7.2 Delay lines requirements

7.2.1 Single sideband processing in a finite bandwidth

Assume that the conversion loss is negligible for the lower sideband. At a given IF2 frequency $\omega_\mathrm{\scriptscriptstyle IF2}$ the directly correlated signal is:

$$V_r = \ensuremath{A_\mathrm{\scriptscriptstyle }}\cos{(\ensuremat... ...scriptstyle }}+\ensuremath{\omega_\mathrm{\scriptscriptstyle IF2}}\Delta \tau)}$$ (7.7)

while the sine correlator would give:

$$V_i = \ensuremath{A_\mathrm{\scriptscriptstyle }}\sin{(\ensuremat... ...scriptstyle }}+\ensuremath{\omega_\mathrm{\scriptscriptstyle IF2}}\Delta \tau)}$$ (7.8)

$$V = V_r+ i V_i = \ensuremath{A_\mathrm{\scriptscriptstyle }}e^{i ... ...scriptstyle }}+\ensuremath{\omega_\mathrm{\scriptscriptstyle IF2}}\Delta \tau)}$$ (7.9)

Assume we use a correlator with a finite bandwidth $\Delta \nu$. The correlator output is obtained by summing on frequency in the IF passband:

$$V = \int \ensuremath{A_\mathrm{\scriptscriptstyle }}e^{i (\ensure... ...{\scriptscriptstyle IF2}}) d\ensuremath{\omega_\mathrm{\scriptscriptstyle IF2}}$$ (7.10)

where $B(\ensuremath{\omega_\mathrm{\scriptscriptstyle IF2}})$ is a complex passband function characteristic of the system: gain of the amplifiers and relative phase factors.

$$V = \ensuremath{A_\mathrm{\scriptscriptstyle }}e^{i\ensuremath{\v... ...{\scriptscriptstyle IF2}}) d\ensuremath{\omega_\mathrm{\scriptscriptstyle IF2}}$$ (7.11)

We have assumed that the source visibility is constant across the band; the source visibility, when the delay error varies, is multiplied by the Fourier transform of the complex passband.

The delay error must be kept much smaller than the inverse of the instantaneous bandwidth to limit the signal loss to a small level. The delays are usually tracked in steps, multiples of a minimum value. To limit the loss to 1%, the minimum delay step must be $\sim 0.25/\Delta \nu$ (0.5 ns for a 500 MHz bandwidth).

7.2.2 Double sideband system

In that case the signals coming from the upper and lower sidebands have similar attenuation in the RF part and similar conversion loss in the mixers. They will have similar amplitudes in the correlator output. The result for the cosine correlator is:

$$V = \ensuremath{A_\mathrm{\scriptscriptstyle U}}e^{i[\ensuremath{\var... ...thrm{\scriptscriptstyle LO2}}\ensuremath{\tau_\mathrm{\scriptscriptstyle G}})]}$$

$$+ \ensuremath{A_\mathrm{\scriptscriptstyle L}}e^{i[-\ensuremath{\va... ...thrm{\scriptscriptstyle LO2}}\ensuremath{\tau_\mathrm{\scriptscriptstyle G}})]}$$ (7.12)

Assuming the same visibility in both sidebands:

$$V = \ensuremath{A_\mathrm{\scriptscriptstyle }}\cos{(\ensuremath{... ...athrm{\scriptscriptstyle LO2}}\ensuremath{\tau_\mathrm{\scriptscriptstyle G}})}$$ (7.13)

If the delays are tracked, and the LO phases rotated as above, the exponential term is 1 and only the real part of the visibility is measured. Some trick is thus needed to separate the signal from the sidebands.