it.grypho.scala.leonardo.transform

AST node for the Fourier transform: fourier(e, t, w) in the grammar.

Computes the unilateral Fourier transform F{e(t)} with frequency variable w via the Laplace-to-Fourier substitution s -> i*w (see fourierOf). Results are generally complex-valued. The binder t is excluded from children; w appears free in the result.

eval applies fourierOf and guards against the fixpoint.

AST node for the inverse Laplace transform: invlaplace(f, s, t) in the grammar.

Computes L^-1{f(s)} as a function of t via the rule table in inverseLaplaceOf. The frequency binder s is excluded from children (dual of _Laplace); t names the output variable and appears free in the result.

eval applies inverseLaplaceOf and guards against the fixpoint.

AST node for the inverse z-transform: invztrans(X, z, n) in the grammar.

Recovers x[n] from a rational X(z) via the rule table in inverseZTransformOf. The frequency binder z is excluded from children (dual of _ZTransform); n names the output index and appears free in the result.

eval applies inverseZTransformOf and guards against the fixpoint.

AST node for the Laplace transform: laplace(e, t, s) in the grammar.

Computes L{e(t)} with Laplace variable s via the rule table in laplaceOf. The integration binder t is excluded from children per the _Functional convention; s names the output variable and appears free in the result.

eval applies laplaceOf and guards against the fixpoint (stays symbolic when the rule table cannot reduce the expression).

AST node for the z-transform: ztrans(x, n, z) in the grammar.

Computes the one-sided (unilateral) transform X(z) = sum(k >= 0) x[k]*z^-k via the rule table in zTransformOf. The summation binder n is excluded from children per the _Functional convention; z names the output variable and appears free in the result.

One-sided rather than bilateral by design, matching laplace, which is itself unilateral. A bilateral transform would have to carry a region of convergence on every result — without one the same X(z) inverts to a causal or an anti-causal signal, so the inverse would have to guess — and that is deferred as future work.

eval applies zTransformOf and guards against the fixpoint (stays symbolic when the rule table cannot reduce the expression), the convention shared with _Laplace.

Computes F{e} with time variable t and frequency variable w.

Picks a fresh internal name for the Laplace frequency variable that cannot collide with any free variable already present in e (or with the binders t and w). Returns _Fourier(e, t, w) when laplaceOf cannot reduce the expression.

Computes L^-1{f} with Laplace variable s and time variable t.

Applies simplifyFully to the result when a rule fires, keeping the output readable.

Recovers x[n] from f = X(z).

Computes L{e} with time variable t and Laplace variable s.

Delegates to laplaceImpl which carries the string name of t for fast pattern matching without re-boxing. Returns _Laplace(e, t, s) when no rule fires.

Computes Z{x} with index variable n and frequency variable z.