# 3Linear and Planar Array Factor Synthesis

The array factor is a function of the amplitude and phase weights, the relative element positions, and the frequency. Values for these variables exist that yield a desirable array factor (as long as the laws of physics are obeyed). This chapter presents analytical, statistical, and numerical techniques to synthesize or optimize an array factor.

## 3.1. SYNTHESIS OF AMPLITUDE AND PHASE TAPERS

The techniques presented in this section are more academic than practical. They provide some insight into the design of low-sidelobe tapers, but the fact that amplitude and phase weighting are required and that the weights can significantly vary from element to element make them very difficult to implement with real hardware.

### 3.1.1. Fourier Synthesis

In Chapter 2, the array weights were shown to be coefficients of a Fourier series. These coefficients come from the inner product of the desired array factor with a single Fourier component.

where *m* is one of the *M* harmonics. The number of elements in the array is 2*M* (even) or 2*M* + 1 (odd). Note that *u* ranges from −1 to +1. The limits on the integral should also span this range. Thus,

If *d* > *λ*/2, then the limits ...