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Identifiers and Marking

Fresh identifier representing memory locations

  • $\SID{S.T}=m$ unique scalar identifier for $S$ in time context $T$ ;
  • $\VID{S.T}=v$ unique vector identifier for $S$ in time context $T$ ;
  • $\TID{T}=t$ unique scalar identifier representing the current time in time context $T$.

Marking recursive definitions

  • $\vmark{Xi.T}=\emptyset$: not yet marked ;
  • $\vmark{Xi.T}<-v$: mark it with identifier $v$ ;
  • $\vmark{Xi.T}= v$: already marked with $v$.

Signal Compilation

Function $\cs{.}:\FS\times\FT->\MS\times\IS$

Number

$$ \inference[(num)]{ }{ \cs{k.T} = k \times \emptyset } $$

User interface

$$ \inference[(ctrl)]{ }{ \cs{u.T} = u \times \emptyset } $$

Inputs

$$ \inference[(input)]{ }{ \cs{\ain{c}.T} = \ain{c} \times \emptyset } $$

Signal Compilation

Numerical Operation

$$ \inference[(nop)]{ \cs{S_1.T}=M_1\times J_1 \\ \cs{S_2.T}=M_1\times J_2 \\ \vdots \\ \VID{\star(M_1,M_2,...)} = m }{ \cs{\star(S_1,S_2,...).T}= m\times{\instr{T}{m}{\star(M_1,M_2,...)} }\bigcup_i J_i } $$

Signal Compilation

Upsampling

The $S_1*>S_2$ upsampling appears at the output of an ondemand. It is necessary to compile $S_1$ into the clock time reference $S_2$ (which is added to the current time reference). The signal $S_1$ must also be stored in a variable to do the upsampling.

$$ \inference[(up)]{ \cs{S_2.T}=M_2\times J_2 \\ \cs{S_1.M_2.T}=M_1\times J_1 \\ \SID{M_1.M_2.T} = m\\ J_3 = {\instr{M_2.T}{m}{M_1}} }{ \cs{(S_1*>S_2).T}= m \times J_1 \cup J_1 \cup J_2 } $$

Signal Compilation

Downsampling

The downsampling $S_1<*S_2$ appears at the entrance of an ondemand. This means that compiling $S_1<*S_2$ into the $M_2.T$ time environment (where $M_2$ is the compiled version of $S_2$) is like compiling $S_1$ into the $T$ time environment and using a variable to do the downsampling.

$$ \inference[(down)]{ \cs{S_1.T}=M_1\times J_1 \\ \cs{S_2.T}=M_2\times J_2 \\ \SID{M_1.T} = m\\ J_3 = {\instr{T}{m}{M_1}} }{ \cs{(S_1<*S_2).M_2.T}= m\times J_3\cup J_1\cup J_2 } $$

Signal Compilation

Delay

$$ \inference[(up)]{ \cs{S_1.T}=M_1\times J_1 \\ \cs{S_2.T}=M_2\times J_2 \\ \VID{M_1.T} = v\\ \TID{T} = t\\ J_3 = {\instr{T}{v[t,0]}{M_1}} \cup {\instr{T}{t}{t+1}} }{ \cs{(S_1@S_2).T}= v[t,M_2]\times J_3 \cup J_1 \cup J_2 } $$

Signal Compilation: recursion

First visit

If it is the first visit, we have $\vmark{X_i.T}=0$: $$ \inference[(r1)]{ \VID{X_i.T} = v\ \vmark{X_i.T}<-v\ \rdef{X}=(...,S_i,...) \ \cs{S_i.T}=M_i\times J_i \ \cs{S_d.T}=M_d\times J_d \ \TID{T} = t \ J_3 = {\instr{T}{v[t,0]}{M_i}} \cup {\instr{T}{t}{t+1}} }{ \cs{(X_i@S_d).T}= v[t,M_d]\times J_3 \cup J_i \cup J_d } $$

Signal Compilation: recursion

Next visits

If it is not the first visit, we have $\vmark{X_i.T}=v$: $$ \inference[(r2)]{ \cs{S_d.T}=M_d\times J_d \ \TID{T} = t \ }{ \cs{(X_i@S_d).T}= v[t,M_d]\times J_d } $$

Global compilation

$\cc{.}:\FS^n-&gt;\IS$

Global compilation

$$ \inference[(comp)]{ \vdots\\ \cs{S_i.1} = M_i\times J_i\\ J_i' = { \instr{1}{\aout{i}}{M_i} } \cup J_i\\ \vdots }{ \cc{(...,S_i,...)} = ... \cup J_i' \cup ... } $$