Digital Filter Structures• LTI Digital Systems are described by:-

– Input x[n] and output y[n] relationship.

– Constant coefficient difference equation.

– Impulse response h[n]. (Convolution sum)

– Frequency response,

– Transfer Function H(z).

• Realization of digital filters via convolution sum of input and impulse response is limited to finite impulse response ,FIR filters only.

• Both the IIR & FIR filters can be directly realized using any of the other forms as described above.

).( ωjeH

Structural Representation via Block

Diagram

][b][ay[n]

-:equation differencet coefficienconstant Linear )2(

k]-h[k]x[nk]-x[k]h[ny[n]

-: sumn Convolutio (1)

-:bygiven are filters digital of iprelationshoutput input domain In time

0

k

1

k knxknyM

k

N

k

kk

−+−−=

==

∑∑

∑∑

==

∞

−∞=

∞

−∞=

Block Diagram of FIR Digital Filters

][by[n]

-:equation differencet coefficienconstant Linear )2(

k]-h[k]x[ny[n]

-: sumn Convolutio From (1)

0

k

0

knxM

k

M

k

−=

=

∑

∑

=

=

1]1[ bh =0]0[ bh =

+

y[n]

x[n]D D D D D

2]2[ bh = 3]3[ bh =MbMh =][

Block Diagram of IIR Filter

][b][ay[n]

-:equation differencet coefficienconstant Linear )2(

k]-h[k]x[nk]-x[k]h[ny[n]

samples). impulse infinite of becauseimplement toal(Impractic -: sumn Convolutio (1)

0

k

1

k knxknyM

k

N

k

kk

−+−−=

==

∑∑

∑∑

==

∞

−∞=

∞

−∞=

3a− 1a−

1b0b

+

y[n]

x[n]D D D D D

2b3b Mb

2a−1−− NaNa−

D D D D D

Implementation of Digital Filters.

• Two ways of Actual Implementation.

– Via Software or computer programming.

– Or via Hardware construction

• Choice depends on application dictated by the sampling period or the Bandwidth/ Nyquist frequency.

Equivalent Structures

• Transpose Operation

– Reverse all paths

– Replace pick-off nodes by adders and vice-

versa

– Interchange input and output nodes.

Direct & Transposed Form of FIR

Digital Filters

1]1[ bh =0]0[ bh = 2]2[ bh =

+

y[n]

x[n]

3]3[ bh = MbMh =][

0]0[ bh =

yn]

x[n]

1]1[ −=− MbMhMbMh =][

+

2]2[ −=− MbMh

+ +1−

z

1−z

1−z

1−z

1−z

1−z

1−z

1−z

Drawback and Limitations of Direct

Form Implementations of Digital

Filters.• All coefficients, input/output and delayed

sequences cannot be represented in an infinite precision but limited to finite word length.

• Quantization of coefficients lead to finite word length effect.

• Finite word length effect due to the roundoff or truncation of coefficients will result in some form of error associated with the actual impulse response or frequency response of the filter required.

Drawback and Limitations of Direct

Form Implementations of Digital

Filters.• Finite word length effect due to the

roundoff or truncation of output of multipliers will also contribute to some form of noise at the output of the filters.

• Furthermore overflow due to multiplications effect on stable IIR filters may result in them becoming unstable i.e. exhibits limit cycle oscillations.

Reducing Drawback and

Limitations of Direct Form

Implementations of Digital Filters.

• Direct Form Implementations suffer from the worst effects due to finite word length.

• This leads us to other favorable forms such as cascade and parallel forms to reduce the effects of finite word length.

Cascade Form of 4th order FIR

Digital Filter.

21β11β

1−z

+

x[n]

12β+

22β

h[0]1−

z 1−z1−

z

).1(]0[][)( 2

2

1

11

0

−−

==

− ++Π==∑ zzhzkhzH kk

P

k

N

k

k ββ

y[n]

Parallel or Frequency-Sampling Structure For FIR Filters.

M

1

1−z

1−z

x[n]

1−z

1−z

2

10

2

or

ej

=

−

α

πα

Mje

/2πα

Mje

/)1(2 απ +

MMje

/)1(2 απ +−

)(αH

)1( α+H

)1( α+−MH

y[n]

+

+

+

+

∑−

=−+

−

−

+−=

1

01/)(2

2

1

)(1)(

M

kMkj

jM

ze

kH

M

ezzH

απ

πα α

Block Diagram of Direct Form I, IIR Filter

][b][ay[n]

-:equation differencet coefficienconstant Linear

0

k

1

k knxknyM

k

N

k

−+−−= ∑∑==

3a− 1a−

1b0b

+

y[n]

x[n]D D D D D

2b3b Mb

2a−1−− NaNa−

D D D D D

Block Diagram of Direct Form I, IIR Filter

N

N

M

M

kN

k

kM

k

kM

k

kN

k

kM

k

kN

k

kM

k

kN

k

M

k

N

k

zazaza

zbzbzbb

z

z

zX

zYzH

zzXzzY

zzXzzYzY

zzXzzY

knxkny

−−−

−−−

−

=

−

=

−

=

−

=

−

=

−

=

−

=

−

=

==

+++

+++=

+

==

=+

=+

+−=

−+−−=

∑

∑

∑∑

∑∑

∑∑

∑∑

.......1

.....

)a1(

b

)(

)()(

)(b)a1)((

)(b)(a)(

)(b)(aY(z)

-:equation above theof transform-z theTaking

][b][ay[n]

-:equation differencet coefficienconstant Linear

2

2

1

1

2

2

1

10

1

k

0

k

0

k

1

k

0

k

1

k

0

k

1

k

0

k

1

k

Block Diagram of Direct Form I, IIR Filter

N

N

M

M

kN

k

kM

k

zazaza

zbzbzbb

z

z

zX

zYzH

−−−

−−−

−

=

−

=

+++

+++=

+

==

∑

∑

.......1

.....

)a1(

b

)(

)()(

2

2

1

1

2

2

1

10

1

k

0

k

3a− 1a−

1b0b

+

Y(z)]

X(z)

2b3b Mb

2a−1−− NaNa−

1−z

1−z 1−

z1−

z 1−z

1−z

1−z

1−z

1−z

1−z

Example 8.3 Direct Form I, Realization

][b][ay[n]

-: tranform- Zinverse theTaking

]2.018.04.0)[(]02.0362.00.44z)[()(

]02.0362.00.44z)[(]2.018.04.01Y(z)[

2.018.04.01

02.0362.00.44z

X(z)

Y(z)

.2.018.04.01

02.0362.00.44z

2.018.04.0

02.0362.00.44zH(z)

-:asgiven isfunction transfer IIRorder A third

0

k

1

k

321321-

321-321

321

321-

321

321-

23

2

knxkny

zzzZYzzzXzY

zzzXzzz

zzz

zz

zzz

zz

zzz

z

M

k

N

k

−+−−=

−+−++=

++=−++

−++

++=

−++

++=

−++

++=

∑∑==

−−−−−

−−−−−

−−−

−−

−−−

−−

3a− 1a−

1b

+

y[n]

x[n]

2b3b

2a−

1−z

1−z

1−z

1−z

1−z

1−z

1−z

=0.44 =0.362 =0.02

=-0.4=-0.18

=0.2

Block Diagram of Direct Form 1 IIR Filter

• We can treat the z-transform of the output as the product of :-

(1) z-transform of input

(2) Non recursive part transfer function.

(3) Recursive part transfer function

}

)a1(

1}{b){()(

1

k

0

k

kN

k

kM

k z

zzXzY−

=

−

=∑

∑+

=

Block Diagram of Direct Form 1 IIR Filter

3a−

1a−1b

0bY(z)

X(z)

2b

3b

Mb

2a−

4a−

Na−

+

+

+

+

+

+

+

+

+

+

+

+

4b

1−z

1−z

1−z

1−z

1−z

1−z

1−z

1−z

1−z

1−z

Non-Recursive Part Recursive Part

Block Diagram of Direct Form 1 Transpose IIR Filter

3a−

1a−1b

0b

Y(z)

X(z)

2b

3b

Mb

2a−

4a−

Na−

+

+

+

+

+

+

+

+

+

+

+

4b

1−z

1−z

1−z

1−z

1−z

1−z

1−z

1−z

1−z

1−z

Recursive Part Non-Recursive Part

Block Diagram of Non-canonic Direct Form 1 IIR Filter

3a−

1a−1b

0by[n]

x[n]

2b

3b

Mb

2a−

4a−

Na−

+

+

+

+

+

+

+

+

+

+

+

+

4b

1−z

1−z

1−z

1−z

1−z

1−z

1−z

1−z

1−z

1−z

Block Diagram of Canonic Direct Form II, IIR Filter

3a−

1a−1b

0by[n]

x[n]

2b

3b

Mb

2a−

4a−

Na−

+

+

+

+

+

+

+

+

+

+

+

+

4b

1−z

1−z

1−z

1−z

1−z

Example 8.3 Block Diagram of Canonic Direct Form II, IIR Filter

2.03 =− a

4.01 −=− a 44.01 =b

y[n]

x[n]

362.02 =b

02.03 =b

18.02 −=− a

+

+

+

+

+

+

+

1−z

1−z

1−z

321

321-

321

321-

23

2

2.018.04.01

02.0362.00.44z

X(z)

Y(z)

.2.018.04.01

02.0362.00.44z

2.018.04.0

02.0362.00.44zH(z)

-:asgiven isfunction transfer IIRorder A third

−−−

−−

−−−

−−

−++

++=

−++

++=

−++

++=

zzz

zz

zzz

zz

zzz

z

Cascade Realizations

sections.order first for 0&..).........1

1()(

sections.order second..).........1

1()(

).s(sectionspolynomialorder secondor first are)()(

)(

)(.......

)(

)(

)(

)(

)(

)(

)(

)()(

221

1

1

1

10

2

2

1

1

2

2

1

1

10

2

3

2

2

1

1

=+

+Π=

++

++Π=

==

−

−

=

−−

−−

=

kk

k

kM

k

kk

kkM

k

kk

M

M

z

zpzH

zz

zzpzH

zDandzPWhere

zD

zP

zD

zP

zD

zP

zD

zP

zX

zYzH

αβα

β

αα

ββ

)(

)(

1

1

zD

zP

)(

)(

2

2

zD

zP

)(

)(

3

3

zD

zP

)(

)(

zD

zP

M

M

Via different poles & zeros pairing, there are other realizations.

X(z) Y(z)

Cascade Realizations

sections.order first for 0&

sections.order second..).........1

1()(

22

2

2

1

1

2

2

1

1

10

=

++

++Π=

−−

−−

=

kk

kk

kkM

k zz

zzpzH

αβ

αα

ββ

1−z

1−z

+

+ +

+

k1α−

k2α−

k1β

k2β

)(zX k

)(zYk

Example 8.4 Cascade Canonic Direct Form II, Realization

)4.01

)(5.08.01

02.0362.00.44(

2.018.04.01

02.0362.00.44z

X(z)

Y(z)

.2.018.04.01

02.0362.00.44z

2.018.04.0

02.0362.00.44zH(z)

-:asgiven isfunction transfer IIRorder A third

1

1

21

21

321

321-

321

321-

23

2

−

−

−−

−−

−−−

−−

−−−

−−

−++

++=

−++

++=

−++

++=

−++

++=

z

z

zz

zz

zzz

zz

zzz

zz

zzz

z

1−z

1−z

+

+ +

+

8.01 −=− kα

5.02 −=− kα

362.01 =kβ

02.02 =kβ

)(zX

)(zY

44.00 =kβ1−

z

+

+

4.01 =− kα

Parallel Form I Realizations of IIR Filters

pole. realfor 0 .)1

()(

)()(

poles. simple assume function, transfer theofexpansion fraction partial Via

21

1 12

2

1

1

1

1000 ==

++

++=+= ∑ ∑

= =−−

−

kk

M

k

M

k kk

kk

k

k

zz

z

zD

zNzH αγ

αα

γγγγ

)(zX

0γ

)(

)(

1

1

zD

zN

)(

)(

2

2

zD

zN

)(

)(

zD

zN

M

M

+

+

+

)(zY

Parallel Form I Realizations of IIR Filters

pole. realfor 0 .)1

()(

poles. simple assume function, transfer theofexpansion fraction partial Via

21

12

2

1

1

1

100 ==

++

++= ∑

=−−

−

kk

M

k kk

kk

zz

zzH αγ

αα

γγγ

1−z

1−z

+

+

+

k1α−

k2α−

k1γ

k0γ)(zYk

Example 8.5 Parallel Form I Realizations of IIR Filters

21

1

1-

1

1

21

21

321

321-

23

2

5.08.01

2.05.0

0.4z-1

0.6-0.1H(z)

-:expansionfraction partial Via

)4.01

)(5.08.01

02.0362.00.44(.

2.018.04.01

02.0362.00.44z

2.018.04.0

02.0362.00.44zH(z)

-:asgiven isfunction transfer IIRorder A third

−−

−

−

−

−−

−−

−−−

−−

++

−−++=

−++

++=

−++

++=

−++

++=

zz

z

z

z

zz

zz

zzz

zz

zzz

z

1−z

1−z

+

+

+

8.01 −=− kα

5.02 −=− kα 2.01 −=kγ

5.00 −=kγ

)(zY

1−z

+ +

0.4

-0.1

0.6X(z)

Other Structures.

• FIR Systems Cascaded Lattice Structures.

• Lattice and Lattice-Ladder Structures for IIR Systems.