10−3 10−2 10−1 100
Probability of false alarm
Probability of detection T c = 1 ms C/N 0 = 30 dB C/N 0 = 33 dB C/N 0 = 36 dB C/N 0 = 39 dB C/N 0 = 42 dB
Figure 3.7. ROC curves for different values of C/N0. Basic acquisition scheme, 1
ms coherent integration time.
3.7 Coherent output SNR
Although the ROC completely characterizes the detector performance [43], it is often use- ful to have a single metric, the output or equivalent coherent SNR, which encapsulates as much information about the detector performance as possible. This parameter character- izes the quality of the cell random variable and, in some sense, summarizes the informa- tion carried by the ROC.
In general, determining the equivalent coherent SNR is a difficult problem, since a gen- eral acquisition block employs non-linear operations for increasing the quality of the de- cision variables and reducing the impact of phase and frequency errors and other signal impairments. Nonlinear operations mix the useful signal and noise components leading to cell and decision variables whose quality cannot be easily determined. The problem of quantifying the equivalent coherent SNR when non-coherent integrations are used has been thoroughly investigated in the literature and will be considered in Chapter5. When considering the basic acquisition scheme reported in Figure3.5, one notices that all the operations before the squaring blocks are linear and thus the variables YI(τ,FD)
distributed. The quality of the GNSS signal is usually measured at this stage [20, 21] by the so called coherent output SNR, defined as
ρc= max
φ0
E2[YI(τ,FD)]
Var [YI(τ,FD)]
(3.36) By using Eqs. (3.28) and (3.21), the coherent output SNR, under ideal conditions, assumes the following expression
ρc= max φ0 A2 4 2N σ2 IF cos2φ0 = A 2 2 N σ2 IF = λ σ2 n = N C N0fs/2 = 2 C N0N Ts= 2 C N0Tc (3.37)
The ROC defined by Eqs. (3.26) and (3.34) is a parametric curve where the decision threshold β is only an intermediate parameter. Thus it is possible to operate the following change of variable
β0 = β
σ2 n
(3.38) In this way the ROC can be parameterized with respect to β0, leading to the following
expression: ( Pf a(β0) = exp n −β20 o Pd(β0) = Q1¡√ρc, √ β0¢ (3.39)
From Eq. (3.39) it clearly emerges that, when only coherent integrations are used, the ROC only depends on ρc, the coherent output SNR. In this case the coherent output SNR
completely characterizes the acquisition performance and corresponds to the equivalent coherent SNR.
The coherent output SNR represents a fundamental metric for characterizing the acquisi- tion performance. In other words, the degradations due to quantization, front-end filter- ing and frequency and delay errors can be directly expressed in terms of losses affecting
ρc. These degradations will be discussed in Chapter6.
In Chapter5 different integration strategies are considered. Also in these cases the co- herent output SNR represents a fundamental parameter for quantifying the acquisition performance, however it is no longer sufficient for completely characterizing the acquisi- tion block and additional parameters have to be introduced.
Chapter 4
Cell and decision probabilities
In Chapter3it was shown that acquisition is a complex process that requires several steps in order to provide a reliable decision variable that can be used for deciding the presence of the signal and providing a rough estimation of its Doppler frequency and code delay. Moreover, it has been recognized that GNSS acquisition is carried out in two different domains, the cell domain and the decision domain. Thus two different sets of probabilities, respectively related to the cell domain and to the decision domain, characterize the ac- quisition performance. The first set is relative to the search space cells that are random variables characterized by their pdfs. The cell pdfs depend on the techniques employed for evaluating the single cell and on the channel model considered. For instance the type of integration used for reducing the noise impact, coherent [2,5], non-coherent [40,42] and differentially non-coherent [9], and the presence or absence of fading [6, 7], strongly im- pact the single cell probabilities. The second probability set refers to the decision statistic provided by the decision unit. In the rest of the thesis the first set of probabilities is called
cell probabilities whereas the second one is called decision probabilities. These two sets
are strongly dependent but they do not generally coincide. In the literature the role of cell probabilities is well assessed and different works analyze these probabilities [47, 48]. Instead the decision probabilities are only marginally considered. The major texts in the GNSS literature [2, 4, 32] usually analyze only the cell probabilities, whereas the decision cells are completely ignored. The decision probabilities allow one to completely quantify the acquisition performance, since they do not only depend on the statistical properties of the CAF but also on the strategy adopted for the signal detection. Indeed two acqui- sition systems can have the same cell probabilities and one can have better performance than the other due to characterization by better decision probabilities. This chapter pro- vides a complete framework for the analysis of decision probabilities, deriving their re- lationship with cell probabilities. The concept of decision probabilities is not new, for instance in [48, 49], the correlation maximum-based strategy is thoroughly analyzed and
in [50, 51] the serial search technique is considered. However no explicit comparison be- tween strategies is made and often the proposed models are not supported by simulation results. In [47] the serial search with double dwell decision and the maximum search technique are analyzed from the decision probabilities point of view. However only the case of Doppler absence is considered and miss-detection and false alarm probabilities are not studied.
Three acquisition algorithms are considered: the typical serial scheme, the maximum search technique and a hybrid strategy [34, 52], formed by the combination of the two other methods. The spread of GNSS receivers employing hybrid structures for signal acquisition is self-imposing because, with the advent of longer spreading codes, a full serial search would be too slow, while a full parallel search would be prohibitively ex- pensive [53]. Furthermore the availability of digital techniques based on the FFT algo- rithm [54] allows a faster computation of the search space so that the development of hybrid algorithms is the natural consequence of the row-by-row structure of these tech- niques. The first part of the chapter establishes a theoretical model describing the re- lationship between cell and decision probabilities whit the three considered acquisition strategies. In this context the cell probabilities are not specified and general formulas, independent from the search space computation method, are derived. In the second part, the theoretical model is tested by simulations. Surprisingly, it is shown that secondary phenomena, such as the imperfect code orthogonality and the presence of secondary cor- relation peaks, strongly impact the decision probabilities. These secondary phenomena are generally neglected in the literature [2, 4, 32], since their impact is not clearly observ- able at the cell probabilities level: the reported simulations allow a better understanding of their role in the acquisition performance. An enhanced model accounting these sec- ondary phenomena has been proposed, finally establishing a good agreement with the theoretical formulas. The simulation tests have been also performed under unrealistic conditions in order to have a complete validation of the theoretical results and in order to clearly observe possible secondary effects.
4.1 Statistical model
Since the search space S(τ,FD) is evaluated over a finite and discrete set of code delays
and Doppler frequencies, τ = τmin+h∆τ and fd= fd, min+l∆f , it can be represented as a
matrix of random cells Xnwith n = 1,2,...,M = HL. Therefore, the basic elements of the
system performance evaluation are the detection and false alarm probabilities of a single cell, hereinafter indicated respectively as Pdand Pf a, and also known as single-trial [2]
probabilities. The cells Xnare distributed according to
4.1 – Statistical model
under the null hypothesis H0 that is verified when the local code delay or the local
Doppler shift do not match the input signal ones. The false alarm probability on a single
cell is given by
Pf a(β) = Z +∞
β
fXn(x)dx (4.2)
where β is a preassigned threshold. The alternative hypothesis H1 implies perfect code
and Doppler shift alignment and the corresponding random variable is distributed ac- cording to
Xn|H1 ∼ fA(x) (4.3)
thus the cell detection probability is given by
Pd(β) =
Z +∞
β
fA(x)dx (4.4)
Even if single cell statistics play a fundamental role in determining the overall perfor- mance, the acquisition decision is taken on the basis of the whole search space. In par- ticular a decision strategy is usually adopted and a decision statistic is derived from the whole search space. Thus the acquisition performances are strongly dependent on the decision statistic and the overall detection and the overall false alarm probabilities, denoted
PD (detection) and PF A(false alarm) should be evaluated.
In the next sections, the expression of PD and PF A are derived for the main searching
strategies described in literature, adopting the following assumptions:
• The alternative hypothesis H1is verified only in one single cell. This means that if
the Doppler shift and the code delay are rightly compensated on the n-th cell, only the n-th random variable is affected by this condition, being distributed according to fA(x), whereas the adjacent cells still remain distributed according to fXn(x).
This condition corresponds to the assumption that the principal lobe of the correla- tion function is tight enough to influence one cell only. The random cell verifying
H1will be denoted by XA.
• Only one random variable XAis present in the search space.
• The variable XAcan be in any cell with a uniform probability M1 = LH1 .
• All the random cells of the search space are assumed to be statistically independent. This condition is justified in AppendixB.
Note that the probabilistic model of the searching process does not depend on the specific expressions for fXn(x) and fA(x): these distributions depend on how the ambi- guity function is evaluated over the search space, on the integration time and on the type of averaging.
4.1.1 Searching strategies
In the acquisition process different strategies can be adopted in order to explore the search space more or less quickly and with a minor or greater accuracy. In this section three strategies are considered.
1) Maximum: the CAF is evaluated all over the search space, for each value of Doppler
shift and code delay. Then the decision is taken only on the maximum of the am- biguity function. If the maximum’s value is greater than the imposed threshold β, the satellite is considered acquired and the estimated Doppler shift and code delay are those corresponding to the maximum position.
2) Serial: this strategy consists in serially evaluating the ambiguity function cell by cell.
Once a value is obtained, it is immediately compared with the threshold and the acquisition process stops at the first threshold crossing. The estimated Doppler shift and code delay are those corresponding to the position of the cell under test. In this way, on average, only half of the search space cells is evaluated.
3) Hybrid: the ambiguity function is evaluated row-by-row (or column-by-column), ex-
ploiting, for example, FFT-based algorithms, and the decision is taken on the max- imum of each row (column). The acquisition process terminates as soon as the maximum in the current row (column) exceeds the threshold.