Time and Amplitude Quantisation Losses in GNSS Receivers

A.R. Pratt, J-A. Avila-Rodriguez

Abstract:	There are several sources of loss through the process of transferring from an analogue IF or base-band signal to a digital representation. There have been several attempts at formulating usable results; the first being by R V Harrington (1955) [1] in radar signal processing, and the most recent by J W Betz [2]. The most widely used performance curves are published in A J Van Dierendonk’s paper [3], but these only report a subset of the results that receiver designers might use. There are two processes involved in quantization loss. The first is due to the imprecision of a digital representation of signal and noise using a finite number of analogue levels. Such losses occur whenever a restriction to the signal precision is made. There are usually several such stages in digital GNSS receivers – at the first analogue to digital converter (ADC) and at each subsequent truncation of the digital representation. Most of these are not significant, unlike the initial digital conversion step. The second source of performance loss is due to a finite sampling rate (quantization in the time domain). Such losses can exceed those due to amplitude quantization. It is clear that the receiver designer can balance the losses from amplitude quantization with those from quantization losses due to sampling rate. There are two approaches to approximate the quantization losses. One of these uses the equivalent of an infinite sampling rate to probe the losses due to amplitude quantization. This is briefly reviewed in the paper. It is interesting to note that this methodology is able to support a (wide) variety of noise (and interference) probability distribution functions and permits of the introduction of non-white interference – for example, sinusoidal or CW interference in addition to background noise. The model includes the possibility to estimate performance when the ADC levels are not uniformly spaced, but we have not used this in the researched here. Overall, this technique provides a method to determine a lower bound for the losses due to amplitude quantization alone. The second approach probes the temporal behaviour of the analogue signal in comparison with threshold/ ADC levels. Some progress can be made in this direction using known theory (but not widely known) due to S O Rice 1946 [4] and A R Pratt [5, 6]. Specifically, the paper derives theoretical results for the rates of threshold crossing for band-limited noise and the average duration of each crossing. These results are applicable to noise spectra matched to BPSK and BOC(m,n) signals. These provide useful insight into the duration of each crossing and provide motivation for the investigation of losses due to sampling rates. In the GNSS case, for the most part, signal levels are significantly below the noise level prior to the code correlation processing gain. Therefore, the statistics due to noise provide a first guide to the expected threshold crossing behaviour at low C/N0 ratios appropriate for GNSS. The addition of received satellite signals does change the threshold crossing statistics by small amounts only. It is clear that most threshold crossing durations are rather short (less than 15% of the reciprocal of the bandwidth). Furthermore, the crossing durations become shorter as the magnitude of ADC level increases away from the noise mean level. Finally, the paper examines the performance losses due to the combined analogue and sample rate quantization through simulation of mainly BPSK spreading symbol signals in white noise. Simulation appears to be the only tractable method available to gain suitable results, as the theoretical investigation of the (ADC) level crossing behaviour (statistics) is very difficult. The parameterized curves obtained so far provide an initial basis for balancing losses due to the number of ADC levels versus the signal sampling rate. Further results will be obtained but are not published.
Published in:	Proceedings of the 22nd International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS 2009) September 22 - 25, 2009 Savannah International Convention Center Savannah, GA
Pages:	3179 - 3197
Cite this article:	Pratt, A.R., Avila-Rodriguez, J-A., "Time and Amplitude Quantisation Losses in GNSS Receivers," Proceedings of the 22nd International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS 2009), Savannah, GA, September 2009, pp. 3179-3197.
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