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Convolution and matched filtering are often used to detect a known signal in the presence of noise. The probability of detection and probability of missed detection are well known and widely used statistics. Oftentimes we are not only interested in the probability of detecting a signal but also accurately estimating when the signal occurred and the error statistics associated with that time measurement. Accurately representing the timing error is important for geolocation schemes, such as Time of Arrival (TOA) and Time Difference of Arrival (TDOA), as well as other applications. The Cramér Rao Lower Bound (CRLB) and other, tighter, bounds have been calculated for the error variance on Time of Arrival estimators. However, achieving these bounds requires an amount of interpolation be performed on the signal of interest that may be greater than computational constraints allow. Furthermore, at low Signal to Noise Ratios (SNRs), the probability distribution for the error is non-Gaussian and depends on the shape of the signal of interest. In this paper we introduce a method of characterizing the localization accuracy of the matched filtering operation when used to detect a discrete signal in Additive White Gaussian Noise (AWGN) without additional interpolation. The actual localization accuracy depends on the shape of the signal that is being detected. We develop a statistical method for analyzing the localization error probability mass function for arbitrary signal shapes at any SNR. Finally, we use our proposed analysis method to calculate the error probability mass functions for a few signals commonly used in detection scenarios. These illustrative results serve as examples of the kinds of statistical results that can be generated using our proposed method. The illustrative results demonstrate our method’s unique ability to calculate the non-Gaussian, and signal shape dependent, error distribution at low Signal to Noise Ratios. The error variance calculated using the proposed method is shown to closely track simulation results, deviating from the Cramér Rao Lower Bound at low Signal to Noise Ratios.
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Low Complexity Robust Data Demodulation for GNSS
In this article, we provide closed-form approximations of log-likelihood ratio (LLR) values for direct sequence spread spectrum (DS-SS) systems over three particular scenarios, which are commonly found in the Global Navigation Satellite System (GNSS) environment. Those scenarios are the open sky with smooth variation of the signal-to-noise ratio (SNR), the additive Gaussian interference, and pulsed jamming. In most of the current communications systems, block-wise estimators are considered. However, for some applications such as GNSSs, symbol-wise estimators are available due to the low data rate. Usually, the noise variance is considered either perfectly known or available through symbol-wise estimators, leading to possible mismatched demodulation, which could induce errors in the decoding process. In this contribution, we first derive two closed-form expressions for LLRs in additive white Gaussian and Laplacian noise channels, under noise uncertainty, based on conjugate priors. Then, assuming those cases where the statistical knowledge about the estimation error is characterized by a noise variance following an inverse log-normal distribution, we derive the corresponding closed-form LLR approximations. The relevance of the proposed expressions is investigated in the context of the GPS L1C signal where the clock and ephemeris data (CED) are encoded with low-density parity-check (LDPC) codes. Then, the CED is iteratively decoded based on the belief propagation (BP) algorithm. Simulation results show significant frame error rate (FER) improvement compared to classical approaches not accounting for such uncertainty.
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- PAR ID:
- 10230646
- Date Published:
- Journal Name:
- Sensors
- Volume:
- 21
- Issue:
- 4
- ISSN:
- 1424-8220
- Page Range / eLocation ID:
- 1341
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3390/s21041341
