Digital
g
Filter Structures
• Isitpracticaltousetheconvolutional sumtoimplementLTIsystems?
f
y [n ] ¦ k f h [k ] x [n k ]
• FIR:Yes. y [n ]
¦
f
k f
h [k ] x [n k ]
• IIR:No itisnotpracticalbecausetheimpulseresponseisofinfinitelength.
Butitcanbeimplementedbydirectcomputationofdifferenceequation
y[n] ¦kN 1 d k y[n k ] ¦kM 0 pk x[n k ]
•
However,theabove
However
directimplementationmethodsmaynot provide
satisfactoryperformanceduetothefiniteprecisionarithmeticofpractical
processors
•
Itisthusofpracticalinteresttodevelopalternaterealizationsandchoosethe
structure that provides satisfactory performance under finite precision
structurethatprovidessatisfactoryperformanceunderfiniteprecision
arithmetic
Block Diagram
g
Representation
p
• FortheimplementationofanLTIdigitalfilter,theinputoutput
relationshipmustbedescribedbyavalidcomputationalalgorithm
• Toillustratewhatwemeanbyacomputationalalgorithm,considerthe
causalfirstorderLTIdigitalfiltershownbelow
• Thefilterisdescribedbythediff.eqn.:
y[n] d1 y[n 1] p0 x[ n] p1x[n 1]
• Usingtheaboveequationwecancomputey[n]
if h i i i l
iftheinitialcondition(y[1])isknown
di i ( [ 1]) i k
y[0] d1 y[1] p0 x[0] p1x[1]
•
y[1] d1 y[0] p0 x[1] p1x[0]
y[2] d1 y[1] p0 x[2] p1x[1]
..
.
•
Each step of the calculation requires a knowledge of
the previously calculated value of the output sample
(delayed value of the output), the present value of
the input sample, and the previous value of the input
sample (delayed value of the input)
As a result, the firstorder difference equation
q
can be
interpreted as a valid computational algorithm
Basic Building Blocks
• ThecomputationalalgorithmofanLTIdigitalfiltercanbe
convenientlyrepresentedinblockdiagramformusingthebasic
building blocks shown below
buildingblocksshownbelow
x[n]
A
y[n]
x[n]
[ ]
w[n]
Multiplier
Adder
z 1
x[n]
y[n]]
y[
x[n]
x[n]
y[n]
Unit delay
x[n]
Pick-off node
Advantagesofblockdiagramrepresentation
g
g
p
(1)Easytowritedownthecomputationalalgorithmbyinspection
(2)Easytoanalyzetheblockdiagramtodeterminetheexplicitrelationbetweentheoutputandinput
(3)Simplifiesderivingother“equivalent”blockdiagramsyieldingdifferentcomputationalalgorithms
(4)Easytodeterminethehardwarerequirements
(5)Easiertodevelopblockdiagramrepresentationsfromthetransferfunctiondirectly
Analysis
y
of Block Diagrams
g
• Example Considerthesingleloopfeedbackstructureshownbelow
• TheoutputE(z)oftheadderis
E( z)
X ( z ) G2 ( z )Y ( z )
• Butfromthefigure Y ( z ) G1 ( z ) E ( z )
• EliminatingE(z)fromtheprevioustwoequationswearriveat
[1 G1 ( z )G2 ( z )]Y ( z ) G1 ( z ) X ( z )
H ( z)
Y ( z)
G1 ( z )
X ( z ) 1 G1 ( z )G2 ( z )
Th DelayThe
Delay
D l -F
Free Loop
L
P
Problem
bl
• For
Forphysicalrealizability
physical realizability ofthedigitalfilterstructure,itisnecessary
of the digital filter structure it is necessary
thattheblockdiagramrepresentationcontainsnodelayfreeloops
• Toillustratethedelayfreeloopproblemconsiderthestructurebelow
• Analysisofthisstructureyields
u[n] w[n] y[n]
y[ n] B (v[n] Au[n])
y[n] B v[n] A( w[n] y[n])
Todeterminethecurrentvalueofy[n]
youneedtoknowthevalueofy[n]!
Thisisphysicallyimpossible
p y
y p
toachieve
Canonic and Noncanonic Structures
• Adigitalfilterstructureissaidtobecanonic ifthenumberofdelaysin
the block diagram representation = theorder
theblockdiagramrepresentation
the order ofthetransferfunction
of the transfer function
• Otherwise,itisanoncanonic structure
• Example
y[ n] d1 y[n 1] p0 x[n] p1x[ n 1]
No.ofDelaysintheblockdiagram=2
Orderofthetransferfunction=1
Î Noncanonic
Equivalent Structures
•
•
Twodigitalfilterstructuresaredefinedtobeequivalent iftheyhavethesametransfer
function
AsimplewaytogenerateequivalentstructuresisusingTransposeOperation
p
y g
q
g
p
p
TransposeOperation
(1)Reverseallpaths
(2)Replacepickoffnodesbyadders,andviceversa
(3)Interchangetheinputandoutputnodes
Equivalent Structures
TransposeOperation
Transpose
Operation
(1)Reverseallpaths
(2)Replacepickoffnodesbyadders,andviceversa
(3)Interchangetheinputandoutputnodes
Equivalent Structures
• Question:Whystudydifferentstructures althoughtheyhavethesame
transferfunction?
•
•
Assuminginfiniteprecisionarithmetic,anygivenrealizationofadigital
filterbehavesidenticallytoanyotherequivalentstructure.
, p
,
g limitations,aspecific
, p
However,inpractice,duetothefinitewordlength
realizationbehavestotallydifferentlyfromitsotherequivalentrealizations
• Thereareliterallyaninfinitenumberofequivalentstructures
realizingthesametransferfunction
• Itisthusimpossibletodevelopallequivalentrealizations.
• Onewaytofindagoodstructureistodeterminealargenumberof
equivalent structures analyze the finite wordlength effectsineach
equivalentstructures,analyzethefinitewordlength
effects in each
case,andselecttheoneshowingtheleast quantization effects
• Inthiscoursewerestrictourattentiontoadiscussionofsome
commonlyusedstructures.
Basic FIR Digital Filter Structures
• AcausalFIRfilteroforderN ischaracterizedbyatransferfunctionH(z)
givenby
b
N
n
H ( z)
¦n 0 h[n]z
• In
Inthetime
the timedomain
domaintheinput
the inputoutput
outputrelationoftheaboveFIRfilteris
relation of the above FIR filter is
givenby
N
y[n]
¦k
0 h[k ]x[n k ]
• AnFIRfilteroforderN ischaracterizedbyN+1coefficientsand,in
g
general,require
q
N+1
multipliersand
N
adders
• Structuresinwhichthemultipliercoefficientsareequaltothe
coefficientsofthetransferfunctionarecalleddirectform structures
Direct Form FIR Digital Filter Structures
• AdirectformrealizationofanFIRfiltercanbereadilydevelopedfrom
theconvolutionsumdescriptionasindicatedbelowforN =4
• Ananalysisofthisstructureyields
y [n ] h [0]x [n ] h [1]x [n 1] h [2]x [n 2] h [3]x [n 3] h [4]x [n 4]
whichispreciselyoftheformoftheconvolutionsum description
• Thedirectformstructureisalsoknownasatapped delayline ora
transversalfilter
Direct Form FIR Digital
g
Filter
Structures
• Thetranspose ofthedirectformstructure
shownearlierisindicatedbelow
• Both
Bothdirectformstructuresarecanonic
direct form structures are canonic
withrespecttodelays
Cascade Form FIR Digital Filter
Structures
• AhigherorderFIRtransferfunctioncan
also be realized as a cascade of second
alsoberealizedasacascadeofsecond
orderFIRsectionsandpossiblyafirstorder
section
• WeexpressH(z)as
H ( z ) h[0]–kK 1(1 E1k z 1 E 2 k z 2 )
where K N if N iseven,andifNis
whereifN
is even and K N21 if N is
2
odd,with E 2 K 0
Cascade Form FIR Digital
Filter Structures
• AcascaderealizationforN =6isshownbelow
• Eachsecondordersectionintheabovestructure
canalsoberealizedinthetransposeddirectform
Polyphase FIR Structures
• Example:consideracausalFIRtransferfunctionH(z)withN =8:
H (z )
h [0] h [1]z 1 h [2]z 2 h [3]z 3 h [4]z 4
h [5]z 5 h [6]z 6 h [7]z 7 h [8]z 8
• H(z)canbeexpressedasasumoftwoterms,withonetermcontainingthe
evenindexedcoefficientsandtheothercontainingtheoddindexedcoeff.:
H (z ) (h [0] h [2]z 2 h [4]z 4 h [6]z 6 h [8]z 8 ) (h [1]z 1 h [3]z 3 h [5]z 5 h [7]z 7 )
H (z ) (h [0] h [2]z 2 h [4]z 4 h [6]z 6 h [8]z 8 ) z 1 (h [1] h [3]z 2 h [5]z 4 h [7]z 6 )
H ( z)
E0 ( z 2 ) z 1E1( z 2 )
Polyphase
Decomposition
Polyphase FIR Structures
• Inasimilarmanner,bygroupingthetermsintheoriginalexpression
forH(z),wecanreexpressitintheform
H ( z ) E0 ( z 3 ) z 1E1( z 3 ) z 2 E2 ( z 3 )
E0 ( z ) h[0] h[3]z 1 h[6]z 2
E1( z ) h[1] h[4]z 1 h[7]z 2
E2 ( z ) h[2] h[5]z 1 h[8]z 2
• The
Thesubfilters
subfilters (theE
(the E filters) inthepolyphase
in the polyphase realizationofanFIR
realization of an FIR
transferfunctionarealsoFIRfiltersandcanberealizedusingany
methodsdescribedsofar
• Theabovepolyphase realizationsarenoncanonic.Toobtaina
canonicrealizationoftheoverallstructure,thedelaysinallsubfilters
mustbeshared
Linear--Phase FIR Structures
Linear
• Thesymmetry(orantisymmetry)propertyofalinearphaseFIRfiltercan
beexploitedtoreducethenumberofmultipliersintoalmosthalfofthat
inthedirectformimplementations
• Consideralength7TypeI FIRtransferfunctionwithasymmetricimpulse
response:
H ( z ) h[0] h[1]z 1 h[2]z 2 h[3]z 3
h[2]z 4 h[1]z 5 h[0]z 6
• RewritingH(z)intheform
H ( z ) h[0](1 z 6 ) h[1]( z 1 z 5 )
h[2]( z 2 z 4 ) h[3]z 3
weobtainthelinearphaserealization
• Inthisexample:
4
• No.ofMultipliersofLinearPhaserealization=
N
f M lti li
f Li
Ph
li ti
• No.ofMultipliersofDirectFormrealization= 7
Linear--Phase FIR Structures
Linear
• A similar decomposition can be applied to a
Type 2 FIR transfer function
• For example,
example a length
length-8
8 Type 2 FIR transfer
function can be expressed as
H ( z ) h[0](1 z 7 ) h[1]( z 1 z 6 )
h[2]( z 2 z 5 ) h[3]( z 3 z 4 )
• The corresponding realization is shown on
th nextt slide
the
lid
Copyright © 2001, S. K. Mitra
Linear--Phase FIR Structures
Linear
• Note: The Type 1 linear-phase structure for
a length-7
length 7 FIR filter requires 4 multipliers,
multipliers
whereas a direct form realization requires 7
multipliers
lti li
Copyright © 2001, S. K. Mitra
Basic IIR Digital Filter Structures
•
•
•
•
•
ThecausalIIRdigitalfiltersweareconcernedwithinthiscourseare
characterizedbyarealrationaltransferfunctionofz1 or,equivalentlybya
constant coefficient difference equation
constantcoefficientdifferenceequation
Fromthedifferenceequationrepresentation,itcanbeseenthatthe
realizationofthecausalIIR digitalfiltersrequiressomeformoffeedback
An Nth
AnN
th orderIIRdigitaltransferfunctionischaracterizedby2N+1unique
order IIR digital transfer function is characterized by 2N+1 unique
coefficients,andingeneral,requires2N+1multipliersand2N twoinputadders
forimplementation
Direct form IIR filters: Filter structures in which the multiplier coefficients are
DirectformIIRfilters:Filterstructuresinwhichthemultipliercoefficientsare
equaltothecoefficientsofthetransferfunction
Example:Considerforsimplicitya3rdorderIIRfilterwithatransferfunction.
p
p y
(seenextslide)
H ( z)
P( z )
D( z )
p0 p1z 1 p2 z 2 p3 z 3
1 d1z 1 d 2 z 2 d3 z 3
Direct Form IIR Digital Filter Structures
H ( z)
P( z )
D( z )
p0 p1z 1 p2 z 2 p3 z 3
1 d1z 1 d 2 z 2 d3 z 3
Direct Form I
Note:ThedirectformIstructureis
noncanonic
i asitemploys6delaysto
it
l
6d l
t
realizea3rdordertransferfunction
Transposed Direct Form I
Direct Form IIR Digital Filter Structures
• Variousothernoncanonic directformstructurescanbederivedby
simpleblockdiagrammanipulationsasshownbelow
Sharingofalldelays
reduces the total number
reducesthetotalnumber
ofdelaysto3resultingina
canonicrealization
Direct Form IIR Digital Filter Structures
DirectFormII
TransposedDirectFormII
Cascade Form IIR Digital Filter Structures
• Byexpressingthenumeratorandthedenominatorpolynomialsofthe
transferfunctionasaproductofpolynomialsoflowerdegree,a
digital filter can be realized as a cascade of loworder
digitalfiltercanberealizedasacascadeoflow
orderfiltersections
filter sections
• Consider,forexample,H(z)=P(z)/D(z)expressedas
H (z )
P (z )
D (z )
P1 (z )P2 (z )P3 (z )
D1 ( z ) D 2 ( z ) D 3 ( z )
• Examplesofcascaderealizationsobtainedbydifferentpolezero
pairingsordifferentorderingofsectionsareshownbelow
• Thereare36differentstructures.Duetofinitewordlength effects,
each such cascade realization behaves differently from others
eachsuchcascaderealizationbehavesdifferentlyfromothers
Cascade Form IIR Digital Filter Structures
• Usually,thepolynomialsarefactoredintoaproductof1st order and
2nd order polynomials:
§ 1 E1k z 1 E 2 k z 2 ·
¸
H ( z ) p0 – ¨¨
1
2 ¸
1
D
D
z
z
k ©
¹
1k
2k
Intheabove,forafirstorderfactor D 2 k E 2 k 0
• Question:Whyprefer1st orderand2nd ordersections?
• It’seasiertocheckandensurestability
1st Order
2nd Order
Sectiontransferfunc.
"
1 cz 1
"
1 az 1 bz 2
Tobestableandcausal
c 1
a 2 AND a 1 b
Cascade Form IIR Digital Filter Structures
• Example Considerthe3rdordertransferfunction
H ( z)
1 E11z 1 ·§ 1 E12 z 1 E 22 z 2 ·
§
p0 ¨
¨
¸
1 ¸¨
1
2 ¸
D
z
1
1
D
z
D
z
© 11 ¹©
¹
12
22
• Onepossiblerealizationisshownbelow
Cascade Form IIR Digital Filter Structures
• Example DirectformIIandcascadeformrealizationsof
0.44 z 1 0.362 z 2 0.02 z 3
H ( z)
1 0.4 z 1 0.18 z 2 0.2 z 3
§ 0.44 0.362 z 1 0.02 z 2 ·§ z 1 ·
¨
¸¨
1
2
1 ¸
© 1 0.8 z 0.5 z
¹© 10.4 z ¹
DirectformII
Cascadeformofa1
Cascade
formofa1st orderand2nd ordersections
Parallel Form IIR Digital Filter Structures
• Apartialfractionexpansionofthetransferfunctioninz1 leadstothe
parallelform structure
• Assumingsimplepoles,thetransferfunctionH(z)canbeexpressedas
J 0 k J 1k z 1 ·
§
H ( z) J 0 ¦ ¨
1
2 ¸
z
z
1
D
D
¹
©
k
k
1
2
k
• Intheaboveforarealpole
D 2k
J 1k
0
P ll l f
Parallelform
Parallel Form IIR Digital Filter Structures
• Example – FindaparallelformIrealization:
H ( z)
0.44 z 1 0.362 z 2 0.02 z 3
1 0.4 z 1 0.18 z 2 0.2 z 3
H ( z ) 0.1 0.6
1 0.4 z
1
0.5 0.2 z 1
1 0.8 z 1 0.5 z 2
Using MATLAB - FIR
• Note: An FIR transfer function can be
treated as an IIR transfer function with a
constant numerator of unityy and a
denominator which is the polynomial
describing the FIR transfer function
Copyright © 2001, S.
K. Mitra
Using MATLAB – Cascade Form
• The cascade form requires the factorization of the transfer function
which can be developed using the M-file zp2sos
• The
Th statement sos = zp2sos(z,p,k)
(
) generates a matrix
i sos
containing the coefficients of each 2nd-order section of the equivalent
transfer function H(z) determined from its pole-zero form
• sos is an Lx6 matrix of the form
sos
ª p 01
« p 02
«
«p
¬ 0L
p 11
p 12
p 21
p 22
d 01
d 02
d 11
d 12
d
d
21
p 1L
p 2L
d 0L
d 1L
d
2L
22
º
»
»
»
¼
• L denotes the number of sections
given byy
• The form of the overall transfer function is g
p0i p1i z 1 p2i z 2
H ( z) – Hi ( z) –
1
2
i 1
i 1 d 0i d1i z d 2i z
L
L
Using MATLAB – Parallel Form
• Parallel forms can be developed using the functions
residuez to obtain the partial fraction expansion