# Algorithms and conventions

## Signal and downconversion

The analytic sampled signal uses

\[
x[n]=A\,c(\tau_n)b(\tau_n)s_{BOC}(\tau_n)
e^{j(\phi_0+2\pi(f_{IF}+f_D)n/f_s+\pi\dot f_D(n/f_s)^2)}+w[n].
\]

The generator defines a positive Doppler as positive complex phase rotation.
Digital downconversion and a candidate Doppler wipeoff therefore use

\[
y[n]=x[n]e^{-j2\pi(f_{IF}+f_{bin})n/f_s}.
\]

The phase-accumulator NCO has modulus \(2^P\), frequency-control word
\(FCW=\operatorname{round}(f_o2^P/f_s)\), and update
\(phase_{n+1}=(phase_n+FCW)\bmod 2^P\).

## Code and FFT acquisition

Fractional code phase advances by \(f_c/f_s\) chips per sample and indexes the
chip with `floor(phase) mod code_length`. The BOC(1,1) sine-phased square wave
is +1 for the first half of a chip and -1 for the second half. This convention
is shared by generation, acquisition, and tracking.

The circular correlation is

\[
R_f[k]=\operatorname{IFFT}\{\operatorname{FFT}(y_f[n])
\operatorname{FFT}(c[n])^*\}[k].
\]

For coherent interval \(T_c\), the complex correlator output is retained before
power formation. Noncoherent accumulation is explicitly

\[
P_f[k]=\sum_{m=0}^{M-1}|R_{f,m}[k]|^2,
\quad T_{obs}=M T_c.
\]

Differential combining uses adjacent complex results before magnitude. E1B
hypothesis mode enumerates primary-epoch signs (limited to eight periods to
bound exponential work). E1C search enumerates secondary phase; known-phase
wipeoff uses phase zero at the first observation epoch. Delay results are
reported over one 4 ms primary period even for longer coherent integrations.

Zero-padding can provide a convenient FFT size or a denser interpolation of a
particular numerical representation, but it does not increase signal
bandwidth, physical information, or fundamental delay resolution.

## Detection

After circularly excluding a guard interval around the main peak, the model
computes

\[
M_2=P_{max}/P_{second},\qquad M_\mu=P_{max}/\bar P_N,
\]

plus `(peak-noise_mean)/(sidelobe-noise_mean)` and a CFAR-style
`(peak-noise_median)/noise_std`. The acquisition C/N0 diagnostic is

\[
\widehat{C/N_0}=10\log_{10}\left((P_{max}-P_N)/(P_NT_c)\right).
\]

It is an estimator, not a calibrated RF measurement. Half-chip competitive BOC
neighbors set a side-peak flag. `flag` is the implemented mitigation; an
unambiguous BOC/CBOC strategy plugs into the detector without changing results.
Parabolic interpolation preserves raw Doppler and sample bins.

## Tracking

Each component channel owns carrier/code NCO phase, loop states, correlation
history, lock counters, C/N0 history, and state-machine state. Phases advance at
every accepted sample and never reset on Python block boundaries.

Carrier discriminators are

\[
e_{PLL,2}=\tan^{-1}(Q/I),\qquad e_{PLL,4}=\operatorname{atan2}(Q,I)
\]

in radians. The cross/phase-difference FLL is

\[
e_{FLL}=\angle(P_kP_{k-1}^*)/(2\pi T)\quad[Hz].
\]

The normalized envelope DLL reports a locally calibrated chip error

\[
e_{DLL}=d\frac{|E|-|L|}{|E|+|L|}\quad[chips],
\]

with power and dot-product alternatives. A BOC false-lock flag compares VEVL
and prompt magnitudes.

For a second-order filter, noise bandwidth and damping generate

\[
\omega_n=\frac{8\zeta B_n}{4\zeta^2+1},\quad
k_p=2\zeta\omega_n,\quad k_i=\omega_n^2.
\]

The third-order starting model places three repeated stable poles derived from
the same \(\omega_n\): correction
\(3\omega_ne+3\omega_n^2\int e+\omega_n^3\iint e\). These are physical
starting coefficients, not final optimized receiver-loop values.

Carrier NCO frequency is corrected in hertz; code NCO frequency is nominal code
rate plus DLL correction and optional carrier aiding
\(f_{code,aid}=f_{carrier}f_c/f_{RF}\). Prompt C/N0 uses coherent versus total
prompt moments over a sliding window and divides the ratio by integration time.