Signal Flow Graphs

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```					           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                                   xk-1
• Scaling: (Weight, Product, Multiplier
A
yk
or
xk                    xk                yk
A
1                     yk = A.xk          Professor A G Constantinides
Signal Flow Graphs
•              +
X+Y
X
+

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).

2                                      Professor A G Constantinides
Signal Flow Graphs
• For example
yk  a1 yk 1  a2 yk  2  bxk
b
xk                                         yk

z-1

yk-1
a1

yk-2
a2

3                                             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)

4                                   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.

5                                            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

6                                    Professor A G Constantinides
Signal Flow Graphs
Direct form 1 : Consider the transfer function
n
 ai .z i
Y ( z)
H ( z)          i 0m
X ( z ) 1   b . z i
i
i 1
• So that             1  m b .z i   X ( z ). n a .z i 
Y ( z ).      i
 i 1                       i
i  0      
                                    
• Set                          n a . z i 
W ( z )  X ( z ).  1
i  0
            

7                                                      Professor A G Constantinides
Signal Flow Graphs
• For which
z-1      z-1                   z-1        n delays

a0      a1       a2                     an

+
+
+       +

W(z)
• Moreover
 m b .z i .Y ( z )
Y ( z)  W ( z)   i
i 1
           

8                                             Professor A G Constantinides
Signal Flow Graphs
• For which
W(z)   + +                  Y(z)
-     -
- -
z-1

b1
z-1

b2
z-1

b3

z-1

bm
m delays
9                                 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

10                                 Professor A G Constantinides
Signal Flow Graphs
• Canonical form: Let H ( z )  H1 ( z ).H 2 ( z )
W ( z)       1                       Y ( z) n
H1 ( z )             m
H 2 ( z)           ai .z i
X ( z ) 1   b . z i               W ( z ) i 0
i
i 1
• ie
 m b .z i .W ( z )
W ( z)  X ( z)   i
i 1
           


• and                n a .z i .W ( z )
Y ( z)   i
i  0
           

11                                               Professor A G Constantinides
Signal Flow Graphs
• Hence SFG (n > m)

a0
a1
a2         +           Y(z)
X(z)                                                         +
+
+
+       -
W(z)                             an
+
-       -

b1
b2

bm

12                                                                    Professor A G Constantinides
Signal Flow Graphs
• Direct form 2 : Reduction in effects due to
finite register can be achieved by factoring
corresponding to factors
• In general H ( z )   H i ( z )
i
with                      1     2
a0i  a1i .z  a2i .z
Hi ( z) 
1  b1i .z 1  b2i .z 2
• or                                 1
a0i  a1i .z
Hi ( z) 
13                           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

14                                        Professor A G Constantinides
Signal Flow Graphs
• Sensitivity: Consider the effect of changing
a multiplier on the transfer function

V(z)                  U(z)

1                                          2
4          3
X(z)                                       Y(z)

Linear T-I Discrete System

• Set          V ( z )  a. X ( z )  b.U ( z )
Y ( z )  c. X ( z )  d .U ( z )
15
• With constraint U ( z )   .V ( z ) A G Constantinides
Professor
Signal Flow Graphs
• Hence V ( z )     a
.X ( z)
1  b.
And Y ( z )                  a
 c  d . .         G( z)
X ( z)             1  b.

 G ( z ) da(1  b )  ad (b)
thus             
             (1  b ) 2
 a  d 
        .        
 1  b   1  b 
16                                          Professor A G Constantinides
Signal Flow Graphs
• Two-ports

X2(z)
X1(z)      Linear
Systems             T(z)
S
Y1(z)
Y2(z)

17                             Professor A G Constantinides
Signal Flow Graphs
• Example: Complex Multiplier   j 
y1  jy 2  ( x1  jx 2 )(  j )
x1(n)                        y1(n)
M
x2(n)                        y2(n)

       
M 
        


18                                  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)                  +      y1(n)

-

            +
x2(n)                  +      y2(n)

+

19                                       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
•   This is the basis for designing
•   i) Oscillators
•   ii) Discrete Fourier Transforms (see later)
20
•   iii) CORDIC operators in SONAR   Professor A G Constantinides
Signal Flow Graphs
• Example: Oscillator
• Consider y(n)  M x(n) and externally
impose the constraint
x(n)  D   y(n)
So that
I  M      D   y(n)  0

• For oscillation
det     I  M   D  0

21                                         Professor A G Constantinides
Signal Flow Graphs
• Set           z 1        0
D             1 
0           z 
• Hence
1   z 1   z 1 
detI  M      D   det        1         1 
 z       1 z 


 1 z          1 2
   z   1 2

 1  2         z     z
1      2  2 2

22                                              Professor A G Constantinides
Signal Flow Graphs
• With  2   2  1 and   cos 0T , 0 the
oscillation frequency
• Set x1 (n)  cos 0 nT    then
y1 (n)   cos 0 nT      x2 (n)
and x1 (n)  y1 (n  1)
• We obtain x2 (n)  sin 0 nT   
• Hence x1(n) and x2(n) correspond to two
sinusoidal oscillations at 90 w.r.t. each
other
23                                    Professor A G Constantinides
Signal Flow Graphs
Alternative SFG with three real multipliers

x1 ( n)                               y1 (n)
+

+

+

+

+

x2 ( n )                                      y2 ( n)
 (   )
+

24                                            Professor A G Constantinides

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