Published: 2026-05-24 Consider an antenna array composed of $N$ elements, where the $n$-th element is located at position $\mathbf{r}n$ and is excited with a complex weight $w_n = |w_n| e^{j\alpha_n}$, where $|w_n|$ is the amplitude and $\alpha_n$ is the phase of the excitation. If the array is illuminated by (or radiating toward) a far-field direction specified by the unit vector $\hat{r}$ (equivalently described by spherical angles $\theta, \phi$), the array factor is defined as: $$ AF(\theta, \phi) = \sum{n=0}^{N-1} w_n \, e^{j k \, \hat{r} \cdot \mathbf{r}n} $$ where $k = 2\pi/\lambda$ is the free-space wavenumber, $\lambda$ is the wavelength, and $\hat{r} \cdot \mathbf{r}_n$ is the projection of the element position onto the observation directionâphysically representing the path length difference (and hence the phase difference) between the $n$-th element and the coordinate origin. The total far-field pattern of the array is then given by the pattern multiplication principle: $$ \mathbf{E}{\text{total}}(\theta, \phi) = \mathbf{E}{\text{element}}(\theta, \phi) \cdot AF(\theta, \phi) $$ where $\mathbf{E}{\text{element}}$ is the radiation pattern of a single element (assumed identical for all elements in a uniform array). This factorization is exact when all elements are identical and identically oriented, and it reveals that the array factor governs the angular selectivity imposed by the array geometry and weighting alone. The complex weight $w_n$ applied to each element is the primary design lever for shaping the array factor's sidelobe behavior, main beam width, and null placement. Uniform Weighting. Setting $w_n = 1$ for all $n$ maximizes the directivity (in the sense of narrowest main beam for a given aperture), but produces the highest sidelobesâapproximately $-13.26\,\text{dB}$ relative to the main beam peak for large $N$. Tapered Amplitude Distributions. Windowing the element amplitudes according to classical tapering functions (Hann, Hamming, Blackman, Taylor, Chebyshev) progressively suppresses sidelobes at the expense of main beam broadening. The Dolph-Chebyshev distribution is optimal in the sense that it achieves the narrowest main beam for a specified uniform sidelobe level, or equivalently the lowest sidelobe level for a specified beamwidth. Taylor distributions are widely used in practice because they provide a practical compromise, with near-in sidelobes controlled to a specified level and far-out sidelobes decaying as $1/u$. Phase Weighting and Beam Steering. By applying a progressive phase shift $\beta_n = -k\,\hat{r}{\text{steer}} \cdot \mathbf{r}_n$, the main beam can be electronically steered to any desired direction $\hat{r}{\text{steer}}$ without physically moving the arrayâthis is the foundation of phased array technology. The array factor becomes: $$ AF(\theta,\phi) = \sum_{n=0}^{N-1} |w_n| \, e^{jk(\hat{r} - \hat{r}_{\text{steer}}) \cdot \mathbf{r}_n} $$ Adaptive Weighting. In adaptive beamforming (e.g., MVDR/Capon, LMS, or SMI algorithms), the weights are computed dynamically to place nulls in the directions of interference sources while maintaining gain toward a desired signal. The array factor in this context becomes data-dependent and time-varying, optimized in real time according to a statistical criterion. The array factor is the cornerstone quantity linking array geometry, element excitation weights, and the resulting directional characteristics of an antenna system. It provides a clean, analytically tractable framework for understanding beam formation, grating lobe conditions, sidelobe control, and beam steering. Its mathematical structureâa weighted sum of complex exponentialsâdirectly reveals the sensitivity of array performance to phase coherence and timing accuracy, making it indispensable for the design, calibration, and operation of modern phased arrays, MIMO systems, and spatially distributed sensors.