kshana 0.35.0

Open, reproducible PNT-resilience simulator with quantum-sensor performance models
Documentation 
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160

// SPDX-License-Identifier: AGPL-3.0-only
//! First-order ionospheric group delay per band, the slant total electron content (TEC) seen
//! on a link to a LEO satellite, and the dual-frequency ionosphere-free combination.
//!
//! * [`group_delay_m`]: the first-order group delay `I = 40.3·STEC/f²` (m), STEC the slant TEC
//!   in electrons/m² (Misra & Enge, *Global Positioning System: Signals, Measurements, and Performance*; the same constant
//!   underlies the IS-GPS-200 group-delay ratio `γ = (f_L1/f_L2)²`).
//! * [`stec_from_klobuchar`]: slant TEC from the engine's Klobuchar model
//!   ([`crate::gnss_sim::klobuchar_delay_m`], IS-GPS-200 § 20.3.3.5.2.5), which describes a
//!   path through the whole ionosphere.
//! * [`fraction_below`]: the fraction of the vertical electron content that lies below a
//!   satellite at height `h`, for a Chapman layer of peak height `h_m` and scale height `H`:
//!   `erfc(√(½·exp(−(h − h_m)/H)))`. A LEO satellite near 500 km sits in the topside, so part
//!   of the electron content is above it and does not delay its signal. This is a modelled
//!   scaling (a single Chapman layer has no plasmasphere); it can be overridden by a stated
//!   fraction.
//! * [`iono_free_coefficients`] and [`iono_free_noise_amplification`]: the combination
//!   `P_IF = a₁P₁ + a₂P₂`, `a₁ = f₁²/(f₁² − f₂²)`, `a₂ = −f₂²/(f₁² − f₂²)`, which removes the
//!   first-order delay and amplifies independent noise by `√(a₁²σ₁² + a₂²σ₂²)`.
//! * [`dll_code_noise_m`]: the thermal code-tracking noise of a non-coherent early-minus-late
//!   power delay-locked loop for a BPSK (binary phase-shift keying) signal,
//!   `σ = T_c·c·√(B_L·d/(2·C/N0)·(1 + 2/(T·C/N0·(2 − d))))` (Kaplan & Hegarty, *Understanding
//!   GPS/GNSS: Principles and Applications*, 3rd ed., the code-tracking thermal-noise
//!   expression for an infinite front-end bandwidth).

use super::C_M_S;

/// First-order ionospheric constant (m³/s²): `I = K·STEC/f²`.
pub const K_IONO: f64 = 40.3;
/// One TEC unit (electrons/m²).
pub const TECU: f64 = 1.0e16;

/// First-order group delay (m) at `f_hz` for slant TEC `stec_el_m2` (electrons/m²).
pub fn group_delay_m(stec_el_m2: f64, f_hz: f64) -> f64 {
    K_IONO * stec_el_m2 / (f_hz * f_hz)
}

/// Slant TEC (electrons/m²) that produces the Klobuchar L1 delay for the given geometry.
pub fn stec_from_klobuchar(
    coeffs: &crate::gnss_sim::KlobucharCoeffs,
    lat_rad: f64,
    lon_rad: f64,
    el_rad: f64,
    az_rad: f64,
    gps_sod: f64,
) -> f64 {
    let d = crate::gnss_sim::klobuchar_delay_m(coeffs, lat_rad, lon_rad, el_rad, az_rad, gps_sod);
    d * crate::gnss_sim::L1_HZ * crate::gnss_sim::L1_HZ / K_IONO
}

/// Single-layer mapping function at elevation `el_rad` for a shell at `shell_m` above a sphere
/// of radius `re_m`: `1/√(1 − (re·cos el/(re + h))²)`.
pub fn single_layer_mapping(el_rad: f64, re_m: f64, shell_m: f64) -> f64 {
    let x = re_m * el_rad.cos() / (re_m + shell_m);
    1.0 / (1.0 - x * x).max(1e-9).sqrt()
}

/// Complementary error function, via the engine's error function.
fn erfc(x: f64) -> f64 {
    1.0 - crate::detection::erf(x)
}

/// Fraction of the vertical electron content below height `h_m` for a Chapman layer with peak
/// at `peak_m` and scale height `scale_m`.
pub fn fraction_below(h_m: f64, peak_m: f64, scale_m: f64) -> f64 {
    let z = (h_m - peak_m) / scale_m.max(1.0);
    let x = 0.5 * (-z).exp();
    erfc(x.sqrt()).clamp(0.0, 1.0)
}

/// Ionosphere-free combination coefficients `(a₁, a₂)` for carriers `f1_hz`, `f2_hz`.
pub fn iono_free_coefficients(f1_hz: f64, f2_hz: f64) -> (f64, f64) {
    let (a, b) = (f1_hz * f1_hz, f2_hz * f2_hz);
    (a / (a - b), -b / (a - b))
}

/// Slant total electron content (TEC units, 1e16 electrons/m²) from the geometry-free code
/// combination of two pseudoranges (or two first-order group delays) `p1_m` on `f1_hz` and
/// `p2_m` on `f2_hz`: `((p2 − p1) + dcb) f1² f2² / (40.3 (f1² − f2²)) / 1e16`. The geometry,
/// clocks and troposphere cancel in `p2 − p1`; `dcb_p1_minus_p2_m` is the receiver plus
/// satellite differential code bias of `P1 − P2` in metres (zero for modelled delays), which the
/// measured difference carries with the opposite sign and is added back here.
pub fn geometry_free_stec_tecu(
    p1_m: f64,
    p2_m: f64,
    f1_hz: f64,
    f2_hz: f64,
    dcb_p1_minus_p2_m: f64,
) -> f64 {
    let (a, b) = (f1_hz * f1_hz, f2_hz * f2_hz);
    ((p2_m - p1_m) + dcb_p1_minus_p2_m) * a * b / (40.3 * (a - b)) / TECU
}

/// Noise amplification of the ionosphere-free combination for per-band noise `sigma1`,
/// `sigma2`: `√(a₁²σ₁² + a₂²σ₂²)`. With `sigma1 = sigma2 = 1` it is the familiar factor
/// (2.98 for GPS L1/L2, 2.59 for L1/L5).
pub fn iono_free_noise_amplification(f1_hz: f64, f2_hz: f64, sigma1: f64, sigma2: f64) -> f64 {
    let (a1, a2) = iono_free_coefficients(f1_hz, f2_hz);
    (a1 * a1 * sigma1 * sigma1 + a2 * a2 * sigma2 * sigma2).sqrt()
}

/// Thermal code-tracking noise (m, one sigma) of a non-coherent early-minus-late power DLL on a
/// BPSK signal: chip rate `chip_rate_hz`, carrier-to-noise density `cn0_dbhz`, loop bandwidth
/// `bl_hz`, early-late spacing `d_chips`, coherent integration `t_s`.
pub fn dll_code_noise_m(
    chip_rate_hz: f64,
    cn0_dbhz: f64,
    bl_hz: f64,
    d_chips: f64,
    t_s: f64,
) -> f64 {
    let cn0 = 10f64.powf(cn0_dbhz / 10.0);
    let var_chips = bl_hz * d_chips / (2.0 * cn0) * (1.0 + 2.0 / (t_s * cn0 * (2.0 - d_chips)));
    var_chips.sqrt() * C_M_S / chip_rate_hz
}

#[cfg(test)]
mod tests {
    use super::*;

    #[test]
    fn delay_scales_as_one_over_f_squared() {
        let stec = 50.0 * TECU;
        let d1 = group_delay_m(stec, 1.2e9);
        let d2 = group_delay_m(stec, 2.4e9);
        assert!((d1 / d2 - 4.0).abs() < 1e-12);
        // 1 TECU at L1 is 0.1624 m.
        assert!((group_delay_m(TECU, crate::gnss_sim::L1_HZ) - 0.162_376).abs() < 1e-5);
    }

    #[test]
    fn chapman_fraction_is_half_at_about_the_peak_and_tends_to_one_high_up() {
        let f_peak = fraction_below(350e3, 350e3, 100e3);
        assert!((f_peak - erfc(0.5f64.sqrt())).abs() < 1e-6);
        assert!(fraction_below(20_000e3, 350e3, 100e3) > 0.999);
        assert!(fraction_below(0.0, 350e3, 100e3) < 1e-6);
        let leo = fraction_below(510e3, 350e3, 100e3);
        assert!((0.5..0.8).contains(&leo), "{leo}");
    }

    #[test]
    fn iono_free_combination_removes_the_first_order_delay() {
        let (f1, f2) = (1575.42e6, 1176.45e6);
        let (a1, a2) = iono_free_coefficients(f1, f2);
        assert!((a1 + a2 - 1.0).abs() < 1e-12);
        let stec = 30.0 * TECU;
        let resid = a1 * group_delay_m(stec, f1) + a2 * group_delay_m(stec, f2);
        assert!(resid.abs() < 1e-9);
    }

    #[test]
    fn code_noise_falls_with_cn0_and_chip_rate() {
        let a = dll_code_noise_m(1.023e6, 45.0, 1.0, 0.5, 0.02);
        let b = dll_code_noise_m(1.023e6, 55.0, 1.0, 0.5, 0.02);
        let c = dll_code_noise_m(10.23e6, 45.0, 1.0, 0.5, 0.02);
        assert!(b < a && c < a);
        assert!((a / c - 10.0).abs() < 1e-9);
        assert!((0.3..2.0).contains(&a), "{a}");
    }
}