Introduction

Carrier-to-noise density (C/N0) estimates are considered the most important

quality control parameter in GNSS receivers (Kaplan 2006). Apart from its role as a significant parameter to accept or reject satellite observations in the position solution, accurate C/N0 estimation is also required for:

i. quantifying the performance of algorithms proposed for weak GNSS signals. Most weak GPS signal tracking algorithms found in the literature are analyzed based on their ability to track signals against the C/N0 levels

measured at the receiver end (Kazemi & O’Driscoll 2008, Lashley & Bevly 2008 etc.);

ii. algorithms that use C/N0 estimates as a measure of thermal noise. This

includes Kalman filter (KF) based tracking algorithms, which use C/N0

estimates as a measure of the noise in the accumulated correlator outputs and adaptive noise bandwidth tuning in phase locked loops, which helps in choosing an optimum bandwidth depending on the environment of the receiver (Psiaki & Jung 2002, Petovello & Lachapelle 2006, Mongrédien et al 2007, Muthuraman et al 2008 etc.).

Although several algorithms are used for C/N0 estimation, the theoretical analysis

of the C/N0 estimation process has been only marginally developed in the GNSS context.

The most widely used C/N0 estimator, which is referred to here as the standard estimator

(SE), relies on the computation of the narrow band power (NBP) versus wide band power (WBP) ratio (Van Dierendonck 1995). The SE uses 1 ms accumulated correlator

(complex) outputs to compute the narrow band power over the data bit period of 20 ms (assuming known bit boundary) as

𝑁𝑁𝐵𝐵𝑃𝑃m = � � 𝐼𝐼𝑘𝑘 𝐾𝐾=20 𝑘𝑘=1 � m 2 + � � 𝑄𝑄𝑘𝑘 𝐾𝐾=20 𝑅𝑅=1 � m 2 (3.1)

where 𝑁𝑁𝐵𝐵𝑃𝑃_𝑚𝑚 is the narrow band power computed over the 𝑚𝑚-th data bit period, 𝐾𝐾 is the number of observations per data bit interval, and 𝑌𝑌_𝑘𝑘 = 𝐼𝐼_𝑘𝑘 + 𝑗𝑗 𝑄𝑄_𝑘𝑘 represents the 1 ms accumulated correlator outputs. Similarly, the wide band power is calculated as the sum of square of 1 ms outputs as 𝑊𝑊𝐵𝐵𝑃𝑃𝑚𝑚 = � � (𝐼𝐼𝑘𝑘2+ 𝑄𝑄𝑘𝑘2) 𝐾𝐾=20 𝑘𝑘=1 � 𝑚𝑚 . (3.2)

The ratio 𝑁𝑁𝑃𝑃_𝑚𝑚 is computed as

𝑁𝑁𝑃𝑃𝑚𝑚 = _{𝑊𝑊𝐵𝐵𝑃𝑃}𝑁𝑁𝐵𝐵𝑃𝑃𝑚𝑚

𝑚𝑚. (3.3)

The mean of this ratio (𝜇𝜇_{𝑁𝑁𝑃𝑃}) is given as

μNP =_{M � NP}1 m M m=1

(3.4)

where 𝑀𝑀 is the number of data bit periods over which the averaging is done. The ratio 𝜇𝜇𝑁𝑁𝑃𝑃 is a function that monotonically increases with respect to C/N0. Thus, C/N0 can be

uniquely estimated by inverting this function. Input blocks of 1 ms, which is the most common configuration, are chosen for ease of implementation in the GPS L1 C/A case (C/A code period = 1 ms). In Figure 3-1, the SE is used for estimating the C/N0 of a

progressively attenuated signal generated using a Spirent GPS hardware simulator. More specifically, the mean and standard deviation of the simulated attenuation, with respect to

a 32 dB-Hz reference level, are shown. Beyond 9 dB of attenuation (C/N0 of 23 dB-Hz

and lower), the SE fails considerably in terms of bias in the estimate and is, therefore, not reliable for weak signals.

Under such scenarios, C/N0 estimation can also benefit from the availability of the

pilot channel usually present in modernized GNSS signals. For instance, the Cramer-Rao lower bound (CRLB) for the C/N0 estimation using pilot channel observations is found

to be lower than that using data channel observations (Alagha 2001). Rather than evaluating the bias and variance of the SE using pilot channel observations and K greater than 20, this chapter takes a different approach towards the problem of C/N0 estimation.

The following approach is adopted:

i. Definition of performance bounds (CRLB) of C/N0 estimation using either

the data or pilot channel and for the joint data/pilot case;

Figure 3-1: Analysis of the reliability of the standard C/N0 estimator

ii. Theoretical bound on the performance gain achievable by using both the data and pilot channels will be provided;

iii. Analysis of the effect of the predetection interval (𝑇𝑇_{𝑐𝑐𝑐𝑐ℎ}) used for C/N0

estimation on the performance bounds and optimum choice of 𝑇𝑇_{𝑐𝑐𝑐𝑐ℎ} to be used with C/N0 estimators;

iv. Derivation of approximated ML estimators for C/N0 for single channels

(either data or pilot) and for the joint data/pilot case. Bias and variance analysis of derived estimators.

v. Iterative solution for joint data/pilot C/N0 estimation, to account for the

nonlinearity, and verification of its reliability for weak signals in the case of the GPS L2C signals.

Although the problem of C/N0 estimation can be viewed as a scaled signal-to-

noise ratio (SNR) estimation and theoretical analysis for SNR estimation is widely available in the communications context (Pauluzzi & Beaulieu 2000), a detailed analysis in the specific context of GNSS signals is still missing in the literature. A comprehensive review of work, available in the literature, related to SNR estimation is provided in the following paragraph.

The CRLB for SNR estimation using pilot symbols alone, with the amplitude and noise variance as unknown parameters, is commonly available in the literature (Kay 1993). Alagha (2001) provides the CRLB using data channel observations and compares it with the estimation process which uses pilot channel observations. The assumptions used in the derivations and results presented by Alagha (2001) include one observation per data bit and equally probable data bits. This was extended to the use of N data

symbols and M pilot symbols by Chen & Beaulieu (2005). The latter also propose an approximate maximum-likelihood (ML) estimator for similar input conditions. Further, it is clearly stated that, although the amplitude estimator, which uses both data and pilot inputs, can be viewed as a linear combination of independent amplitude estimators on each channel, the optimal SNR estimator using both channels is a nonlinear combination of the individual SNR estimators. Here again, one observation per data bit was assumed and the results were analyzed using the noise variance at the estimation output. But the approximation used in deriving the ML estimator fails at low SNR, thus leading to biased SNR estimates. Li et al (2002) proposes an iterative search algorithm for the amplitude estimate on the data channel, which tries to solve the ML equation by searching through the values within a given range of normalized amplitudes. As mentioned before, although the above work provides an intuitive insight into the performance of C/N0 estimation, an

analysis of theoretical bounds and achievable performance (in terms of bias and variance) specific to the context of GNSS signals is required.

Ramasubramanian & Nadig (2006) address the theoretical analysis by giving the CRLB for C/N0 estimation in GPS receivers for one particular case where 1 ms

accumulated correlator outputs (20 observations per data bit) from the data channel (GPS L1 C/A) were used. Apart from this, C/N0 estimators derived with an analytical approach

can be found in Groves (2005), Schmid & Neubauer (2005), Pany & Eisfeller (2006) and Muthuraman et al (2008). These approaches make use of the statistics of the accumulated correlator outputs to derive the estimator.