Signal Flow Graphs
Linear Time Invariant Discrete Time Systems
can be made up from the elements
{ Storage, Scaling, Summation }
• Storage: (Delay, Register) T or z
-1
xk
• Scaling: (Weight, Product, Multiplier
xk-1
A
yk
xk
or
xk
yk
A
1
yk = A.xk
Professor A G Constantinides
Signal Flow Graphs
• Summation: (Adder, Accumulator)
•
+
X
X+Y
+
Y
• 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).
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Professor A G Constantinides
Signal Flow Graphs
• For example
yk  a1 yk 1  a2 yk 2  bxk
xk
b
yk
z-1
yk-1
a1
a2
3
yk-2
Professor A G Constantinides
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)
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Professor A G Constantinides
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.
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Professor A G Constantinides
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
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Professor A G Constantinides
Signal Flow Graphs
Direct form 1 : Consider the transfer function
n
i
a
.
z
 i
Y ( z)
H ( z) 
 i 0m
X ( z ) 1   b . z i
• So that
• Set
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i 1
i
m
n


i 
Y ( z ). 1   bi .z  X ( z ).  ai .z i 
 i 1

i 0

n

W ( z )  X ( z ).  a1.z i 
i 0

Professor A G Constantinides
Signal Flow Graphs
• For which
a0
z-1
a1
z-1
z-1
a2
an
+
+
• Moreover
n delays
+
+
W(z)
m

Y ( z )  W ( z )   bi .z i .Y ( z )
i 1

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Professor A G Constantinides
Signal Flow Graphs
• For which
W(z)
Y(z)
+ +
-
z-1
b1
b2
z-1
z-1
b3
z-1
9
bm
m delays
Professor A G Constantinides
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
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Professor A G Constantinides
Signal Flow Graphs
• Canonical form: Let H ( z )  H1 ( z ).H 2 ( z )
W ( z)
1
H1 ( z ) 

m
X ( z) 1  
b . z i
• ie
i 1
Y ( z) n
H 2 ( z) 
  ai .z i
W ( z ) i 0
i
m

W ( z )  X ( z )   bi .z i .W ( z )
i 1

• and
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n

Y ( z )   ai .z i .W ( z )
i 0

Professor A G Constantinides
Signal Flow Graphs
• Hence SFG (n > m)
a0
a1
a2
X(z)
+
+
-
-
W(z)
+
+
an
+
Y(z)
+
b1
b2
bm
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Professor A G Constantinides
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 general H ( z )   H i ( z )
i
with
1
2
• or
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a0i  a1i .z  a2i .z
Hi ( z) 
1  b1i .z 1  b2i .z 2
1
a0i  a1i .z
Hi ( z) 
1  b1i .z 1
Professor A G Constantinides
Signal Flow Graphs
k
• Parallel form: Let H ( z )  g   H i ( z )
i 1
• 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
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Professor A G Constantinides
Signal Flow Graphs
• Sensitivity: Consider the effect of changing
a multiplier on the transfer function

U(z)
V(z)
2
1
X(z)
4
3
Y(z)
Linear T-I Discrete System
• Set
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V ( z )  a. X ( z )  b.U ( z )
Y ( z )  c. X ( z )  d .U ( z )
• With constraint U ( z )   .V ( z )Professor A G Constantinides
Signal Flow Graphs
a
• Hence V ( z ) 
.X ( z)
1  b.
And Y ( z )
a
 c  d . .
 G( z)
X ( z)
1  b.
thus
 G ( z ) da(1  b )  ad (b)

 
(1  b ) 2
 a  d 

.

 1  b   1  b 
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Professor A G Constantinides
Signal Flow Graphs
• Two-ports
X1(z)
Y1(z)
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X2(z)
Linear
Systems
T(z)
S
Y2(z)
Professor A G Constantinides
Signal Flow Graphs
• Example: Complex Multiplier   j 
y1  jy2  ( x1  jx2 )(  j )
x1(n)
y1(n)
M
x2(n)
y2(n)

M 

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
 
Professor A G Constantinides
Signal Flow Graphs
• So that y (n)  M
x ( n)
y1 (n)   x1 (n)   x2 (n)
y2 (n)   x1 (n)   x2 (n)
• Its SFD can be drawn as
 +
x1(n)
x2(n)
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+


-
y1(n)
+

+
+
y2(n)
Professor A G Constantinides
Signal Flow Graphs
• Special case  2   2  1
• We have a rotation of x(n) t o y (n) by an angle
1  

  tan  
 
• We can set   cos 0 so that   sin 0 and
  0
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•
•
•
•
This is the basis for designing
i) Oscillators
ii) Discrete Fourier Transforms (see later)
iii) CORDIC operators in SONARProfessor A G Constantinides
Signal Flow Graphs
• Example: Oscillator
• Consider y (n)  M x(n) and externally
impose the constraint
x ( n)  D
So that
I  M
• For oscillation
det
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D
I  M
y ( n)
y ( n)  0
D  0
Professor A G Constantinides
Signal Flow Graphs
• Set
• Hence
detI  M
 z 1
D
0
0
1 
z 
1   z 1
 z 1 
D  det 
1
1 
1 z 
 z

   z 
1
2
2 2
z     z
 1 z
 1  2
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1 2
1 2
Professor A G Constantinides
Signal Flow Graphs
• With  2   2  1 and   cos 0T , 0 the
oscillation frequency
• Set x1 (n)  cos 0nT    then
y1 (n)   cos 0nT      x2 (n)
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and x1 (n)  y1 (n  1)
• We obtain x2 (n)  sin0nT   
• Hence x1(n) and x2(n) correspond to two
sinusoidal oscillations at 90 w.r.t. each
other
Professor A G Constantinides
Signal Flow Graphs
Alternative SFG with three real multipliers
 
x1 (n)
y1 (n)
+
+
+

+
+
x2 (n)
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 (   )
+
y2 ( n )
Professor A G Constantinides