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A Fully Digital Model for Kalman Filters

“GNSS Solutions” is a regular column featuring questions and answers about technical aspects of GNSS. Readers are invited to send their questions to the columnist, Dr. Mark Petovello, Department of Geomatics Engineering, University of Calgary, who will find experts to answer them.

What about fully digital state models for Kalman filters? Columnist Mark Petovello asks Letizia Lo Presti, Marco Rao and Simone Savasta for the answer.

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Q: Is it possible to define a fully digital state model for Kalman filtering?

A: The Kalman filter is a mathematical method, purpose of which is to process noisy measurements in order to obtain an estimate of some relevant parameters of a system. It represents a valuable tool in the GNSS area, with some of its main applications related to the computation of the user position/velocity/time (PVT) solution and to the integration of GNSS receivers with an inertial navigation system (INS) or other sensors.

The Kalman filter is based on a state space representation that describes the analyzed system as a set of differential equations that establishes the connections between the inputs, the outputs, and the state variables of the analyzed system.

Although the state space differential equations are expressed in the continuous time domain, the filter itself is implemented in the discrete time domain, as required by the periodic availability of data/measurements. The typical approach to this problem is to linearize the continuous time system using a Taylor series and then obtain a discrete time approximation therefrom. However, it can be helpful to approach the problem from a discrete time point of view directly.

Several such approaches have previously been developed in the signal processing field and can be extended to the Kalman filter. In the following, we compare the classical method based on the Taylor approximation with a method based on the Laplace-domain (s-domain) to z-domain transformations.

Our purpose is to give some simple rules and methods with which to write the state equations and to prove that the results of the classical methods are only a special case of the more general class of s-z transformations, beause the already known results will be obtained with the presented method.

The Position-Velocity-Acceleration (PVA) Model
For illustration purposes, we consider the three-state (position, velocity and acceleration) model in the Laplace domain (s-domain).

. . .

Method based on Taylor expansion
The typical approach to obtain the state evolution in the discrete time-domain consists of sampling (3) by t = nTs, where Ts = 1/fs is a proper sampling interval that satisfies the Nyquist theorem and fs is the sampling frequency. Then, the exponential in Equation (3) is expanded using a Taylor series expansion, truncated to the second order.

. . .

Method Based on s-z Transformations
We present now an alternative approach, which is based on the idea of representing the signals and the systems in Figure 1 in the discrete-time domain, where the continuous-time t becomes the digital time n and the complex plane s becomes z.

These transformations are ruled by some well-known methods of the theory of digital signal processing. We first need to recall two important results of this discipline to find the way to transform the analog systems of Figure 1 into an all-digital system: the concept of a white sequence and the simulation theorem.

The White Sequence. In order to prevent aliasing of the white noise process, it is common to prefilter the signal prior to sampling. This eliminates the frequencies that cannot be represented in the sampled signal (i.e., those outside the Nyquist bandwidth) and avoids impairing the frequencies that can be represented.

. . .

The Simulation Theorem. To obtain a numerical version H(z) of a generic analog transfer function Ha(f), the Papoulis simulation theorem has to be considered: a discrete representation of an analytical version Ha(f) can be simulated if a generic input x[n] = x(nTs) provides an output discrete signal that is a sampled version of the analog output y(t) of the system Ha(f).

. . .

From the s Plane to the z Plane
Having defined the discrete-time forcing function and the conditions to design the digital transfer functions, the next step is to convert a transfer function Ha(s) from the s domain to the z domain, so as to satisfy the requirements imposed by the simulation theorem.

A unique method to perform this transformation does not exist. In fact, we can obtain the transfer function H(z) from Ha(s) by different mappings of the s plane on the unit circle of the z plane.

. . .

Digital representation of the PVA system
Starting from the previous example, we can derive a new form of (5), starting from a discrete-time driving function η[n] and considering three different mappings of the integrators of Figure 1.  Specifically, the first through third integrators are respectively replaced by the rectangular method, the bilinear transform, and the Cavalieri-Simpson method.

We should point out that the order of these transformations is only required to obtain the results already known in the literature, but it is not mandatory. In fact, any other order or transformation will lead to equally valid results, which are based on different approximations and implementation complexity.

. . .

The equivalence of the results provided by the two different methods shows how the fully digital method is equivalent to the classical procedure, limiting the calculations to discrete equations and vectors operations. The presented discrete time method can be useful for a novice of Kalman filter theory, for anyone who has to deal with complicated model definitions, and/or for those more familiar with discrete systems.

The fully digital approach is easily applied to any kind of H(s); for example, a first order Gauss-Markov process can be modeled in the digital domain applying one among the transformations shown in Figure 3, with different levels of approximation. Even more complex systems such as INS/GPS integrated systems can be described using the fully digital method, obtaining also different results from the ones already described in the literature.

(For Letizia lo Presti, Marco Rao, and Simone Savasta’s complete answer to this question,
including formulas and tables, please download the full article using
the pdf link above.)

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