2.5 Fourier Transform and Related Approximations

The Complex Visibility is

$$V = \vert V\vert e^{i\Phi_V} = \int_{Sky} A(\ensuremath{\text{\bo... ...\unboldmath }}}.{\ensuremath{\text{\boldmath$\sigma$\unboldmath }}}/c)} d\Omega$$ (2.31)

Let $(u,v,w)$ be the coordinate of the baseline vector, in units of the observing wavelength $\nu$, in a frame of the delay tracking vector ${\ensuremath{\text{\boldmath $s$\unboldmath }}}_0$, with $w$ along ${\ensuremath{\text{\boldmath $s$\unboldmath }}}_0$. $(x,y,z)$ are the coordinates of the source vector $\ensuremath{\text{\boldmath $s$\unboldmath }}$ in this frame. Then

$$\nu {\ensuremath{\text{\boldmath$b$\unboldmath }}}.{\ensuremath{\text{\boldmath$s$\unboldmath }}}/c = ux + vy + wz$$

$$\nu {\ensuremath{\text{\boldmath$b$\unboldmath }}}.{\ensuremath{\text{\boldmath$s$\unboldmath }}_0}/c = w$$

$$z = \sqrt{1 - x^2 - y^2}$$

$${\rm and   } d\Omega = \frac{dx dy}{z}   =   \frac{dx dy}{\sqrt{1-x^2-y^2}}$$ (2.32)

Thus,

$$V(u,v,w) = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} A(x,... ...) e^{-2i\pi (ux + vy + w(\sqrt{1-x^2-y^2} -1) )} \frac{dx dy}{\sqrt{1-x^2-y^2}}$$ (2.33)

with $I(x,y) = 0$ when $x^2+y^2 \geq 1$.

If $(x,y)$ are sufficiently small, we can make the approximation

$$(\sqrt{1-x^2-y^2} - 1) w \simeq \frac{1}{2} (x^2+y^2) w \simeq 0$$ (2.34)

and Eq.2.33 becomes

$$V(u,v) = \ensuremath{\int\!\!\int}A'(x,y) I(x,y) e^{-2i\pi (ux+vy)} e^{-i\pi(x^2+y^2)w} dx dy$$ (2.35)

$$\mathrm{with   } A'(x,y) = \frac{A(x,y)}{\sqrt{1-x^2-y^2}}$$ (2.36)

i.e. basically a 2-D Fourier Transform of $A I$, but with a phase error term $\pi (x^2+y^2) w$. Hence, on limited field of views, the relationship between the sky brightness (multiplied by the antenna power pattern) and the visibility reduces to a simple 2-D Fourier transform.

There are other approximations related to field of view limitations. Let us quantify these approximations.

• 2-D Fourier Transform
  We can further neglect the phase error term in Eq.2.36, if the condition

  $$\vert \pi (x^2+y^2) w \vert « 1$$ (2.37)

  
  
  is fulfilled. Now, note that

  $$w < w_{\mathrm{max}} \simeq \frac{b_{\mathrm{max}}}{\lambda} \simeq \frac{1}{\ensuremath{\theta_\mathrm{\scriptscriptstyle s}}}$$ (2.38)

  
  
  where $\ensuremath{\theta_\mathrm{\scriptscriptstyle s}}$ is the synthesized beam width. Thus, if $\ensuremath{\theta_\mathrm{\scriptscriptstyle f}}$ is the field of view to be synthesized, the maximum phase error, obtained at the field edges $\ensuremath{\theta_\mathrm{\scriptscriptstyle f}}/2$, is

  $$\Delta \phi = \frac{\pi\ensuremath{\theta_\mathrm{\scriptscriptstyle f}}^2}{4 \ensuremath{\theta_\mathrm{\scriptscriptstyle s}}}$$ (2.39)

  
  
  Using $\Delta \phi < 0.1$ radian (6$^\circ$) as an upper limit (note that this is the maximum phase error, i.e. the mean phase error is much smaller) result in the condition (with all angles in radian...):

  $$\ensuremath{\theta_\mathrm{\scriptscriptstyle f}}< \frac{1}{3} \sqrt{\ensuremath{\theta_\mathrm{\scriptscriptstyle s}}}$$ (2.40)

  
  

• Bandwidth Smearing
  Assume $u,v$ are computed for the center frequency $\nu_0$. At frequency $\nu_0$, we have

  $$V(u,v) \ensuremath{\rightleftharpoons}A I(x,y)$$ (2.41)

  
  
  The similarity theorem on Fourier pairs give

  $$V(\frac{\nu_0}{\nu} u, \frac{\nu_0}{\nu} v) = (\frac{\nu}{\nu_0})^2 I(\frac{\nu}{\nu_0} x ,\frac{\nu}{\nu_0} y)$$ (2.42)

  
  
  Averaging over the bandwidth $\Delta \nu$, there is a radial smearing equal to

  $$\sim \frac{\Delta \nu}{\nu_0} \sqrt{x^2+y^2}$$ (2.43)

  
  
  and hence the constraint

  $$\sqrt{x^2+y^2} \leq 0.1 \frac{\ensuremath{\theta_\mathrm{\scriptscriptstyle s}}\nu_0}{\Delta \nu}$$ (2.44)

  
  
  if we want that smearing to be less than 10% of the synthesized beam.

• Time Averaging
  Assume for simplicity that the interferometer observes the Celestial Pole. The baselines cover a sector of angular width $\Omega_e \Delta t$, where $\Omega_e$ is the Earth rotation speed, and $\Delta t$ the integration time. The smearing is circumferential and of magnitude $\Omega_e \Delta t \sqrt{x^2+y^2}$, hence the constraint

  $$\sqrt{x^2+y^2} \leq 0.1 \frac{\ensuremath{\theta_\mathrm{\scriptscriptstyle s}}}{\Omega_e \Delta t}$$ (2.45)

  
  
  For other declinations, the smearing is no longer rotational, but of similar magnitude.

To better fix the importance of such approximations, the relevant values for the Plateau de Bure interferometer are given in Table 2.1.

Table 2.1: Field of view limitations as function of angular resolution and observing frequency for the Plateau de Bure interferometer.

Config.	Resolution	Frequency	2-D	0.5 GHz	1 Min Time	Primary
		(GHz)	Field	Bandwidth	Averaging	Beam
Compact	5$''$	80 GHz	5$'$	80$''$	2$'$	60$''$
Standard	2$''$	80 GHz	3.5$'$	30$''$	45$''$	60$''$
Standard	2$''$	220 GHz	3.5$'$	1.5$'$	45$''$	24$''$
High	0.5$''$	230 GHz	1.7$'$	22$''$	12$''$	24$''$

Note that these fields of view correspond to a maximum phase error of 6$^\circ$ only, or to a (one dimensional) distortion of a tenth of the synthesized beam, and thus are not strict limits. In particular, atmospheric errors often results in larger errors (which are independent of the field of view, however).