**2. Signal model**

The steered beam adaptive arrays are based on the concept of maximizing the desired-to-undesired (interference plus thermal white noise) signal ratio at the output of the array [2, 3]. Conceivably, this leads to placing nulls in the directions of the interference and noise while placing a maximum gain in the direction of the desired signal.

Consider the *N*-element adaptive array shown in **Figure 1**, receiving complex analytic signals<sup>1</sup> where the *<sup>i</sup>*th complex analytic signal is denoted by *xi*ð Þ*<sup>t</sup>* and the *<sup>i</sup>*th complex weight is denoted by *wi* . We define the signal vector,

$$X = \left[ \overline{\mathbf{x}}\_1(t), \overline{\mathbf{x}}\_2(t), \overline{\mathbf{x}}\_3(t), \dots, \overline{\mathbf{x}}\_N(t) \right] \tag{1}$$

This signal vector can be split into a desired signal term, an interference term, and a thermal noise term,

$$X = X\_d + X\_I + X\_n \tag{2}$$

The output signal of the array may be split in a similar way,

$$
\overline{\mathbf{S}}\_o(t) = \mathbf{X}^T \boldsymbol{w} = \overline{\mathbf{S}}\_d(t) + \overline{\mathbf{S}}\_l(t), \ + \overline{\mathbf{S}}\_n(t) \tag{3}
$$

The desired signal power, interference power, and noise power at the output of the array are then given as

$$P\_d = E\{ \left| \overline{\mathbf{S}}\_d(t) \right|^2 \} \tag{4}$$

$$P\_I = E\{ \left| \overline{\mathbf{S}}\_I(t) \right|^2 \} \tag{5}$$

$$P\_n = E\{ \left| \overline{\mathbf{S}}\_n(t) \right|^2 \}\tag{6}$$

where *E*{.} denotes expectation with respect to time. The steering vector adaptive array is based on the concept of maximizing the ratio of the desired signal

<sup>1</sup> Complex analytic signals correspond to real passband signals, as a result of complex baseband signal processing.

**Figure 1.** *Block diagram of an adaptive antenna array system.*

power *Pd* to the undesired signals powers *Pu* ¼ *PI* þ *Pn*; that is, to adjust the weights so that the quantity

$$\text{SINR} = \frac{P\_d}{P\_u} = \frac{P\_d}{P\_I + P\_n} \tag{7}$$

is maximized. In his paper [3], Applebaum proved that the feedback loop shown in **Figure 2** maximizes the output *SINR* of the array. More details on this are given in the next section.

In **Figure 2**, it should be noted that the *N*-element array has *N* feedback loops, where *w*0*<sup>j</sup>* is the *j*th component of the steering vector *w*0, and *k* is the feedback loop gain. In the absence of any incoming signals, the weight vector *w* is a scalar complex-valued multiple of *w*0. In this case, the array has a pattern determined by *w*0, and this pattern is called the quiescent pattern of the array [2, 3]. *w*<sup>0</sup> can be chosen to obtain a desired pattern from the array (i.e., it can be determined by the user's knowledge of the arrival angle of the desired signal). Because the vector *w*<sup>0</sup> steers the main beam of the quiescent antenna pattern, it is usually called the steering vector. This steering vector is vulnerable to errors. One kind of error is the pointing error, which is studied extensively in this research in order to mitigate its effect on the performance of the adaptive array.
