Array Response and Steering Vectors

For a ULA with $N$ elements spaced $d$ apart, the steering vector for a plane wave arriving at angle $\theta$ from broadside is:

$$\mathbf{a}(\theta) = \begin{bmatrix} 1 \\ e^{j2\pi d\sin\theta/\lambda} \\ \vdots \\ e^{j2\pi(N-1)d\sin\theta/\lambda} \end{bmatrix}$$

def steering_vector(N, d_lambda, theta):
    """ULA steering vector. d_lambda = d/wavelength, theta in radians."""
    n = np.arange(N)
    return np.exp(1j * 2 * np.pi * d_lambda * n * np.sin(theta))

The array factor is the weighted sum of element responses:

$$AF(\theta) = \mathbf{w}^H\mathbf{a}(\theta)$$

For uniform weights ($w_n = 1/N$), the beam pattern is:

$$|AF(\theta)|^2 = \frac{\sin^2(N\pi d\sin\theta/\lambda)} {N^2 \sin^2(\pi d\sin\theta/\lambda)}$$

The main beam has width $\approx 2/(Nd\cos\theta_0/\lambda)$ radians.

A UPA with $N_h \times N_v$ elements has 2D steering:

$$\mathbf{a}(\theta, \phi) = \mathbf{a}_h(\theta) \otimes \mathbf{a}_v(\phi)$$

where $\mathbf{a}_h$ and $\mathbf{a}_v$ are ULA steering vectors for the horizontal and vertical directions, and $\otimes$ is the Kronecker product.

def upa_steering(Nh, Nv, d_lambda, theta, phi):
    ah = steering_vector(Nh, d_lambda, theta)
    av = steering_vector(Nv, d_lambda, phi)
    return np.kron(ah, av)

The half-power beamwidth (HPBW) of a ULA with $N$ elements at half-wavelength spacing is approximately:

$$\text{HPBW} \approx \frac{0.886 \lambda}{N d \cos\theta_0} \approx \frac{102°}{N}$$

for broadside ($\theta_0 = 0$) and $d = \lambda/2$.

The array gain of $N$ elements with coherent combining is:

$$G_{\text{array}} = N$$

or $10\log_{10}(N)$ dB. This is because the signal adds coherently (power $\propto N^2$) while noise adds incoherently (power $\propto N$).