#### 3.1. Received Data Model and Covariance Model

[19] In this section a single polarization point source telescope signal model is described [*Leshem et al.*, 2000]. This model includes a description of astronomical sources, additive interfering signals and noise. Assume that there are *p* telescope antennae, and suppose that the antenna signals *x*_{i}(*t*) are composed of *q*_{s} astronomical source signals, *q*_{r} interfering sources, and noise. Let the telescope output signals *x*_{i}(*t*) be stacked in a vector **x**(*t*)

[20] Further let **x**_{ℓ}^{s}(*t*) be the telescope array output signal corresponding to the ℓth astronomical source in the direction **s**_{ℓ}, let **x**_{k}^{r}(*t*) be the telescope array output signal corresponding to the *k*th interfering source in the direction **s**_{k}^{r}, and let **x**^{n}(*t*) be the noise vector. The resulting array output signal then can be expressed by

[21] The noise **x**^{n}(*t*) is independent identically distributed (i.i.d.) Gaussian noise, so it is uncorrelated between the array elements, or in other words spatially white at the aperture plane. The astronomical source signals also are assumed to be identically distributed Gaussian noise signals. The sources are assumed to be independent, or in other words spatially white at the celestial sphere.

[22] For the LOFAR ITS telescope experiments, we assume that the narrowband interferer model [*Leshem et al.*, 2000; *Whalen*, 1971] holds. This means that for narrowband signals with bandwidth Δ*f*, the condition

is valid, where τ denotes the geometrical signal time delay differences between the antenna elements. This condition implies that geometric time delay differences can be represented by phase shifts. For practical reasons, the frequency resolution for the observations discussed in this paper is 10 kHz, although the frequency resolution of LOFAR will be of the order of 1 kHz. The maximum geometric time delay across the array τ, is determined by the array size (200 m) and observation direction.

[23] Assume that there is an interferer with index k, with signal *y*_{k}^{r}(*t*). Because the narrowband condition holds, the telescope output signal **x**_{k}^{r}(*t*) can be written in terms of the array response vector **a**_{k}^{r}. Let the array response vector **a**_{k}^{r} be defined by

where **b**_{i0} is the location of the *i*th antenna with respect to an arbitrary reference location, λ the wavelength of the impinging signal, and *a*_{i}^{r} are the antenna gains in the direction **s**_{k}^{r}. This yields

Let the antenna directional gain vector **a**_{k}^{rg} be defined by **a**_{k}^{rg} = (*a*_{1}^{r}, ⋯ , *a*_{p}^{r})^{t}, and define ℛ = (**b**_{10}, ⋯ , **b**_{p0})^{t}, then **a**_{k}^{r} can be compactly written as

Here the vector represents the geometrical delay (phase) vector for the telescope antenna locations ℛ and the source direction **s**_{k}^{r}.

[24] Assume there are *q*_{r} interferers, and define **x**^{r}(*t*) by **x**^{r}(*t*) = Σ_{k}**x**_{k}^{r}(*t*), which also can be written as

[25] The interferer signal power σ_{k}^{2} is given by ℰ{*y*_{k}^{r}(*t*)} = σ_{k}^{2}, which leads to the following expression for the interference array covariance matrix **R**_{r}, dropping the time index t for **R**_{r}:

[26] The array response vector and the covariance matrix **R**_{s} for astronomical sources can be expressed in a similar way. Concerning the system noise, it can be represented by a diagonal noise matrix **D**, and is given by

where the σ_{i}^{2} is the noise power of the *i*th antenna without source or interferer contributions.

[27] The covariance matrix

can be expressed as

[28] Let **A**_{r} be defined by stacking the array response vectors for the interferers in a matrix

and stack the interfering source powers in a diagonal matrix **B**_{r}. For the astronomical sources the same definitions for **A**_{s} and **B**_{s} can be made. Using these definitions, the covariance matrix **R** can be expressed in a more compact form:

#### 3.2. Imaging and Beam Forming

[29] Traditionally [*Perley et al.*, 1994], the synthesized sky images are generated by fourier transforming the correlation data, here represented by the covariance matrix **R**. For the ITS station, the observed snapshots contain only a fairly limited number of spatial sample points. This implies that making sky images with the ITS station using beam forming is more practical than using spatial fourier transforms. Therefore the beam forming approach, used for ITS imaging, is discussed next.

[30] Assume that the complex gain of the array antenna elements can be adjusted by a multiplicative complex weight number *w*_{i} for each of the antenna elements *i*. Given the array output signal vector **x**(*t*) and a weight vector for the array **w** = (*w*_{1}, ⋯ , *w*_{p})^{t}, then the weighted-summed array output signal *y*(*t*) is given by

The beam former output power *P* is then given by

[31] For a classical or capon beam former we can define the weight vector, in terms of sky direction cosine coordinates (l,m):

where **a**(*l*, *m*) is the array response to signals from direction (*l*, *m*). The classical beam former is equivalent to direct fourier transforming or taking the fourier transform of all u,v data points without weighting. The sidelobe pattern of this beam former is shown in Figure 2.