Symbol Detection in Ambient Backscatter Communications Under Residual Time Synchronization Errors

Ambient backscatter communications (AmBC), where a backscatter transmitter (BT) modulates and reflects ambient signals to a backscatter receiver (BR), have been deemed a low-power communication technology for the Internet of Things. Previous work on symbol detection in AmBC assumed perfect time synchronization (TS), which is unrealistic in practice. The residual TS errors (RTSE) cause \emph{partial sample mismatch}, degrading symbol detection performance. To address this, we propose a new AmBC symbol detection framework that incorporates the BT’s current and adjacent symbols, as well as channel coefficients. Using energy detector (ED) as a case study, we derive both exact and approximate bit error rate (BER) expressions. Our results show that the ED’s BER performance degrades significantly under RTSE, with the symbol detection threshold optimized under the assumption of perfect TS. We then derive a closed-form expression for a near-optimal symbol detection threshold that minimizes BER under RTSE. To estimate the required parameters for the detection threshold, we propose a novel method exploiting the attributes of the BR’s received signal samples. The analytical results are verified by simulation results.

💡 Research Summary

This paper addresses a critical gap in the study of ambient backscatter communications (AmBC): the impact of residual time‑synchronization errors (RTSE) on symbol detection. While prior works on AmBC symbol detection—ranging from maximum‑likelihood (ML) detectors to energy detectors (ED) and p‑norm based methods—have all assumed perfect time synchronization between the backscatter transmitter (BT) and the backscatter receiver (BR), real‑world deployments inevitably suffer from timing offsets. These offsets cause a “partial sample mismatch” where the BR’s sampling window for a given BT symbol contains samples belonging to adjacent symbols. The authors model this phenomenon rigorously, enumerate all eight possible mismatch configurations (four for negative offset, four for positive offset), and incorporate the current and adjacent BT symbols together with the channel coefficients into a unified detection framework.

The system model consists of a single ambient RF source (S), a single‑antenna BT, and a single‑antenna BR. The direct link (S→BR) and the backscatter link (S→BT→BR) are characterized by complex channel gains h, f, and g, respectively, assumed to be independent block‑fading. The BT employs on‑off keying (OOK) modulation, and each BT symbol spans N consecutive ambient source samples. Under perfect synchronization, the received sample at the BR is either (y_{pk}(n)=hs(n)+\omega(n)) (BT symbol 0) or (y_{pk}(n)=\mu s(n)+\omega(n)) (BT symbol 1) where (\mu = h+\eta fg) and (\eta) denotes the BT’s internal attenuation.

When a timing offset (\Delta n = b_n - n) exists, the BR’s sampling interval for the k‑th BT symbol includes (|\Delta n| = n_a) samples from the preceding or succeeding symbol. The authors express the received sample in this case as either (Y_h(n)=hs(n)+\omega(n)) or (Y_\mu(n)=\mu s(n)+\omega(n)) depending on the symbol values of the current and adjacent BT symbols. This leads to a composite energy metric (\Gamma_{ip,k}) that aggregates contributions from two symbols with possibly different variances (\sigma_0^2) and (\sigma_1^2).

The paper first derives an exact BER expression for the ED under RTSE (Equation 10), which involves four integrals over the probability density functions of (\Gamma_{ip,k}) conditioned on the four possible symbol pairs ((i,j)). While exact, this expression is computationally intensive. To obtain a tractable form, the authors invoke the Lyapunov central limit theorem, approximating (\Gamma_{ip,k}) as Gaussian with mean and variance that are explicit functions of the channel gains, transmit power (P_s), noise power (N_\omega), and the offset magnitude (n_a). This yields a concise approximate BER formula.

Leveraging the approximate BER, the authors analytically differentiate with respect to the detection threshold (\gamma_{ip,th}) and solve for the near‑optimal threshold that minimizes BER. The resulting closed‑form expression (a function of (\sigma_0^2, \sigma_1^2) and (n_a)) reduces to the classic optimal threshold (\gamma_{p,th}^{*}=2N\sigma_0^2\sigma_1^2/(\sigma_0^2+\sigma_1^2)) when (n_a=0) (perfect synchronization). Importantly, the new threshold adapts to the degree of sample mismatch: larger offsets shift the threshold toward the variance of the dominant component, thereby mitigating the BER degradation caused by RTSE.

A practical challenge is that the optimal threshold requires knowledge of the channel coefficients, ambient source power, noise power, and the offset magnitude—all of which are generally unavailable at the BR. To overcome this, the authors propose a novel parameter‑estimation method that relies solely on the received signal samples. By computing sample means and variances over multiple symbol intervals and exploiting the statistical differences between energy distributions of the two OOK states, the BR can estimate (\sigma_0^2) and (\sigma_1^2). The offset magnitude (n_a) is inferred from the autocorrelation of consecutive energy measurements: a systematic shift in the correlation pattern reveals how many samples belong to the neighboring symbol. This estimation procedure works without dedicated pilot symbols and remains valid under RTSE.

The analysis is further extended to the case where the ambient source transmits a phase‑shift keying (PSK) signal rather than a complex Gaussian one. The authors show that the derived BER and threshold expressions remain applicable after substituting the appropriate average power of the PSK constellation, confirming the robustness of the framework.

Simulation results validate the theoretical findings. With typical parameters (e.g., (N=20), (P_s=0) dBm, (N_\omega=-100) dBm, (\eta=-10) dB), the BER using the classic threshold designed for perfect synchronization deteriorates sharply as the normalized offset (\Delta n/N) grows (e.g., from (10^{-3}) to (10^{-1}) for (\Delta n/N=0.2)). In contrast, employing the proposed near‑optimal threshold restores BER to within a factor of two of the perfect‑synchronization benchmark, even when the offset occupies up to 30 % of a symbol duration. The parameter‑estimation method introduces only a modest performance loss (≤ 0.5 dB) when estimation errors are within 5 %. Similar trends are observed for QPSK ambient signals.

In summary, the paper makes four major contributions: (1) a comprehensive detection model that captures all eight partial‑sample‑mismatch scenarios caused by RTSE; (2) exact and approximate BER expressions for an energy detector under these conditions; (3) a closed‑form near‑optimal detection threshold that explicitly accounts for RTSE; and (4) a practical, pilot‑free method to estimate the required channel and offset parameters from received samples. These results provide a solid theoretical foundation and a feasible implementation pathway for low‑power IoT devices operating in realistic, imperfectly synchronized AmBC environments. Future work may explore multi‑antenna receivers, multiple backscatter tags, and adaptive threshold tracking in time‑varying channels.