These are the stochastic processes available in the model selection panel.

• WN - White Noise : independent sample-to-sample random noise.
• RW - Random Walk : integrated white noise with long-term growth.
• GM - Gauss-Markov : correlated process with exponential memory.
• DR - Drift : slowly varying deterministic trend component.
• QN - Quantization Noise : high-frequency measurement resolution noise.

1) Continuous-time model

$$x(t) = w(t)$$

$$\mathbb{E}[w(t)] = 0, \qquad \mathbb{E}[w(t)w(s)] = q\,\delta(t-s)$$

where:

• \(q\): continuous-time noise intensity (power spectral density).
• \(\delta(\cdot)\): Dirac delta function.
• \(t,s\): time instants.

2) Sampling at frequency \(f\)

$$t_k = k\Delta t, \qquad \Delta t = \frac{1}{f}$$

We define the discrete-time sample over the interval \([t_k, t_k+\Delta t]\) as:

$$x_k := x_{t_k} = \frac{1}{\Delta t}\int_{t_k}^{t_k+\Delta t} w(t)\,dt.$$

Using the covariance definition of white noise, the variance of the sampled process is

$$\mathrm{Var}(x_k) = \frac{1}{\Delta t^2} \int_{t_k}^{t_k+\Delta t}\int_{t_k}^{t_k+\Delta t} q\,\delta(t-s)\,dt\,ds = \frac{q}{\Delta t}. $$

Therefore the sampled process is

$$x_k \sim \mathcal{N}(0,\sigma^2), \qquad \sigma^2 = \frac{q}{\Delta t} = q f.$$

3) Parameter conversion

• Estimated parameter: \(\sigma^2\) (discrete-time variance).
• Continuous intensity: \(q = \sigma^2\Delta t = \sigma^2/f\).
• Returned WN parameter in the KF table: \(\sqrt{q}\).

Using \([\cdot]\) to denote the units of a quantity, and \(\diamond\) to denote the base unit of the signal.

Examples: for a gyroscope, \(\diamond\) can be \(\frac{\mathrm{deg}}{\mathrm{s}}\) or \(\frac{\mathrm{rad}}{\mathrm{s}}\); for an accelerometer, \(\diamond\) can be \(\frac{\mathrm{mm}}{\mathrm{s}}\) or \(\frac{\mathrm{m}}{\mathrm{s}^2}\).

Units: if \([x] = \diamond\), then

$$[q] = \frac{\diamond^2}{\mathrm{Hz}}$$

and therefore

$$\left[\sqrt{q}\right] = \frac{\diamond}{\sqrt{\mathrm{Hz}}}. $$

1) Continuous-time model

A continuous-time random walk is defined as the integral of white noise:

$$\frac{dx(t)}{dt} = w(t)$$

$$\mathbb{E}[w(t)] = 0, \qquad \mathbb{E}[w(t)w(s)] = q\,\delta(t-s)$$

Equivalently,

$$x(t) = \int_0^t w(\tau)\,d\tau.$$

2) Sampling at frequency \(f\)

$$t_k = k\Delta t, \qquad \Delta t = \frac{1}{f}$$

The increment between two samples is

$$x(t_{k+1}) - x(t_k) = \int_{t_k}^{t_{k+1}} w(t)\,dt.$$

Using the covariance definition of white noise, the variance of this increment is

$$\mathrm{Var}\big(x(t_{k+1})-x(t_k)\big) = \int_{t_k}^{t_{k+1}}\int_{t_k}^{t_{k+1}} q\,\delta(t-s)\,dt\,ds = q\Delta t.$$

Therefore the discrete-time model is

$$x_{k+1} = x_k + \eta_k, \qquad \eta_k \sim \mathcal{N}(0,\gamma^2)$$

with

$$\gamma^2 = q\Delta t = \frac{q}{f}. $$

3) Parameter conversion

• Estimated parameter: \(\gamma^2\) (variance of the increments).
• Continuous intensity: \(q = \gamma^2 / \Delta t = \gamma^2 f\).
• Returned RW parameter in the KF table: \(\sqrt{q}\).

Units follow the same convention introduced in the WN card.

$$[\gamma^2] = \diamond^2, \qquad [q] = \frac{\diamond^2}{\mathrm{s}}, \qquad [\sqrt{q}] = \frac{\diamond}{\sqrt{\mathrm{s}}} = \frac{\diamond}{\mathrm{s}\sqrt{\mathrm{Hz}}}. $$

1) Continuous-time model (Ornstein–Uhlenbeck / FOGM)

$$\frac{dx(t)}{dt} = -\beta\,x(t) + w(t)$$

$$\mathbb{E}[w(t)] = 0, \qquad \mathbb{E}[w(t)w(s)] = q\,\delta(t-s)$$

where \(\beta>0\) is the decay rate and \(q\) is the continuous-time noise intensity.

2) Sampling at frequency \(f\) and AR(1) reparametrization

$$t_k = k\Delta t, \qquad \Delta t = \frac{1}{f}, \qquad x_k := x(t_k).$$

Over one sampling interval, the solution can be written as a deterministic decay term plus a driven term:

$$x_{k+1} = e^{-\beta\Delta t}\,x_k + \int_{t_k}^{t_{k+1}} e^{-\beta(t_{k+1}-s)}\,w(s)\,ds.$$

Define the AR(1) parameters

$$\phi := e^{-\beta\Delta t}, \qquad \eta_k := \int_{t_k}^{t_{k+1}} e^{-\beta(t_{k+1}-s)}\,w(s)\,ds,$$

which yields the discrete-time AR(1) form

$$x_{k+1} = \phi x_k + \eta_k, \qquad \eta_k \sim \mathcal{N}(0,\sigma^2).$$

Using \(\mathbb{E}[w(t)w(s)]=q\delta(t-s)\), the innovation variance is

$$\sigma^2 = \mathrm{Var}(\eta_k) = \frac{q}{2\beta}\left(1-e^{-2\beta\Delta t}\right).$$

3) Parameter conversion

• Estimated parameters (discrete): \(\phi\) and \(\sigma^2\).
• Continuous decay rate: $$\beta = -\frac{\log(\phi)}{\Delta t}.$$
• Continuous intensity: $$q = \frac{2\beta\sigma^2}{1-e^{-2\beta\Delta t}}.$$
• Returned GM parameters in the KF table: \(\sqrt{q}\) and \(1/\beta\).

The quantity \(1/\beta\) is often called the correlation time, commonly denoted by \(\tau\).

Units

$$[\beta] = \frac{1}{\mathrm{s}}, \qquad \left[\frac{1}{\beta}\right]=\mathrm{s}.$$

$$[q] = \frac{\diamond^2}{\mathrm{s}}, \qquad \left[\sqrt{q}\right] = \frac{\diamond}{\sqrt{\mathrm{s}}} = \frac{\diamond}{\mathrm{s}\sqrt{\mathrm{Hz}}}. $$

1) Continuous-time model

A deterministic drift is defined as a linear trend in time:

$$x(t) = x(0) + \omega\,t$$

where \(\omega\) is the drift rate.

2) Sampling at frequency \(f\)

$$t_k = k\Delta t, \qquad \Delta t = \frac{1}{f}, \qquad x_k := x(t_k).$$

This gives

$$x_k = x(0) + \omega\,k\Delta t.$$

Equivalently, the discrete-time recursion is

$$x_{k+1} = x_k + \mu,$$

with the per-sample drift increment

$$\mu := \omega\Delta t = \frac{\omega}{f}. $$

3) Parameter conversion

• Estimated parameter: \(\mu\) (drift increment per sample).
• Continuous drift rate: \(\omega = \mu/\Delta t = \mu f\).
• Returned DR parameter in the KF table: \(\omega\).

Units

$$[\omega] = \frac{\diamond}{\mathrm{s}}, \qquad [\mu] = \diamond.$$

1) Measurement model

Quantization noise arises from the finite resolution of digital sensors.

A continuous signal \(x(t)\) is recorded with a quantization step \(Q\).

The measured value is

$$x_k^{(m)} = Q\,\mathrm{round}\!\left(\frac{x(t_k)}{Q}\right).$$

The quantization error is

$$e_k = x_k^{(m)} - x(t_k), \qquad e_k \in [-Q/2,\,Q/2].$$

Under the usual assumption that the signal varies sufficiently between samples, the error is modeled as

$$e_k \sim U(-Q/2,\,Q/2).$$

2) Discrete-time stochastic representation used in GMWM

To reproduce the theoretical Allan variance / wavelet variance of quantization noise, the process is modeled as

$$x_k = \sqrt{12Q^2}\,(Y_k - Y_{k-1}),$$

with

$$Y_k \sim U(0,1).$$

This construction generates a process whose wavelet variance corresponds to quantization noise.

3) Parameter conversion

• Estimated parameter: \(Q^2\).
• Returned QN parameter in the KF table: \(Q\).
• The parameter does not depend on the sampling frequency \(f\).

Units

Since \(Y_k\) is dimensionless, the signal and the quantization step share the same unit.

$$[Q] = \diamond, \qquad [Q^2] = \diamond^2.$$

This application helps you model stochastic sensor noise from Inertial Measurement Unit time series using the Generalized Method of Wavelet Moments.

The workflow is: inspect the wavelet variance, select candidate processes, fit the model, and compare raw estimated parameters with frequency-aware transformed parameters for Kalman-filter usage.

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How to use

1. Select a dataset (library or custom).

2. Choose a sensor channel and model components (WN, RW, GM, DR, QN).

3. Click “Fit Model” to estimate the selected model.

4. Review the Summary tab (left: estimated parameters, right: transformed Kalman-filter parameters).

Application developed by: