Lecture - Fundamental Concepts

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

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