delay is constant and is given by

• In the general case we have

where now the amplitude response is of theform

27)( =ωτ

)()( 2/ ω= ω−ω HjeeH jNj ~

)(ωH~ ∑ −ω−=+

=

+2/)1(

121

21 ))(sin(][2

N

n

N nnh



General Form of Frequency Response• In each of the four types of linear-phase FIR

filters, the frequency response is of the form

• The amplitude response for each ofthe four types of linear-phase FIR filters canbecome negative over certain frequencyranges, typically in the stopband

)()( 2/ ω= βω−ω HeeeH jjNj ~

~)(ωH



• The magnitude and phase responses of thelinear-phase FIR are given by

• The group delay in each case is

)(|)(| ω=ω HeH j

=ωθ )(

0)(for,

0)(for,

2

2

<ωπ−β+−

≥ωβ+−ω

ω

H

HN

N

~

~

~

2)( N=ωτ



• Note that, even though the group delay isconstant, since in general is not aconstant, the output waveform is not areplica of the input waveform

• An FIR filter with a frequency response thatis a real function of ω is often called a zero-phase filter

• Such a filter must have a noncausal impulseresponse

|)(| ωjeH


Zero Locations of Linear-Zero Locations of Linear-Phase FIR Transfer FunctionsPhase FIR Transfer Functions

• Consider first an FIR filter with a symmetricimpulse response:

• Its transfer function can be written as

• By making a change of variable ,we can write

∑∑=

−

=

− −==N

n

nN

n

n znNhznhzH00

][][)(

][][ nNhnh −=

nNm −=

∑∑∑=

−

=

+−

=

− ==−N

m

mNN

m

mNN

n

n zmhzzmhznNh000

][][][


Zero Locations of Linear-Zero Locations of Linear-Phase FIR Transfer FunctionsPhase FIR Transfer Functions• But,

• Hence for an FIR filter with a symmetricimpulse response of length N+1 we have

• A real-coefficient polynomial H(z)satisfying the above condition is called amirror-image polynomial (MIP)

)()( 1−−= zHzzH N

)(][ 10

−= =∑ zHzmhN

mm


Zero Locations of Linear-Zero Locations of Linear-Phase FIR Transfer FunctionsPhase FIR Transfer Functions• Now consider first an FIR filter with an

antisymmetric impulse response:

• Its transfer function can be written as

• By making a change of variable ,we get

][][ nNhnh −−=

∑∑=

−

=

− −−==N

n

nN

n

n znNhznhzH00

][][)(

)(][][ 1

00

−−

=

+−

=

− −=−=−− ∑∑ zHzzmhznNh NN

m

mNN

n

n

nNm −=


Zero Locations of Linear-Zero Locations of Linear-Phase FIR Transfer FunctionsPhase FIR Transfer Functions• Hence, the transfer function H(z) of an FIR

filter with an antisymmetric impulseresponse satisfies the condition

• A real-coefficient polynomial H(z)satisfying the above condition is called aantimirror-image polynomial (AIP)

)()( 1−−−= zHzzH N


Zero Locations of Linear-Zero Locations of Linear-Phase FIR Transfer FunctionsPhase FIR Transfer Functions• It follows from the relation

that if is a zero of H(z), so is• Moreover, for an FIR filter with a real

impulse response, the zeros of H(z) occur incomplex conjugate pairs

• Hence, a zero at is associated with azero at

)()( 1−−±= zHzzH N

oz ξ=

oz ξ=

oz ξ/1=

*oz ξ=


Zero Locations of Linear-Zero Locations of Linear-Phase FIR Transfer FunctionsPhase FIR Transfer Functions• Thus, a complex zero that is not on the unit

circle is associated with a set of 4 zeros givenby

• A zero on the unit circle appear as a pair

as its reciprocal is also its complex conjugate

,φjrez ±= φjr ez ±= 1

φjez ±=


Zero Locations of Linear-Zero Locations of Linear-Phase FIR Transfer FunctionsPhase FIR Transfer Functions• Since a zero at is its own reciprocal,

it can appear only singly• Now a Type 2 FIR filter satisfies

with degree N odd• Hence

implying , i.e., H(z) must have azero at

1±=z

)()( 1−−= zHzzH N

1−=z

)()()()( 1111 −−=−−=− − HHH N

01 =− )(H


Zero Locations of Linear-Zero Locations of Linear-Phase FIR Transfer FunctionsPhase FIR Transfer Functions• Likewise, a Type 3 or 4 FIR filter satisfies

• Thusimplying that H(z) must have a zero at z = 1

• On the other hand, only the Type 3 FIRfilter is restricted to have a zero atsince here the degree N is even and hence,

)()( 1−−−= zHzzH N

)()()()( 1111 HHH N −=−= −

1−=z

)()()()( 1111 −−=−−−=− − HHH N



• Typical zero locations shown below

1− 1

Type 2Type 1

1− 1

1− 1

Type 4Type 3

1− 1


Zero Locations of Linear-Zero Locations of Linear-Phase FIR Transfer FunctionsPhase FIR Transfer Functions• Summarizing

(1) Type 1 FIR filter: Either an even numberor no zeros at z = 1 and(2) Type 2 FIR filter: Either an even numberor no zeros at z = 1, and an odd number ofzeros at(3) Type 3 FIR filter: An odd number ofzeros at z = 1 and

1−=z

1−=z

1−=z



(4) Type 4 FIR filter: An odd number ofzeros at z = 1, and either an even number orno zeros at

• The presence of zeros at leads to thefollowing limitations on the use of theselinear-phase transfer functions for designingfrequency-selective filters

1−=z1±=z


Zero Locations of Linear-Zero Locations of Linear-Phase FIR Transfer FunctionsPhase FIR Transfer Functions• A Type 2 FIR filter cannot be used to

design a highpass filter since it always has azero

• A Type 3 FIR filter has zeros at both z = 1and , and hence cannot be used todesign either a lowpass or a highpass or abandstop filter

1−=z

1−=z


Zero Locations of Linear-Zero Locations of Linear-Phase FIR Transfer FunctionsPhase FIR Transfer Functions• A Type 4 FIR filter is not appropriate to

design a lowpass filter due to the presenceof a zero at z = 1

• Type 1 FIR filter has no such restrictionsand can be used to design almost any typeof filter


Bounded Real TransferBounded Real TransferFunctionsFunctions

• A causal stable real-coefficient transferfunction H(z) is defined as a bounded real(BR) transfer function if

• Let x[n] and y[n] denote, respectively, theinput and output of a digital filtercharacterized by a BR transfer function H(z)with and denoting theirDTFTs

)( ωjeX )( ωjeY

1|)(| ≤ωjeH for all values of ω



• Then the condition implies that

• Integrating the above from to π, andapplying Parseval’s relation we get

1|)(| ≤ωjeH22

)()( ωω ≤ jj eXeY

∑∑∞

−∞=

∞

−∞=≤

nnnxny 22 ][][

π−



• Thus, for all finite-energy inputs, the outputenergy is less than or equal to the inputenergy implying that a digital filtercharacterized by a BR transfer function canbe viewed as a passive structure

• If , then the output energy isequal to the input energy, and such a digitalfilter is therefore a lossless system

1|)(| =ωjeH



• A causal stable real-coefficient transferfunction H(z) with is thuscalled a lossless bounded real (LBR)transfer function

• The BR and LBR transfer functions are thekeys to the realization of digital filters withlow coefficient sensitivity

1|)(| =ωjeH

Linear-Phase FIR Transfer Functionssip.cua.edu/res/docs/courses/ee515/chapter04/ch4-4.pdf ·...

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