professor a g constantinides 1 signal flow graphs linear time invariant discrete time systems can be...

1 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs

Linear Time Invariant Discrete Time Systems can be made up from the elements

{ Storage, Scaling, Summation }

• Storage: (Delay, Register)

• Scaling: (Weight, Product, Multiplier

T or z-1

xk xk-1

xk

A

ykor

xk

A

yk

yk = A.xk



• Summation: (Adder, Accumulator)•

• A linear system equation of the type considered so far, can be represented in terms of an interconnection of these elements

• Conversely the system equation may be obtained from the interconnected components (structure).

X

Y

X + Y+

+



• For example

kkkk bxyayay 2211

xk

byk

a1

a2

z-1

yk-1

yk-2



• A SFG structure indicates the way through which the operations are to be carried out in an implementation.

• In a LTID system, a structure can be:

i) computable : (All loops contain delays)

ii) non-computable : (Some loops contain no delays)



• Transposition of SFG is the process of reversing the direction of flow on all transmission paths while keeping their transfer functions the same.

• This entails: – Multipliers replaced by multipliers of same value

– Adders replaced by branching points

– Branching points replaced by adders

• For a single-input / output SFG the transpose SFG has the same transfer function overall, as the original.


Structures Structures

• STRUCTURES: (The computational schemes for deriving the input / output relationships.)

• For a given transfer function there are many realisation structures.

• Each structure has different properties w.r.t.

• i) Coefficient sensitivity

• ii) Finite register computations



Direct form 1 : Consider the transfer function

• So that

• Set

m

i

ii

n

i

ii

zb

za

zXzY

zH

1

0

.1

.

)()(

)(

n

i

ii

m

i

ii zazXzbzY

01.).(.1).(

n

i

izazXzW0

1.).()(



• For which

• Moreover

z-1 z-1 z-1

a0 a1 a2 an

n delays

W(z)

+ +

++

)(..)()(1

zYzbzWzYm

i

ii



• For whichW(z)

++ Y(z)- -

- -

b1

z-1

z-1

b2 z-1

b3

bm

z-1

m delays



• This figure and the previous one can be combined by cascading to produce overall structure.

• Simple structure but NOT used extensively in practice because its performance degrades rapidly due to finite register computation effects



• Canonical form: Let

• ie

• and

)().()( 21 zHzHzH

m

i

ii zbzX

zWzH

1

1

.1

1)()(

)(

n

i

ii za

zWzY

zH0

2 .)()(

)(

)(..)()(1

zWzbzXzWm

i

ii

)(..)(0

zWzazYn

i

ii



• Hence SFG (n > m)

++X(z) Y(z)

+

- --

+++

W(z)

a0

a1

a2

an

b1

b2

bm



• Direct form 2 : Reduction in effects due to finite register can be achieved by factoring H(z) and cascading structures corresponding to factors

• In general

with

• or

i

i zHzH )()(

22

11

22

110

..1

..)(

zbzb

zazaazH

ii

iiii

11

110

.1

.)(

zb

zaazH

i

iii



• Parallel form: Let

• with Hi(z) as in cascade but a0i = 0

• With Transposition many more structures can be derived. Each will have different performance when implemented with finite precision

k

ii zHgzH

1)()(



• Sensitivity: Consider the effect of changing a multiplier on the transfer function

• Set

• With constraint

X(z)

14 3

2

V(z) U(z)

Y(z)

Linear T-I Discrete System

)(.)(.)( zUbzXazV )(.)(.)( zUdzXczY

)(.)( zVzU



• Hence

And

thus

)(..1

)( zXba

zV

)(.1

..)()(

zGba

dczXzY

2)1(

)()1()(

b

badbdazG

bd

ba

1.

1



• Two-ports

X1(z)

Y1(z)

X2(z)

Y2(z)

T(z)

LinearSystems

S



• Example: Complex Multiplier

x1(n)

x2(n)

M

y1(n)

y2(n)

M

j

))(( 2121 jjxxjyy



• So that

• Its SFD can be drawn as

)()( nxMny

)()()( 211 nxnxny )()()( 212 nxnxny

x1(n)

x2(n)

y1(n)

y2(n)

+

+

+

+

-

+



• Special case• We have a rotation of t o by an angle

• We can set so that and

• This is the basis for designing• i) Oscillators• ii) Discrete Fourier Transforms (see later) • iii) CORDIC operators in SONAR

122 )(nx )(ny

1tan

0cos

0 0sin



• Example: Oscillator

• Consider and externally impose the constraint

So that

• For oscillation

)()( nxMny

)()( nyDnx

0)( nyDMI

0det DMI



• Set

• Hence

1

1

0

0

z

zD

11

11

1

1detdet

zz

zzDMI

21211 zz

222121 zz



• With and , the oscillation frequency

• Set then

and

• We obtain

• Hence x1(n) and x2(n) correspond to two

sinusoidal oscillations at 90 w.r.t. each other

122 T0cos 0

nTnx 01 cos)(

)(cos)( 201 nxnTny

)1()( 11 nynx

nTnx 02 sin)(



Alternative SFG with three real multipliers

+

+

+

+

+

+ )(2 ny

)(1 ny)(1 nx

)(2 nx

)(



• Example: Oscillator

)()( nxMny

professor a g constantinides 1 signal flow graphs linear time invariant discrete time systems can be...

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