Why Use a Kalman Filter?

The Kalman filter is a recursive algorithm for estimating the state of a dynamical system in the presence of noise. Invented by Rudolf E. Kálmán in the 1960s, it underpins guidance systems, robotics, and signal processing. The beauty of the filter lies in its efficiency and optimality for linear systems with Gaussian noise. It combines prior knowledge of a system's state with noisy measurements to produce an improved estimate. In this calculator we consider the simplest case: a one-dimensional state undergoing a prediction step and an update with a single measurement.

The Prediction Step

Before receiving a new measurement, we predict the next state using a model. For a constant or slowly changing variable, the prediction may simply carry over the previous estimate. We represent the predicted state as ${\mathrm{\backslash hat\{x\}}}_{\mathrm{p}}$ and its variance as $P_{\mathrm{p}}$. If the process has variance $Q$, we update the variance by $P_{\mathrm{p}}=P_0+Q$. This prediction accounts for uncertainty in the system's evolution.

The Update Step

Once a measurement $z$ becomes available with variance $R$, we blend it with the predicted state. The Kalman gain $K$ balances trust between the prediction and the measurement:

$K=\frac{P_{\mathrm{p}}}{P_{\mathrm{p}}+R}$

The updated estimate is then

${\mathrm{\backslash hat\{x\}}}_1={\mathrm{\backslash hat\{x\}}}_{\mathrm{p}}+K\left(z-{\mathrm{\backslash hat\{x\}}}_{\mathrm{p}}\right)$

The updated variance becomes

$P_1=\left(1-K\right)P_{\mathrm{p}}$

This iterative process repeats as new measurements arrive. Over time, the estimate converges, providing a smoothed representation of the underlying variable even when individual measurements are noisy.

Using the Calculator

Enter your prior estimate and its variance, followed by the latest measurement and associated variance. Choose a process variance that reflects how much the true state might drift between updates. Press Update to compute the Kalman gain, the updated estimate, and the new variance. The calculator displays these values with four decimal places of precision. Because the calculation involves only basic arithmetic, you can quickly explore different scenarios and see how the filter balances prediction versus measurement.

Applications and Intuition

Although this example is one-dimensional, the same principles extend to multidimensional systems. Aircraft navigation filters the position and velocity in three dimensions, while economics models smooth noisy financial data. The Kalman gain effectively weights how much we trust the latest measurement compared to our predicted value. When the measurement variance $R$ is large, the gain is small and the filter leans on the prediction. When $R$ is small, the measurement dominates. Over repeated updates, the filter tends to reduce variance, yielding a more confident estimate than any single measurement alone.

Example Scenario

Imagine tracking the temperature of a chemical reactor with a sensor that has a standard deviation of 2 degrees. Our prior estimate from the previous time step is 100 degrees with a variance of 1. If we assume the process variance is 0.5, our prediction variance becomes 1.5. A new sensor reading of 97 degrees arrives. Plugging these numbers into the formula gives a Kalman gain of approximately $K=\frac{1.5}{1.5+4}$, which is about 0.2727. The updated temperature estimate is around 99.2 degrees, with a variance reduced to about 1.09. Repeating this process with subsequent measurements progressively refines our knowledge of the true temperature.

From satellite tracking to smartphone sensors, the Kalman filter remains a lightweight yet powerful tool for state estimation. By experimenting with different variances and measurements in this calculator, you can see how quickly the filter adapts and how measurement noise influences the outcome. Whether you are a student learning control theory or an engineer prototyping a sensor fusion algorithm, this single-step calculator provides a concrete demonstration of the Kalman filter's mechanics.

The algorithm owes its success to an elegant derivation rooted in linear algebra and probability theory. Kálmán recognized that minimizing the mean squared error of the estimate leads to recursive equations blending prediction and observation. These equations scale gracefully to high dimensions and continuous-time settings, forming the basis for extended and unscented versions used in complex applications.

By experimenting with the numbers here, you can see how the filter behaves when the process variance grows or the sensor becomes less reliable. The formulas may appear simple, but they encapsulate a profound insight: weighting information according to its uncertainty yields the best possible estimate in this linear Gaussian framework. This calculator lets you experience that insight firsthand.