2ρth-Order Subspace Constrained Optimization

Fewer Sensors,
More Targets

With M=8 sensors, COP resolves up to 28 sources — 4x beyond the classical M-1=7 limit. Underdetermined high-resolution DOA estimation for radar, sonar, and 5G.

Max Sources (M=8)

Beyond M-1 Limit

M=4

Min Sensors

0.1°

Resolution

Baselines Compared

📡

Underdetermined DOA

4th-order cumulant subspace enables resolving K > M-1 sources. Classical MUSIC/ESPRIT fail here.

🎯

High Resolution

Subspace constrained optimization yields super-resolution peaks. 0.1° separation achieved in simulation.

📈

Robust to Noise

Noise subspace projection + constrained optimization provides strong SNR resilience down to -5dB.

🚀

Real-Time Tracking

COP-DOA integrated with EKF/UKF/PHD filter for multi-target birth-death tracking scenarios.

💻

Cortex-M7 Ready

Optimized C implementation runs on STM32H7 @ 480MHz for embedded radar applications.

📚

IEEE TSP 2026

Full theoretical analysis with CRLB, convergence proof, and extensive Monte Carlo validation.

Interactive DOA Estimation

Adjust parameters and see how COP resolves more sources than sensors

Sensors (M) 8 Sources (K) 10 SNR (dB) 10 Rho (ρ) 2 Snapshots 500

Detection Rate

RMSE (°)

Max Capacity

Algorithm Comparison

COP vs classical and sparse DOA estimation methods (M=8, ULA)

Algorithm	Type	Underdetermined	Max Sources (M=8)	Resolution	Complexity
COP-4th (Proposed)	4th-order subspace	Yes	28	Super	O(M⁴)
MUSIC	2nd-order subspace	No	7	Super	O(M³)
ESPRIT	2nd-order subspace	No	7	Super	O(M³)
Capon/MVDR	Beamforming	No	Limited	Standard	O(M³)
L1-SVD	Sparse recovery	Yes	Grid-dep.	Grid-dep.	O(G·M²)
LASSO	Sparse recovery	Yes	Grid-dep.	Grid-dep.	O(G·M²)
COP-CBF	COP + Beamforming	Yes	28	Enhanced	O(M⁴)
COP-MVDR	COP + Adaptive BF	Yes	28	Super	O(M⁴)

Detection Rate vs K (Sources)

RMSE vs SNR

COP Pipeline

From raw sensor data to tracked DOA estimates

M Sensors

ULA / UCA

→

4th-Order
Cumulant

C_4x matrix

→

COP
Optimization

Subspace constrained

→

Peak
Detection

DOA estimates

→

EKF/PHD
Tracker

Multi-target

COP

COP-4th Order

Proposed Method

max J(θ) = a H (θ) U s U s H a(θ) s.t. a H (θ) U n U n H a(θ) \leq ε

✅

Virtual aperture expansion — 4th-order cumulant creates M²×M² virtual array.

✅

2ρ(M-1) max sources — 28 sources with M=8 at ρ=2.

✅

Noise-immune — Gaussian noise vanishes in 4th-order domain.

✅

Gridless — No grid discretization bias.

Classical MUSIC

2nd-Order Subspace

P(θ) = 1 / (a H (θ) U n U n H a(θ)) Requires K < M

❌

M-1 source limit — Only 7 sources with M=8.

❌

Fails underdetermined — Signal subspace estimation breaks down.

✅

Low complexity — O(M³) EVD only.

✅

Well understood — Mature theory since 1986.

Applications & Deploy

From defense radar to consumer audio — COP scales across domains

🛰

Missile Defense

Iron Dome-class radar: resolve dense missile swarms with minimal antenna elements.

Defense

📡

5G/6G Beamforming

Massive MIMO DOA for user localization with fewer RF chains. Cost reduction 4x.

Telecom

🎤

Smart Audio

4-mic array resolves 10+ speakers. Conference systems, smart home, hearing aids.

Consumer

🚢

Autonomous Vehicles

FMCW radar DOA for pedestrian/vehicle detection with compact antenna arrays.

Automotive

🌊

Sonar / Underwater

Submarine sonar with limited hydrophone arrays. Detect more vessels than elements.

Maritime

💻

Edge MCU

Cortex-M7 real-time implementation. STM32H7 @ 480MHz, INT16 fixed-point.

Embedded

        # Quick Start

        pip install numpy scipy matplotlib

        from iron_dome_sim.doa import SubspaceCOP, MUSIC

        from iron_dome_sim.signal_model.array import UniformLinearArray

        ula = UniformLinearArray(M=8)

        cop = SubspaceCOP(ula, rho=2)

        spectrum = cop.estimate(X)  # K=28 sources from M=8 sensors!

Fewer Sensors,More Targets