Baseband simulation, correlation, and acquisition for the GPS L1 C/A signal.

This module builds the "generate a replica, correlate, and acquire a PRN" layer on top of Orbis.GNSS.Signal.CA. It works entirely in complex baseband (the carrier has already been removed down to a residual Doppler), using the standard time-domain signal-processing model described in the GNSS acquisition literature:

• Kaplan & Hegarty, Understanding GPS/GNSS: Principles and Applications (3rd ed.), Ch. 5-8 (signal acquisition, coherent correlation, the sinc Doppler-mismatch loss, and the C/N0 to post-correlation SNR relation).
• Misra & Enge, Global Positioning System: Signals, Measurements, and Performance (2nd ed.), Ch. 10-11.
• Borre, Akos, Bertelsen, Rinder & Jensen, A Software-Defined GPS and Galileo Receiver (the 2D code-phase by Doppler search and the peak-to-noise acquisition metric).

Sampled-code replica (zero-order-hold / nearest-chip)

The 1023-chip C/A code is sampled to a sampling rate fs over an integration time T (so N = round(fs * T) samples). At sample n the code phase, in chips, advances at the code rate:

code_rate = f_chip * (1 + fd_code / f_L1)
sampled[n] = chip(prn, floor((code_phase + n * code_rate / fs) mod 1023))

with f_chip = 1_023_000 cps and f_L1 = 1_575_420_000 Hz. code_phase is the initial offset in chips and fd_code is an optional code-rate Doppler scaling (defaults to 0.0; code Doppler is negligible over one 1 ms period but the parameter is exposed). This is a clean nearest-chip (zero-order-hold) sampler: it picks the chip the sample instant falls within. CA.chip/2 already wraps its index modulo 1023, so the sampler wraps for free.

Coherent correlation

For a complex baseband record x[n] = xI[n] + j*xQ[n] (a real-valued test signal has xQ = 0), correlation against a local carrier wipe-off exp(-j 2*pi*f_d*n/fs) times the real bipolar code c[n] is the coherent sum over the integration window:

S = sum_{n=0}^{N-1} x[n] * c[n] * exp(-j 2*pi*f_d*n/fs)
I = sum ( xI[n]*c[n]*cos(th_n) + xQ[n]*c[n]*sin(th_n) )
Q = sum ( xQ[n]*c[n]*cos(th_n) - xI[n]*c[n]*sin(th_n) ),  th_n = 2*pi*f_d*n/fs
power = I^2 + Q^2

The amplitude is recovered as sqrt(I^2 + Q^2); for a real input the carrier wipe-off still spreads energy across both I and Q so the magnitude is the meaningful quantity.

Acquisition metric

acquire/3 performs a 2D search over code-phase bins (at sample resolution) and Doppler bins, computing the correlator power on each cell. The detection metric is the standard peak-to-noise ratio used in software receivers:

metric = peak_power / mean(off_peak_powers)

where the off-peak set excludes the peak cell and an exclusion zone of one code-phase bin on either side of the peak (at the peak's Doppler bin), so the main correlation lobe is not mistaken for the noise floor. For a clean signal this metric is large (order of the number of samples); for noise or the wrong PRN it is close to one. The alternative peak-to-second-peak ratio is also a standard choice; this module exposes peak-to-mean-off-peak as metric.

Coherent integration loss and post-correlation SNR

A residual frequency error f over a coherent integration time T attenuates the correlation by the sinc-squared Doppler-mismatch loss (Kaplan & Hegarty, the residual-carrier loss whose discrete form |sin(N*pi*df*Ts) / (N*sin(pi*df*Ts))| tends to sinc(pi*f*T) in the continuous limit):

coherent_loss(f, T) = sinc^2(pi*f*T) = ( sin(pi*f*T) / (pi*f*T) )^2

which is 1 (0 dB) at f = 0 and has its first null at f = 1/T. The correlation amplitude scales as |sinc(pi*f*T)| = sqrt(coherent_loss).

For the post-correlation signal-to-noise ratio this module exposes only the relation it can state cleanly from a standard reference: coherent integration over T seconds gives the predetection SNR

snr_post_db(cn0_dbhz, T) = cn0_dbhz + 10*log10(T)

i.e. the carrier-to-noise-density ratio C/N0 (dB-Hz) plus the processing gain 10*log10(T) corresponding to a 1/T effective noise bandwidth (Kaplan & Hegarty; Misra & Enge). The exact noise-bandwidth convention (1/T versus 1/(2T)) differs between texts; this module uses the 1/T form. No second, competing gain formula is exposed.