The Fourier Transform Approach to Array Patterns

So far we have obtained the field pattern of an array by directly adding the electric field contributions from different elements. Now, it is well established that for a given aperture, if the electric field distribution across the aperture is known, then the radiation pattern can be obtained from a Fourier Transform of this distribution (see, for example, Christiansen & Hogbom 1985). This principle can also be used for computing the field pattern of an array. Consider the case of the array pattern for the 2-element array discussed earlier, as an example. The electric field distribution across the aperture can be taken to be zero at all points except at the location of the two elements, where it is a delta function for isotropic point sources. The Fourier Transform of this gives the sinusoidal oscillations in $\sin\theta$, which have also been inferred from eqn 6.2.2.

Using the Fourier Transform makes it easy to understand the principle of pattern multiplication described above. When the isotropic array elements are replaced with directional elements, it corresponds to convolving their delta function electric field distribution with the electric field distribution across the finite apertures of these directional elements. Since convolution of two functions maps to multiplication of their Fourier Transforms in the transform domain, the total field pattern of the array is naturally the product of the field pattern of the array with isotropic elements with the field pattern of a single element. The computational advantages of the Fourier Transform makes this approach the natural way to obtain the array pattern of two dimensional array telescopes having a complicated distribution of elements.