The Array Factor
We'll now derive the most important function in array theory  the Array Factor. Consider a set
of N identical antennas oriented in the same direction, each with radiation pattern given by:
Assume that element i is located at position given by:
Suppose (as in Figure 4 here) that the signals from the elements in the antenna array are each multiplied by a complex weight () and then summed together to form the phased array output, Y. The output of the antenna array will vary based on the angle of arrival of an incident plane wave (as described here). In this manner, the array itself is a spatial filter  it filters incoming signals based on their angle of arrival. The output Y is a function of , the arrival angle of a wave relative to the array. In addition, if the array is transmitting, the radiation pattern will be identical in shape to the receive pattern, due to reciprocity. Y can be written as:
where k is the wave vector of the incident wave. The above equation can be factor simply as:
The quantity AF is the Array Factor. The Array Factor is a function of the positions of the antennas in the array and the weights used. By tayloring these parameters the antenna array's performance may be optimized to achieve desirable properties. For instance, the antenna array can be steered (change the direction of maximum radiation or reception) by changing the weights. Using the steering vector, the Array Factor can be written compactly as:
In the above Array Factor equation, T is the transpose operator. We'll now move on to weighting methods (selection of the weights) used in antenna arrays, where some of the versatility and power of antenna arrays will be shown.
Side Note: If the elements are identical (antenna array made up of all the same type of antennas), and have the same physical orientation (all point or face the same direction), then the radiation (or reception) pattern for an antenna array is simply the Array Factor multiplied by the radiation pattern . This concept is known as pattern multiplication.
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