Page 250 - DSP Integrated Circuits

6.4 SFGs in Precedence Form 235

1. Collapse unnecessary nodes in the fully specified signal-flow graph by
removing all branches with transmittance = 1. Transmittances of -1
can often be propagated into adjacent branches. This will give an
equivalent, and potentially simpler, signal-flow graph with fewer
nodes and branches.
2. Assign node variables to all nodes in the fully specified signal-flow
graph according to
• Input and output nodes with variables, Xi(n) and yi(n), respectively.
• Contents of the delay elements (outputs) with variables, vi(n).
• Outputs from the basic operations, i.e., all the remaining nodes, with
variables, HI
The computational order for the branches, corresponding to arithmetic
operations, is determined in steps 3 to 7.
3. Remove all branches with delay elements in the signal-flow graph.
Let 7' = 1.
4. Choose all initial nodes in the (remaining) signal-flow graph and
denote this set of nodes by Nj, as shown in Figure 6.14.
5. Delete the branches that correspond to basic operations that are
executable (that is, operations for which all inputs are initial nodes).
Remove all initial nodes that no longer have any outgoing branches.
6. If there are any nodes left, let j <— j + 1 and repeat from step 4. The
algorithm is not sequentially computable if there are some operations
but no initial nodes left. Hence, the precedence form does not exist.
7. Connect nodes with branches (basic operations) according to the
signal-flow graph. Note that the node variables vi(n) always belong to
the first set of nodes, while the node variables HI belong to the other
sets.

Box 6.1 Procedure for deriving the precedence form of a signal-flow graph.

flow graph. Hence, the algorithm is not sequentially computable. Such algorithms
have no practical use.

EXAMPLE 6.1
Derive the signal-flow graph in precedence form for the second-order section in
direct form II shown in Figure 6.15.
The necessary quantization in the recursive loops is shown but not the order-
ing of the additions. Hence, the signal-flow graph is not fully specified. We get a
fully specified signal-flow graph by ordering the additions according to
Figure 6.16. Note that there are several ways to order the additions.