Block Diagram
Representation of
Linear Systems
Suppose we wish to synthesize a filter given a
transfer function? We will do an analog filter
example followed by a digital filter.
Example: Synthesize a filter based upon the
following transfer function:
1
H ( s) 
.
s 1
First, let us write the relationship between the input
and the output.
Y (s)
1
H (s) 

.
X (s) s  1
Y ( s )[s  1]  X ( s ).
dy(t )
 y (t )  x(t ).
dt
Suppose we constructed a block diagram of the
relationship between y(t) and dy(t)/dt:
dy (t )
dt
1
s
y(t )
This simple block diagram can be the cornerstone
for a realization (implementation) of this filter.
From the relation
dy (t )
 y (t )  x(t ),
dt
or,
dy (t )
 x(t )  y (t ),
dt
we can complete the block diagram for this system:
dy (t )
dt
+
+
-
1
s
y(t )
Example: Synthesize a filter based upon the
following transfer function:
1
H ( z) 
.
1
1 z
We proceed as in the analog filter case by writing a
relationship between the input and the output.
Y ( z)
1
H ( z) 

.
1
X ( z) 1  z
1
Y ( z )[1  z ]  X ( z ).
y[ n]  y[ n  1]  x[ n].
y[ n]  x[ n]  y[ n  1].
We start with a simple block diagram showing the
relationship between y[n] and y[n-1]:
y[n]
z
1
y[n 1]
y[n]
y[n]
+
x[n]
+
+
z
1
y[n 1]
Example: Synthesize a filter based upon the
following transfer function:
1
z
H ( z) 
.
1
1 z
We can use the results of the previous example to
construct the new block diagram. The new output is
equal to z-1 times the old output (from the old block
diagram).
1
Y ( z)
z
1
H ( z) 

 z H old ( z ).
1
X ( z) 1  z
y[n]  yold [n  1].
x[n]
+
z
1
y[n]
We can follow this method to find a block diagram
realization of the following transfer function:
1
2
3
b0  b1 z  b2 z  b3 z
H ( z) 
1
2
3
1  a1 z  a2 z  a3 z
we would obtain the following
+
+
+
y[n]
b0
b1
b2
b3
x[n]
+
z-1
z-1
z-1
a1
a2
+
+
a3
There is also an alternative method based upon
Tellegen’s Theorem:
Given a block diagram realization for a system, an
equivalent realization can be constructed by
replacing the inputs with outputs, replacing the
summing nodes with junctions and replacing the
junctions with summing nodes.
Tellegen’s Theorem:
Inputs  Outputs
Summing Nodes  Junctions
We can apply this theorem to the previous block
diagram, repeated on the next slide, and obtain the
block diagram on the two slides following the next
slide.
+
+
+
y[n]
b0
b1
b2
b3
x[n]
+
z-1
z-1
z-1
a1
a2
+
+
a3
x[n]
b0
b1
b2
b3
y[n]
+
z-1
+
a1
z-1
+
a2
z-1
+
a3
x[n]
b3
b2
b1
b0
y[n]
+
a3
z-1
+
a2
z-1
+
a1
z-1
+
We implement a similar transfer function by
expanding it according to its poles:
1
H ( z) 
1  a1 z 1  a2 z  2  a3 z 3
1

1
1
1
(1  p1 z )(1  p2 z )(1  p3 z )
R3
R1
R2



.
1
1
1
(1  p1 z ) (1  p2 z ) (1  p3 z )
x[n]
y[n]
+
+
p1 z 1
+
p2 z 1
p3 z 1
R1
+
p1 z
x[n]
1
R2
+
p2 z 1
R3
+
p3 z 1
+
y[n]