# Demo — Phased Array Simulator
This is the interactive phased-array simulator we use in class, embedded here for self-study. It builds the array factor for a linear or planar array, steers the beam with element phasing, and visualizes the result five ways: a pattern plot, a 2D heatmap, the element phase fronts, a polar cut, and a 3D hemisphere. Use it to *see* the ideas from the reading — beam steering, beamwidth, side lobes, and grating lobes.
## The idea
Each element radiates the same signal with a programmed phase. Sloping the phase across the array tilts the wavefront and points the main lobe at the steering angle $\theta_0$:
$$
\Delta\phi = \frac{2\pi}{\lambda}\,d\,\sin\theta_0.
$$
Keep the spacing $d \le \lambda/2$ to avoid grating lobes at broadside.
## Interactive demo
Open in full screen
## Walkthrough
Work the **Pattern** tab first, then **Polar Cut**, then **Planar**, then **3D Hemisphere**.
1. **Baseline.** Linear, $M = 8$, $d_x/\lambda = 0.50$, $\theta_0 = 0$. Note the main lobe at boresight and the first side lobes near $-13$ dB (uniform illumination).
2. **Steer it.** Push $\theta_0$ to $30^\circ$, then $60^\circ$. The main lobe moves to the steering angle and **broadens** by about $1/\cos\theta_0$.
3. **Read the numbers.** Switch to **Polar Cut** and read HPBW and SLL directly. At $N = 8$, $d = \lambda/2$, boresight, HPBW is about $12.7^\circ$ — the in-class MATLAB anchor.
4. **Break it.** Raise $d_x/\lambda$ from $0.50$ to $0.80$. **Grating lobes** appear. EW implication: a poorly spaced array leaks full-strength energy in unintended directions.
5. **Go planar.** Switch to **Planar** ($M = 8$, $N = 8$, $d_x = d_y = \lambda/2$) and steer $\theta_0$ and $\phi_0$. The **2D Heatmap** shows the beam wandering across the hemisphere.
6. **See the cone.** Open the **3D Hemisphere** tab. Drag to rotate and confirm the focused cone with its surrounding side-lobe rings.
7. **Add the element factor.** Toggle **Element factor (cos θ)**. The distant side lobes pull back toward boresight — real elements have their own pattern that multiplies the array factor.
## Key observations
- **Uniform illumination → ≈ −13 dB side lobes.** Tapering would lower them at the cost of a wider main beam.
- **HPBW broadens as $1/\cos\theta_0$.** Steering to $60^\circ$ roughly doubles the beamwidth and costs about 3 dB of gain.
- **Grating lobes appear for $d > \lambda/2$ at broadside** — the angular-domain Nyquist limit.
- **A planar array's beam is a 3D cone**, steerable in both azimuth and elevation.
## Source
MATLAB · code/L6_ArrayFactorAndSteering.m↓
The in-class script builds the array factor for an $N$-element linear array, plots it in polar dB, steers the beam to $0^\circ$, $30^\circ$, and $60^\circ$ to watch the HPBW broaden, and marks the first side-lobe level to confirm the $-13.2$ dB rule.