International Journal of Navigation and Observation

Volume 2015, Article ID 295029, 15 pages

http://dx.doi.org/10.1155/2015/295029

## Enhancing Weak-Signal Carrier Phase Tracking in GNSS Receivers

Institute for the Protection and Security of the Citizen, European Commission’s Joint Research Centre (JRC), Ispra, Italy

Received 30 June 2015; Revised 29 September 2015; Accepted 20 October 2015

Academic Editor: Olivier Julien

Copyright © 2015 James T. Curran. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Examining the performance of the GNSS PLL, this paper presents novel results describing the statistical properties of four popular phase estimators under both strong- and weak-signal conditions when subject to thermal noise, deterministic dynamics, and typical pedestrian motion. Design routines are developed which employ these results to enhance weak-signal performance of the PLL in terms of transient response, steady-state errors, and cycle-slips. By examining both single and data-pilot signals, it is shown that appropriate design and tuning of the PLL can significantly enhance tracking performance, in particular when used for pedestrian applications.

#### 1. Introduction

Despite the military origins of Global Navigation Satellite Systems (GNSS), the most widespread use of GNSS receivers is civilian and the single most common receiver platform is the cellular handset. Although the civilian user is, generally, less demanding in terms of position, velocity, and timing accuracy, signal processing for civilian applications is not a simple task. Severe attenuation experienced in the indoor environment, multipath propagation through urban environments, and the limitations of consumer-grade receivers are all obstacles to maintaining acceptable receiver performance.

While many receivers can adequately track carrier frequency under most operating conditions, including in the indoor environment, reliable carrier phase tracking still proves challenging. Owing to a very short wavelength, when subject to any appreciable attenuation, the dynamics of pedestrian motion can induce carrier phase cycle-slips or even loss of phase-lock. Despite these challenges, the ability to track carrier phase is desirable for many reasons including enhanced bit-synchronization, reduced bit-error-rate, enhanced range estimation, improved velocity estimation, and, ultimately, provision for carrier-based positioning.

In response to this challenge, this paper focuses on the process of carrier phase tracking in a scalar phase-lock-loop (PLL). The primary weakness of the PLL when operating on attenuated signals is the process of phase error estimation or phase discrimination. The performance of phase discriminator functions typically degrades rapidly with reduced signal strength and their behavior under weak-signal conditions is generally unique to each discriminator function. To best design a PLL, therefore, this behavior must be understood. This work aims to develop a thorough mathematical model for the carrier phase discriminator and, from this model, to infer best practices for GNSS PLL design. In particular, two case studies are investigated: pedestrian navigation using the GPS L1 C/A signal and data-pilot tracking of the Galileo E1 B/C signal.

Two classes of phase discriminator will be examined, those which employ pure-PLL discriminators and those which employ Costas discriminators. Pure-PLL discriminators are those which are designed to capture the entire phase error on the interval and are therefore useful for synchronization with continuous wave signals or those with smooth modulation, such as frequency-modulation. They represent the earliest form of PLL, dating back to the 1930s [1] and over the last decade have seen applications in GNSS receivers for modernized signals which include a pilot signal-component. By the 1950s, the use of suppressed-carrier modulation required the development of PLLs which were insensitive to carrier-modulation of which the most notable is the Costas PLL [2]. This type of PLL, capturing the phase error on the interval , is widely used in GNSS receivers for BPSK modulated signals, such as GPS L1 C/A or Galileo E1B. Strictly speaking, the Costas PLL is that which performs phase estimation via the product of the in-phase and quadrature base-band channels; however, the term Costas PLL or Costas discriminator has become synonymous with the class of all modulation-insensitive phase discriminators.

The paper is organized as follows: Section 2 introduces the GNSS signal, the PLL architecture, and the linearized PLL model. A statistical analysis of four popular carrier phase estimators is developed in Section 3. Weak-signal effects on the transient and steady-state performance of the PLL are considered in Section 4 and Section 5 presents the application of the theory developed here to the problem of PLL design.

#### 2. Receiver Model and PLL Architecture

To facilitate the following analysis, the PLL is modeled as a simplified linear, time-invariant (LTI) system. A model of the received signal and the corresponding correlator values are developed and a general description of the classical PLL is introduced. A selection of discriminator functions are examined and equivalent linear models are provided, including an assessment of the operating region over which the linearization is accurate. These component models are then combined to yield a linearized system describing the PLL operation. Through these models, it is proposed that the PLL behavior under weak-signal conditions can be described as the superposition of the response of an equivalent linear model of the PLL to various stimuli, including that of thermal noise and of phase variations, where the particular linear model is a function of the prevailing signal strength.

##### 2.1. Downconversion and IF Signal Processing

The correlation of the local replica signals with the incoming digital intermediate frequency (IF) signal over the interval can be approximated by the well-known expressions for the in-phase, , and quadrature, , values [3, 4]:where , , and denote the mean code phase, carrier frequency, and carrier phase errors, respectively, and is the spreading code autocorrelation function. The variable denotes the coherent integration period and also defines the interval between successive updates of the tracking loop. It is assumed that the coherent integration period is aligned with the data modulation symbol boundaries, such that the variable denotes the data sign, which is constant during correlation interval. Under normal PLL operation, the code phase and carrier frequency are reasonably well tracked by the receiver, such that , and so they have a negligible effect on (1). The propagation of the thermal noise to the correlator values is modeled as additive white Gaussian noise (AWGN):where represents the one-sided thermal noise floor in W/Hz. An estimate of the carrier phase tracking error, , is then made by applying a carrier phase discriminator to the values and . This estimation procedure is discussed in more detail in Section 3.

##### 2.2. The Phase-Lock Loop

The standard phase-lock loop is a feedback control loop which tracks the carrier phase using estimates of the carrier phase tracking error. Although all realizable phase error estimators are nonlinear, if the estimate is linearized around zero phase error and normalized such that the noise-free estimate has unity gain, the phase error estimate, denoted by , can be approximated by [4]where represents the linear region of the discriminator. That is, the phase error estimate is approximately equal to a constant times the true phase error, plus a zero mean, white noise, . The constant gain, , is referred to as the discriminator gain and depends on the chosen discriminator function and the prevailing signal-to-noise-ratio. The variance of is also dependent on the phase discriminator used and the received signal-to-noise ratio. The two-sided spectral density of is denoted here by . The linear region is defined as the interval , over which this discriminator approximation is valid. The exact details of the linearization of this phase error estimate and the values of the PSD of for various discriminators will be given in Section 3.

The remainder of the PLL is linear and can be represented by a system of -domain transfer functions, where the update interval of the system is . Such a linearized loop model is useful as it facilitates the estimation of loop stability and tracking performance. Of particular interest are the transfer functions between the carrier phase, , and the carrier phase estimate, , between the carrier phase, , and the tracking error, , and between the thermal noise, , and the tracking error, . These quantities are depicted in a linearized loop model in Figure 1. The transfer functions of interest are given bywhere uppercase symbols represent the -transform of the corresponding lowercase time series. The functions and represent the -transform of the loop filter and the numerically controlled oscillator, respectively. The numerically controlled oscillator is defined as [4]