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Transcript of LinearPhase FIR Transfer Functionssip.cua.edu/res/docs/courses/ee515/chapter04/ch44.pdf ·...
1Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
• It is nearly impossible to design a linearphase IIR transfer function
• It is always possible to design an FIRtransfer function with an exact linearphaseresponse
• Consider a causal FIR transfer function H(z)of length N+1, i.e., of order N:
∑ =−= N
nnznhzH 0 ][)(
2Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
• 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],[][
3Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
• Since the length of the impulse response canbe either even or odd, we can define fourtypes of linearphase 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
4Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
Type 1: N = 8 Type 2: N = 7
Type 3: N = 8 Type 4: N = 7
5Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
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 ][][][][][
6Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
• 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 +++++ −−
7Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
• 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+
8Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
• 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
9Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
• In the general case for Type 1 FIR filters,the frequency response is of the form
where the amplitude response , alsocalled the zerophase response, is of theform
)()( 2/ ω= ω−ω HeeH jNj ~
)(ωH~
)(ωH~∑ ω−+==
2/
1 22 )cos(][2][N
n
NN nnhh
10Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
• Example  Consider
which is seen to be a slightly modifiedversion of a length7 movingaverage FIRfilter
• The above transfer function has asymmetric impulse response and therefore alinear phase response
][)( 62154321
21
61
0−−−−−− ++++++= zzzzzzzH
11Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
• A plot of the magnitude response ofalong with that of the 7point movingaverage 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 filtermovingaverage
12Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
• Note the improved magnitude responseobtained by simply changing the first and thelast impulse response coefficients of amovingaverage (MA) filter
• It can be shown that we an express
which is seen to be a cascade of a 2point MAfilter with a 6point MA filter
• Thus, has a double zero at , i.e.,(ω = π)
)()()( 54321611
21
0 11 −−−−−− +++++⋅+= zzzzzzzH
1−=z)(zH0
13Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
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 ][][][][
14Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
• 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
15Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
• 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
16Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
• 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
17Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
• 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 ~
18Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
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
19Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
• The corresponding frequency response isgiven by
• 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)(
20Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
• 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
21Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
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
22Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
• The corresponding frequency response isgiven by
• 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)(
23Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
• 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
24Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
General Form of Frequency Response• In each of the four types of linearphase FIR
filters, the frequency response is of the form
• The amplitude response for each ofthe four types of linearphase FIR filters canbecome negative over certain frequencyranges, typically in the stopband
)()( 2/ ω= βω−ω HeeeH jjNj ~
~)(ωH
25Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
• The magnitude and phase responses of thelinearphase 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
26Copyright © 2001, S. K. Mitra
LinearPhase FIR TransferLinearPhase FIR TransferFunctionsFunctions
• 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 zerophase filter
• Such a filter must have a noncausal impulseresponse
)( ωjeH
27Copyright © 2001, S. K. Mitra
Zero Locations of LinearZero Locations of LinearPhase 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
][][][
28Copyright © 2001, S. K. Mitra
Zero Locations of LinearZero Locations of LinearPhase FIR Transfer FunctionsPhase FIR Transfer Functions• But,
• Hence for an FIR filter with a symmetricimpulse response of length N+1 we have
• A realcoefficient polynomial H(z)satisfying the above condition is called amirrorimage polynomial (MIP)
)()( 1−−= zHzzH N
)(][ 10
−= =∑ zHzmhN
mm
29Copyright © 2001, S. K. Mitra
Zero Locations of LinearZero Locations of LinearPhase 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 −=
30Copyright © 2001, S. K. Mitra
Zero Locations of LinearZero Locations of LinearPhase FIR Transfer FunctionsPhase FIR Transfer Functions• Hence, the transfer function H(z) of an FIR
filter with an antisymmetric impulseresponse satisfies the condition
• A realcoefficient polynomial H(z)satisfying the above condition is called aantimirrorimage polynomial (AIP)
)()( 1−−−= zHzzH N
31Copyright © 2001, S. K. Mitra
Zero Locations of LinearZero Locations of LinearPhase 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=
32Copyright © 2001, S. K. Mitra
Zero Locations of LinearZero Locations of LinearPhase 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 ±=
33Copyright © 2001, S. K. Mitra
Zero Locations of LinearZero Locations of LinearPhase 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
34Copyright © 2001, S. K. Mitra
Zero Locations of LinearZero Locations of LinearPhase 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
35Copyright © 2001, S. K. Mitra
Zero Locations of LinearZero Locations of LinearPhase FIR Transfer FunctionsPhase FIR Transfer Functions
• Typical zero locations shown below
1− 1
Type 2Type 1
1− 1
1− 1
Type 4Type 3
1− 1
36Copyright © 2001, S. K. Mitra
Zero Locations of LinearZero Locations of LinearPhase 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
37Copyright © 2001, S. K. Mitra
Zero Locations of LinearZero Locations of LinearPhase FIR Transfer FunctionsPhase FIR Transfer Functions
(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 theselinearphase transfer functions for designingfrequencyselective filters
1−=z1±=z
38Copyright © 2001, S. K. Mitra
Zero Locations of LinearZero Locations of LinearPhase 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
39Copyright © 2001, S. K. Mitra
Zero Locations of LinearZero Locations of LinearPhase 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
40Copyright © 2001, S. K. Mitra
Bounded Real TransferBounded Real TransferFunctionsFunctions
• A causal stable realcoefficient 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
41Copyright © 2001, S. K. Mitra
Bounded Real TransferBounded Real TransferFunctionsFunctions
• 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−
42Copyright © 2001, S. K. Mitra
Bounded Real TransferBounded Real TransferFunctionsFunctions
• Thus, for all finiteenergy 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
43Copyright © 2001, S. K. Mitra
Bounded Real TransferBounded Real TransferFunctionsFunctions
• A causal stable realcoefficient 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