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Taking a more realistic approach, we must relax the assumption of quasi-monochromatic radiation. In reality, interferometers observe over a finite bandwidth or frequency range ∆ν. To compensate we must integrate the interferometer response over this bandwidth ∆ν centred at frequency ν0 as follows:

Rc= Iν Z ν0+ν/2 ν0−ν/2 cos(2πντg)dν = Iνsinc(∆ντg) cos(2πντg)dν (2.9)

The introduction of the sinc(x) function, often referred to as the ‘bandwidth pattern’, attenuates the source emission (Thompson, 1999). This attenuation will be significant if the source offset from the phase centre is comparable to the interferometer resolution divided by the fractional bandwidth. Thus when observing over a large bandwidth it is best to observe over multiple, narrow channels in order to avoid such ‘bandwidth attenuation’. A further consequence of observing over a finite bandwidth is that the sensitivity of the interferometer is not uniform over the sky, i.e. sources not on the plane orthogonal to the interferometer baseline will not be observed with full coherence. Therefore, it is necessary to add a time delay in order to shift the maximum of the ‘sinc’ pattern (or fringe attenuation pattern) to the centre of the source. For reference Figure 2.3 shows a schematic of the additional time delay τ0 in the simplest case of the two element interferometer. In running

Figure 2.3: A schematic illustration of a pair of antennas separated by a baseline length b receiving a wavefront from the general direction described by the unit vector ~s as shown in Figure 2.2. This schematic also shows the additional time delay, τ0 added to the received signal at each antenna

to shift the fringe attenuation pattern to the reference (delay) direction described by the unit vector ~s0.

through the same steps in deriving the complex visibility function as before but with this additional time delay gives:

V (τg) = Rc− iRs = Z Ω Iν(τg)e−i2πν(τ0−τg)dΩ (2.10)

It is necessary to continuously track the source by changing this time delay τ0 such that

the fringe pattern moves along with the source as the Earth rotates. This must be done to an accuracy of δτ << 1/∆ν in order to minimise the effect of bandwidth attenuation. In expressing the complex visibility function defined in Equation2.10 as a function of the u, v, w and l, m, n coordinate systems where the w axis points towards the reference centre ~s0, we find that:

ντ0 =

ν~b · ~s0

2.1. An introduction to radio interferometric theory 93

Inserting this expression, in combination with Equation2.7, into Equation2.10means that the 3-D Fourier transform equation becomes:

V (u, v, w) = Z l Z m Z n Iν(l, m, n)e−i2π[ul+vm+w( √ 1−l2_−m2_−1)] dldm √ 1 − l2_{− m}2 (2.12)

In order to simplify the inverse Fourier transform of this equation, whereby Iν(l, m), the

source brightness on the sky is obtained from the measured visibility, it is often reduced into the form of a 2-D Fourier transform. This is done under the condition that w = 0 when the baselines are considered to be coplanar, or such that l and m are sufficiently small such that w(√1 − l2_{− m}2_{− 1) ≈ 0. Thus, Equation 2.12reduces to:}

V (u, v) = Z +∞ −∞ Z +∞ −∞ Iν(l, m)e−i2π(ul+vm)dldm (2.13)

The interferometer observes visibilities Vν(u, v) from which one can inverse Fourier trans-

form Equation2.13to obtain the observed sky brightness of the source in question, Iν(l, m).

However, note that the technique used in radio interferometry is known as ‘aperture syn- thesis imaging’ because the interferometer synthesises an aperture of diameter equal to that of its largest baseline. Thus, when an interferometer observes a source it is only sampling part of the sky, since the visibility function is measured at particular places in the u, v plane. This sampling is described by the sampling function S(u, v) that equates to zero at those u, v coordinates where no data is taken. The Fourier transform of the sampled visibility data is known as the ‘dirty image’ and is written mathematically as:

I_νD(l, m) = Z Z

Vν(u, v)S(u, v)e2iπ(ul+vm)dudv

= Iν(l, m)

O

B(l, m)

(2.14)

where the bottom half of above equation states that the true image of the sky I_ν(l, m), when convolved with the synthesised beam B(l, m) (note that the symbolN denotes convolution) gives the dirty image I_νD(l, m). The synthesised beam, B(l, m) is often referred to as the Point Spread Function (PSF) or indeed the dirty beam. It follows that whilst the inverse

Fourier transform of Equation2.13yields the true sky image I_ν(l, m), the synthesised beam B(l, m) is obtained via the inverse Fourier transform of the sampling function, i.e.

B(l, m) = Z +∞

−∞

Z +∞

−∞

S(u, v)e−π(ul+vm)dudv (2.15)

The rotation of the Earth is what is used to improve the u, v coverage of the sampling function. As the Earth rotates around its axis, the orientation of each baseline associated with the interferometer changes with respect to the source, filling out the u, v plane. This results in lower side-lobes (see Figure 2.1) within the sampling function and produces a better image on the sky. The most common method used for the imaging and deconvolution process is the CLEAN algorithm (see Section2.4.1) although others such as the Maximum Entropy Method (MEM) also exist.

Another assumption in need of relaxing is that our antennas are equally sensitive to all parts of the sky. In reality, antennas are point probes that simply measure the voltage of the incoming electromagnetic waves at a specific point on the sky. Since, they are dishes, they are directional elements and in order to produce accurate images of the sky, it is necessary to include a factor, Aν(l, m) that describes the normalised antennas response

pattern as a function of the directional cosines l, m (i.e. the primary beam pattern, see Figure 2.1). If all the antennas of the interferometer are identical and their reception patterns are constant in time, the antennas response A_ν(l, m) can be inserted into the 2-D Fourier transform equation (i.e. Equation2.13):

Vν(u, v) =

Z +∞

−∞

Z +∞

−∞

Aν(l, m)Iν(l, m)e−2iπ(ul+vm)dldm (2.16)

Thus, the final sky image is obtained by dividing the true image of the sky Iν(l, m), by the

antenna response pattern Aν(l, m). The above outlines the theoretical framework behind

synthesis imaging. The following sections will describe the more specific steps taken in the treatment, calibration and imaging of the e-MERLIN COBRaS L-band data set.