commonly used in practice. The effects of mutual coupling between elements will be ignored, and it is assumed that individual phase and amplitude, control of each element is available. There are various ways of implementing this control; this "beam-forming" process can be carried out at radio frequency or intermediate frequency or baseband. The Radiation Pattern of a Linear Array Figure 9.8 shows a linear array of equispaced antenna elements. First we will consider the radiation pattern of an array of (hypothetical) isotropic radiators, F(0). This pattern is called the array factor. Note that the angle, 0, is defined relative to the array axis, rather than the normal to the aperture. This is because the array line is an axis of symmetry for the array factor, which is then a function of O only. The array in Figure 9.8 contains N elements with spacing d. In direction O the path from the nth element to a distant point is nd COS O shorter than that from the 0th (reference) element to the same distant point. If the nth element is excited with amplitude lanl and phase On, the contribution from the element to the field at a great distance at the angle Ois proportional to lan exp(jQn). The total field is then given by equation 9.15, where an is a complex number equal to lanl exp(jQn). Fig. 9.8 Linear array of equispaced antenna elements. Adaptive Arrays F=Ya.exp(jnkd.cos8) (9.15) A useful method of analyzing linear arrays, from Schelkunoff, consists of associating a polynomial with an array. Let Z = exp(jkd cos 0). Then equation 9.15 is as shown in equation 9.16. (9.16) The radiation pattern (array factor) is F(z) = F(exp(jkd cos 0)). The complex coefficients of the polynomial F(z) are the excitations of the individual elements in the array. In the general case, equation 9.15 is often used directly to evaluate F(0). As an example, consider an array where all the elements are excited with unit amplitude and zero phase. This is the sampled equivalent of a uniformly excited linear aperture. The radiation pattern for N eleme