Array Fundamentals#

This section derives the key equations for phased array analysis.

Array Factor Derivation#

Consider an array of N identical elements located at positions \(\vec{r}_n = (x_n, y_n, z_n)\). For a plane wave arriving from direction \((\theta, \phi)\), the array factor is:

\[AF(\theta, \phi) = \sum_{n=1}^{N} w_n \exp\left(jk \vec{r}_n \cdot \hat{u}\right)\]

where:

  • \(w_n\) is the complex weight for element n

  • \(k = 2\pi/\lambda\) is the wavenumber

  • \(\hat{u} = (\sin\theta\cos\phi, \sin\theta\sin\phi, \cos\theta)\) is the unit vector toward the observation direction

For a planar array in the xy-plane (\(z_n = 0\)):

\[AF(\theta, \phi) = \sum_{n=1}^{N} w_n \exp\left(jk(x_n u + y_n v)\right)\]

where \(u = \sin\theta\cos\phi\) and \(v = \sin\theta\sin\phi\) are direction cosines.

Uniform Linear Array#

For a uniform linear array (ULA) along the x-axis with spacing d:

\[x_n = (n - 1) d, \quad n = 1, 2, \ldots, N\]

With uniform weights (\(w_n = 1\)):

\[AF(u) = \sum_{n=0}^{N-1} \exp(jknd \cdot u) = \frac{\sin(Nkdu/2)}{\sin(kdu/2)}\]

This is the classical array factor with:

  • Main beam at \(u = 0\) (broadside)

  • First null at \(u = \pm \lambda/(Nd)\)

  • Half-power beamwidth \(\approx 0.886\lambda/(Nd)\)

Beam Steering#

To steer the main beam to direction \((u_0, v_0)\), apply progressive phase shifts:

\[w_n = \exp\left(-jk(x_n u_0 + y_n v_0)\right)\]

This cancels the phase variation at the desired direction, making all elements add coherently. The resulting array factor becomes:

\[AF(u, v) = \sum_{n=1}^{N} \exp\left(jk[x_n(u-u_0) + y_n(v-v_0)]\right)\]

The pattern shifts in (u, v) space to center on \((u_0, v_0)\).

Pattern Multiplication#

The total radiation pattern of an array is the product of:

  1. Element pattern \(E_e(\theta, \phi)\): Individual element’s directivity

  2. Array factor \(AF(\theta, \phi)\): Interference pattern from element positions

\[E_{total}(\theta, \phi) = E_e(\theta, \phi) \cdot AF(\theta, \phi)\]

This separability assumes:

  • Identical elements

  • Negligible mutual coupling

  • Elements small compared to wavelength

Grating Lobes#

Grating lobes are additional main beams that appear when the array factor is periodic in \((u, v)\) space. For a rectangular grid with spacing \((d_x, d_y)\), grating lobes appear at:

\[(u_{gl}, v_{gl}) = \left(u_0 + \frac{m\lambda}{d_x}, v_0 + \frac{n\lambda}{d_y}\right)\]

for integers m, n.

A grating lobe enters the visible region (\(u^2 + v^2 \leq 1\)) when:

\[d > \frac{\lambda}{1 + |\sin\theta_{max}|}\]

For \(d = \lambda/2\), grating lobes stay outside the visible region for all scan angles.

Directivity#

Array directivity is the ratio of peak radiation intensity to average intensity:

\[D = \frac{4\pi |AF_{max}|^2}{\int_0^{2\pi}\int_0^{\pi} |AF(\theta,\phi)|^2 \sin\theta \, d\theta \, d\phi}\]

For a uniform rectangular array with N elements at \(\lambda/2\) spacing:

\[D_{max} \approx \frac{4\pi A}{\lambda^2} = \pi N\]

where A is the physical aperture area.

Beamwidth#

The half-power beamwidth (HPBW) for a uniform rectangular aperture of dimensions \(L_x \times L_y\):

\[\theta_{HPBW} \approx \frac{0.886\lambda}{L} \text{ (radians)}\]

For an N-element array with spacing d:

\[\theta_{HPBW} \approx \frac{0.886\lambda}{Nd} = \frac{0.886}{N} \cdot \frac{\lambda}{d}\]

At \(\lambda/2\) spacing:

\[\theta_{HPBW} \approx \frac{1.77}{N} \text{ radians} = \frac{101°}{N}\]

Scan Loss#

When the beam is steered away from broadside, two effects reduce gain:

  1. Projected aperture: Effective area decreases as \(\cos\theta\)

  2. Element pattern: Element gain typically decreases off-axis

Combined scan loss:

\[L_{scan}(\theta) \approx \cos^n\theta\]

where n depends on the element pattern (typically 1.3-1.5 for patch elements).

FFT-Based Computation#

For a uniform rectangular array, the array factor can be computed efficiently using the 2D FFT. If weights are arranged on a grid \(W[m, n]\):

\[AF(u, v) = \text{FFT2}\{W[m, n]\}\]

This is \(O(N \log N)\) instead of \(O(N \cdot M)\) for direct computation over M observation angles.

The FFT output corresponds to direction cosines:

\[u_k = \frac{k \lambda}{N_x d_x}, \quad v_l = \frac{l \lambda}{N_y d_y}\]