Integrated Modeling Using Finite State Machines and Dataflow Graphs - Signal Processing Systems - page 991

( )

to denote
the set of actor input and output ports of the CAL actor a . Furthermore, we will use
Γ (

To begin with, we will first introduce some notation. We use ports a

)

to denote the set of actions of the actor a . To refer to the set of ports which are
modified by an action l

a

∈ Γ ( )

a , that is, the action will consume or produce tokens on

) l is used. The number of tokens which are transferred
on a port p by an action l will be denoted by trans

(

that port, the notation ports a

( ) l , p .Furthermore, CAL actors

a

σ ∈ Σ

themselves can have a state , e.g., representing state variables or states from

a schedule FSM controlling action execution. We will call an action l state-guarded
if its execution depends on the value of the actor state . Moreover, an action l is

called state-changing if the execution of the action may modify the state

σ

σ of the

CAL actor.
With the previous, notation we can define a statically related group of ports Z of
an actor a . This set of ports Z

⊆ ( )

ports a is a maximal subset of the ports of a which

conforms to the following conditions:

∀ ∈ Γ ( ) ( ⊆ ( ) l ) ∨ ( ∩ ( ) l = )

l a : Z ports a Z ports a 0 (1)

∀ ∈ ∀ , ∈ Γ ( ) , ∈ ( ) l ∩ ( ) ( ) l , p = ( )

p Z : l m a p ports a ports a m :trans a trans a m , p

andtrans ( a ) l , p isalsoaconstantinteger (2)

The conditions ensure that ( 1 ) all the ports in the statically related group are
processed in an atomic action, e.g., like SDF actors consume and produce tokens
atomically, and if multiple actions are working on these ports that ( 2 ) all these
actions produce and consume the same number of tokens. Indeed, there may be
multiple statically related groups of ports Z . For example, consider the actor a 1
depicted in Fig. 23 a . This actor has two statically related groups of ports Z 1 =
{ . This is also the port partitioning along which the

actor has been split into its two constituent SDF actors as depicted in Fig. 23 b .
However, the conditions are not able to recognize CSDF actors nor SDF actors
where the consumption and production of the actor is distributed over more than
one action. For example, consider actor a 2 as depicted in Fig. 24 a . The actor is
undoubtedly a CSDF actor but no statically related group of ports can be found
as both ports violate Eq. ( 2 ) . This stems from the fact that the analysis to derive
statically related groups does not consider the actor state

i 1 , i 2 , o 1 } and Z 2 = { i 3 , i 4 , o 2 }

. Therefore, the analysis
cannot recognize the cyclic action execution of a 2 . This is also the reason why in
contrast to the analysis presented in Sect. 3 the algorithm to derive statically related
groups cannot recognize SDF actors where the consumption and production of the
actor is distributed over more than one action.
We now consider the dataflow graph depicted in Fig. 25 b . The dataflow graph has
seven statically related groups Z 1 - Z 7 which can be transformed into SDF actors for
quasi-static scheduling. The statically related groups induce two subgraphs (in fact
there are three but one subgraph consists only of one actor alone). We will denote the
subgraph containing the statically related groups Z 1 ,

σ

Z 3 , and Z 5 with g 1 and the other

subgraph consisting of the groups Z 2 , Z 4 and Z 6 with g 2 . Scheduling of subgraph g 2