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Analytical Model for GNSS Receiver Implementation Losses* © The MITRE Corporation. All rights reserved. Analytical Model for GNSS Receiver Implementation Losses* Christopher J. Hegarty, The MITRE Corporation intensive and do not provide much insight into the various BIOGRAPHY1 loss mechanisms. Dr. Christopher J. Hegarty is a director with The MITRE Earlier analytical studies of DSSS receiver SNR losses Corporation, where he has worked mainly on aviation due to quantization include derivations in [1] for applications of GNSS since 1992. He is currently the quantization losses of 1.96 dB and 0.55 dB, respectively, Chair of the Program Management Committee of RTCA, for 1- and 2-bit uniform quantizers. These derivations are Inc., and co-chairs RTCA Special Committee 159 simplified in that they assume that the received signal is (GNSS). He served as editor of NAVIGATION: The sampled, but with independent noise upon each sample Journal of the Institute of Navigation from 1997 – 2006 and ignoring bandlimiting effects on the desired signal and as president of the Institute of Navigation in 2008. He component. A more thorough treatment of bandlimiting, was a recipient of the ION Early Achievement Award in sampling, and quantization effects in digital matched 1998, the U.S. Department of State Superior Honor filters is provided in [2]. The results in [2] were later Award in 2005, the ION Kepler Award in 2005, and the inferred to apply to GNSS receiver losses in [3, 4]. As Worcester Polytechnic Institute Hobart Newell Award in detailed later in this paper, however, the results in [2] are 2006. He is a Fellow of the ION, and co-editor/co-author truly not directly applicable to GNSS or other DSSS of the textbook Understanding GPS: Principles and receivers because the losses derived therein presume Applications, 2nd Ed. coherent integration over only one symbol of the desired signal by the receiver, whereas DSSS receivers coherently ABSTRACT integrate over many spreading symbols. Global Navigation Satellite System (GNSS) receivers Monte Carlo simulation results for GNSS receiver SNR suffer signal-to-noise ratio (SNR) losses due to losses for a variety of modulation types, receiver bandlimiting, quantization, and sampling. This paper configurations, and in the presence of both white noise presents an analytical model for GNSS receiver losses and non-white interference are provided in [5] and [6]. applicable to a wide variety of hardware configurations. These results, although useful, have several limitations. The model addresses digitization of the received signal by First, if loss results are required for a scenario not yet a uniform quantizer with an arbitrary (even or odd) investigated, additional simulations must be run, and integer number of output levels. The model provides SNR these can be extremely time intensive. A simulation tool loss values for GNSS signals in the presence of both developed by the second author of [5] and augmented in additive white Gaussian noise and interference, provided capability by others at MITRE takes over 6 hours on a that the interference can be accurately modeled as a non- desktop personal computer to yield results for a typical white, Gaussian wide sense stationary process. run. Second, simulations provide limited insights into the various mechanisms contributing towards implementation INTRODUCTION losses. Lastly, the loss results presented in [5, 6] are all for receivers using sampling rates that are commensurate Direct sequence spread spectrum (DSSS) receivers, with the desired GNSS signal symbol rates, i.e., with an including those used for the Global Navigation Satellite integer number of samples per spreading symbol. As System (GNSS), suffer signal-to-noise ratio (SNR) losses discussed within this paper, with commensurate sampling, due to bandlimiting, quantization, and sampling. Previous receiver SNR losses are highly dependent on the phasing research into such losses has included: (1) low-fidelity of the sampling epochs relative to the symbol transitions. analytical models that predict SNR loss values with many Well-designed GNSS receivers avoid commensurate simplifying assumptions, (2) extensive simulation sampling rates, because of errors that can arise in campaigns that provide useful loss values, but are time pseudorange measurements when such rates are used [7, 8]. *The contents of this material reflect the views of the author. Neither the Federal Aviation Administration nor the Department This paper presents an analytical model for GNSS of the Transportation makes any warranty or guarantee, or receiver losses applicable to a wide variety of hardware promise, expressed or implied, concerning the content or configurations. The model addresses digitization of the accuracy of the views expressed herein. received signal by a uniform quantizer with an arbitrary © The MITRE Corporation. All rights reserved. r()()()() t Cs t ej n t i t r()()()() t Cs t ej n t i t r r q (kM 1) 1 z n Complex n 1 k Lowpass Filter × Quantizer M Sampler n kM (fs Hz) se* j n Figure 1. GNSS Receiver Signal Processing Model (even or odd) integer number of output levels. The model effective signal power and effective noise density at the includes the effects of sampling and bandlimiting, without output of the correlator, which when combined yield SNR limitations on the allowable sample rates or filter transfer losses. functions. Both commensurate and non-commensurate sampling rates are considered, although as mentioned Signal Model above the former are mostly of interest in comparing results with those reported previously in earlier studies. The baseband normalized desired signal, s(t), is modeled very generally as: As observed in the previous literature, losses are s t c p()ll t lT j d q() t lT dependent on the desired signal modulation type. () l (c ) 1 l ()c (1) ll Furthermore, different losses are incurred for the three where is the fraction of power in the signal‟s in-phase components of the received signal considered: (1) desired component, Tc is the symbol period, {cl} and {dl} are GNSS signal, (2) noise, and (3) Gaussian non-white independent, white pseudorandom sequences from the interference (e.g., inter-/intra-system interference from a (l) (l) binary alphabet [-1,+1], and p (t) and q (t) are, number of GNSS signals of the same or different respectively, the l-th spreading symbols used for the modulation type). The analytical formulation provides inphase and quadraphase components. The spreading insight into how implementation losses result from a symbols are real-valued and selected from finite sets, combination of attenuation of the desired signal, and p()l ( t ) p ( t ), , p ( t ) , q()l ( t ) q ( t ), , q ( t ) with attenuation and/or enhancement of the noise/interference. 1 N p 1 Nq probabilities P p()l ()() t p t P and This paper is organized as follows. The analytical model ipi is first derived and described in the following section, ()l P q()() t q t P . This definition is broad enough to with certain cumbersome steps in the derivation relegated iqi to appendices. The subsequent section provides apply to simple GNSS signals such as the GPS C/A code, implementation loss results from the analytical model and for which there is only an in-phase component with all compares pertinent results against those found in the symbols taking on the same, rectangular shape, or to more previous literature. The final section provides a short complicated GNSS signals such as the GPS L1 civil summary and conclusions. signal (L1C) that include in- and quadra-phase components with a mixture of different symbols (the L1C ANALYTICAL MODEL pilot component uses two different symbol types that are time multiplexed). A simplified block diagram of signal processing within a GNSS receiver is shown in Figure 1. The received signal, The signal defined in equation (1) is cyclostationary, with r(t), is modeled as the sum of a desired signal, noise n(t), a two-parameter autocorrelation function that is periodic and interference i(t). The baseband normalized (unity with time, t, with period Tc: power) desired signal is denoted s(t), with received phase (radians) and received power level of C watts. Note R t E s t s* t s ;()() that Figure 1 is highly simplified and omits certain N p receiver processing functions that are considered lossless P p()() t lT p t lT (2) pn n c n c this analysis, e.g., down-conversion of the original L-band nl1 received signal to its in-phase (I) and quadra-phase (Q) Nq 1 P q()() t lT q t lT components (treated here as a single complex signal). qn n c n c nl1 The following subsections provide statistical models for and its time-average may be defined as the signal, noise, and interference, and then follow these received signal components through the processing shown in Figure 1. The final subsection derives expressions for © The MITRE Corporation. All rights reserved. T 1 c It is assumed that within the receiver passband, the power Rss R(;) t dt of the desired signal is much lower than the combined T c 0 (3) power of the noise and interference, e.g., with ideal low- NNpq pass filtering, PRp p 1 PRq q n n n n nn11 BB where C Sfdf() NBI Sfdf() (9) T sI0 c 1 f B f B R p()() t p t dt pn T n n c 0 (4) T where Sf()is the Fourier transform of R () . 1 c s s R q()() t q t dt qn T n n c 0 Sampling Noise and Interference Models The low-pass filtered received signal is uniformly sampled at rate fs. After sampling, the received signal may The noise, n(t), and interference, i(t), are assumed to be be expressed as: mutually independent, complex Gaussian wide sense rnn r() t stationary random processes.
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