Signal Flow Graphs

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

2 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs •  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+

+

3 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs • For example

kkkk bxyayay 2211

xkb yk

a1

a2

z-1

yk-1

yk-2

4 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs • 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)

5 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs • 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.

6 Professor A G Constantinides

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

7 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs Direct form 1 : Consider the transfer function

• So that

• Set

m

i

ii

n

i

ii

zb

za

zXzYzH

1

0

.1

.

)()()(

n

i

ii

m

i

ii zazXzbzY

01.).(.1).(

n

i

izazXzW0

1.).()(

8 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs • For which

• Moreover

z-1 z-1 z-1

a0 a1 a2 an

n delays

W(z)

+ +

++

)(..)()(1

zYzbzWzYm

i

ii

9 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs • For which

W(z)++ Y(z)

- -- -

b1

z-1

z-1

b2 z-1

b3

bm

z-1

m delays

10 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs • 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

11 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs • Canonical form: Let

• ie

• and

)().()( 21 zHzHzH

m

i

ii zbzX

zWzH

1

1.1

1)()()(

n

i

ii za

zWzYzH

02 .

)()()(

)(..)()(1

zWzbzXzWm

i

ii

)(..)(0

zWzazYn

i

ii

12 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs • Hence SFG (n > m)

++X(z) Y(z)

+

- --

+++

W(z)

a0

a1

a2

an

b1

b2

bm

13 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs • Direct form 2 : Reduction in effects due to

finite register can be achieved by factoring H(z) and cascading structures corresponding to factors

• In generalwith

• or

i

i zHzH )()(

22

11

22

110

..1..)(

zbzbzazaazH

ii

iiii

11

110

.1.)(

zbzaazH

i

iii

14 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs • 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)()(

15 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs • 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

16 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs • Hence

And

thus

)(..1

)( zXbazV

)(.1

..)()( zG

badc

zXzY

2)1()()1()(

bbadbdazG

bd

ba

1.

1

17 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs • Two-ports

X1(z)

Y1(z)

X2(z)

Y2(z)

T(z)Linear

SystemsS

18 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs • Example: Complex Multiplier

x1(n)

x2(n)M

y1(n)

y2(n)

M

j

))(( 2121 jjxxjyy

19 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs • So that

• Its SFD can be drawn as

)()( nxMny

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

x1(n)

x2(n)

y1(n)

y2(n)

+

+

+

+

-

+

20 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs • 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

21 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs • Example: Oscillator• Consider and externally

impose the constraint

So that

• For oscillation

)()( nxMny

)()( nyDnx

0)( nyDMI

0det DMI

22 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs • Set

• Hence

1

1

00

zzD

11

11

11detdet

zzzzDMI

21211 zz

222121 zz

23 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs • 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)(

24 Professor A G Constantinides

Signal Flow Graphs Signal Flow Graphs Alternative SFG with three real multipliers

+

+

+

+

+

+ )(2 ny

)(1 ny)(1 nx

)(2 nx

)(