LN10 Imlementation Structures v3

7/25/2019 LN10 Imlementation Structures v3

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ilolu 1

IMPLEMENTATION STRUCTURES FOR DISCRETE-TIME SYSTEMS

FINITE PRECISION NUMERICAL EFFECTS-NUMBER REPRESENTATIONS

QUANTIZATION IN IMPLEMENTING SYSTEMS

REALIZABLE POLE LOCATIONS

DIRECT FORM-I, DIRECT FORM-II

CASCADE FORM

PARALLEL FORMS

TRANSPOSED FORMS

FIR STRUCTURES

GENERALIZED LINEAR PHASE FIR STRUCTURES

DETERMINATION OF THE SYSTEM FUNCTION FROM A FLOW GRAPH


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ilolu 2

Although two structures may be equivalent with regard to their input-output

characteristics for infinite precision representations of coefficients and

variables, they may have vastly different behavior when the numerical precision

is limited.

Oppenheim, Schafer, 3rded., p. 403


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ilolu 3

FINITE PRECISION NUMERICAL EFFECTS

NUMBER REPRESENTATIONS

A real number in twos complement form (infinite precision)

( + 2= )

: arbitrary scale factor

0 0 1 0


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ilolu 4

Quantized form ( +1bits, finite precision )

( + 2

= )

Quantization step size,

2


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ilolu 5

THE ROLE OFIn A/D conversion

[ , ] volts 1 1 binary numbersEx: A 14 bit A/D converter is specified to have a dynamic range of 5volts.Assuming uniform quantization what are the values of 14 binary bits when its

input is 3.111 Volt?

Solution:

5 1 3 2 5 23.111

5097.1

5 0 9 7 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 0

1

0

MATLAB code to check

x = 5*(2^12+2^9+2^8+2^7+2^6+2^5+2^3+2^0)*2^-13d = 5*2^-13(3.111-x)/d

Result

x = 3.110961914062500

d = 6.103515625000000e-04

ans = 0.062400000000343


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ilolu 6

In fixed-point arithmetic, it is common to assume that each binary number has a

scale factor of

2

For example

2 .

In floating-point arithmetic,


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ilolu 7

QUANTIZATION IN IMPLEMENTING SYSTEMS


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ilolu 8

REALIZABLE POLE LOCATIONS


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ilolu 9

{0,18 ,

28 ,

38 ,

48 ,

58 ,

68 ,

78}

0 , 18 , 12 , 38 , 12 , 58 , 32 , 78 cos {0, 18 , 28 , 38 , 48 , 58 , 68 , 78}


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ilolu 10


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ilolu 11


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ilolu 12


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ilolu 13


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ilolu 14

DIRECT FORM-I, DIRECT FORM-II

N

k

k

k

M

k

k

k

N

N

M

M

za

zb

zazaza

zbzbzbbzH

0

0

2

2

1

1

2

2

1

10

...1

...

with 10 a ,


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ilolu 15

Direct Form - I

0b

1b

2b

1Nb

Nb

1a

2a

1

Na

Na

1z

1

z

1z

1z

1

z

1z

nx ny nv

Direct Form - II

2a

1

Na

Na

nx ny0b

1b

2

b

1Nb

Nb

1z

1z

1z

nw

1 a


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ilolu 16

Ex:

21

21

125.075.01

21

zz

zzzH

Direct Form - I

Direct Form - II

2 75.0

125.0

1z

1

z1

z

1z

nx ny

75.0

125.0

nx ny

2

1z

1z

1a


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ilolu 17

CASCADE FORM

SN

k kk

kkk

N

k

kk

N

k

k

M

k

kk

M

k

k

N

N

MM

zaza

zbzbb

zdzdzc

zgzgzf

A

zazaza

zbzbzbbzH

1

2

2

1

1

2

2

1

10

1

11

1

1

1

11

1

1

2

2

1

1

2

2

1

10

1

111

111

...1

...

21

21


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ilolu 18

Ex: Cascade form of a 6thorder system.

2ndorder subsystems have Direct Form-II realizations.

11a

21a

nx ny01b

11b

21b

1z

1

z

nw1

12a

22a

02b

12b

22b

1z

1

z

nw2

13a

23a

03b

13b

23b

1z

1

z

nw3

ny1 ny2 ny3


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ilolu 19

Ex: Cascade form of a 2ndorder system.

11

11

21

21

25.015.01

11

125.075.01

21

zz

zz

zz

zzzH

nx ny

5.0

1z

1z

1z

1z

25.0


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ilolu 20

PARALLEL FORMS

21

111

1

1

1

0

1

11

1

1

N

kkk

kk

N

k k

k

N

k

k

zdzd

zeB

zc

AzCzH

P

21 2NNNNM

P


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ilolu 21

11a

21a

nx ny

01e

11e

1z

1z

nw1

ny1

12a

22a

02e

12e

1z

1z

nw2

ny2

13a

23a

03e

13e

1

z

1z

nw3

ny3

0C


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ilolu 22

Ex:

21

1

21

21

125.075.01

878

125.075.01

21

zz

z

zz

zz

zH

nx ny

75.0

125.0

7

8

1z

1z

8


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ilolu 23

Ex:

11

21

21

25.01

25

5.01

188

125.075.01

21

zz

zz

zzzH

nx

ny

5.0

18

1z

8

25.0

25

1z


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ilolu 24

TRANSPOSED FORMS

For a single input, single output (SISO) linear flow graph: Reverse all branch

directions, interchange the input and output node assignments, keep

transmittences the same, then the system function remains unchanged


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ilolu 25

Ex:

21

21

125.075.01

21

zz

zzzH

Direct Form II

Transposed Direct Form II

Transposed Direct Form II, redrawn.

75.0

125.0

nx ny

2

1z

1z

nx ny

75.0

125.0

2

1z

1z

nx ny

75.0

125.0

2

1z

1z


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ilolu 26

FIR STRUCTURES

M

k

knxkhny0

Direct Form

Transposed Direct Form

nx

ny

0h

1z

1z

1z

1z

1h 2h 1Mh Mh

nx

ny

Mh

1z

1z

1z

1z

1Mh 2Mh 1h 0h


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ilolu 27

GENERALIZED LINEAR PHASE FIR STRUCTURES

Odd Length Filters (Type-I and Type-III)

: even (filter order)

Note that 02

Mh for Type-III filters!

-1 multiplications in parentheses are for Type-III (odd symmetry) filters!

1z

0h 1h 2h 12

Mh

2

Mh

nx

ny

1z

1z

1z

1z

1z

1z

1z

1 1 1 1


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ilolu 28

EVEN LENGTH FILTERS (TYPE-II AND TYPE-IV)

: odd (filter order)

-1 multiplications in parentheses are for Type-IV (odd symmetry) filters!

1z

0h 1h 2h 2

1Mh

nx

ny

1z

1z

1z

1

z1

z1

z

1 1 1 1


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ilolu 29

DETERMINATION OF THE SYSTEM FUNCTION FROM A FLOW GRAPH

nxnwnw 41

zXzWzW 41 (a)

zWzW12

(b)

zXzWzW 23 (c)

zWzzW3

1

4

(d)

zWzWzY 42 (e)

a b zXzWzW 42 (f)

cd zXzWzzW 21

4 (g)

f,g

zX

z

zzW

1

1

2 1

1

(h)

f,g

zXz

zzW

1

1

4

1

1

(i)

h,ie

zXzz

zXz

zzzY

1

1

1

11

1

1

11

1z

1 nw1

nw3

nw2

nw4

nx ny


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ilolu 30

Ex: a) Given the following flow graph of an LTI filter, determine its transfer

function H(z).

b)Plot the Direct Form II structure for the filter H1(z)=(1-2z-1)H(z), where H(z) is

the filter in part-a.

x[n] y[n]

z-1z-1

-1

-0.5

-0.5


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Ex: Consider the following system function with real valued coefficients

22

11

2112

1)(

zbzb

zzbbzH

a)Find and plot the direct form II structure for H(z). Determine the number

of multiplications, additions and delay terms.

b)

Find and plot the signal flow graph of a new filter structure such that thereare two multiplications only. You can have more delay terms than those in

part a. (multiplication by 1 or -1 does not count).

a) num. of multiplications=4

num. of additions=4

num. of delay terms = 2

b)

x[n]

y[n]z-1

z-1

-b1

-b2

b1

b2

x[n]y[n]

z-1

z-1

-b1

-1

-1

b2

z-2

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LN10 Imlementation Structures v3