5.2.1 Single side band processing in a finite bandwidth

Assume that the conversion loss is negligible for the lower side band. At a given IF2 frequency the directly correlated signal is:

$$V_r = \ensuremath{A_\mathrm{\scriptscriptstyle }}\cos{(\ensur... ...nsuremath{\omega_\mathrm{\scriptscriptstyle IF2}}\Delta \tau)}$$ (5.7)

while the sine correlator would give:

$$V_i = \ensuremath{A_\mathrm{\scriptscriptstyle }}\sin{(\ensur... ...nsuremath{\omega_\mathrm{\scriptscriptstyle IF2}}\Delta \tau)}$$ (5.8)

$$V = V_r+ i V_i = \ensuremath{A_\mathrm{\scriptscriptstyle }} ... ...nsuremath{\omega_\mathrm{\scriptscriptstyle IF2}}\Delta \tau)}$$ (5.9)

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

$$V = \int \ensuremath{A_\mathrm{\scriptscriptstyle }} e^{i (\e... ...le IF2}} ) d\ensuremath{\omega_\mathrm{\scriptscriptstyle IF2}}$$ (5.10)

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

$$V = \ensuremath{A_\mathrm{\scriptscriptstyle }} e^{i\ensurema... ...le IF2}} ) d\ensuremath{\omega_\mathrm{\scriptscriptstyle IF2}}$$ (5.11)

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).