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The benefits of the Gaussian sum filter can be illustrated using a simple example. Consider the following one‐dimensional navigation scenario. A radio transmitter broadcasts a ranging signal from a fixed location, x_t. A ranging receiver is mounted on a vehicle that is free to move in the x‐direction. The vehicle motion can be represented using a first‐order Gauss–Markov velocity model [2] with uncertainty σ_v and time constant τ_v. The resulting state vector is given by

(36.52)

where p_k and v_k are the position and velocity of the vehicle at time k. The dynamics of the vehicle are given by

(36.53)

where

(36.54)

and w_k is a zero‐mean Gaussian random vector with

(36.55)

The ranging signal consists of both a noise‐corrupted measurement of the true range along with a measurement of the integrated carrier phase. The integrated carrier phase is a high‐precision measurement, but is corrupted by an unknown integer ambiguity. The observation model is

Figure 36.2 MMAE filter implementation. The MMAE filter constructs the state estimate by combining results from individual Kalman filters tuned to a parameter realization [7].

(36.56)

(36.57)

where λ is the carrier wavelength, and N is the integer ambiguity. Both observations are corrupted by zero‐mean white Gaussian noise sequences with

(36.58)

(36.59)

(36.60)

Our goal is to use the MMAE estimator to accurately represent the (non‐Gaussian) posterior pdf, thus maintaining a consistent overall state estimate and uncertainty, while incorporating all available information.

In this example, the integer ambiguity is the unknown parameter set, which in the previous development we designated as the vector a. We choose a range of J plausible integers based upon any a priori knowledge or even the initial range observation itself, which results in the following unknown parameter vector:

(36.61)

with overall joint probability density

(36.62)

From this point forward, the implementation proceeds as outlined in the previous section. A total of J weighted Kalman filters are constructed, each with the assumption that N^[j] is the correct integer ambiguity. The joint posterior density is given by

(36.63)

In order to demonstrate the performance of the Gaussian sum filter, the above scenario was implemented in a simulation environment. A trajectory and measurement set is randomly generated using the parameters specified in Table 36.1.

Note the carrier phase wavelength is 0.2 m, and the carrier phase measurement uncertainty is 0.1 cycles, which results in a measurement precision of 0.02 m, which is an improvement of 50 times over the pseudorange measurement errors.

The resulting trajectory, range observations, and phase observations are shown in Figure 36.3.

Table 36.1 Simulation parameters

Parameter	Value	Units
σ ρ	0.5	m
σ ϕ	0.1	cycles
λ	0.2	m
σ v	0.2	m/s
τ v	500	s
x t	0	m
Δt	1.0	s

The MMAE global state estimate and density function of position after one observation (t = 1 s) are shown in Figure 36.4. The probability density function is clearly multi‐modal, which accurately represents the range of solutions associated with the phase observation. As expected, the peaks are located at integer multiples of the carrier wavelength which corresponds to the most likely values of the unknown integer ambiguity. These peaks indirectly indicate the relative likelihood of the associated ambiguity being correct by exhibiting influence on the overall position density.

After 22 cycles, the position density shows a reduced number of peaks (see Figure 36.5). This indicates that the filter is incorporating sensor observations and the statistical dynamics model to effectively eliminate a number of potential ambiguity possibilities.

After 100 cycles (Figure 36.6), the filter has converged to a single ambiguity.

The global state estimate and associated standard deviation result for this simulation are shown in Figure 36.7. The shape of the uncertainty bound clearly shows the effects described above. As the likelihood of each integer ambiguity realization changes, the overall uncertainty changes and eventually collapses to the centimeter level.

Finally, the associated normalized filter weights for a subset of the integer ambiguity realizations are shown in Figure 36.8. As expected, the highly unlikely edge integers quickly collapse. The integers closer to the mean take longer to resolve. It is important to note that the resulting uncertainty is dependent on the actual measurement realization sequence received; thus, each realization would produce a different uncertainty (Table 36.2). This is a notable difference from the standard linear Kalman filter, where the uncertainty is independent of the observed measurements. Finally, it is important to note that, in this example, the state estimate and uncertainty of the MMAE filter are truly optimal (i.e. minimum mean square error). This would not be the case if the integer ambiguity were resolved using a more traditional approach (e.g. float estimate with an ad hoc fixing stage). This is an interesting property of the Gaussian sum filter and sets the stage for us to investigate additional nonlinear estimation techniques.

Table 36.2 Summary of filter classes

Linear and extended Kalman filter
Strengths	Weaknesses	Use case
Optimal for linear Gaussian systemsComputationally simple	Suboptimal approximation for nonlinear systems, can be prone to divergence	Linear, or close‐to‐linear, Gaussian problems
Gaussian sum filter
Strengths	Weaknesses	Use case
Optimal for linear Gaussian systems with discrete parameter vector	If parameter vector is not discrete, the differences must be observableConservative tuning can mask difference between models and reduce performanceIncreased computation requirements over simple Kalman filter	Linear, or close‐to‐linear, Gaussian problems with discrete parameters
Grid particle filter
Strengths	Weaknesses	Use case
Optimal solution when state space consists of discrete elementsSuitable for wide range of nonlinear conditions	Computational requirements can be excessiveProcessing requirements scale geometrically with the number of dimensionsDiscretizing continuous state space results in suboptimal performance	Nonlinear problems with lower dimensionality
Sampling particle filter
Strengths	Weaknesses	Use case
Can produce nearly optimal solution for nonlinear problemsComputational requirements can be reduced over a grid particle filter via importance sampling strategies	Maintaining good particle distribution can be difficultLack of repeatability from run to runComputational requirements can still be large	Nonlinear problems with higher dimensionality

Figure 36.3 Sample vehicle trajectory and observations. Note that the range observations are accurate but not precise and the phase observations are precise but not accurate. Our goal is to accurately estimate the joint pdf of this system.

Figure 36.4 MMAE initial state estimate and position density function. Note the position density function is extremely multi‐modal due to the limited information available at this point.

Figure 36.5 MMAE state estimate (after 22 observations). Range observations combined with the vehicle dynamics model are eliminating unlikely integer ambiguity values.

Figure 36.6 MMAE state estimate (after 100 observations). Note the state estimate is almost completely unimodal and has converged to the correct integer ambiguity.