3.1. Satellite/Aircraft Relative Geometry and Position Errors

As described in Part 1 of this paper (Sabatini et al., Reference Sabatini, Moore and Hill2012), in order to generate CIF/WIF associated to critical antenna masking conditions, a dedicated analysis is required which takes into account the variation of the attitude angles. Therefore, using the CATIA 3D model of the aircraft, a series of Antenna Obscuration Matrices (AOMs) are collected with pitch angles varying between −90° and 90° and with roll angles varying between −90° and 90°. An example of the resulting Global Masking Profile (GMP) obtained for the TORNADO Interdiction and Strike (IDS) is shown in Figure 3. The obscuration integrity flag criteria are the following:

• • When the current aircraft manoeuvre will lead to less than four satellites being in view, the CIF shall be generated.

• • When less than four satellites are in view, the WIF shall be generated.

Additionally, if only four satellites are in view:

• • When one (or more) satellite(s) elevation angle(s) (antenna frame) is (are) less than 10 degrees, the Caution Integrity Flag shall be generated.

• • When one (or more) satellite(s) elevation angle(s) is (are) less than 5 degrees, the Warning Integrity Flag shall be generated.

Figure 3.

TORNADO-IDS global masking profile (B = bank and P = pitch – in degrees).

Using the definition of Dilution of Precision (DOP) factors, GNSS accuracy can be expressed as (Kaplan and Hegarty, Reference Kaplan and Hegarty2006):

(8)$$\sigma _{\rm P} = {\rm DOP} \times \sigma _{{\rm UERE}} $$

where:

σ _P

is the standard deviation of the positioning accuracy.

σ _UERE

is the standard deviation of the satellite pseudorange measurement error.

For the C/A-code σ _UERE is in the order of 33·3 m. Therefore, the 1-σ Estimated Position Error (EPE), Estimated Horizontal Error (EHE) and Estimated Vertical Errors (EVE) of a GNSS receiver are calculated using the Positional DOP (PDOP), the Horizontal DOP (HDOP) or the Vertical DOP (VDOP) respectively. In order to generate CIFs and WIFs that are consistent with current GNSS Required Navigation Performance (RNP), we need to introduce the Horizontal and Vertical Accuracy (HA/VA) requirements applicable to the different flight phases. Table 1 shows the GNSS signal-in-space alert requirements in terms of HA/VA for En-route, Non-Precision Approach (NPA) and for the three categories of Precision Approach. Table 2 shows the GNSS signal-in-space protection requirements. The Horizontal Alert Limit (HAL) is the radius of a circle in the horizontal plane, with its centre being at the true position, which describes the region within which the indicated horizontal position must be contained with the required probability for a particular navigation mode. Similarly, the Vertical Alert Limit (VAL) is half the length of a segment on the vertical axis, with its centre being at the true position, which describes the region within which the indicated vertical position must be contained with the required probability for a particular navigation mode.

Table 1.

GNSS signal-in-space alert requirements (ICAO, 2006).

Table 2.

GNSS signal-in-space protection requirements (CAA, 2003).

As a result of our discussion, the DOP integrity flags criteria are the following:

• • When the EHE exceeds the HA 95% or the VA 95% alert requirements, the CIF shall be generated.

• • When the EHE exceeds the HAL or the EVE exceeds the VAL, the WIF shall be generated.

During the landing phase, a GNSS Landing System (GLS) has to be augmented by GBAS in order to achieve the RNP, as well as Lateral and Vertical Protection Levels (LPL and VPL). LPL/VPL are defined as the statistical error values that bound the Lateral/Vertical Navigation System Error (NSE) with a specified level of confidence. In particular, for the case of a Local Area Augmentation System (LAAS), which allows for multiple Differential GPS (DGPS) reference receivers (up to four) to be implemented, two different hypotheses are made regarding the presence of errors in the measurements. These hypotheses are (RTCA, 2004):

• • H0 Hypothesis – No faults are present in the range measurements (includes both the signal and the receiver measurements) used in the ground station to compute the differential corrections.

• • H1 Hypothesis – A fault is present in one or more range measurements and is caused by one of the reference receivers used in the ground station.

Consequently, LPL and VPL are computed as follows:

(9)$${\rm LPL} = {\rm MAX}\; \{ LPL_{H0}, \,\; LPL_{H1} \} $$

(10)$${\rm VPL} = {\rm MAX}\; \{ VPL_{H0}, \,\; VPL_{H1} \} $$

The lateral and vertical accuracy (NSE 95%) and alert limits required by a GLS in the presence of a LAAS, considering the continuously varying position of the aircraft with respect to the Landing Threshold Point (LTP), are also given in (RTCA, 2004). Additionally, this RTCA standard provides the so-called Continuity of Protection Levels in terms of Predicted Lateral and Vertical Protection Levels (PLPL and PVPL). Although the definition in (RTCA, 2004) is quite comprehensive, a generic statement is made that the PVPL and PLPL computations shall be based on the ranging sources expected to be available for the duration of the approach. In other terms, it is implied that the airborne subsystem shall determine which ranging sources are expected to be available, including the ground subsystem's declaration of satellite differential correction availability (satellite setting information). Unfortunately, this generic definition does not address the various conditions for satellite signal losses associated to specific aircraft manoeuvres (including curved GLS precision approaches). Therefore, it is suggested that an extended definition of PLPL and PVPL be developed which takes into account the continuously varying satellite/aircraft relative geometry. In particular, when the current aircraft manoeuvre will lead to less than four satellites in view or unacceptable accuracy degradations, the CIF shall be generated. Following our discussion, the additional integrity flags criteria adopted for GLS in the presence of a LAAS are the following:

• • When the PLPL exceeds LAL or PVPL exceeds the VAL, the CIF shall be generated.

• • When the LPL exceeds the LAL or the VPL exceeds the VAL, the WIF shall be generated.

3.2. Radio Frequency Link Errors

Multipath integrity flags were defined using the Early-Late Phase (ELP) observable and the range error. In a GPS receiver having three correlators (early, prompt and late), the phase of a correlator is given by (Mubarak and Dempster, Reference Mubarak and Dempster2009):

(11)$${\rm \Phi} _{\rm C} (\rm t) = \tan ^{ - 1} \left( {\displaystyle{{{\rm Q}_{\rm C}} \over {{\rm I}_{\rm C}}}} \right)$$

where the subscript C can refer to early (E), prompt (P) and late (L) respectively.

The prompt phase is always kept close to zero by the carrier tracking loop. Early and late correlators are then placed on each side of the prompt, which means one of the phases of the correlator is positive and the other is negative. So, the phase difference between the two is increased in the presence of multipath. ELP is simply the phase difference between the early and late correlator outputs, where the phase of a correlator output is equal to the inverse tangent of the Q channel output divided by the I channel output. Mathematically, the ELP is calculated by (Mubarak and Dempster, Reference Mubarak and Dempster2009):

(12)$${\rm ELP}({\rm t}) = \tan ^{ - 1} \left[ {\displaystyle{{{\rm Q}_{\rm L} ({\rm t})} \over {{\rm I}_{\rm L} ({\rm t})}} - \displaystyle{{{\rm Q}_{\rm E} ({\rm t})} \over {{\rm I}_{\rm E} ({\rm t})}}} \right]$$

The probability of multipath detection is approximately 80% by setting the ELP threshold at 0·1 radians (Fig. 4).

Figure 4.

Probability of detecting multipath for varying ELP thresholds. The symbol α indicates the ratio of multipath to direct signal amplitude (i.e., α = A_m/A_d). Adapted from Mubarak and Dempster (Reference Mubarak and Dempster2010).

Additionally, as shown in Figure 5, with an ELP threshold of 0·1 radians, the Probability of False Alarm (PFA) is 0·3 when ${\textstyle{{\rm C} \over {{\rm N}_0}}} $ is 42 dB-Hz. This is an acceptable compromise for the ABIA caution flag implementation.

Figure 5.

PFA for varying ELP thresholds. Adapted from Mubarak and Dempster (Reference Mubarak and Dempster2010).

Multipath range error experimental results are also used to trigger the warning integrity flag associated to multipath. As discussed in Part 1 (Sabatini et al., Reference Sabatini, Moore and Hill2012), the effect of ground-echo signals translates into a sudden increase of the multipath ranging error of up to two orders of magnitude with respect to the airframe multipath errors alone. However, the region of potential ground-echo multipath for the L1 frequency only extends up to 448·5 metres Above Ground Level (AGL) (theoretical value applicable to all aircraft types). As a result of our analysis, the multipath integrity flags criteria are the following:

• • When the ELP exceeds 0·1 radians, the CIF shall be generated.

• • When the multipath ranging error shows a sudden increase with the aircraft flying in proximity of the ground (below 448·5 metres), the WIF shall be generated.

Based on results of the GPS-based Time and Space Position Information (TSPI) flight test activities (Sabatini and Palmerini, Reference Sabatini and Palmerini2008), an additional (practical) criteria was developed for the TORNADO-IDS aircraft. During TORNADO-IDS flight trials it was observed that flying above 500 ft AGL (even with pitch/bank angles exceeding 45°), the ground-echo multipath was not a factor and the multipath range error (due to the airframe only) never exceeded 2 metres. Consequently, the additional criteria applicable to the TORNADO-IDS aircraft is:

• • When the multipath ranging error exceeds 2 metres and the aircraft flies in proximity of the ground (below 500 ft AGL), the WIF shall be generated.

As introduced in Part 1 (Sabatini et al., Reference Sabatini, Moore and Hill2012), avionics GNSS receivers are resistant to dynamic stress errors. A robust avionics GNSS receiver design typically employs a Frequency Lock Loop (FLL) as a backup to the Phase Lock Loop (PLL) during initial loop closure and during high-dynamic stress with loss of phase lock; it will revert to pure PLL for the steady-state low to moderate dynamics in order to produce the highest carrier Doppler phase measurements. However, flight test activities performed with avionics GPS receivers showed that the reacquisition time after loss of one or more satellite signals could be up to 40 seconds, depending on flight conditions and satellite constellations (Sabatini and Palmerini, Reference Sabatini and Palmerini2008). Additionally, the analysis of receiver data, recorded during several flights and at speeds up to 500 kn, highlighted that the Doppler effect causes a frequency shift with respect to the L1 carrier frequency, reaching a maximum value of about 15 KHz (Sabatini and Palmerini, Reference Sabatini and Palmerini2008). This value is relatively low if compared with the GPS frequency bandwidth (i.e., about 30 MHz) and the high dynamic characteristics of the carrier tracking loops internal to the avionics receiver guarantee that neither the data accuracy is degraded nor the carrier phase is lost because of Doppler shift. Nevertheless, the coupling between such a frequency shift and the signal reacquisition strategy of the receiver can significantly affect the time necessary to get data after a signal loss.

Figure 6 shows the fitting functions and associated curves relative to the acquisition time experimental data presented in (Sabatini et al., Reference Sabatini, Moore and Hill2012). These functions were conveniently used to estimate GNSS acquisition time based on satellite/aircraft relative velocity Line Of Sight (LOS). With reference to Figure 6, it is evident that for C/N₀ below 28 dB-Hz the acquisition time exceeds 1 second even in low dynamics conditions. Therefore, in these conditions, the required acquisition times might become unacceptable in case of satellite signal losses during certain mission- and safety-critical GNSS applications (military weapon aiming, missile guidance, precision approach, etc.).

Figure 6.

Doppler shift and signal acquisition in avionics GPS receivers.

Consequently, the following criteria were defined for the Doppler integrity flags thresholds:

• • When the C/N₀ is below 28 dB-Hz and the signal is lost, the CIF shall be generated if the estimated acquisition time is less than the application-specific TTA requirements.

• • When the C/N₀ is below 28 dB-Hz and the signal is lost, the WIF shall be generated if the estimated acquisition time exceeds the application-specific TTA requirements.

3.3. Receiver Tracking Errors

As discussed before, in order to define additional robust criteria for the ABIA integrity thresholds associated with dynamic stress errors, signal fading (C/N₀ reduction) and interferences (J/S increase), a dedicated analysis of the GNSS receiver tracking performance was required. When the GNSS code and/or carrier tracking errors exceed certain thresholds, the receiver loses lock to the satellites. Since both the code and carrier tracking loops are nonlinear, especially near the threshold regions, only Monte Carlo simulations of the GNSS receiver in different dynamics and Signal to Noise Ratio (SNR) conditions can determine the receiver tracking performance (Ward, Reference Ward1997). Nevertheless, some conservative ‘rules-of-thumb’ approximating the measurement errors of the GNSS tracking loops were used for the ABIA initial design. Numerous sources of measurement errors affect the Carrier Tracking Loops and Code Tracking Loops. However, for our purposes, it was sufficient to analyze the dominant error sources in each type of tracking loop. Consider a typical avionics GNSS receiver employing a two-quadrant arctangent discriminator; the PLL threshold is given by (Kaplan and Hegarty, Reference Kaplan and Hegarty2006):

(13)$$3\sigma _{{\rm PLL}} = 3\sigma _{\rm j} + \theta _{\rm e} \le 45{\rm ^{\circ}}$$

where:

σ _j=

1-sigma phase jitter from all sources except dynamic stress error.

θ _e=

dynamic stress error in the PLL tracking loop.

Expanding Equation (13), the 1-sigma threshold for the PLL tracking loop becomes (Kaplan and Hegarty, Reference Kaplan and Hegarty2006):

(14)$$\sigma _{{\rm PLL}} = \sqrt {\sigma _{{\rm tPLL}}^2 + \sigma _\upsilon ^2 + \theta _{\rm A}^2} + \displaystyle{{\theta _{\rm e}} \over 3} \le 15{\rm ^{\circ}}$$

where:

σ _tPLL=

1-sigma thermal noise.

σ _υ=

vibration-induced oscillator phase noise.

θ _A=

Allan variance–induced oscillator jitter.

The PLL thermal noise is often thought to be the only carrier tracking error, since the other sources of PLL jitter may be either transient or negligible. The PLL thermal noise jitter is computed as follows:

(15)$$\sigma _{{\rm tPLL}} = \displaystyle{{360} \over {2\pi}} \sqrt {\displaystyle{{{\rm B}_{\rm n}} \over {\left( {\displaystyle{{\rm c} \over {{\rm n}_0}}} \right)\left( {1 + \displaystyle{1 \over {2{\rm T}{\displaystyle{{\rm c} \over {{\rm n}_0}}}}}} \right)}}} \quad ({\rm degrees})$$

where:

B_n=

carrier loop noise bandwidth (Hz).

${\displaystyle{{\rm c} \over {{\rm n}_0}}} $$=

carrier to noise power ratio $({\displaystyle{{\rm c} \over {{\rm n}_0}}} = 10^{\textstyle{{{\textstyle{{\rm C} \over {{\rm N}_0}}}} \over {10}}}\,{\rm for}\,{\textstyle{{\rm C} \over {{\rm N}_0}}}\, {\rm expressed\,in\,dB} {\mbox{\hbox -}} {\rm Hz)}$.

T=

pre-detection integration time (seconds).

B_n and ${\textstyle{{\rm C} \over {{\rm N}_0}}} $ can be derived from the SNR model as described before. Determination of the vibration-induced oscillator phase noise is a complex analysis problem. In some cases, the expected vibration environment is so severe that the reference oscillator must be mounted using vibration isolators in order that the GPS receiver can successfully operate in PLL. The equation for vibration induced oscillator jitter is:

(16)$$\sigma _\upsilon = \displaystyle{{360{\rm f}_{\rm L}} \over {2\pi}} \sqrt {\int_{{\rm f}_{\min}} ^{{\rm f}_{\max}} {\displaystyle{{{\rm S}_\upsilon ^2 ({\rm f}_{\rm m} )({\rm P}({\rm f}_{\rm m} ))} \over {{\rm f}_{\rm m}^2}} {\rm df}_{\rm m} \quad ({\rm degrees})}} \; \; \; $$

where:

f_L=

L-band frequency (Hz).

S_υ(f_m)=

oscillator vibration sensitivity of $\Delta {\rm f}/{\rm f}_{\rm L}}}} $ per g as a function of f_m.

f_m=

random vibration modulation frequency (Hz).

P(f_m)=

power curve of the random vibration as a function of f_m${\displaystyle\left({{{\rm g2}} \over {{\rm Hz}}}\right)}$

g=

gravity acceleration.

Usually the oscillator vibration sensitivity S_υ(f_m), is not variable over the range of the random vibration modulation frequency, then Equation (16) can be simplified to:

(17)$${\rm \sigma} _{\rm \upsilon} = \displaystyle{{360{\rm f}_{\rm L} {\rm S}_{\rm \upsilon}} \over {2{\rm \pi}}} \sqrt {\int_{{\rm f}_{\min}} ^{{\rm f}_{\max}} {\displaystyle{{{\rm P}({\rm f}_{\rm m} )} \over {{\rm f}_{\rm m}^2}} {\rm df}_{\rm m}}} \ ({\rm degrees})$$

The equations used to determine Allan deviation phase noise are empirical. They are stated in terms of what the requirements are for the short-term stability of the reference oscillator as determined by the Allan variance method of stability measurement. The equation for second-order loop, short-term Allan deviation is:

(18)$$\theta _{{\rm A}2} = 144\displaystyle{{\sigma _{\rm A} (\tau )*{\rm f}_{\rm L}} \over {{\rm B}_{\rm n}}} \; \,\; ({\rm rad})$$

The equation for third-order loop, short-term Allan deviation for PLL is:

(19)$$\theta _{{\rm A}3} = 160\displaystyle{{\sigma _{\rm A} (\tau )*{\rm f}_{\rm L}} \over {{\rm B}_{\rm n}}} \,\; ({\rm rad})$$

where:

σ _A(τ)=

Allan deviation-induced jitter (degrees).

f_L =

L-band input frequency (Hz).

τ=

short-term stability gate time for Allan variance measurement (seconds).

B_n=

noise bandwidth.

Usually σ _A(τ) can be determined for the oscillator and it changes very little with gate time τ. In our research, the loop filter is assumed as a third-order with a noise bandwidth B_n = 18 Hz and the gate time ${\rm \tau} = {\textstyle{1 \over {{\rm B}_{\rm n}}}} = 56\;{\rm ms}$. The Allan deviation is specified to be σ _A(τ) = 10⁻¹⁰. The dynamic stress error depends on the loop bandwidth and order. In a third-order loop, the dynamic stress error is (Kaplan and Hegarty, Reference Kaplan and Hegarty2006):

(20)$$\theta _{{\rm e}3} = \displaystyle{{{\textstyle{{{\rm d}^3 {\rm R}} \over {{\rm dt}^3}}}} \over {\omega _0^3}} = \displaystyle{{{\textstyle{{{\rm d}^3 {\rm R}} \over {{\rm dt}^3}}}} \over {\left( {{\textstyle{{{\rm B}_{\rm n}} \over 0 \cdot 7845}}} \right)^3}} = 0 \!\cdot\! 4828\displaystyle{{{\textstyle{{{\rm d}^3 {\rm R}} \over {{\rm dt}^3}}}} \over {{\rm B}_{\rm n}^3}} \,({\rm degrees})$$

where:

R=

LOS range to the satellite.

${\textstyle{{{\rm d}^2 {\rm R}} \over {{\rm dt}^2}}} $$=

maximum LOS acceleration dynamics $ (^{{\scale70%\circ}}/{{\rm s}^2}}} )$.

ω₀=

loop filter natural radian frequency.

B_n=

noise bandwidth.

For the L1 frequency we have: ${{\rm d}^3 {\rm R}} / {\rm dt}^3 = ({98} / {\rm s}^3}) \times \left(360{\rm ^{\circ}}/{\rm cycle}\right)\times{({{{1575 \cdot 42 \times 10^6\, {{{{\rm cycles}}/ {\rm s})/ {\rm c}}} = 185398^{\circ} /{\rm s}^3} $. Frequency jitter due to thermal noise and dynamic stress error are the main errors in a GNSS receiver FLL. The receiver tracking threshold is such that the 3-sigma jitter must not exceed one-fourth of the frequency pull-in range of the FLL discriminator. Therefore, the FLL tracking threshold is (Kaplan and Hegarty, Reference Kaplan and Hegarty2006):

(21)$$3\sigma _{{\rm FLL}} = 3\sigma _{{\rm tFLL}} + {\rm f}_{\rm e} \le \displaystyle{1 \over {4{\rm T}\;}} \,({\rm Hz})$$

where:

3σ _tFLL=

3-sigma thermal noise frequency jitter.

f_e=

dynamic stress error in the FLL tracking loop.

Equation (21) shows that the dynamic stress frequency error is a 3-sigma effect and is additive to the thermal noise frequency jitter. The reference oscillator vibration and Allan deviation–induced frequency jitter are small-order effects on the FLL and are considered negligible. The 1-sigma frequency jitter threshold is 1/(12 T) = 0·0833/T Hz. The FLL tracking loop jitter due to thermal noise is:

(22)$$\sigma _{{\rm tFLL}} = \displaystyle{1 \over {2\pi {\rm T}}}\sqrt {\displaystyle{{4{\rm FB}_{\rm n}} \over {{\textstyle{{\rm C} \over {{\rm n}_0}}}}} \left[ {1 + \displaystyle{1 \over {{\textstyle{{{\rm T}_{\rm C}} \over {{\rm n}_0}}}}}} \right]} \,(\rm Hz)$$

where F is 1 at high ${\textstyle{{\rm C} \over {{\rm N}_0}}} $ and 2 near the threshold.

σ _tFLL is independent of C/A or P(Y) code modulation and loop order. Since the FLL tracking loop involves one more integrator that the PLL tracking loop of the same order (4, 12), the dynamic stress error is:

(23)$${\rm f}_{\rm e} = \displaystyle{{\rm d} \over {{\rm dt}}}\left( {\displaystyle{1 \over {360{\rm \omega} _0^{\rm n}}} \displaystyle{{{\rm d}^{\rm n} {\rm R}} \over {{\rm dt}^{\rm n}}}} \right) = \displaystyle{1 \over {360{\rm \omega} _0^{\rm n}}} \displaystyle{{{\rm d}^{{\rm n} + 1} {\rm R}} \over {{\rm dt}^{{\rm n} + 1}}} {\rm \;} \,({\rm Hz})$$

Regarding the code tracking loop, a conservative rule-of-thumb for the Delay Lock Loop (DLL) tracking threshold is that the 3-sigma value of the jitter due to all sources of loop stress must not exceed the correlator spacing (d), expressed in chips. Therefore (4, 13):

(24)$$3{\rm \sigma} _{{\rm DLL}} = 3{\rm \sigma} _{{\rm tDLL}} + {\rm R}_{\rm e} \le {\rm d\; \;} ({\rm chips})$$

where:

σ _DFLL=

1-sigma thermal noise code tracking jitter.

R_e=

dynamic stress error in the DLL tracking loop.

The DLL thermal noise code tracking jitter is given by:

(25)$$\sigma _{{\rm tDLL}} = \sqrt {\displaystyle{{4{\rm F}_1 {\rm d}^2 {\rm B}_{\rm n}} \over {{\textstyle{{\rm c} \over {{\rm n}_0}}}}} [ {2(1 - {\rm d}) + \displaystyle{{4{\rm F}_2 {\rm d}} \over {{\textstyle{{{\rm {\bf T}c}} \over {{\rm n}_0}}}}}}]} \; \,(\rm Hz)$$

where:

F₁=

DLL discriminator correlator factor (1 for time shared tau-dithered early/late correlator and 0·5 for dedicated early and late correlators).

d=

Correlator spacing between early, prompt and late.

B_n=

Code loop noise bandwidth.

F₂=

DLL dicriminator type factor (1 for early/late type discriminator and 0·5 for dot product type discriminator).

The DLL tracking loop dynamic stress error is given by:

(26)$${\rm R}_{\rm e} = \displaystyle{{{\textstyle{{{\rm dR}^{\rm n}} \over {{\rm dt}^{\rm n}}}}} \over {\omega _0^{\rm n}}} \,\; ({\rm chips})$$

where dRⁿ/dtⁿ is expressed in chips/secⁿ.

The PLL, FLL and DLL error equations described above allow to determine the (C/N₀) corresponding to the tracking threshold of the receiver. A generic criteria applicable to the ABIA system is:

(27)$$\left( {\displaystyle{{\rm C} \over {{\rm N} _ 0 }}} \right)_{{\rm Threshold}} = \max \left[ {\left( {\displaystyle{{\rm C} \over {{\rm N}_0 }}} \right)_{{\rm PLL}} , \left( {\displaystyle{{\rm C} \over {{\rm N} _0}}} \right)_{{\rm FLL}}, \left( {\displaystyle{{\rm C} \over {{\rm N}_0}}} \right)_{{\rm DLL}}} \right]$$

where:

$\left({\displaystyle{{\rm C} \over {\rm N}}}_0\right)_{{\rm PLL}} $$=

Minimum carrier-to-noise ratio for PLL carrier tracking.

$\left({\displaystyle{{\rm C} \over {\rm N}}}_0\right)_{{\rm FLL}} $$=

Minimum carrier-to-noise ratio for FLL carrier tracking.

$\left({\displaystyle{{\rm C} \over {\rm N}}}_0\right)_{{\rm DLL}} $$=

Minimum carrier-to-noise ratio for DLL code tracking.

Numerical solutions of Equations (13), (21) and (24) show that the weak link in unaided avionics GNSS receivers is the PLL due to its greater sensitivity to dynamics stress. Therefore, the $\left[\left({\textstyle{{\rm C} \over {\rm N}}}\right]_0\right)_{{\rm PLL}} $ threshold can be adopted in these cases. In general, when the PLL loop order is made higher, there is an improvement in dynamic stress performance. This is why third order PLL are widely adopted in avionics GNSS receivers (e.g., TORNADO-IDS). Assuming 15 to 18 Hz noise bandwidth and 5 to 20 msec pre-detection integration time (typical values for avionics receivers), the ‘rule-of-thumb’ tracking threshold for the PLL gives 25 to 28 dB-Hz. Additionally, in aided avionics receiver applications, the PLL tracking threshold can be significantly reduced by using external velocity aiding in the carrier tracking loop. With this provision, a tracking threshold of approximately 15 to 18 dB-Hz can be achieved. Using these theoretical and experimental threshold values, we can also calculate the receiver J/S performance for the various cases of practical interest, as described in (Sabatini et al., Reference Sabatini, Moore and Hill2012). When available, flight test data collected in representative portions of the aircraft operational flight envelope (or the results of Monte Carlo simulation) will be used. Taking an additional 5% margin on the 3-sigma tracking thresholds for the CIF, the following additional criteria are introduced for the ABIA integrity thresholds:

• • When 42·25°≤3σ _PLL≤45° or 0·2375 T≤3σ _FLL≤0·25 T or 0·05d≤3σ _DLL≤3σ _DLL d the CIF shall be generated.

• • When 3σ _PLL>45° or 3σ _FLL>${\textstyle{1 \over {4T}}}$ or 3σ _PLL>d the WIF shall be generated.

Additionally, taking a 1 dB margin on the maximum J/S performance of the receiver to generate the CIF, the following criteria were adopted for the CIF and WIF flags associated to interference:

• • When the difference between the received (incident) jammer power (dBw) and the received (incident) signal power (dBw) is 1 dB below the J/S performance of the receiver at its tracking threshold, the CIF shall be generated.

• • When the difference between the received (incident) jammer power (dBw) and the received (incident) signal power (dBw) is above the J/S performance of the receiver at its tracking threshold, the WIF shall be generated.

During the GPS-TSPI flight test activities performed on TORNADO-IDS with unaided L1 C/A code avionics receivers (Sabatini and Palmerini, Reference Sabatini and Palmerini2008), it was also found that, in all dynamics conditions explored, a ${\textstyle{{\rm C} \over {{\rm N}_0}}} $ of 25 dB-Hz was sufficient to continue tracking of the satellites (Sabatini and Palmerini, Reference Sabatini and Palmerini2008). Consequently, taking the usual 1 dB margin for the CIF, the following additional criteria were adopted for the TORNADO-IDS C/N₀ integrity flags:

• • When the ${\textstyle{{\rm C} \over {{\rm N}_0}}} $ is less than 26 dB-Hz the CIF shall be generated.

• • When the ${\textstyle{{\rm C} \over {{\rm N}_0}}} $ is less than 25 dB-Hz the WIF shall be generated.