p. 1 dsp-ii digital signal processing ii lecture 3: filter realization marc moonen dept....

p. 1DSP-II

Digital Signal Processing II

Lecture 3: Filter Realization

Marc MoonenDept. E.E./ESAT, K.U.Leuven

[email protected]/~moonen/

DSP-II Version 2006-2007 Lecture-3 Filter Realization p. 2

PART-I : Filter Design/Realization

• Step-1 : define filter specs (pass-band, stop-band, optimization criterion,…)

• Step-2 : derive optimal transfer funcion FIR or IIR • Step-3 : filter realization (block scheme/flow graph)

direct form realizations, lattice realizations,… • Step-4 : filter implementation (software/hardware)

accuracy issues, …

question: implemented filter = designed filter ?

Lecture-2

Lecture-3

Lecture-4


Lecture 3 : Filter Realizations

• FIR Filter Realizations

• IIR Filter Realizations

PS: We will assume real-valued filter coefficients


FIR Filter Realizations

FIR Filter Realization =Construct (realize) LTI system (with delay elements,

adders and multipliers), such that I/O behavior is given by..

Several possibilities exist…1. Direct form2. Transposed direct form 3. Lattice (LPC lattice)4. Lossless lattice5. Frequency-domain realization: see Part-2

][....]1[.][.][ 10 Nkubkubkubky N



1. Direct formu[k]

u[k-4]u[k-3]u[k-2]u[k-1]

xbo

+

xb4

xb3

+

xb2

+

xb1

+y[k]

][....]1[.][.][ 10 Nkubkubkubky N



2. Transposed direct form Starting point is direct form :

`Retiming’ = select subgraph (shaded) remove delay element on all inbound arrows add delay element on all outbound arrows

u[k]

u[k-4]u[k-3]u[k-2]u[k-1]

xbo

+

xb4

xb3

+

xb2

+

xb1

+y[k]



`Retiming’ : results in...

u[k]

u[k-1]

xbo

+

xb1

+y[k]

u[k-3]u[k-2]

xb4

xb3

+

xb2

+



`Retiming’ : repeated application results in...

i.e. `transposed direct form’

u[k]

xbo

+y[k]

xb1

+

xb2

+

xb3

+

xb4

(=different software/hardware, same i/o-behavior)



3. Lattice form : Derived from combined realization of

with `flipped’ version of H(z)

Reversed (real-valued) coefficient vector results in...

i.e. - same magnitude response - different phase response

][....]1[.][.][~ :)(.)(~01

1 NkubkubkubkyzHzzH NNN

212)(...)(~).(~)(~

jjj ezezezzHzHzHzH

][....]1[.][.][ :)( 10 NkubkubkubkyzH N



Hence starting point is…

u[k] u[k-1]

u[k-2]

xb1

+

xb2

+

x

+

x

+

b3

u[k-3]

xb3

+

b2 x

+

xbo

+

y[k]

b4x

+

u[k-4]

xb4b1 x bo

y[k]~



With , this is rewritten as

20

10332

0

20222

0

3011,00

0123

32101

0

0

0123

3210

0

0

1' ,

1' ,

1' '

]1[]...[.''''''''

.0

01.

11

][]...[.''''00''''

11

][~][

bbbbbbbbbbb

Nkukubbbbbbbb

z

Nkukubbbb

bbbbkyky

T

T

TNkukubbbbbbbbbb

kyky

][]...[.][~][

01234

43210

1 , 00

40 bb

`sloppy notation’


FIR Filter RealizationsThis is equivalent to...

…Now repeat procedure for shaded graph (=same structure as the one we started from)

y[k]

u[k] u[k-3]

u[k-2]

xb’o

+

xb’1

+

x

+

x

+

b’3

u[k-2]

xb’2

+

b’2 x

+

b’3

xb’1 x b’o

+

+

x

xko

y[k]~



Repeated application results in `lattice form’

u[k]

y[k]

+

+

x

xko

+

+

x

xk1

+

+

x

xk2

+

+

x

xk3

xbo

y[k]~

= different software/hardware, same i/o-behavior



Lattice form : • Also known as `LPC Lattice’ (`linear predictive coding lattice’) Ki’s are so-called `reflection coefficients’ Every set of bi’s corresponds to a set of Ki’s, and vice versa.

• Procedure for computing Ki’s from bi’s corresponds to the well-known `Schur-Cohn’ stability test (from control theory):

problem = for a given polynomial B(z), how do we find out if all the zeros of B(z) are stable (i.e. lie inside unit circle) ? solution = from bi’s, compute reflection coefficients Ki’s (following procedure on previous slides). Zeros are stable if and only if all reflection coefficients statisfy |Ki|<1



Lattice form :

• Procedure breaks down if |Ki|=1 is encountered. Means at least one root of B(z) lies on or outside the unit circle (cfr Schur-Cohn). Then have to select other realization (direct form, lossless lattice, …) for B(z).

• Lattice form is often applied to `minimum phase’ filters, i.e. filter with only stable zeros (roots of B(z) strictly inside unit circle). Then design procedure never breaks down.



4. Lossless lattice : Derived from combined realization of (possibly scaled) H(z)

with which is such that

(*)

PS : interpretation ?… (see next slide)

][.~~...]1[.

~~][.~~][~~ :)(

~~10 NkubkubkubkyzH N

][....]1[.][.][ :)( 10 NkubkubkubkyzH N

1)(~~).(

~~)().( 11 zHzHzHzH


FIR Filter RealizationsPS : Interpretation ? When evaluated on the unit circle, this formula is equivalent to (for filters with real-valued coefficients)

i.e. and are `power complementary’ (= form a 1-input/2-output `lossless’ system, see also below)

1)(~~)(

22

j

j

ezez

zHzH

)(~~ zH)(zH

2

)(~~ jeH

2)( jeH

1


FIR Filter RealizationsPS : How is computed ?

Note that if `a’ is a root of R(z), then `1/a’ is also a root of R(z), hence

Note that ai’s can be selected such that all of them lie inside the unit circle. Then is a minimum-phase FIR filter.

This is referred to as spectral factorization, =spectral factor.

)(~~ zH

)(

11 )().(1)(~~).(

~~

zR

zHzHzHzH

iii azazzHzH ))(()(

~~).(~~ 11

iiazzH )()(

~~ 1

)(~~ zH

)(~~ zH



Hence starting point is…

u[k] u[k-1]

u[k-2]

xb1

+

xb2

+

x

+

x

+

u[k-3]

xb3

+

x

+

xbo

+

y[k]

x

+

u[k-4]

xb4

x

y[k]~

0

~~b 1

~~b 2

~~b 4

~~b3

~~b



From (*) (page 16), it follows that

Hence there exists a theta_0 such that

T

T

Nkukubbbb

bbbbz

Nkukubbbb

bbbbkyky

]1[]...[.'

~~'~~'

~~'~~

''''.

001

.cossinsincos

][]...[.'

~~'~~'

~~'~~0

0''''cossinsincos

][~~][

3210

32101

00

00

3210

3210

00

00

TNkukubbbbb

bbbbbkyky

][]...[.~~~~~~~~~~][~~][

43210

43210

4

4

0

04040 ~~~~ 0

~~.~~.

b

b

b

bbbbb

`sloppy notation’



This is equivalent to...

…Now repeat procedure for shaded graph (=same structure as the one we started from!)

(why?)

u[k] u[k-3]

u[k-2]

xb’o

+

xb’1

+

x

+

x

+

u[k-2]

xb’2

+

x

+

b’3

xx

y[k]

0cos

+

+

xx

xx

0sin

0cos

y[k]~~

0sin



Repeated application results in `lossless lattice’

u[k]

x

3sin

2cos

+

+

xx

xx

2sin

1cos

+

+

xx

xx

1sin

3cos

+

+

xx

xx

3cos2cos1cos

2sin1sin 3sin

y[k]

0cos

+

+

xx

xx

0sin

0cos

y[k]~~

0sin



Lossless lattice :

• also known as `paraunitary lattice’ (see also Lecture 6)

• each section is based on an orthogonal transformation, which preserves norm/energy/power

(= forms a 2-input/2-output `lossless’ system) hence overall structure preserves the input energy, i.e. is `lossless’ (see also next slides)

22

21

22

21

2

1

2

1 )()()()(.cossinsincos

OUTOUTININININ

OUTOUT



• State-space description Example: 2nd-order system:

- R is `realization matrix’ (A-B-C-D-matrix) for 1-input/2-output system - Orthogonality of R implies `losslessness’ (see next slide)

2

][][2

][1

2

][~~][

]1[2

]1[1

2

1

11

11

00

00

2

1

hence ,2

1 if 100010001

.

][][][

.

010cos.0sinsin.0cos

00

.

cossin00sincos0000100001

][~~][

]1[]1[

kukxkx

kykykxkx

T

R

RR

kukxkx

kykykxkx

~



• Orthogonality of R implies `losslessness’ : assume initial internal state is x1[0], x2[0] (2nd order example) input sequence is u[0],u[1],u[2],…,u[L] corresponding output sequences are y[0],y[1],… and y[0],y[1],... then orthogonality implies that ènergy’ in initial states + inputs = ènergy’ in final states + outputs

• Equivalent transfer function based property is… i.e. H(z) is èmbedded’ in 1-input/2-output lossless system (see page 17)

][~~...]0[

~~][...]0[]1[]1[

][...]1[]0[]0[]0[22222

221

22222

21

LyyLyyLxLx

Luuuxx

1)(~~)(

22

j

j

ezez

zHzH

~~~~



PS : Relevance of lattice realizations : robustness example : o = original transfer function + = transfer function after 8-bit truncation of lattice filter parameters (angles theta) - = transfer function after 8-bit truncation of direct-form coefficients (bi’s)

PS : A 2x2 orthogonal transformation (rotation) may be implemented in hardware based on a so-called `CORDIC’ (`COR’ for CO-ordinate Rotation’) architecture, which is more efficient (and more robust) than a

multiply-add based implementation


IIR Filter Realizations

Construct LTI system such that I/O behavior is given by..

Several possibilities exist… 1. Direct form 2. Transposed direct form 3. Lattice-ladder form 4. Lossless lattice 5. Parallel and cascade realization 6. State space realization

NN

NN

zazazbzbb

zAzBzH

...1...

)()()( 1

1

110



1. Direct form Starting point is… u[k]

xbo

xb4

xb3

xb2

xb1

+ +++y[k]

+ +++

x-a4

x-a3

x-a2

x-a1

)(1zA

)(zB

NN

NN

zazazbzbb

zAzBzH

...1...

)()()( 1

1

110


IIR Filter Realizationswhich is equivalent to...

PS : If all a_i=0 (hence H(z) is FIR), then this reduces to a direct form FIR

xbo

xb4

xb3

xb2

xb1

+ +++y[k]

u[k] + +++

x-a4

x-a3

x-a2

x-a1

Direct form for B(z)

)(1zA

)(zB


IIR Filter Realizations -State-space description:

-N delay elements (=minimum number =max of numerator and denominator polynomial order = max(p,q)) -2N+1 multipliers, 2N adders

u[k]

xbo

xb4

xb3

xb2

xb1

+ +++y[k]

+ +++

x-a4

x-a3

x-a2

x-a1

x1[k] x2[k] x3[k] x4[k]

][.

][][][][

.....][

][.

0001

][][][][

.

010000100001

]1[]1[]1[]1[

0

4

3

2

1

044033022011

4

3

2

13321

4

3

2

1

kub

kxkxkxkx

babbabbabbabky

ku

kxkxkxkxaaaa

kxkxkxkx



2. Transposed direct form Starting point is…

+ +++

x-a4

x-a3

x-a2

x-a1

)(1zA

y[k]

u[k]

xbo

xb4

xb3

xb2

xb1

+ +++

)(zB



which is equivalent to...

)(1zA

)(zB

x-a4

x-a3

x-a2

x-a1

y[k]

u[k]

xbo

xb4

xb3

xb2

xb1

+ +++



Transposed direct form is obtained after retiming ...

PS : If all a_i=0, then this reduces to a transposed direct form FIR

Transposeddirect form B(z)

x-a4

x-a3

x-a2

x-a1

y[k]

u[k]

xbo

xb4

xb3

xb2

xb1

+ +++

)(zB

)(1zA



- State-space description

i.e. direct forms state space matrices are `transpose’ of each other (which justifies the name `transposed dir.form’).

][.

][][][][

.0001][

][.

.

.

..

][][][][

.

000100010001

]1[]1[]1[]1[

0

4

3

2

1

044

033

022

011

4

3

2

1

4

3

2

1

4

3

2

1

kub

kxkxkxkx

ky

ku

babbabbabbab

kxkxkxkx

aaaa

kxkxkxkx

u[k]

x-a4

x-a3

x-a2

x-a1

y[k]

xbo

xb4

xb3

xb2

xb1

+ +++x1[k] x2[k] x3[k] x4[k]

T

DCBA

DCBA

formdirect TransposedformDirect



3. Lattice-ladder form Derived from combined realization of

with...

- numerator polynomial is denominator polynomial with reversed coefficient vector (see also page 9)

- hence is an `all-pass’ (=`SISO lossless’) filter :

NN

NN

zazazbzbb

zAzBzH

...1...

)()()( 1

1

110

NN

NNN

zazazzaa

zAzAzH

...1

.1...)()(~

)(~1

1

11

1)(~)(

)(~2

22

j

j

j

ez

ezez zA

zAzH

)(~ zH


IIR Filter RealizationsStarting point is…

u[k]

+ +

x xb1

+

b2 x

+

b0x b3 x b4x xa3 xa2 xa1 x1

xa1 xa2 xa3 xa4

+ + ++

+ + ++

y[k]y[k]~

a4

x[k]

-



This is equivalent to…

+ +

x x

+

x xx x x x

+ + +

x’[k-1]

x

xu[k]

b0

y[k]

y[k]~

x

+

x

+

0cos

4.411.43

aaaaa

4.412.42

aaaaa

4.413.41

aaaaa

1

4.411.01aaabb

4.412.02aaabb

4.413.03aaabb

4.414.04aaabb

+

x

x x x

+ ++ -

4.413.41

aaaaa

4.412.42

aaaaa

4.411.43

aaaaa

0sin

0sin

0cos

040sin ka

(prove it!)



Right-hand part has the same structure as what we started from, hence repeated application leads to `lattice-ladder form’

u[k]

y[k]y[k]~

x

x

x

x

+

x

+

0cos

+

x

0cos

0sin

0sin

b0 =los

sles

s se

ctio

n



Lattice-Ladder form :

• Ki’s are `reflection coefficients’• Procedure for computing Ki’s (=sin(theta_i) !!) from ai’s again

corresponds to `Schur-Cohn’ stability test (see lecture-2): all zeros of A(z) are stable (i.e. lie inside unit circle) iff all reflection coefficients statisfy |Ki|<1 (i=1,…,N-1) (ps: procedure breaks down if |Ki|=1 is encountered)• Orthogonal transformations correspond to `lossless’ sections

see next slide…

22

21

22

21

2

1

2

1 )()()()(.cossinsincos

OUTOUTININININ

OUTOUT


IIR Filter Realizations• State-space description Example: 2nd-order system:

- R is `realization matrix’ (A-B-C-D-matrix) for u[k] -> y[k] (SISO) - R is orthogonal matrix (R^T.R=I), which implies `losslessness’ , i.e.

][][][

.][

and

][][][

.1000sincos0cossin

.sincos0cossin0

001

][~]1[]1[

2

1

012

2

1

11

11

00

002

1

kukxkx

ky

kukxkx

kykxkx

R

~

1)(~ 2

jezzH



PS : `All-pass’ part (u[k]->y’[k]) is known as `Gray-Markel’ structure

PS : Note that the all-pass part corresponds to A(z) (i.e. N angles theta_i correspond to N coefficients a_i), while the ladder section corresponds to B(z). If all a_i=0 (hence H(z) is FIR !), then all theta_i=0, hence the all-pass part reduces to a delay line, and the lattice-ladder form reduces to a direct-form FIR.

x

x

x

+

0cos

+

x

0cos

0sin

0sin

in

out



4. Lossless-lattice : Derived from combined realization of

with...

- numerator polynomial is C(z) is such that

i.e. and are `power complementary’ (p.17-18)

NN

NN

zazazbzbb

zAzBzH

...1...

)()()( 1

1

110

NN

NN

zazazczcc

zAzCzH

...1

....)()()(

~~1

1

110

)(~~ zH )(zH

)().()().()().(

1)(~~).(

~~)().(111

11

zAzAzCzCzBzB

zHzHzHzH



`Lossless-lattice’ : similar derivation (but more complicated -hence skipped) leads to…

PS: Gray-Markel is a special case of this (ki’=0)

x

x

x

x

+

+

x

u[k]

y[k]y[k]~

x

x

x

+

x +

x=l

ossl

ess

sect

ion

'0

'0 sink

'0cos

'0cos

'0sin

0cos

0cos

00 sink

0sin

N~sin

N~cos



• Orthogonal transformations correspond to `lossless sections

• State-space description with orthogonal realization matrix R (try it) implies overall lossless characteristic

• Note that 2N+1 filter coefficients (a_i’s and b_I’s) define 2N+1 rotation angles. If all a_i=0 (hence H(z) is FIR), rotation angles will be such that all feedback paths are cancelled (in a magic way)

Example: (draw it/try it!)

23

22

21

23

22

21

3

2

1

3

2

1

)()()()()()(

.'cos0'sin

010'sin0'cos

.1000cossin0sincos

OUTOUTOUTINININ

INININ

OUTOUTOUT

62,

21,

31

21

21)(

~~21

21)( '

1'00

11 kkkzzHzzH

1)(~~)(

22

j

j

ezez

zHzH



5. Parallel & Cascade Realization: Parallel real. obtained from partial fraction decomposition, e.g. for simple poles:

– similar for the case of multiple poles– each term realized in, e.g., direct form– same number of delay elements, additions and multiplications (if

p=q) as direct forms– transmission zeros are realized iff signals from different sections

exactly cancel out. =Problem in finite word-length implementation

21

121

1

11

110 11)(

)()(N

i N

iN

i i

i

zzz

zc

zAzBzH

u[k]

y[k]

+

+



Cascade realization obtained from pole-zero factorization of H(z) e.g. for N even:

– similar for N odd– each section realized in, e.g., direct form– second-order sections are called `bi-quads’– same number of delay elements, additions and multiplications as

direct forms

2/

121

21

0 ..1..1.

)()()(

N

i ii

ii

zzzzb

zAzBzH

u[k]

y[k]



• Cascade realization is non-unique: - multiple ways of pairing poles and zeros - multiple ways of ordering sections in cascade

• ``Pairing procedure’’ : - pairing of little importance in high-precision (e.g. floating point implementation), but important in fixed-point implementation (with short word-lengths) - principle = pair poles and zeros to produce a frequency response for each section that is as flat as possible (i.e. ratio of max. to min. magnitude response close to unity) - obtained by pairing each pole to a zero as close to it as possible - procedure : start with pole pair nearest to the unit circle, and pair this to nearest complex zeros. remove pole-zero pair, and repeat, etc….

u[k]

y[k]


IIR Filter Realizations 6. State-space Realization :

• State-space description:

• State-space realization = realization such that all the elements of A,B,C,D are the multiplier coefficients in the structure

• Example: (transposed) direct form is NOT a state-space realization, cfr. elements (bi-bo.ai) in B or C

][.][.][][.][.]1[

kuDkxCkykuBkxAkx

+

+

CBy[k]

u[k]

D

A



• Example: bi-quad `coupled realization’ - direct form realization of bi-quad may be unsatisfactory, e.g. for short word-lengths and poles near z=1 or z=-1 (see lecture-4) - alternative realization:

state space description :

][][][

.][

][.01

][][

.]1[]1[

2

1

2

1

2

1

kukxkx

ky

kukxkx

kxkx

21

21

..1..1)(

zzzzzH

ii

iii

poles : .j

+

+ +

-

y[k]

u[k]


IIR Filter Realizations • State-space description/realization is non-unique: any non-singular matrix T transforms A,B,C,D into an alternative state-space

description/realization

with

• State-space realization with orthogonal realization matrix is referred to as `orthogonal filter’ :

Relevance : see lecture-4

][.~][~.~][

][.~][~.~]1[~

kuDkxCky

kuBkxAkx

][.][~100

..100

~~~~

1

1

kxTkx

TDCBAT

DCBA

IRRRR

DCBA

R

TT

..

p. 1 dsp-ii digital signal processing ii lecture 3: filter realization marc moonen dept....

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