Prototyping sonar equations in progress

scripts/prototype.md

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+# Sonar Equations Implementation Prototyping
+
+```julia
+using ModelingToolkit
+```
+
+```julia
+@variables begin
+    SL # [dB // μPa² m²]
+    PL # [dB // m²]
+    NL # total noise power level in the processing badn [dB // μPa²]
+    NLf # noise spectrum level [dB // μPa² / Hz] (aka noise level in a 1-Hz band)
+    AG # [dB]
+    DT # [dB]
+end
+```
+
+```julia
+-NL + 10log10(Tp * B) == -NLf + 10log10(Tp)
+-NL + 10log10(Tp) + 10log10(B) == -NLf + 10log10(Tp)
+NL - 10log10(B) == NLf
+```
+
+```julia
+abstract type GenericScenario end
+
+abstract type ReverberationLimitedScenario <: GenericScenario end
+abstract type NoiseLimitedScenario <: GenericScenario end
+```
+
+```julia
+struct NumOfTx{N} end
+NumOfTx(N::UInt8) = NumOfTx{N}()
+
+struct NumOfRx{N} end
+NumOfRx(N::UInt8) = NumOfRx{N}()
+```
+
+```julia
+abstract type SonarType end
+
+abstract type Passive <: SonarType end
+abstract type Active{
+    NumTx <: NumOfTx,
+    NumRx <: NumOfRx,
+    IS <: GenericScenario
+} <: SonarType end
+
+abstract type Broadband <: Passive end
+abstract type Narrowband <: Passive end
+abstract type Intercept <: Passive end
+```
+
+## Passive Sonar
+
+```julia
+let
+    SNRᵃ = SL - PL - NL + AG
+    SNRᵈ = SNRᵃ
+    SE = SNRᵈ - DTᵈ
+end
+```
+
+```julia
+signal_to_noise_ratio(::Type{<:Passive}, SL, PL, NL, AG) = SL - PL - NL + AG
+```
+
+### Broadband
+
+TODO.
+
+```julia
+let
+    SNRᵃ = SL - PL - NL + AG
+    SNRᵈ = SNRᵃ
+    SE = SNRᵈ - DTᵈ
+end
+```
+
+### Narrowband
+
+TODO.
+
+```julia
+let
+    SNRᵃ = SL - PL - NL + AG
+    SNRᵈ = SNRᵃ
+    SE = SNRᵈ - DTᵈ
+end
+```
+
+## Active Sonar
+
+### Monostatic Geometry
+
+```julia
+let
+    RL = SL - 2PL + TSr
+    SNRᵃ = SL - 2PL + TS - (NL ⊕ RL) + AG
+    SNRᵈ = SNRᵃ + 10log10(Tₚ * B)
+    SE = SNRᵈ - DTᵈ
+end
+```
+
+```julia
+function signal_to_noise_ratio(::Type{ActiveType},
+    SL, PL, TS, NL, RL, AG, Tp, B
+) where {
+    ActiveType <: Active{
+        <: NumOfTx{1},
+        <: NumOfTx{1},
+        <: GenericScenario
+    }
+}
+    SL - 2PL + TS - (NL ⊕ RL) + AG + 10log10(Tp * B)
+end
+```
+
+#### Noise-Limited Scenario
+
+```julia
+let
+    NL ⊕ RL ≈ NL
+
+    SNRᵃ = SL - 2PL + TS - (NL ⊕ RL) + AG
+         ≈ SL - 2PL + TS - NL + AG
+
+    SNRᵈ = SNRᵃ + 10log10(Tₚ * B)
+         ≈ SL - 2PL + TS - NL + AG + 10log10(Tₚ * B)
+         = SL - 2PL + TS - NLf + AG + 10log10(Tp)
+
+    SE = SNRᵈ - DTᵈ
+end
+```
+
+Pith: Increasing `SL` or `Tp` directly improves `SNRᵈ` irrespective of `B`.
+
+```julia
+function signal_to_noise_ratio(::Type{ActiveType},
+    SL, PL, TS, NLf, AG, Tp
+) where {
+    ActiveType <: Active{
+        <: NumOfTx{1},
+        <: NumOfTx{1},
+        <: NoiseLimitedScenario
+    }
+}
+    SL - 2PL + TS - NLf + AG + 10log10(Tp)
+end
+```
+
+#### Reverberation-Limited Scenario
+
+```julia
+let
+    RL = SL - 2PL + TSr
+
+    TSr = Sb + 10log10(2π * r * (cw * Tp / 2))
+
+    NL ⊕ RL ≈ RL
+
+    SNRᵃ = SL - 2PL + TS - (NL ⊕ RL) + AG
+         ≈ SL - 2PL + TS - RL + AG
+         = SL - 2PL + TS - (SL - 2PL + TSr) + AG
+         = SL - 2PL + TS - SL + 2PL - TSr + AG
+         = TS - TSr + AG
+         = TS - (Sb + 10log10(2π * r * (cw * Tp / 2))) + AG
+
+    SNRᵈ = SNRᵃ + 10log10(Tₚ * B)
+         ≈ TS - (Sb + 10log10(2π * r * (cw * Tp / 2))) + AG + 10log10(Tₚ * B)
+         = TS - (Sb + 10log10(2π * r * (cw / 2))) + AG + 10log10(B)
+
+    SE = SNRᵈ - DTᵈ
+end
+```
+
+Pith:
+
+* Increasing `SL` or `Tp` does not improve `SNRᵃ`.
+* Increasing `B` improves `SNRᵈ` irrespective of `Tp`.
+* CW have inversely proportional `B * Tp = 1`.
+* FM transmit waveform preferred over CW.
+
+```julia
+function signal_to_noise_ratio(::Type{ActiveType},
+    TS, Sb, r, cw, AG, B
+) where {
+    ActiveType <: Active{
+        <: NumOfTx{1},
+        <: NumOfTx{1},
+        <: ReverberationLimited
+    }
+}
+    TS - (Sb + 10log10(2π * r * (cw / 2))) + AG + 10log10(B)
+end
+```