5.1.4 Signal phase

One antenna is affected by the geometrical delay , and by the phase ( in the upper side band, in the lower side band), which is the quantity to be measured. We apply a compensating delay in the second IF, as well as a phase to the first LO and a phase on the second LO. We note $\Delta\tau = \ensuremath{\tau_\mathrm{\scriptscriptstyle I}} +\ensuremath{\tau_\mathrm{\scriptscriptstyle G}}$ the delay tracking error. In a 2-antenna system, we may assume that the signal path through the first antenna suffers no delay of phase offset terms. Obviously the compensating delay in the second antenna may need to be negative, if the second antenna is closer to the source: in that case one will apply the positive delay $-\ensuremath{\tau_\mathrm{\scriptscriptstyle I}}$ on the first antenna. In a N antenna system, one will apply phase and delay commands to all the antennas; a common delay will be applied to all the antennas since no negative delay can be built with current technology.

Let us first consider the upper side band of the first LO (second LO conversion is assumed upper side band for simplicity):

	USB	LSB
HF Frequency
HF Phase	$\ensuremath{\varphi_\mathrm{\scriptscriptstyle U}} +\ensuremath{\omega_\mathrm{\scriptscriptstyle U}}\ensuremath{\tau_\mathrm{\scriptscriptstyle G}}$	$\ensuremath{\varphi_\mathrm{\scriptscriptstyle L}} +\ensuremath{\omega_\mathrm{\scriptscriptstyle L}}\ensuremath{\tau_\mathrm{\scriptscriptstyle G}}$
LO1 Frequency
LO1 Phase
IF1 Frequency	$\ensuremath{\omega_\mathrm{\scriptscriptstyle IF1}} =\ensuremath{\omega_\mathrm{\scriptscriptstyle U}} -\ensuremath{\omega_\mathrm{\scriptscriptstyle 1}}$	$\ensuremath{\omega_\mathrm{\scriptscriptstyle IF1}} =\ensuremath{\omega_\mathrm{\scriptscriptstyle 1}} -\ensuremath{\omega_\mathrm{\scriptscriptstyle L}}$
IF1 Phase	$\ensuremath{\varphi_\mathrm{\scriptscriptstyle U}} +\ensuremath{\omega_\mathrm{... ...hrm{\scriptscriptstyle G}} -\ensuremath{\varphi_\mathrm{\scriptscriptstyle 1}}$	$-\ensuremath{\varphi_\mathrm{\scriptscriptstyle L}} -\ensuremath{\omega_\mathrm... ...hrm{\scriptscriptstyle G}} +\ensuremath{\varphi_\mathrm{\scriptscriptstyle 1}}$
LO2 Frequency
LO2 Phase
IF2 Frequency	$\ensuremath{\omega_\mathrm{\scriptscriptstyle IF2}} =\ensuremath{\omega_\mathrm... ...thrm{\scriptscriptstyle 1}} -\ensuremath{\omega_\mathrm{\scriptscriptstyle 2}}$	$\ensuremath{\omega_\mathrm{\scriptscriptstyle IF2}} =\ensuremath{\omega_\mathrm... ...thrm{\scriptscriptstyle L}} -\ensuremath{\omega_\mathrm{\scriptscriptstyle 2}}$
IF2 Phase	$\ensuremath{\varphi_\mathrm{\scriptscriptstyle U}} +\ensuremath{\omega_\mathrm{... ...hrm{\scriptscriptstyle 1}} -\ensuremath{\varphi_\mathrm{\scriptscriptstyle 2}}$	$-\ensuremath{\varphi_\mathrm{\scriptscriptstyle L}} -\ensuremath{\omega_\mathrm... ...hrm{\scriptscriptstyle 1}} -\ensuremath{\varphi_\mathrm{\scriptscriptstyle 2}}$
after	$\ensuremath{\varphi_\mathrm{\scriptscriptstyle U}} +\ensuremath{\omega_\mathrm{... ...mathrm{\scriptscriptstyle IF2}}\ensuremath{\tau_\mathrm{\scriptscriptstyle I}}$	$-\ensuremath{\varphi_\mathrm{\scriptscriptstyle L}} -\ensuremath{\omega_\mathrm... ...mathrm{\scriptscriptstyle IF2}}\ensuremath{\tau_\mathrm{\scriptscriptstyle I}}$
Final	$\ensuremath{\varphi_\mathrm{\scriptscriptstyle U}} +\ensuremath{\omega_\mathrm{\scriptscriptstyle IF2}}\Delta\tau$	$-\ensuremath{\varphi_\mathrm{\scriptscriptstyle L}} +\ensuremath{\omega_\mathrm{\scriptscriptstyle IF2}}\Delta\tau$
	$-(\ensuremath{\varphi_\mathrm{\scriptscriptstyle 1}} +\ensuremath{\omega_\mathrm{\scriptscriptstyle 1}}\ensuremath{\tau_\mathrm{\scriptscriptstyle G}} )$	$+(\ensuremath{\varphi_\mathrm{\scriptscriptstyle 1}} +\ensuremath{\omega_\mathrm{\scriptscriptstyle 1}}\ensuremath{\tau_\mathrm{\scriptscriptstyle G}} )$
	$-(\ensuremath{\varphi_\mathrm{\scriptscriptstyle 2}} +\ensuremath{\omega_\mathrm{\scriptscriptstyle 2}}\ensuremath{\tau_\mathrm{\scriptscriptstyle G}} )$	$-(\ensuremath{\varphi_\mathrm{\scriptscriptstyle 2}} +\ensuremath{\omega_\mathrm{\scriptscriptstyle 2}}\ensuremath{\tau_\mathrm{\scriptscriptstyle G}} )$

To stop the fringes in both sidebands we need the following conditions:

$$\Delta\tau = \ensuremath{\tau_\mathrm{\scriptscriptstyle I}} +\ensuremath{\tau_\mathrm{\scriptscriptstyle G}}$$ = 0 (5.3)

$$\ensuremath{\varphi_\mathrm{\scriptscriptstyle 1}} +\ensuremath{\... ..._\mathrm{\scriptscriptstyle 1}}\ensuremath{\tau_\mathrm{\scriptscriptstyle G}}$$ = 0 (5.4)

$$\ensuremath{\varphi_\mathrm{\scriptscriptstyle 2}} +\ensuremath{\... ..._\mathrm{\scriptscriptstyle 2}}\ensuremath{\tau_\mathrm{\scriptscriptstyle G}}$$ = 0 (5.5)

One sees that delay tracking in the second IF imposes a phase tracking on the first and second oscillators. The delay error $\Delta\tau$ appears as a phase term proportional to frequency in the IF2 band .

The condition that e.g. $\ensuremath{\varphi_\mathrm{\scriptscriptstyle 1}} = -\ensuremath{\omega_\mathrm{\scriptscriptstyle 1}}\ensuremath{\tau_\mathrm{\scriptscriptstyle G}}$ means that must be commanded to vary at a rate

$$\dot{\ensuremath{\varphi_\mathrm{\scriptscriptstyle 1}} } = -... ...riptstyle G}} } \sim 2\pi\frac{b}{\lambda_1}\frac{2\pi}{86400}$$ (5.6)

which is about 10 turns per second for $\lambda_1=1$mm and b=1km. The condition is much easier for the second LO. In practice the phase is commanded typically every second, as well as its rate of change during the next second (the real curve is approximated by a piecewise linear curve). Note that a linear drift with time of the phase is strictly equivalent to a small frequency offset.