Abstract
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.
Highlights
Reliable position, navigation, and timing information is a demanded feature in new applications such as intelligent transportation systems (ITSs), automated aircraft landing, or autonomous unmanned ground/air vehicles
We reformulate the problem of obtaining the likelihood ratio (LLR) values by first computing the joint pdf of symbols and estimated variance, which is marginalized in order to compute the desired LLRs used at the decoder. To implement such marginalization in practice, we propose to impose a conjugate prior distribution that allows for an analytic closed-form approximation that enables a reduced complexity implementation when compared to the maximum likelihood (ML) solution
As an example, we provide the frame error rate (FER) performance for GPS L1C Subframe 2 [47] (N = 1200), which is based on an irregular low-density parity-check (LDPC) code of rate 1/2 and decoded by the sum-product algorithm [48]
Summary
Navigation, and timing information is a demanded feature in new applications such as intelligent transportation systems (ITSs), automated aircraft landing, or autonomous unmanned ground/air vehicles. The effects of interference, whether intentional or unintentional, can degrade GNSS receiver performance, sometimes to the point of causing denial of service or even counterfeit transmissions to control the receiver positioning solution. Several of these effects have been reported in the state-of-the-art [5,6,7,8,9,10,11,12]. The latter has been long disregarded, with only a few articles [32,33,34,35]
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