2.2.2 Finite Bandwidth

Integrating over $d\nu$,

$$R = \frac{1}{\Delta \nu} \int_{\nu_0-\Delta \nu / 2}^{\nu_0+\Delta \nu /2} \vert V\vert \cos(2 \pi \nu \tau_g - \Phi_V) d\nu$$ (2.18)

Using $\nu = \nu_0 + n$

 

$$\frac{1}{\Delta \nu} \int_{-\Delta \nu /2}^{\Delta \nu / 2} \vert V\vert \cos(2 \pi \nu_0 \tau_g - \Phi_V + 2 \pi n \tau_g) dn$$ R = (2.19)

$$\frac{1}{\Delta \nu} [ \int_{-\Delta \nu /2}^{\Delta \nu / 2} \vert V\vert \cos(2 \pi \nu_0 \tau_g - \Phi_V) \cos(2 \pi n \tau_g) dn$$ =

$$- \int_{-\Delta \nu /2}^{\Delta \nu / 2} \vert V\vert \sin(2 \pi \nu_0 \tau_g - \Phi_V) \sin(2 \pi n \tau_g) dn ]$$ (2.20)

$$\frac{1}{\Delta \nu} \vert V\vert \cos( 2 \pi \nu_0 \tau_g - \Phi... ... \pi n \tau_g) \right]_{-\Delta \nu /2}^{\Delta \nu / 2} \frac{1}{2 \pi \tau_g}$$ =

$$+ \frac{1}{\Delta \nu} \vert V\vert \sin( 2 \pi \nu_0 \tau_g - \P... ... \pi n \tau_g) \right]_{-\Delta \nu /2}^{\Delta \nu / 2} \frac{1}{2 \pi \tau_g}$$ (2.21)

$$\vert V\vert \cos(2 \pi \nu_0 \tau_g - \Phi_V) \frac{\sin(\pi \Delta \nu \tau_g)} {\pi \Delta \nu \tau_g}$$ = (2.22)

The fringe visibility is attenuated by a ${\rm sin}(x)/x$ envelope, called the bandwidth pattern, which falls off rapidly. A 1% loss in visibility is obtained for $\vert\Delta\nu\tau_g\vert \simeq 0.078$, or with $\Delta\nu = 500$MHz and a baseline length b = 100m, when the zenith angle $\theta$ (defined in Fig.2.3) is 2 arcmin only. Thus, the ability to track a source for a significant hour angle coverage requires proper compensation of the geometrical delay when a finite bandwidth is desired.