On GNSS Synchronization Performance Degradation under Interference Scenarios: Bias and Misspecified Cramér-Rao Bounds

Lorenzo Ortega; Corentin Lubeigt; Jordi Vilà-Valls,; Eric Chaumette

doi:10.33012/navi.606

Abstract

Global navigation satellite systems (GNSSs) play a key role in a plethora of applications, ranging from navigation and timing to Earth observation and space weather characterization. For navigation purposes, interference scenarios are among the most challenging operation conditions, with a clear impact on the maximum likelihood estimates (MLEs) of signal synchronization parameters. While several interference mitigation techniques exist, an approach for theoretically analyzing GNSS MLE performance degradation under interference, which is fundamental for system/receiver design, is lacking. The main goal of this contribution is to provide such analysis, by deriving closed-form expressions of the misspecified Cramér–Rao (MCRB) bound and estimation bias, for a generic GNSS signal corrupted by interference. The proposed bias and MCRB expressions are validated for a linear frequency-modulation chirp signal interference.

Keywords

1 INTRODUCTION

Global navigation satellite systems (GNSSs) (Teunissen & Montenbruck, 2017) appear in a plethora of applications, ranging from navigation and timing to Earth observation, attitude estimation, and space weather characterization. Indeed, reliable position, navigation, and timing information is fundamental in new applications such as intelligent transportation systems and autonomous unmanned ground/air vehicles, for which GNSSs have become the cornerstone source of positioning data, and this dependence can only but grow in the future. However, GNSSs were originally designed to operate in clear sky nominal conditions, and their performance clearly degrades under harsh environments. Among non-nominal operation conditions, multipath, interference (i.e., intentional [jamming] or unintentional), and spoofing conditions are the most challenging, presenting a key issue in safety-critical scenarios (Amin et al., 2016). Interference degrades GNSS performance and can lead to a denial of service or even counterfeit transmissions to control the receiver positioning solution. These effects have been reported in the state of the art, and several interference mitigation countermeasures have been proposed (Amin et al., 2017; Arribas et al., 2019; Borio & Gioia, 2021; Chien, 2015 2018; Fernández-Prades et al., 2016; Liu et al., 2022; Morales-Ferre et al., 2020; Pirayesh & Zeng, 2022).

It is well known that interference impacts the maximum likelihood estimator (MLE) of signal synchronization parameters (i.e., delay, Doppler, phase), which plays a key role in baseband signal processing in standard two-step GNSS receivers (Teunissen & Montenbruck, 2017). While several interference mitigation techniques exist (Morales-Ferre et al., 2020), an approach for theoretically analyzing the GNSS MLE performance degradation induced by an interference (or a set of interferences) is lacking, yet fundamental for system/receiver design. From an estimation perspective, because the system of interest can be formulated as a Gaussian conditional signal model (CSM) under nominal conditions, it is sound to obtain the corresponding Cramér–Rao bound (CRB) (Trees & Bell, 2007). Indeed, the CRB gives an accurate estimation of the mean square error (MSE) of the MLE in the asymptotic region of operation, i.e., in the large sample and/or high signal-to-noise ratio (SNR) regimes of the CSM (Renaux et al., 2006; Stoica & Nehorai, 1990). Even if CRBs for different GNSS receiver architectures under nominal conditions are available in the literature (see (Medina et al., 2020), (Medina et al., 2021), (McPhee et al., 2023a) and references therein), such performance bounds have not been studied for the interference case of interest in this contribution.

The main hypothesis is that the receiver is not aware that an interference is present, and therefore, it assumes that the received signal is only corrupted by additive Gaussian noise as under nominal conditions. This assumption implies that the signal model at the receiver input and the assumed signal model do not coincide, that is, there exists a model mismatch. In this case, the MLE is no longer unbiased, and theoretical characterization leads to closed-form expressions of i) the estimation bias induced by the interference (this result was first presented in Ortega et al. (2022)) and ii) the corresponding misspecified CRB (MCRB) (Richmond & Horowitz, 2015), (Fortunati et al., 2017), (Lubeigt et al., 2023), (McPhee et al., 2023b). The proposed bias and MCRB expressions are validated for a representative linear frequency-modulation (LFM) chirp signal interference. Notably, once a compact MCRB form is derived, this form can be used for i) the derivation of metrics that allow one to compare the robustness of different GNSS signals to interference and to assess the design of new GNSS signals and ii) the design of next-generation interference countermeasures.

2 TRUE AND MISSPECIFIED SIGNAL MODELS

2.1 Correctly Specified Signal Model

A GNSS band-limited signal s(t) with bandwidth B is transmitted over a carrier frequency f_c (λ_c = c/f_c, ω_c = 2πf_c). The synchronization parameters to be estimated are the delay and Doppler shift, η = (τ, b)^⊤. Under the narrowband assumption, the influence of the Doppler parameter on the baseband signal samples is negligible, s((1 – b)(t – τ)) ≈ s(t – τ) (Dogandzic & Nehorai, 2001). For short observation times, a good approximation of the baseband output of the receiver’s Hilbert filter (GNSS signal + interference) is given as follows (Skolnik, 1990):

where I(t) is a band-limited unknown interference (or set of interferences) within the frequency band of interest, n(t) is complex white Gaussian noise with an unknown variance , and α = ρe^jΦ is a complex gain. The discrete vector signal model is built from N = N₁ − N₂ + 1 samples at T_s = 1/F_s ≤ 1 / B:

with x = (…,x(kT_s),…)^⊤, I = (…,I(kT_s),…)^⊤, n = (…, n(kT_s),…)^⊤, N₁ ≤ k ≤ N₂ signal samples, and

The unknown deterministic parameters can be gathered in vector , with , 0 ≤ Φ ≤ 2π. The correctly specified signal model is represented by a probability density function (pdf) denoted as p_∈ (x; ∈), which follows a complex circular Gaussian distribution, .

2.2 Misspecified Signal Model

The misspecified signal model represents the case in which interference is not considered, i.e., when a mismatched MLE (MMLE) is implemented at the receiver. This nominal case leads to the definition of the misspecified parameter vector η′ = [τ′,b′]^⊤ and the complete set of unknown parameters , yielding the following signal model at the output of the Hilbert filter:

where n′(t) is complex white Gaussian noise with an unknown variance and α′ = ρ′e^jΦ′. Again, we can build the discrete vector signal model from N samples at T_s = 1/F_s:

The misspecified signal model is represented by a pdf denoted as f_∈′, (x; ∈′) that follows a complex circular Gaussian distribution, . We then have the following:

Note that considering the misspecified signal model induces a bias to the corresponding MMLE. These biased estimated parameters are commonly referred to as pseudotrue parameters, . For this particular contribution, we are not interested in the noise variance parameter.

3 MMLE BIAS COMPUTATION VIA KULLBACK–LEIBLER DIVERGENCE

Pseudotrue parameters are simply those that give the minimum Kullback–Leibler divergence (KLD) (Fortunati et al., 2017), D(p_∈ || f_∈′) = E_{p_∈} [ln p_∈ (x; ∈) – ln f_∈′, (x′;∈′)], between the true and assumed models, where E_{p_∈} [·] is the expectation with respect to the true model’s pdf:

We aim to compute the pseudotrue parameters, . We must then minimize Equation (8) with respect to the argument θ′, and the equation can be simplified as follows:

We define the orthogonal projector with π_A = A (A^HA)⁻¹ A^H, which leads to the following:

Then, the parameters that minimize the KLD are as follows:

Here, α_pt = ρ_pte^jΦpt and . This result may be connected to the asymptotic MMLE behavior (Fortunati et al., 2017):

Because the pseudotrue parameters, obtained as the MMLE without noise, are those that give the minimum KLD between the true and assumed models, the bias is defined as Δα = α_pt − α, Δη = η_pt − η.

4 CLOSED-FORM MCRB EXPRESSIONS FOR A BAND-LIMITED SIGNAL UNDER INTERFERENCE

In Richmond & Horowitz (2015), the MCRB was derived as an extension of the Slepian–Bangs formulas, a result that was later expressed as a combination of two information matrices (A(θ_pt) and B(θ_pt)) in Fortunati et al. (2017):

with the following relations:

Here, δm = ≜ αa(η −α_ptμ(η_pt) = αμ(η + I − α_ptμ(η_pt) is the mean difference between the true and misspecified models.

4.1 Single-Source Fisher Information Matrix

In B(θ_pt), one can recognize the Fisher information matrix (FIM) of a single-source CSM. A compact expression of this FIM, which depends only on the baseband signal samples, was recently derived in Medina et al. (2020). For completeness, we recall the following:

with

Here, the elements of W can be expressed with respect to the baseband signal samples as follows:

s, the baseband sample vector, D, V^Δ,1(·), and V^Δ,2(·) are defined as follows:

14a

14b

14c

14d

We refer the reader to Appendix B for details on the closed-form expressions of V^Δ,1(q) and V^Δ,2(q).

4.2 Model Mismatch Information Matrix

The matrix A(θ_pt) accounts for the model misspecification. Its elements can also be expressed in a compact form as a function of the baseband signal and interference samples as follows:

with , and for l∈ (1, ⋯, 6). Here, [Q_q]_p,. is the p-th row of the matrix Q_q (refer to Appendix A for Q_q). With Δτ = τ − τ_pt and Δb = b − b_pt, W^A is obtained from the following:

16a

16b

16c

16d

16e

16f

16g

16h

16i

16j

16k

16l

16m

with

Proof. See Appendices A and B.

4.3 Implementation of the Bias and MCRB Expressions

In this section, we provide a step-by-step explanation of how to calculate the bias and MCRB of the synchronization parameters of the received signal:

First, we must calculate the parameters α_pt = ρ_pte^jΦpt and from Equation (10).
Then, we compute the bias of the synchronization parameters as Δα = α_pt – α, Δη = η_pt – η.
To compute the MCRB, we first compute the single-source FIM B(θ_pt). This process is described in Section 4.1.
Then, we compute the model mismatch information matrix A(θ_pt). To do this, we apply the following steps:
- – We compute from Equation (15).
- – To compute W^A, we define Δτ = τ−τ_pt and Δb = b−b_pt. Then, we compute the elements of the matrix given by Equations (16a)–(16m).
- – We next compute the matrices Q_q, with q = {1, 2, 3, 4}, which are included in Appendix A.
- – Finally, we compute
The MCRB is then computed as MCRB (θ_pt) = A(θ_pt)⁻¹B(θ_pt)A(θ_pt)⁻¹.

5 VALIDATION

Let us consider the case in which a global positioning system (GPS) L1 C/A signal experiences interference from a jammer that is generating an LFM chirp signal, which is defined as follows:

where α_c is the chirp rate, A_i is the amplitude, and T = NT_s is the waveform period. The instantaneous frequency is and therefore, the waveform bandwidth is B = α_cT. We consider the case in which, after the Hilbert filter, the chirp is located at the baseband frequency, i.e., the central frequency of the chirp is f_i = 0. Then, the chirp equation can be rewritten as follows:

The MSE and bias results for the parameters of interest, θ^T = [ρ, Φ, η^T], are shown in Figures 1–4, with respect to the SNR at the output of the matched filter (i.e., SNR_OUT) and considering the following setup: a GNSS receiver with F_s = 4 MHz and a chirp bandwidth equal to 2 MHz, with initial phase ϕ = 0 and amplitude A_i = 10. The number of Monte Carlo iterations is set to 1000. In the results, one can observe that i) the root MSE of the true parameter converges to , ii) of the pseudotrue parameter converges to , and iii) is always higher than (refer to (Medina et al., 2020)), which represents the asymptotic estimation performance of the parameters without any source of interference. Such results validate and prove the exactness of the proposed MCRB and bias expressions. Finally, we emphasize that the MCRB characterizes the MMLE asymptotically and is therefore unable to evaluate any occurrences prior to the convergence region. Therefore, the calculation of the MSE of the MMLE also indicates the threshold from which the MCRB theoretically characterizes the MSE of the MMLE.

FIGURE 1

MMLE root MSE for the time-delay τ estimation with respect to the true and pseudotrue parameters and the corresponding bounds. The interference is a chirp signal with B = 2 MHz, A_i = 10, and initial phase ϕ = 0. The integration time is set to 2 ms.

FIGURE 2

MMLE root MSE for the Doppler F_d estimation with respect to the true and pseudotrue parameters and the corresponding bounds. The interference is a chirp signal with B = 2 MHz, A_i = 10, and initial phase ϕ = 0. The integration time is set to 2 ms.

FIGURE 3

MMLE root MSE for the amplitude ρ estimation with respect to the true and pseudotrue parameters and the corresponding bounds. The interference is a chirp signal with B = 2 MHz, A_i = 10, and initial phase ϕ = 0. The integration time is set to 2 ms.

FIGURE 4

E root MSE for the phase Φ estimation with respect to the true and pseudotrue parameters and the corresponding bounds. The interference is a chirp signal with B = 2 MHz, A_i = 10, and initial phase ϕ = 0. The integration time is set to 2 ms.

In a second example, we evaluate the degradation caused by a single tone located at frequency f_i = 0.5 MHz. For this particular case, the interference samples are given by I = I(⋯, A_ie^{j2πf_ikT_s+jϕ},…), which is a complex function, and can be rewritten as follows:

where ϕ is the initial phase of the tone and A_i is the amplitude of the tone. For our particular scenario, we set the initial phase to π / 2 and A_i = 10. In Figures 5, 7, 9, and 11, we illustrate the MSE and bias results for the parameters of interest, θ^T = [ρ, Φ, η^T], as a function of the SNR at the output of the match filter, SNR_OUT. We set F_s = 4 MHz and the integration time to 2 ms. Note that the MSE converges to the theoretical result, re-validating the closed-form expressions. Moreover, in Figures 6, 8, 10, and 12, we also include one scenario in which the integration time is set to 4 ms. Note that for this particular case, the bias is lower and the Doppler estimation performance is improved. This result can be proved theoretically owing to the closed-form expressions of the FIM, which allow us to assess how the different design parameters affect the calculation of the MSE of the MLE. For this particular case, increasing the integration time increases the dimension of the matrices D and D², which are related to the Fisher matrix parameters of the Doppler parameter. As the integration time increases, the estimation performance improves.

FIGURE 5

MMLE root MSE for the time-delay τ estimation with respect to the true and pseudotrue parameters and the corresponding bounds. The interference is a tone signal with f_i = 0.5 MHz, A_i = 10, and initial phase ϕ = π / 2. The integration time is set to 2 ms.

FIGURE 6

FIGURE 7

MMLE root MSE for the Doppler estimation with respect to the true and pseudotrue parameters and the corresponding bounds. The interference is a tone signal with f_i = 0.5 MHz, A_i = 10, and initial phase ϕ = π / 2. The integration time is set to 2 ms.

FIGURE 8

MMLE root MSE for the Doppler estimation with respect to the true and pseudotrue parameters and the corresponding bounds. The interference is a tone signal with f_i = 0.5 MHz, A_i = 10, and initial phase ϕ = π / 2. The integration time is set to 4 ms.

FIGURE 9

MMLE root MSE for the amplitude ρ estimation with respect to the true and pseudotrue parameters and the corresponding bounds. The interference is a tone signal with f_i = 0.5 MHz, A_i = 10, and initial phase ϕ = π / 2. The integration time is set to 2 ms.

FIGURE 10

MMLE root MSE for the amplitude p estimation with respect to the true and pseudotrue parameters and the corresponding bounds. The interference is a tone signal with f_i = 0.5 MHz, A_i = 10, and initial phase ϕ = π / 2. The integration time is set to 4 ms.

FIGURE 11

MMLE root MSE for the phase Φ estimation with respect to the true and pseudotrue parameters and the corresponding bounds. The interference is a tone signal with f_i = 0.5 MHz, A_i = 10, and initial phase ϕ = π / 2. The integration time is set to 2 ms.

FIGURE 12

6 CONCLUSION

It is well documented in the literature that interference signals can have a substantial impact on the performance of GNSS receivers, but to the best of the authors’ knowledge, from an estimation perspective, an approach for theoretically analyzing the impact of such interference on the first GNSS receiver stage (i.e., time-delay and Doppler estimation) is lacking. In practice, at the receiver, there exists a model mismatch, and interference induces both i) an estimation bias and ii) a variance degradation. In this contribution, we provided theoretical closed-form expressions that characterize the MSE for the MLEs of the GNSS synchronization parameters, that is, bias and MCRB. Comparing these results with the standard CRB, associated with the unbiased MLEs without any interference, allows one to theoretically characterize the performance degradation of the time-delay and Doppler estimation. The exactness of the proposed expressions was validated for a representative case of a chirp interference jamming a GPS L1 C/A signal. Results were provided to demonstrate this validity and the impact on both time-delay and Doppler estimation. Importantly, such analyses may provide a starting point for deriving robustness metrics or new GNSS signals and for designing interference countermeasures.

How to cite this article:

Ortega, L., Lubeigt, C., Vilà-Valls, J., & Chaumette, E. (2023). On GNSS synchronization performance degradation under interference scenarios: Bias and misspecified Cramér-Rao bounds. NAVIGATION, 70(4). https://doi.org/10.33012/navi.606

CONFLICT OF INTEREST

The authors declare no potential conflicts of interest.

ACKNOWLEDGMENTS

This work was partially supported by DGA/AID projects 2022.65.0082 and 2021.65.0070.00.470.75.01 and TéSA. Part of this work was previously presented at the ION GNSS+ 2022 conference (Ortega et al., 2022).

APPENDIX

A ON THE COMPUTATION OF A (θ_PT)

To compute A(θ_pt), continuous time expressions are considered: μ(t; η) = s(t −τ)e^{−jω_cb(t−τ)}, , with Ã(t) = [μ(t;η), I(t), μ(t;η_pt)] and , which leads to the discrete expression . The second derivative of interest can be written in matrix form as follows:

with

where s⁽¹⁾(·) and s⁽²⁾(·) refer to the first and second time derivatives, respectively. The product of the mean difference term and the Hessian matrix, under its discrete form, can be written as follows:

This product can also be written as follows:

with

When the number of samples tends to infinity, each β_l is the sum of three integrals:

This result leads to the expression in Equation (15):

Then, the computation of A(θ_pt) is reduced to three sets of integrals. The first set of integrals is as follows:

The corresponding closed-form expressions are given in Equations (16a)–(16f). The derivation of , , and can be found in Lubeigt et al. (2020) (Equations (A.27), (A.28), and (A.29), respectively). The remaining terms are derived in Appendix B. The second set of integrals is as follows:

The corresponding closed-form expressions are given in Equations (16g)–(16l). The derivation of these terms is given in Appendix B. For the last set, we have the following:

B DERIVATION OF INTERFERENCE CONVOLUTION TERMS USING FOURIER TRANSFORM PROPERTIES

B.1 Prior Considerations

First, we evaluate the Fourier transform of a set of functions. Remembering that the signal is band-limited by band B ≤ F_s, we have the following:

To address any issue that may arise from the spectral shift due to the Doppler effect, one must simply set F_s to be sufficiently large such that .

A first expression is a simple application of the frequency shift relation that is obtained when using the Fourier transform of a signal multiplied by a complex time-varying exponential:

Then, with s₁ defined as s₁(t; b) = s(t)e^j2πf_cbt, we have the following:

Therefore, we obtain the following:

Similarly, we obtain the following relation:

With the superscript ⁽¹⁾ referring to the first time derivative, we have the following:

We have the Fourier transform of the k-th time derivative of a function as follows:

Thus, one directly obtains the following relation:

Now, if s₂ is defined as s₂(t; b) = ts(t)e^j2πf_cbt, we have the following:

Therefore, we obtain the following relation:

Finally, we take s₁ as s₁(t; b) = s(t)e^j2πf_cbt as follows:

Consequently, we obtain the following:

B.2 Evaluation of the Integrals

B.2.1 Derivation of Integral

We apply the Fourier transform properties over the Hermitian product:

Hence, we obtain the following:

B10

with U(p) defined in Equation (17) and V^Δ,0(q) defined in Equation (18). Note the following relations:

B11

B12

B13

B.2.2 Derivation of Integral

Therefore, we have the following relation:

Hence, we obtain the following:

B14

with U, V^Δ,0, and V^Δ,1(q) defined in Equations (17), (18), and (14c), respectively. Note that we have the following relations:

B15

B16

B.2.3 Derivation of Integral

Therefore, we have the following:

Hence, we have the following relation:

B17

with U(·) defined in Equation (17), V^Δ,0(·) defined in Equation (18), V^Δ,1 defined in Equation (14c), and V^,2(·) defined in Equation (14d). Note the following relations:

B18

B19

B.2.4 Derivation of Integral

We apply the Fourier transform properties over the Hermitian product:

Hence, we obtain the following:

B20

B.2.5 Derivation of Integral

Therefore, we have the following:

Hence, we have the following relation:

B21

with U and V^Δ,0 defined in Equations (17) and (18), respectively, and D defined in Equation (14b).

B.2.6 Derivation of Integral

Therefore, we obtain the following:

Hence, we have the following relation:

B22

with U, V^Δ,0, and D defined in Equations (17), (18), and (14b), respectively.

B.2.7 Derivation of Integral

Therefore, we have the following:

Hence, we obtain the following relation:

B23

with U, V^Δ,0, and V^Δ,1 defined in Equations (17), (18), and (14c), respectively.

B.2.8 Derivation of Integral

Therefore, we have the following:

Hence, we obtain the following relation:

B24

with U, V^Δ,0, V^Δ,1, and D defined in Equations (17), (18), (14c), and (14b), respectively.

B.2.9 Derivation of Integral

Therefore, we obtain the following:

Hence, we have the following relation:

B25

with U, V^Δ,0, and V defined in Equations (17), (18), and (14c), respectively.

B.3 Matrix Properties

Based on the definitions of matrices V^Δ,0, V^Δ,1, V^Δ,2, and U, we have the following relations:

(V^Δ,0(q))^H = V^Δ,0(−q)
(V^Δ,1(q))^H = −V^Δ,1(−q)
(V^Δ,2(q))^H = V^Δ,2(−q)
(U^p))^H = U(−p)

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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2. Zeng, H.
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↵
1. Renaux, A.,
2. Forster, P.,
3. Chaumette, E., &
4. Larzabal, P.
(2006). On the high-SNR conditional maximum likelihood estimator full statistical characterization. IEEE Trans. Signal Process., 54(12), 4840 – 4843. https://doi.org/10.1109/TSP.2006.882072
↵
1. Richmond, C. D., &
2. Horowitz, L. L.
(2015). Parameter bounds on estimation accuracy under model misspecification. IEEE Trans. Signal Process., 63(9), 2263–2278. https://doi.org/10.1109/TSP.2015.2411222
↵
1. Skolnik, M. I.
(1990). Radar Handbook (3rd ed.). McGraw-Hill. https://doi.org/10.1109/MAES.2008.4523916
↵
1. Stoica, P., &
2. Nehorai, A.
(1990). Performances study of conditional and unconditional direction of arrival estimation. IEEE Trans. Acoustics, Speech, Signal Processing, 38(10), 1783–1795. https://doi.org/10.1109/29.60109
↵
1. Teunissen, P. J. G., &
2. Montenbruck, O. (Eds.).
(2017). Handbook of global navigation satellite systems. Springer. https://doi.org/10.1007/978-3-319-42928-1
↵
1. Trees, H. L. V., &
2. Bell, K. L. (Eds.).
(2007). Bayesian Bounds for Parameter Estimation and Nonlinear Filtering/Tracking. Wiley/IEEE Press. Retrieved from https://ieeexplore.ieee.org/servlet/opac?bknumber=5263120

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On GNSS Synchronization Performance Degradation under Interference Scenarios: Bias and Misspecified Cramér-Rao Bounds

Abstract

1 INTRODUCTION

2 TRUE AND MISSPECIFIED SIGNAL MODELS

2.1 Correctly Specified Signal Model

2.2 Misspecified Signal Model

3 MMLE BIAS COMPUTATION VIA KULLBACK–LEIBLER DIVERGENCE

4 CLOSED-FORM MCRB EXPRESSIONS FOR A BAND-LIMITED SIGNAL UNDER INTERFERENCE

4.1 Single-Source Fisher Information Matrix

4.2 Model Mismatch Information Matrix

4.3 Implementation of the Bias and MCRB Expressions

5 VALIDATION

6 CONCLUSION

How to cite this article:

CONFLICT OF INTEREST

ACKNOWLEDGMENTS

APPENDIX

A ON THE COMPUTATION OF A (θPT)

B DERIVATION OF INTERFERENCE CONVOLUTION TERMS USING FOURIER TRANSFORM PROPERTIES

B.1 Prior Considerations

B.2 Evaluation of the Integrals

B.2.1 Derivation of Integral <img class="highwire-embed" alt="Embedded Image" src="https://navi.ion.org/sites/default/files/highwire/navi/70/4/navi.606/embed/inline-graphic-31.gif"/>

B.2.2 Derivation of Integral <img class="highwire-embed" alt="Embedded Image" src="https://navi.ion.org/sites/default/files/highwire/navi/70/4/navi.606/embed/inline-graphic-32.gif"/>

B.2.3 Derivation of Integral <img class="highwire-embed" alt="Embedded Image" src="https://navi.ion.org/sites/default/files/highwire/navi/70/4/navi.606/embed/inline-graphic-33.gif"/>

B.2.4 Derivation of Integral <img class="highwire-embed" alt="Embedded Image" src="https://navi.ion.org/sites/default/files/highwire/navi/70/4/navi.606/embed/inline-graphic-34.gif"/>

B.2.5 Derivation of Integral <img class="highwire-embed" alt="Embedded Image" src="https://navi.ion.org/sites/default/files/highwire/navi/70/4/navi.606/embed/inline-graphic-35.gif"/>

B.2.6 Derivation of Integral <img class="highwire-embed" alt="Embedded Image" src="https://navi.ion.org/sites/default/files/highwire/navi/70/4/navi.606/embed/inline-graphic-36.gif"/>

B.2.7 Derivation of Integral <img class="highwire-embed" alt="Embedded Image" src="https://navi.ion.org/sites/default/files/highwire/navi/70/4/navi.606/embed/inline-graphic-37.gif"/>

B.2.8 Derivation of Integral <img class="highwire-embed" alt="Embedded Image" src="https://navi.ion.org/sites/default/files/highwire/navi/70/4/navi.606/embed/inline-graphic-38.gif"/>

B.2.9 Derivation of Integral <img class="highwire-embed" alt="Embedded Image" src="https://navi.ion.org/sites/default/files/highwire/navi/70/4/navi.606/embed/inline-graphic-39.gif"/>

B.3 Matrix Properties

REFERENCES

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Keywords

A ON THE COMPUTATION OF A (θ_PT)

B.2.1 Derivation of Integral

B.2.2 Derivation of Integral

B.2.3 Derivation of Integral

B.2.4 Derivation of Integral

B.2.5 Derivation of Integral

B.2.6 Derivation of Integral

B.2.7 Derivation of Integral

B.2.8 Derivation of Integral

B.2.9 Derivation of Integral