Lecture - Fundamental Concepts

Learning Objectives:

In order to estimate whether a source can be detected by a radio telescope or array, the so-called signal-to-noise ratio (SNR) has to be calculated. For a single telescope the signal is given by the measured antenna temperature $T_\text{A}$ and the noise is given by the radiometer equation

$\displaystyle \Delta T = \frac{C \cdot T_\text{sys}}{\sqrt{\Delta \nu \cdot \tau}} \text{,}$

in which $C$ is a dimensionless constant depending on the receiver system used, $T_\text{sys}$ is the system temperature, $\Delta \nu$ is the bandwidth of the receiver equipment and $\tau$ is the integration time.
In radio interferometry, the signal is given by the brightness distribution $B(\xi, \eta)$ of a source that must be calculated by the Fourier integral of the measured visibility $V(u, v)$:

$\displaystyle B(\xi, \eta) = \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} V(u,v) \cdot \text{e}^{\text{i}\cdot 2\pi \cdot (u \cdot \xi + v \cdot \eta)} \text{d}u ~\text{d}v \text{.}$

Therefore, to calculate the SNR in radio interferometry, one has to calculate the uncertainty of the brightness distribution $B$ from the uncertainty of the complex visibility $V$. As in every measurement, the measured data need to be sampled, meaning that the visibility is only measured at discrete locations $(u_\text{l},v_\text{l})$ along the tracks in the $(u,v)$-plane. Let: $\tau_\text{a}$ be the integration time of the individual samples in the $(u,v)$-plane, $\tau_0$ be the total integration time and $n_\text{a}$ be the total number of antennas. Then the total number of measured visibilities $n_\text{d}$ is given by

$\displaystyle n_\text{d} = n_\text{p} \cdot \frac{\tau_0}{\tau_\text{a}} \text{,}$

in which

$\displaystyle n_\text{p} = \frac{n_\text{a}\cdot (n_\text{a} - 1)}{2}$

is the number of antenna pairs or rather the number of individual two-element interferometers. Since this brightness distribution suffers from incomplete $(u,v)$-coverage, the resulting image is called a "dirty image", which is also indicated by the superscript "D" of the brightness distribution $B^\text{D}(\xi_\text{m}, \eta_\text{m})$. The visibilities are measured at discrete locations $(u_\text{l},v_\text{l})$, so the brightness distribution is given by the discrete Fourier transform

$\displaystyle B^\text{D}(\xi_\text{m}, \eta_\text{m}) = K \cdot \sum_{\text{l}=0}^{2\cdot n_\text{d}} V(u,v) \cdot S(u,v) \cdot \text{e}^{\text{i}\cdot 2\pi \cdot (u \cdot \xi_\text{m} + v \cdot \eta_\text{m})} \text{,}$

in which $K = \frac{1}{2\cdot n_\text{d} + 1}$ is a normalization constant and $S(u,v)$ is a sampling function which is given by

$\displaystyle S(u,v) = \sum_{\text{l}=0}^{2\cdot n_\text{d}} \delta^2(u - u_\text{l}, v - v_\text{l}) \text{.}$

This sampling function is only non-zero where the visibility is measured in the $(u,v)$-plane.
To suppress sidelobes, the beam shape of the observing array can be controlled by applying appropriate weights to the measured visibilities. Furthermore, if the observing array consists of telescopes that have different collecting areas $A_\text{eff}$, and different receivers with different system temperatures $T_\text{sys}$, frequency bandwidths $\Delta \nu_\text{IF}$ and integration times $\tau_\text{a}$, one can also apply weights to control these differences. The weights can be written as

$\displaystyle W(u,v) = \sum_{\text{l}=0}^{2\cdot n_\text{d}} R_\text{l} \cdot T_\text{l} \cdot D_\text{l} \cdot \delta^2(u - u_\text{l}, v - v_\text{l}) \text{,}$

in which $R_\text{l}$ accounts for different telescope properties,  $T_\text{l}$ is a taper which controls the beam shape and $D_\text{l}$ weights the density of the measured visibilities (more detailed information on the weights, especially on the $D_\text{l}$-factor, will be given in the next section 2.4.3). With these weights, the brightness distribution can then be written as

$\displaystyle B^\text{D}(\xi_\text{m}, \eta_\text{m}) = K \cdot \sum_{\text{l}=0}^{2\cdot n_\text{d}} V(u,v) \cdot S(u,v) \cdot W(u,v) \cdot \text{e}^{\text{i}\cdot 2\pi \cdot (u \cdot \xi_\text{m} + v \cdot \eta_\text{m})} \text{.}$

Because there is no measurement in the center of the $(u,v)$-plane, the $l=0$ terms of the dirty image $B^\text{D}(\xi_\text{m}, \eta_\text{m})$ vanish, leading to

$\displaystyle \begin{array}{rl} B^\text{D}(\xi_\text{m}, \eta_\text{m}) & = K \cdot \sum_{\text{l}=0}^{2\cdot n_\text{d}} V_\text{l} \cdot W_\text{l} \cdot \text{e}^{\text{i}\cdot 2\pi \cdot (u_\text{l} \cdot \xi_\text{m} + v_\text{l} \cdot \eta_\text{m})} \\ & = 2 \cdot K \cdot \sum_{\text{l}=1}^{n_\text{d}} W_\text{l} \cdot [ \Re V_\text{l} \cdot \cos(2\pi \cdot (u_\text{l} \cdot \xi_\text{m} + v_\text{l} \cdot \eta_\text{m})) - \Im V_\text{l} \cdot \sin(2\pi \cdot (u_\text{l} \cdot \xi_\text{m} + v_\text{l} \cdot \eta_\text{m}))] \text{,} \end{array}$

in which

$\displaystyle V_\text{l} = V(u,v) \cdot S(u,v) = \Re V_\text{l} + \text{i}  \cdot \Im V_\text{l}$

and

$\displaystyle W_\text{l} = W(u,v) \text{.}$

Note that the two additional terms $\Re V_\text{l} \cdot \sin[2\pi \cdot (u_\text{l} \cdot \xi_\text{m} + v_\text{l} \cdot \eta_\text{m})]$ and $\Im V_\text{l} \cdot \cos[2\pi \cdot (u_\text{l} \cdot \xi_\text{m} + v_\text{l} \cdot \eta_\text{m})]$ vanish under the integral of $B(\xi, \eta)$ and can therefore also be neglected for the sum of $B^\text{D}(\xi_\text{m}, \eta_\text{m})$.

To calculate the sensitivity in the image plane, a point source at the phase and image center is considered. For such a point source, the visibility $V$ is real and constant across the entire $(u,v)$-plane and only varies due to random noise. Furthermore, because $\xi_\text{m} = \eta_\text{m} = 0$, the $\sin[2\pi \cdot (u_\text{l} \cdot \xi_\text{m} + v_\text{l} \cdot \eta_\text{m})]$-term vanishes and $\cos[2\pi \cdot (u_\text{l} \cdot \xi_\text{m} + v_\text{l} \cdot \eta_\text{m})] = 1$. Therefore, the dirty image of such a point source is given by

$\displaystyle \begin{array}{rl} B^\text{D}(0,0) & = 2 \cdot K \cdot \sum_{\text{l}=1}^{n_\text{d}} W_\text{l} \cdot \Re V_\text{l} \\& = 2 \cdot K \cdot \sum_{\text{l}=1}^{n_\text{d}} W_\text{l} \cdot S_\text{l} \\& = 2 \cdot K \cdot S \cdot \sum_{\text{l}=1}^{n_\text{d}} W_\text{l} \text{,} \end{array}$

in which $S = S_\text{l} = \Re V_\text{l} = \Re V$ is the total flux density of the source. Note that here

$\displaystyle K = \frac{1}{2 \cdot \sum_{\text{l}=1}^{n_\text{d}} W_\text{l}} \text{,}$

so that

$\displaystyle B^\text{D}(0,0) = S \text{.}$

Assuming that the uncertainty of the flux density $\Delta S_\text{l} = \Delta S = \text{const}$, the noise in the dirty image is given by

$\displaystyle \Delta B^\text{D} = 2 \cdot K \cdot \Delta S \cdot \sqrt{\sum_{\text{l}=1}^{n_\text{d}} W_\text{l}^2} \text{.}$

Furthermore, assuming an observing array that consists of $n_\text{a}$ identical telescopes $(R_\text{l} = 1)$ and a naturally weighted $(D_\text{l} = 1)$, untapered $(T_\text{l} = 1)$ image, $\Delta B^\text{D}$ simplifies to

$\displaystyle \Delta B^\text{D} = \frac{\Delta S}{\sqrt{n_\text{d}}} \text{.}$

Since $S = \Re V = \Re V_\text{l}$, the uncertainty of the flux density $\Delta S$ is given by the rms noise $\sigma_\text{V}$ of the measured visibilities which is, based on the radiometer equation, given by

$\displaystyle \sigma_\text{V} = \frac{\sqrt{2} \cdot k \cdot T_\text{sys}}{A_\text{eff} \cdot \eta_\text{Q} \cdot \sqrt{\Delta \nu_\text{IF} \cdot \tau_\text{a}}} \text{,}$

in which $k$ is the Boltzmann constant and

$\displaystyle \eta_\text{Q} = \frac{\text{digital correlator sensitivity}}{\text{analogue correlator sensitivity}}$

is the quantization efficiency accounting for the quantization noise due to the conversion of analogue signals to digital signals.
Inserting $\Delta S = \sigma_\text{V}$ into $\Delta B^\text{D}$, the sensitivity for the synthesized image is finally given by

$\displaystyle \Delta B^\text{D} = \frac{\sqrt{2} \cdot k \cdot T_\text{sys}}{A_\text{eff} \cdot \eta_\text{Q} \cdot \sqrt{\frac{n_\text{a} \cdot (n_\text{a} - 1)}{2} \cdot \Delta \nu_\text{IF} \cdot \tau_0}} \text{.}$

Sampling

As previously mentioned, the visibilities are only measured at discrete locations $(u_\text{l}, v_\text{l})$. Therefore, the measured visibilities are given by the multiplication of the true visibility function $V$ and a sampling function $S$. Hence, the dirty image $B^\text{D}$ is then given by the convolution of their Fourier transforms:

$\displaystyle B^\text{D} = \mathcal{FT}[V^\text{S}] = \mathcal{FT}[V \cdot S] = \mathcal{FT}[V] \star \star \mathcal{FT}[S] \text{,}$

in which $\mathcal{FT}[x]$ denotes the Fourier transform of $x$ and the double-star $\star \star$ indicates a two-dimensional convolution.

Weighting

Furthermore, as also previously mentioned, the visibilities can be weighted by multiplying a weighting function $W$ to the visibilities $V^\text{S}$:

$\displaystyle V^\text{W} = W \cdot V^\text{S} = W \cdot V \cdot S \text{.}$

This weighting function $W$ consists of three individual factors:

• $R_\text{l}$ accounts for different telescope properties within an array (e.g. different $A_\text{eff}$, $T_\text{sys}$, $\Delta \nu$ and $\tau_\text{a}$),
• $T_\text{l}$ is a taper that controls the beam shape,
• $D_\text{l}$ weights the density of the measured visibilities.

The $D_\text{l}$-factor affects the angular resolution and sensitivity of the array. On the one hand side, the highest angular resolution can be achieved by weighting the visibilities as if they had been measured uniformly over the entire $(u,v)$-plane. Therefore, this weighting scheme is called uniform weighting. Since the density of the measured visibilities is higher at the center of the $(u,v)$-plane, the visibilities in the outer part are over-weighted leading to the highest angular resolution. On the other hand, the highest sensitivity is achieved if all measured visibilities are weighted by identical weights. This weighting scheme is called natural weighting. The main properties of both schemes are summarized in the following:

• natural weighting:
  • $D_\text{l} = 1$
  • broader synthesized beam
  • highest sensitivity

• uniform weighting:
  • $D_\text{l} = \frac{1}{n(\text{l})}$, in which $n(\text{l})$ is the number of visibilities occurring within an area of constant size around the weighted visibility
  • highest resolution
  • lowest sensitivity

With this weighting function $W$, the dirty image $B^\text{D}$ is given by

$\displaystyle B^\text{D} = \mathcal{FT}[V^\text{W}] = \mathcal{FT}[W \cdot V^\text{S}] = \mathcal{FT}[W] \star \star \mathcal{FT}[V^\text{S}] = \mathcal{FT}[W] \star \star (\mathcal{FT}[V] \star \star \mathcal{FT}[S]) \text{.}$

Gridding

To use the time advantage of the fast Fourier transform (FFT) algorithm, the visibilities must be interpolated onto a regular grid of size $2^{\text{M}_\text{V}}\times 2^{\text{N}_\text{V}}\,(\text{e.g.}\,256\times 256,\,512\times 512,\, 1024 \times 1024,\,...)$, resulting in an image of size $2^{\text{M}_\text{B}}\times 2^{\text{N}_\text{B}}$. This interpolation, also called gridding, is done at first by convolving the weighted discrete visibility $V^\text{W}$ with an appropriate function $C$ to obtain a continuous visibility distribution. This continuous visibility distribution is then resampled at points of the regular grid with spacings $\Delta u$ and $\Delta v$ by multiplying a two-dimensional Shah-function $G$, given by

Fig. 2.32 Illustration of the gridding process. The real $(u,v)$-tracks measured by the IRAM interferometer at Plateau de Bure in France (black dots) have to be interpolated onto a regular grid of $2^N\times2^N$ pixels (red dots). Taken from: U. Klein

$\displaystyle G(u,v) = \sum_{\text{j} = -\infty}^{\infty} \sum_{\text{k} = -\infty}^{\infty} \delta^2\left(j-\frac{u}{\Delta u}, k-\frac{v}{\Delta v}\right) \text{.}$

After these modifications, the visibility $V^\text{G}$ is given by

$\displaystyle V^\text{G} = G \cdot (C \star \star V^\text{W}) = G \cdot [C \star \star (W \cdot V^\text{S})] = G \cdot [C \star \star (W \cdot V \cdot S)]$

and the dirty image $\tilde{B^\text{D}}$ reads

$\displaystyle \tilde{B^\text{D}} = \mathcal{FT}[G] \star \star (\mathcal{FT}[C] \cdot \mathcal{FT}[V^\text{W}]) =  \mathcal{FT}[G] \star \star \{ \mathcal{FT}[C] \cdot [\mathcal{FT}[W] \star \star (\mathcal{FT}[V] \star \star \mathcal{FT}[S])]\} \text{.}$ 

This gridding process is illustrated in the Fig. 2.29, in which real $(u,v)$-tracks measured by the IRAM interferometer at Plateau de Bure in France are shown as black dotted ellipses. Furthermore, a regular grid, onto which the visibilities have been interpolated, is shown as red dots.

Since the sampling intervals $\Delta u$ and $\Delta v$ in the $(u,v)$-plane are inversely proportional to the sampling intervals $\Delta \xi$ and $\Delta \eta$ in the image plane ( $\Delta u^{-1} = M\Delta \xi$ and $\Delta v^{-1} = N\Delta \eta$ for a grid of size $M\times N$), the maximum map size in one domain is given by the minimum sampling interval in the other. Therefore, it is important to choose appropriate sampling intervals $\Delta u$ and $\Delta v$. If $\Delta u$ and $\Delta v$ are chosen to be too large for example, this will result in artefacts in the image plane produced by reflections of structures from the map edges, which is called aliasing.

The most effective way to deal with this aliasing is to use a convolution function $C$ for which the Fourier transform in the image plane $\mathcal{FT}[C]$ decreases rapidly at the image edges and is nearly constant over the image. Therefore, the simplest choice for the convolution function $C$ is a rectangular function, but with this choice the aliasing would be strongest. A better choice would be a sinc function, however a Gaussian-sinc function (product of a Gaussian and a sinc function) leads to the best suppression of aliasing.

Finally, the gridding modifications must be corrected after the Fourier transform by dividing the dirty image $\tilde{B^\text{D}}(\xi_\text{m},\eta_\text{m})$ by the inverse Fourier transform of the gridding convolution function $C(u,v)$. Therefore, the so-called grid-corrected image $\tilde{B^\text{D}_\text{C}}(\xi_\text{m},\eta_\text{m})$ is given by

$\displaystyle \tilde{B^\text{D}_\text{C}}(\xi_\text{m},\eta_\text{m}) = \frac{\tilde{B^\text{D}}(\xi_\text{m},\eta_\text{m})}{\mathcal{FT}[C](u,v)} \text{.}$

As already seen in Chapt. 2 Sect. 4.2, the sensitivity of radio-interferometrical measurements depends on the effective area of the telescopes $A_\text{eff}$ and the bandwidth $\Delta \nu_\text{IF}$. The larger $\Delta \nu_\text{IF}$, the higher the sensitivity. However, using a large bandwidth is problematic, since the Fourier relation between brightness distribution and visibility is only valid for monochromatic signals. Therefore, the effects of the finite bandwidth on the images obtained from radio-interferometrical observations, especially the inevitable effect called bandwidth smearing or chromatic aberration, will be investigated in this section, following the textbook of Taylor et al. (1999).

For a single observing frequency $\nu_0$ the brightness distribution is given by

$\displaystyle \tilde{B}(\xi,\eta) = \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\tilde{V}(u_0,v_0)\cdot \text{e}^{2\pi\text{i}\cdot(u_0\xi+v_0\eta)}\text{d}u_0\text{d}v_0\text{,}$

in which the "~" indicates the influence of a bandpass. The frequency-independent coordinates $u_0$ and $v_0$ are given by

$\displaystyle \begin{array}{rcl} u_0 = \frac{\nu_0}{\nu}u & \text{and} & v_0 = \frac{\nu_0}{\nu}v \text{,} \end{array}$

where $u$ and $v$ are the actual spatial coordinates of the visibility at the frequency $\nu$.

Therefore, using the generalized similarity theorem for Fourier transformations in $n$ dimensions, which reads

$\displaystyle \mathcal{FT}^n[f(\alpha\vec{x})] = \frac{1}{|\alpha|^n}\cdot F\left(\frac{\vec{s}}{\alpha}\right)\text{,}$

the two-dimensional Fourier relation between visibility and brightness distribution is given by

$\displaystyle B(\xi,\eta)=B\left(\xi_0\frac{\nu_0}{\nu},\eta_0\frac{\nu_0}{\nu}\right)\Leftrightarrow\left(\frac{\nu}{\nu_0}\right)^2\cdot V\left(u_0\frac{\nu}{\nu_0},v_0\frac{\nu}{\nu_0}\right)=\left(\frac{\nu}{\nu_0}\right)^2\cdot V(u,v)\text{.}$

Furthermore, since the smeared visibility $\tilde{V}(u_0,v_0)$ is obtained by rescaling and weighting the true visibility $V(u,v)$ by a normalized bandpass function $G_\text{n}(\nu')$, where $\nu'=\nu-\nu_0$, and then integrating over the frequency band, another important effect must be taken into account. There will be a delay error of

$\displaystyle \Delta\tau=\frac{u_0\xi+v_0\eta}{\nu_0}$

for signals arriving from a direction $(\xi,\eta)$ at frequency $\nu$. Therefore, the phase is shifted by

$\displaystyle 2\pi(\nu-\nu_0)\Delta\tau=2\pi\nu'\frac{u_0\xi+v_0\eta}{\nu_0}$

and the smeared visibility is given by

$\displaystyle \tilde{V}(u_0,v_0)=\int_{-\infty}^{\infty}V\left(u_0\frac{\nu}{\nu_0},v_0\frac{\nu}{\nu_0}\right)\left(\frac{\nu}{\nu_0}\right)^2G_\text{n}(\nu')\text{e}^{2\pi\text{i}\frac{\nu'}{\nu_0}(u_0\xi+v_0\eta)}\text{d}\nu'\text{.}$

For simplicity, a point source with unit amplitude located at $(\xi_0,0)$ is assumed without restricting generality. Then, the true visibility is given by

$\displaystyle V(u,v)=\text{e}^{-2\pi\text{i}u\xi_0}$

and the smeared visibility reads

$\displaystyle \tilde{V}(u_0,v_0)=\int_{-\infty}^{\infty}\text{e}^{-2\pi\text{i}u_0\frac{\nu}{\nu_0}\xi_0}\left(\frac{\nu}{\nu_0}\right)^2G_\text{n}(\nu')\text{e}^{2\pi\text{i}\frac{\nu'}{\nu_0}(u_0\xi+v_0\eta)}\text{d}\nu'\text{.}$

Furthermore, assuming that the bandwidth is sufficiently small, so that $(\frac{\nu}{\nu_0})^2\approx1$ (in practice, $0.95\lesssim(\frac{\nu}{\nu_0})^2\lesssim0.998$), the bandwidth-smeared brightness distribution is given by

$\displaystyle \begin{array}{rl} \tilde{B}(\xi,\eta) & =\int_{-\infty}^{\infty}\left[\int_{-\infty}^{\infty}{e}^{-2\pi\text{i}u_0\frac{\nu}{\nu_0}\xi_0}G_\text{n}(\nu')\text{e}^{2\pi\text{i}u_0\frac{\nu'}{\nu_0}\xi}\text{d}\nu'\right]\text{e}^{2\pi\text{i}u_0\xi}\text{d}u_0\delta(\eta) \\                                                           & =\int_{-\infty}^{\infty}\left[\int_{-\infty}^{\infty}{e}^{-2\pi\text{i}u_0\left(1+\frac{\nu'}{\nu_0}\right)\xi_0}G_\text{n}(\nu')\text{e}^{2\pi\text{i}u_0\frac{\nu'}{\nu_0}\xi}\text{d}\nu'\right]\text{e}^{2\pi\text{i}u_0\xi}\text{d}u_0\delta(\eta) \\                                                                                                                      & =\int_{-\infty}^{\infty}\text{e}^{2\pi\text{i}u_0(\xi-\xi_0)}\left[\int_{-\infty}^{\infty}G_\text{n}(\nu')\text{e}^{2\pi\text{i}u_0\frac{\nu'}{\nu_0}(\xi-\xi_0)}\text{d}\nu'\right]\text{d}u_0\delta(\eta)\text{,} \end{array}$

using $\frac{\nu}{\nu_0}=\frac{\nu_0+\nu-\nu_0}{\nu_0}=1+\frac{\nu'}{\nu_0}$. Here, one can see that the term in squared brackets is the Fourier transform  of the normalized bandpass function over $\nu'$, to an argument $\tau=\frac{u_0}{v_0}(\xi-\xi_0)$ that represents a delay corresponding to the positional offset $\xi-\xi_0$. Therefore, it is helpful to define a delay function $d(\tau)$, given by

$\displaystyle d(\tau) = \int_{-\infty}^{\infty} G_\text{n}(\nu')\text{e}^{2\pi\text{i}\tau\nu'}\text{d}\nu'\text{.}$

Using this delay function, the bandwidth-smeared brightness distribution then reads

$\displaystyle \tilde{B}(\xi,\eta)=\int_{-\infty}^{\infty}\text{e}^{2\pi\text{i}u_0(\xi-\xi_0)}d(\tau)\text{d}u_0\delta(\eta)\text{.}$

Therefore, the bandwidth-smeared brightness distribution is the Fourier transform over $u_0$ of the product of the true visibility with the delay function. Using the convolution theorem, the brightness distribution is given by

Fig. 2.33 Illustration of the effect of bandwidth smearing. Two signals that arrive from slightly different directions, $\xi_0$ and $\xi_0-\Delta\xi$, with slightly different frequencies can positively interfere, if their delay difference matches.

$\displaystyle \tilde{B}(\xi,\eta)=\int_{-\infty}^{\infty}\text{e}^{2\pi\text{i}u_0(\xi-\xi_0)}\text{d} u_0\delta(\eta) \star \int_{-\infty}^{\infty}d(\tau)\text{e}^{2\pi\text{i}u_0 \xi}\text{d} u_0\text{,}$

which is the convolution of the true image with a position-dependent bandwidth distortion function $D(\xi)$, which is the Fourier transform of the delay function

$d(\tau)$ over $u_0$. Since this distortion function varies with the radial distance $\xi_0$ from the phase center and is always oriented along the radius to the phase center, the final image of an extended source can be interpreted as a radially-dependent convolution. 

The effect of finite bandwidth is illustrated in Fig. 2.30. Here it is assumed that the observed emission of a source at position $\xi_0$ positively interferes exactly at frequency $\nu_0$.  A signal from a neighboring position shifted by $\Delta\xi$ with respect to $\xi_0$ measured by one telescope at a frequency deviating from $\nu_0$ by an amount of $\Delta\nu/2$, for example, can then also produce maximum interference with a signal from $\xi_0$ measured by the other telescope at frequency $\nu_0$, if the delay difference matches. Therefore, a source observed with finite bandwidth can also produce structures away from its nominal position. 

This finite bandwidth effect cannot be removed, because it is described by a mathematical functional. However, there are two methods to minimize the effects of bandwidth smearing. The first method is to limit the field size by dividing the observed area into subfields and produce separate images for all such subfields. To cover the complete area, the images of the subfields can then be fitted together. This technique is called mosaicing. The second method is to split the full frequency band into narrower sections, which will be summed up after imaging each individual data set. This method is called bandwidth synthesis.

The true visibility of a target source $V_\text{ij}^\text{target}(u,v)$ observed by two telescopes $i$ and $j$ of an array is related to the observed visibility $V_\text{ij}^\text{obs}(u,v)$ by

$\displaystyle V_\text{ij}^\text{obs}(u,v)=G_\text{ij}(t)\cdot V_\text{ij}^\text{target}(u,v)\text{,}$

in which $G_\text{ij}(t)$ is the time-variable and complex gain factor of the correlation product of the two telescopes $i$ and $j$. This gain factor includes variations caused by different effects, e.g., weather effects, effects of the atmospheric paths to the telescopes, and instrumental effects. Therefore, to determine the true visibility, the gain factor has to be calculated. This can be done via frequent observations of calibrator sources with known flux densities $S_\text{cal}$ and accurately known positions. Furthermore, these sources have to be point sources, meaning that their angular size $\Theta_\text{cal}$ is

$\displaystyle \Theta_\text{cal} \ll \frac{\lambda}{D_\text{max}}\text{.}$

Then, the complex gain $G_\text{ij}(t)$ of each interferometer can be determined by

$\displaystyle V_\text{ij}^\text{cal}(u,v)=G_\text{ij}(t)\cdot S_\text{cal}\text{,}$

in which $V_\text{ij}^\text{cal}(u,v)$ is the observed visibility of a calibrator source. This complex gain factor can then be used to calibrate the observed visibility of the target source. The true visibility of a target source $V_\text{ij}^\text{target}(u,v)$ is then given by

$\displaystyle V_\text{ij}^\text{target}(u,v)=\frac{V_\text{ij}^\text{obs}(u,v)}{G_\text{ij}(t)}=V_\text{ij}^\text{obs}(u,v)\cdot\frac{S_\text{cal}}{V_\text{ij}^\text{cal}(u,v)}\text{.}$

Note that the gains $G_\text{ij}$ can also be expressed by the voltage gain factors $g_\text{i}$ and $g_\text{j}$ of the individual i-th and j-th antennas by

$\displaystyle G_\text{ij} = g_\text{i}\cdot g_\text{j}^\star \,\text{.}$

This reduces the amount of calibration data, since there are many more correlated antenna pairs than antennas in large arrays. Furthermore, this makes the calibration procedure more flexible, since, for example, a source that is resolved at the longest baselines of an array can be used to compute the gain factors of the individual antennas from measurements made only at shorter baselines, at which the source appears unresolved.

Gibb's phenomenon

Fig. 2.34 Real and imaginary part of a frequency power spectrum for an idealized rectangular bandpass.

In continuum observations the correlation of the signals is usually measured for zero time lags, meaning that the correlation of the measured signal is averaged over the bandwidth, which is defined via a bandpass. In spectral line observations (see Chapt. 3 Sect. 4), measurements at different frequencies across the bandpass are required, which can be implemented by correlating the signals as a function of time lags. After Fourier transforming this correlation output, this yields the so-called cross power spectrum. This cross power spectrum is obtained for $K$ frequency channels (for more detailed information see Chapt. 3 Sect. 4).

To calibrate an interferometer over the bandpass, which represents the filter characteristics of the last IF stage, a continuum point source is used to determine the complex gains of the interferometer, which yields the true cross-correlation spectrum of the observed target source. For calibrator sources with a constant continuum spectrum over the bandpass, the behavior of the complex gains $g_\text{i}$ and $g_\text{j}$ across the bandpass $\Delta\nu$ are given by the correlated power as a function of frequency for each interferometer of the observing array. The real and imaginary parts of this correlated power, measured for a continuum source with a flat spectrum, are shown in Fig. 2.31 for an idealized rectangular bandpass. 

The observed visibility of an unknown target source is given by 

$\displaystyle V_\text{ij}^\text{obs}(\nu) = f(\nu)\star\left[g_\text{i}(\nu)\cdot g_\text{j}^\star(\nu)\cdot V(\nu)\right]\,\text{,}$

in which $f(\nu)$ is the Fourier transform of the truncating function in the frequency domain, which is a sinc function given by

$\displaystyle f(\nu) = \frac{\sin(\pi\cdot\nu\cdot K/\Delta\nu)}{\pi\cdot\nu}\,\text{.}$

Here, $V_\text{ij}^\text{obs}$ is given by a convolution with $f(\nu)$, since the cross-correlation has only a finite maximum lag of $2K\cdot\Delta\nu$. The observed visibility $V_\text{ij}^\text{cal}$ of a calibrator source with $|V| = 1$ and $\varphi_\text{V} = 0$ is then given by

$\displaystyle V_\text{ij}^\text{cal}(\nu)= f(\nu)\star\left[g_\text{i}(\nu)\cdot g_\text{j}^\star(\nu)\right]\,\text{.}$ 

Finally, the true visibility $V_\text{ij}^\text{true}$ of the observed target source is given by 

$\displaystyle  V_\text{ij}^\text{true}(\nu) = \frac{V_\text{ij}^\text{obs}(\nu)}{V_\text{ij}^\text{cal}(\nu)} = \frac{f(\nu)\star\left[g_\text{i}(\nu)\cdot g_\text{j}^\star(\nu)\cdot V(\nu)\right]}{f(\nu)\star\left[g_\text{i}(\nu)\cdot g_\text{j}^\star(\nu)\right]}\,\text{,}$

in which the gains do not cancel out, due to the convolution. Therefore, the original frequency power spectrum, shown in Fig. 2.31, is convolved with the sinc function $f(\nu)$, due to the finite lag, which will lead to large errors, if the target source has a spectral line near $\nu=0$. This is known as Gibbs' phenomenon.

Since the cross-correlation function has a complex power spectrum, Gibbs' phenomenon has strong influence on the imaginary part of the true visibility, because of the ratio in the equation for $V_\text{ij}^\text{true}$, which leads to non-negligible errors, especially for strong line emission near the step at $\nu=0$. Since the step in the imaginary part is large, these errors have a strong effect on the phases, producing strong ripples.