Phased antenna array design using Hamiltonian Monte Carlo

Phased antenna arrays allow a radio or microwave beam to be steered electronically. They are of increasing practical importance. Phased array optimisation problems are non-convex with multiple local maxima and saddle points. Traditional gradient based optimisation methods fail badly. On the other hand heuristic approaches such as genetic algorithms (GA) work well. They have been widely used in practice.

Here I explore an alternative approach to phased array design using Hamiltonian Monte Carlo (HMC). “Good” designs can be found relatively quickly even for large arrays. Further optimisation can be carried out using GA if required.

For definiteness, assume the array elements are arranged on a $D \times D$ square grid oriented in the xy plane, with grid spacing $a$ equal to one quarter of the radar wavelength. Beam power far from the array in direction $\theta,\phi$ is proportional to the square modulus of the array factor:

(1) $\begin{equation*}A(\theta,\phi) = \sum_{\mathbf r} e^{i ({\bf k} \cdot {\bf r} + \psi_{\mathbf r}) }\end{equation*}$

${\mathbf r}$ are the array element locations and ${\bf k} = \frac{\pi}{2 a} (\sin(\theta) \cos(\phi), \sin(\theta) \cos(\phi))$ is the xy-component of the emitted beam wave-vector. Equation (1) produces complicated interference patterns. The angular distribution of emitted power $|A(\theta,\phi)|^2$ can be altered by adjusting phases $\psi_{\bf r} \in (0,2\pi)$ electronically. For example, power in direction $\theta_0,\phi_0$ can be maximised by choosing $\psi_{\bf r} = -{\bf k} \cdot {\bf r}$ . This gives $|A(\theta_0,\phi_0)| = D^2$ the maximal possible value.

Suppose that the array operator wants to concentrate power in $N$ directions simultaneously e.g. for $N=3$ maximise $| A(\theta_0,\phi_0)| \sim| A(\theta_1,\phi_1)| \sim|A(\theta_2,\phi_2)|$ . Non-linearity of Equation (1) means that a solution to the $N$ -target problem is not a simple superposition of solutions for each direction independently. Some kind of optimisation search over $\{\psi_{\mathbf r} \}$ is needed.

Assume $N=3$ . Introduce an “energy” function,

(2) $\begin{equation*}E(\{\psi_{\bf r}}\})= E_0- \left( | A(\theta_0,\phi_0)| | A(\theta_1,\phi_1)| | A(\theta_2,\phi_2)|\right)^{\frac{1}{3}}\end{equation*}$

and a probability distribution function

(3) $\begin{equation*}P(\{\psi_{\bf r}}\}) \propto \exp(-\beta E(\{\psi_{\bf r}}\}))\end{equation*}$

Equations (1),(2) & (3) convert 3-target optimisation into a statistical physics problem with some complicated implied interactions between the phase parameters. It can be tackled using Markov Chain Monte Carlo (MCMC). When $\beta$ is large, thermalised samples obtained from MCMC are low energy, corresponding to high geometric mean of power in the target directions.

How well does MCMC work in practice? Representing phases by arrows, the best HMC results achieved using stan are shown below for a large array of 4096 elements ( $D=64$ ).

HMC optimised arrayfactor is shown below. Plots on the left are on a linear scale and those on the right display the same data on a log scale (dB).

These HMC results were obtained after only 400 warmup (thermalisation + adaptation) steps!

The best geometric mean obtained using finite temperature HMC for the $D=64$ array was 2033 or a ratio 2033/ $64^2$ = 0.496. Applying GA to the HMC result improved this ratio to 0.512. The HMC+GA optimised phased array is shown below. It differs slightly from the HMC optimised array with overlap 0.96.

GA and rstan R packages were used.

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