Summary
The rise in drone usage—from agriculture and delivery to surveillance and racing—has introduced major privacy and security challenges. Modern drones often use OFDM (Orthogonal Frequency Division Multiplexing) with Zadoff-Chu (ZC) sequences for synchronization. While powerful, detecting these sequences blindly (without knowing their parameters) remains a challenge.
Aim
This article presents a low-complexity solution to blindly detect ZC sequences used by unknown drones. The approach uses a novel double differential method that works without large correlation banks, making it efficient and real-time capable.
ZC Sequence Fundamentals
A ZC sequence of prime length P and root u is defined by:
x_u[n] = exp(-jπu·n(n+1)/P), 0 ≤ n < P- P is an odd prime (e.g., 599, 1021).
- Each root
ucreates a distinct ZC sequence. - ZC sequences have zero autocorrelation and low cross-correlation—ideal for synchronization.
Time Domain Detection with Unknown Root
The paper proposes using single and double differential operators to estimate the ZC root:
y_d[n] = y[n] * conj(y[n-1])This is equivalent to a conjugate delay multiply. It’s a way to estimate and remove frequency offset by exploiting the phase difference between adjacent samples.
y_dd[n] = y_d[n] * conj(y_d[n-1]) Another conjugate delay product — isolates residual modulation, often connected with root index in ZC sequences. These steps eliminate frequency offset and isolate the root information. The detection metric is:
γ_td[n] = ∑ y_dd[n] / sqrt(∑ |y_dd[n]|²)
û = round(P × angle(γ_td[n*]) / 2π)
Conjugate Delay Product
The conjugate delay multiplication is defined as:
\( y_d[n] = y[n] \cdot \text{conj}(y[n-1]) \)
Expanding assuming \( y[n] = A e^{j\omega n} \):
\( y_d[n] = A e^{j\omega n} \cdot A e^{-j\omega(n-1)} = A^2 e^{j\omega} \)
Key Idea:
- The result has a constant phase \( \omega \): this is the frequency.
- This operation isolates the frequency difference between adjacent samples.
Second Conjugate Delay (Double Differencing)
Applying the same idea again:
\( y_{dd}[n] = y_d[n] \cdot \text{conj}(y_d[n-1]) \)
This Second Differencing:
- Removes residual modulation — small leftover variations in phase.
- The remaining structure is often linked to properties like the Zadoff-Chu sequence’s root index \( u \).
SNR Bounds
The effective SNR of this method is:
SNR_td = (P - 2) · |h|⁸ / E[|w_dd[n]|²]Asymptotic bounds:
SNR_td ≥ (P - 2)/(8 + 2√10)Low SNR:
SNR_td ≥ (P - 2)·|h|⁸ / (2σ⁸)
Frequency Domain ZC Without Guard Band
When ZC sequences occupy all subcarriers, the detection is similar. Using inverse DFT properties and root inverses:
γ_fd1[n] = ∑ Y_dd[n] / sqrt(∑ |Y_dd[n]|²)
û⁻¹ = round(P × angle(γ_fd1[n*]) / 2π)
û = modular_inverse(û⁻¹, P)
SNR bounds remain the same as time domain.
Frequency Domain ZC with Guard Bands
In practical systems, edge subcarriers are unused. ZC detection must then work with only the middle subcarriers:
Y[k] = x_u[k - (P-1)/2] · exp(-j2πΔk/N) + W[k]
Y_dd[k] = exp(-2jπu/P)
γ_fd2 = ∑ Y_dd[k]
û = round(P × angle(γ_fd2[(P-1)/2]) / 2π)
CFO (Carrier Frequency Offset) has a stronger negative effect here than in time domain methods.
Simulation Results
Monte Carlo simulations (30,000 runs) verify theoretical predictions:
- Time and frequency domain methods reach 1% error rate at 15 dB SNR.
- CFO has minimal impact in time domain and full-bandwidth frequency domain.
- Guard-banded frequency domain suffers significantly under CFO.
Conclusion
The proposed double differential blind ZC detection method is a major step toward low-cost, real-time drone detection for consumer applications. It's lightweight, scalable, and resilient to synchronization challenges—ideal for future smart cities and surveillance tools.