Cannonical Triplets and Component Sets

Cannonical Triplets

A cannonical triplet Tuple{Function,Function,Real} is a tuple consisting of an instantaneous amplitude (IA) Function, an instantaneous frequency (IF) Function, and a phase reference Real.

\[\mathscr{C}\triangleq\left\{a(t),\omega(t), \phi\vphantom{0^0}\right\}\]

Defining a Cannonical Triplet

We can define a cannonical triplet as follows.

julia> a₀(t) = exp(-t^2)
a₀ (generic function with 1 method)

julia> ω₀(t) = 2.0
ω₀ (generic function with 1 method)

julia> φ₀ = 0.0
0.0

julia> 𝐶₀ = (a₀,ω₀,φ₀)
(a₀, ω₀, 0.0)

Component Sets

A component set Array{Tuple{Function,Function,Real},1} is a set of cannonical triplet Tuple{Function,Function,Real} .

\[\mathscr{S}\triangleq\left\{\mathscr{C}_0,\mathscr{C}_1,\cdots,\mathscr{C}_{K-1}\vphantom{0^0}\right\}\]

Defining a Component Set

We can define a component set as follows. First, we define several cannonical triplet.

a₀(t) = exp(-t^2)
ω₀(t) = 2.0
φ₀ = 0.0
𝐶₀ = (a₀,ω₀,φ₀)

a₁(t) = 1.0
ω₁(t) = 10*t
φ₁ = 0.1
𝐶₁ = (a₁,ω₁,φ₁)

a₂(t) = 0.8*cos(2t)
ω₂(t) = 10 + 7.5*sin(t)
φ₂ = π
𝐶₂ = (a₂,ω₂,φ₂)

Then, store the cannonical triplets in an array.

julia> 𝑆 = [𝐶₀,𝐶₁,𝐶₂]
3-element Array{Tuple{Function,Function,Real},1}:
 (a₀, ω₀, 0.0)
 (a₁, ω₁, 0.1)
 (a₂, ω₂, π)