Direction of Arrival (DoA) Online Simulator

Interactive DOA Simulator

X-axis

XY angle (deg): 45

XZ angle (deg): 30

Noise: 0.05

Y-axis

XY angle (deg): 60

YZ angle (deg): 45

Noise: 0.05

Z-axis

XZ angle (deg): 60

YZ angle (deg): 30

Noise: 0.05

Estimated DOA (deg): 0

Simulation Workflow and Mathematical Background

This simulator demonstrates Direction of Arrival (DOA) estimation using three-axis sensor signals (X, Y, Z), Maximal Ratio Combining (MRC), and the MUSIC algorithm. It allows interactive control of signal angles and noise for teaching purposes.

1. Signal Generation

A pure sinewave signal of frequency f is projected onto three axes using user-defined angles in different planes:

• X-axis: θ_XY, θ_XZ
• Y-axis: θ_XY, θ_YZ
• Z-axis: θ_XZ, θ_YZ

Mathematically, for each time sample t:

x(t) = s(t) * cos(θ_xy_x) * cos(θ_xz_x) + n_x(t)
y(t) = s(t) * sin(θ_xy_y) * cos(θ_yz_y) + n_y(t)
z(t) = s(t) * sin(θ_xz_z) * sin(θ_yz_z) + n_z(t)
  

where n_x, n_y, n_z are zero-mean noise signals (Gaussian or uniform) added to simulate real-world conditions.

2. Maximal Ratio Combining (MRC)

MRC combines the three sensor signals into a single enhanced signal. Each component is weighted proportionally to its power:

w_x = |x|^2 / (|x|^2 + |y|^2 + |z|^2)
w_y = |y|^2 / (|x|^2 + |y|^2 + |z|^2)
w_z = |z|^2 / (|x|^2 + |y|^2 + |z|^2)

rx_MRC(t) = w_x * x(t) + w_y * y(t) + w_z * z(t)
  

This emphasizes components with higher signal strength, improving SNR before DOA estimation.

3. DOA Estimation Concept

The simulator estimates the DOA based on projections of the sensor signals in different planes. In the simplest form:

θ_DOA = atan((y component) / (z component))
  

For vector sensors, this is generalized to 3D using azimuth and elevation angles.

4. MUSIC Algorithm

The MUSIC (MUltiple SIgnal Classification) algorithm estimates DOA with high resolution by exploiting the subspace of the covariance matrix:

R = E[x x^H]  // sample covariance matrix of received signals
R = E_s Λ_s E_s^H + E_n Λ_n E_n^H  // eigen-decomposition
P_MUSIC(θ) = 1 / (a^H(θ) E_n E_n^H a(θ))  // pseudo-spectrum

Peaks in P_MUSIC(θ) correspond to estimated DOA angles.

5. Simulation Workflow

1. User sets angles θ_XY, θ_XZ, θ_YZ for X, Y, Z axes.
2. Pure sinewave signals are projected onto each axis with optional noise.
3. MRC combines the signals into one enhanced signal.
4. Covariance matrix of combined signals is computed.
5. MUSIC algorithm scans angles and produces pseudo-spectrum.
6. DOA is estimated at the angle corresponding to the pseudo-spectrum peak.
7. All signals and pseudo-spectrum are plotted in real-time for teaching and visualization.

This interactive approach helps students understand:

• How sensor orientation affects received signals.
• How noise impacts DOA estimation.
• How MRC improves SNR before estimation.
• How MUSIC provides high-resolution DOA detection.