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An Efficient Blind Detection of OFDM Signals with Unknown Parameters

Ata Bartu Soyuer, Gokhan M. Guvensen

arXiv:2610.23026v1eess.SP

TL;DR

Wideband OFDM sensing must detect signals with unknown power, bandwidth, and carrier frequency, while noise can dominate energy detection. The paper derives single- and multi-antenna GLRTs that use CUSUM for one-dimensional search; simulations report performance close to a genie-aided detector and significantly better than conventional approaches.

  • Problem

    In wideband sensing, noise from largely unoccupied spectrum can dominate energy detection, while OFDM signal power, bandwidth, and carrier frequency may be unknown.

    Source: §I · pp. 1–2

  • Method

    The study uses covariance analysis to derive known- and unknown-noise-power GLRTs for single- and multi-antenna receivers, with CUSUM enabling a one-dimensional search.

    Source: §I · p. 2

  • Results

    The proposed detectors perform close to a genie-aided detector while significantly outperforming conventional approaches.

    Source: §I · p. 3

  • Takeaways & Limitations

    For bandwidth comparisons, the study defines SNR using total signal power over total noise power and says sampling-rate comparisons should hold Ps/N0 fixed.

    Source: C. Discussion on the SNR Definition · p. 4

Abstract

from arXiv · show

This study investigates efficient detector designs for the blind detection of orthogonal frequency-division multiplexing (OFDM) signals with unknown power, bandwidth, and carrier frequency in wideband spectrum sensing. We first describe the received signal model and conduct an asymptotic covariance analysis to obtain a tractable discrete-time representation. Based on this model, generalized likelihood ratio test (GLRT) detectors are derived for both known and unknown noise power cases in a single-antenna receiver setting. The proposed framework is subsequently extended to multi-antenna receivers, and the corresponding multi-antenna GLRT detectors are developed for both known and unknown noise power cases. Importantly, using the cumulative-sum (CUSUM) operations, the resulting detectors reduce to an efficient one-dimensional (1D) search in both single-antenna and multi-antenna settings. We also provide theoretical analysis for the conventional energy detector (Conv-ED) and the hypothetical uniformly most powerful (UMP) test to reveal the impact of system parameters on detection performance. In addition, computational complexity analysis is presented for both the proposed and conventional detectors. Finally, comprehensive simulation results are provided to evaluate the probability of detection. The results demonstrate that the proposed detectors achieve performance close to that of a genie-aided detector while significantly outperforming conventional detection approaches.

I. INTRODUCTION

Wideband sensing can make energy detection noise-dominated; this study develops blind OFDM detectors for unknown signal parameters, with efficient search and performance near a genie-aided benchmark.

I. INTRODUCTION

  • Wideband energy detection can degrade because energy from noise across largely unoccupied spectrum dominates the calculation.Source: §I · p. 1
  • Unlike, the study defines SNR using total signal and noise power, reports degradation as bandwidth increases, and extends the framework to multi-antenna receivers.It also notes that compares candidate bandwidths against separate thresholds, whereas a GLRT is expected to use one detection threshold.Source: §I · p. 2
  • Covariance analysis supports tractable blind sensing, and CUSUM reduces single- and multi-antenna GLRT detection to a one-dimensional search.The GLRT designs cover known and unknown noise power; the study also analyzes Conv-ED and a hypothetical UMP test.Source: §I · p. 2
  • The proposed detectors perform close to a genie-aided detector while significantly outperforming conventional approaches.Source: §I · p. 3

A. Received Signal Model

  • The received signal model treats signal power, bandwidth, and carrier frequency as unknown and samples at the observed-bandwidth rate to avoid correlated noise samples.The study focuses on detecting a single target signal within the observed band.Source: A. Received Signal Model · p. 3, §II · p. 3

B. Asymptotical Covariance Analysis

The asymptotic covariance analysis makes a frequency-domain representation tractable: DFT bins become uncorrelated, and under a Gaussian assumption, asymptotically independent.

B. Asymptotical Covariance Analysis

  • As N grows, the Toeplitz covariance is approximated as circulant, making the frequency-domain covariance asymptotically diagonal and DFT bins uncorrelated.Under the Gaussian assumption, the bins are consequently asymptotically independent.Source: B. Asymptotical Covariance Analysis · pp. 3–4
  • The model represents the signal band as K consecutive DFT bins, with K = ⌊WT⌋, and uses a rectangular PSD assumption for the bin power levels.The adopted PSD model is illustrated in Fig. 2.Source: B. Asymptotical Covariance Analysis · p. 4, Figure 2 · p. 4
  • Equating time- and frequency-domain covariance traces relates total signal power to in-band DFT-bin power and noise power.The derivation uses Parseval’s relation and normalized signal samples.Source: B. Asymptotical Covariance Analysis · p. 4

C. Discussion on the SNR Definition

The paper defines SNR using total signal power and noise power, with a different fairness criterion when comparing sampling rates. For known noise power, its GLRT searches for the highest-power subband using CUSUM-based statistics and a dyadic set of interval lengths.

C. Discussion on the SNR Definition

  • For comparisons across sampling rates, keep Ps/N0 fixed rather than SNR fixed; for a given sampling rate, define SNR as Ps/σ² = Ps/(N0fs).Holding SNR fixed while increasing signal bandwidth W implicitly increases total received signal power under the cited signal-band definitions.Source: C. Discussion on the SNR Definition · p. 4

A. GLRT Detector with Known Noise Power

  • For known noise power, the GLRT metric increases with the average power of a candidate K-sample subband, so detection reduces to finding the highest-power subband.The metric is increasing with estimated subband power when that power is at least σ².Source: A. GLRT Detector with Known Noise Power · p. 5
  • The known-noise-power implementation uses a dyadic search over K values from 1 to 2^⌊log2 N⌋.The paper describes this candidate set as near-optimal, citing and.Source: A. GLRT Detector with Known Noise Power · p. 5

B. GLRT Detector with Unknown Noise Power

For unknown noise power, the GLRT estimates signal and noise parameters under the signal-present hypothesis and compares the resulting likelihoods. Its metric is monotonic in candidate subband power, enabling a CUSUM-based search over interval lengths.

B. GLRT Detector with Unknown Noise Power

  • For unknown noise power, the GLRT combines the signal-present and no-signal likelihoods after estimating the parameters under the feasible region.The likelihood ratio is formed as L2 = Ln − Ld.Source: B. GLRT Detector with Unknown Noise… · p. 5
  • For a given K, the unknown-noise GLRT metric increases monotonically with candidate subband power, yielding its final search form.The corresponding implementation is given in Algorithm 2 and considers dyadic K values.Source: B. GLRT Detector with Unknown Noise… · pp. 5–6

C. Discussion on the CUSUM

The CUSUM is a sufficient statistic for the proposed GLRT detectors and provides a smooth, normalized view of spectral power. Its slope change is more pronounced for narrower signal bandwidths, reflecting the greater difficulty of detecting spread signals at fixed received power.

C. Discussion on the CUSUM

  • The CUSUM constitutes a sufficient statistic for the proposed GLRT detectors; Fig. 3 shows squared magnitudes of DFT bins alongside the corresponding normalized CUSUM, with N = 104.Source: C. Discussion on the CUSUM · p. 6, Figure 3 · p. 6
  • Smaller K produces higher spectral density and a more pronounced CUSUM slope change; at fixed received power, spread signals are generally harder to detect.The paper says this bandwidth–detectability relationship is quantified in Table II and supported by simulations.Source: C. Discussion on the CUSUM · p. 6
  • Because it is bounded between 0 and 1 and smoother than fluctuating DFT-bin samples, the CUSUM is proposed as a potential input to ML/DL detection frameworks.Using it for multi-target detection is identified as future work.Source: C. Discussion on the CUSUM · p. 6

IV. EXTENSION TO MULTI-ANTENNA CASE

The multi-antenna extension models frequency-domain observations using an array response and develops likelihoods across DFT bins. Independence across bins yields a product-form joint likelihood, while the Sherman–Morrison formula simplifies covariance determinant and inverse calculations.

IV. EXTENSION TO MULTI-ANTENNA CASE

  • For a ULA with M antennas, the model uses an array response parameterized by θ = sin(AoA), with antenna spacing d and wavelength λ.The noise vector is modeled as circular complex Gaussian with covariance σ²IM.Source: §IV · p. 6
  • The multi-antenna derivation calculates the covariance and log-likelihood under the signal-present hypothesis from the received-signal model.The covariance is calculated using equations (33) and (35).Source: §IV · p. 6
  • The Sherman–Morrison formula provides the determinant and inverse of Ryk used in the multi-antenna likelihood calculation.The resulting likelihood terms are substituted into the log-likelihood expression.Source: §IV · p. 6
  • Independence of the DFT-bin observations makes each hypothesis’s joint likelihood the product of its per-bin likelihoods.The product is taken over the DFT bins.Source: §IV · p. 6

A. Multi-antenna GLRT with Known Noise Power

The known-noise multi-antenna GLRT replaces a coupled search over signal parameters with a one-dimensional search. A spatial DFT and accumulated power over contiguous frequency segments enable this reduction.

A. Multi-antenna GLRT with Known Noise Power

B. Multi-antenna GLRT with Unknown Noise Power

The unknown-noise multi-antenna GLRT uses the same maximum-power search principle as the known-noise detector. Its maximum bPy is computed efficiently with CUSUM, allowing an analogous implementation.

B. Multi-antenna GLRT with Unknown Noise Power

V. THEORETICAL ANALYSIS ON DETECTION

The theoretical analysis focuses on the conventional energy detector and a hypothetical UMP test because analyzing the proposed GLRT detectors is intractable given their search. A Gaussian-statistic lemma relates detection probability to false-alarm rate and normalized distance.

V. THEORETICAL ANALYSIS ON DETECTION

  • The proposed GLRT detectors are intractable to analyze theoretically because their implementation involves a search, so the analysis examines Conv-ED and hypothetical UMP instead.Source: §V · p. 8
  • For a Gaussian test statistic under both hypotheses, Lemma 1 gives detection probability for a constant false-alarm rate.Source: §V · p. 8
  • The lemma's normalized distance DG is interpreted as the effective detection SNR for the test statistic.Source: §V · p. 8

A. Conventional Energy Detector (Conv-ED)

The section analyzes Conv-ED and a hypothetical UMP test using Gaussian approximations to characterize detection probability. It also relates Conv-ED's performance to whether the signal fills the observed band.

A. Conventional Energy Detector (Conv-ED)

B. Hypothetical Uniformly Most Powerful (UMP) Test

C. Impact of System Parameters on Detection

The analysis characterizes how system parameters affect Conv-ED and UMP detection, then compares detector complexity and evaluates detection performance across single- and multi-antenna simulations. The simulations report that the proposed GLRT performs close to the genie-aided detector and substantially outperforms Conv-ED, particularly at small bandwidths.

C. Impact of System Parameters on Detection

  • The analysis finds that increasing SNR or observation duration improves both Conv-ED and UMP, while bandwidth reduction improves UMP but leaves Conv-ED largely insensitive.Increasing sampling rate degrades Conv-ED, whereas UMP remains largely insensitive.Source: C. Impact of System Parameters on… · p. 9

VI. COMPLEXITY ANALYSIS

  • Proposed GLRT complexity is O(NM log NM) for multi-antenna receivers and O(N log N) for single-antenna receivers; Conv-ED and Conv-PD scale as O(MN) and O(MN log N), respectively.The Conv-PD complexity is approximately O(MN log N).Source: §VI · p. 9, Table III · p. 10

VII. SIMULATION RESULTS

  • The proposed GLRT performs close to the genie-aided GLRT and provides a substantial SNR gain over Conv-ED, particularly at small bandwidths.In multi-antenna simulations, the proposed GLRT also remains close to the genie-aided detector.Source: §VII · pp. 10, 12, Figure 4 · p. 10
  • When interval length K increases, GLRT performance approaches Conv-ED, and the required SNR rises 5.4 dB when K increases by a factor of 16.Conv-ED is largely insensitive to bandwidth, while Conv-PD degrades as K grows.Source: §VII · pp. 10–11, Figure 6 · p. 11
  • When sample count N increases tenfold, required SNR decreases by 5–6 dB for Conv-ED and the proposed GLRT.As observation duration increases, N and K increase proportionally, keeping N/K constant.Source: §VII · p. 11, Figure 7 · p. 11
  • Increasing sampling rate has almost no effect on proposed-GLRT detection, while Conv-ED performance is significantly affected as wider observations add noise.The proposed GLRT shows only marginal performance changes as N increases.Source: Figure 8 · p. 12, §VII · p. 12
  • When antenna count rises from 8 to 32, required SNR falls approximately 3 dB for Conv-ED and 6 dB for proposed GLRT detectors.The proposed GLRT’s advantage over Conv-ED widens with antenna count, and its performance remains close to the genie-aided GLRT.Source: §VII · p. 12, Figure 9 · p. 12

VIII. CONCLUSION

The study introduces efficient GLRT detectors for blind OFDM detection with unknown parameters in wideband sensing. Their performance follows a trend similar to the hypothetical UMP test, and their importance increases in wideband and high-dimensional spatial regimes.

VIII. CONCLUSION

  • The proposed GLRT detectors follow a performance trend similar to the hypothetical UMP test.Source: §VIII · p. 12
  • The study introduces efficient GLRT detectors for blind detection of OFDM signals with unknown parameters in wideband spectrum sensing.Source: §VIII · p. 12
  • The proposed detectors become increasingly important when N ≫ K or M ≫ 1.These conditions describe wideband sensing and high-dimensional spatial regimes, respectively.Source: §VIII · pp. 12–13
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