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Precision Timing, Frequency References & Coherent Synchronization
1. Clock error model
Write an oscillator as $v(t)=A\cos[2\pi f_0 t+\phi(t)]$. Instantaneous fractional frequency error is
\[y(t)=\frac{1}{2\pi f_0}\frac{d\phi(t)}{dt}.\]Two oscillators with relative frequency offset $\Delta f$ accumulate phase difference $\Delta\phi(t)=2\pi\Delta f t$ in the ideal constant-offset approximation. Time deviation and phase deviation must be distinguished when reporting synchronization.
For an ADC sampling a sine wave at input frequency $f_\mathrm{in}$ with uncorrelated rms aperture jitter $\sigma_t$, the jitter-limited SNR approximation is
\[\mathrm{SNR}_\mathrm{jitter}\approx-20\log_{10}(2\pi f_\mathrm{in}\sigma_t).\]2. Allan deviation and phase-noise spectra
Allan deviation characterizes fractional-frequency stability versus averaging time $\tau$. It does not by itself quantify absolute timing offset or guarantee low phase noise at every Fourier offset. For RF measurements, state the oscillator reference, offset-frequency phase-noise spectrum, measurement bandwidth and integration limits.
3. Instrument synchronization architecture
| Layer | Purpose | Possible implementation |
|---|---|---|
| Frequency reference | Common long-term rate | 10 MHz or other distributed frequency standard |
| Epoch reference | Common time origin | PPS, trigger, hardware timestamp |
| Sampling clock | ADC/DAC timing | Shared or deterministically aligned clock |
| RF LO phase | Relative RF carrier phase | Shared LO, deterministic synchronization, calibration |
| Delay calibration | Remove channel skew | Known tone/chirp and phase-versus-frequency fit |
For an array receiving $x_k(t)=a_ke^{j(\omega t+\phi_k)}$, unknown interchannel $\phi_k$ biases the beam steering or direction-of-arrival result. Relative phase calibration must be checked after reconnects and clock resynchronization.
Worked check: a 1 ppb relative offset at 1 GHz corresponds to 1 Hz frequency error, accumulating $2\pi$ rad phase in 1 s if uncompensated.
Hands-on exercise: use coherent_timing_demo.py to compare frequency drift, static phase offset and fixed sample delay.