Kalman Filter Algorithm: Core Principles, Advantages, Applications, and C Code Implementation

This article provides a comprehensive breakdown of the Kalman Filter algorithm, covering everything from its core concepts to practical applications, and serves as a complete reference for both engineering development and theoretical learning. It first clarifies the recursive nature of the Kalman Filter—centered on the “fusion of prediction and observation”—then analyzes its key advantages in detail, such as efficient real-time processing, optimal estimation under Gaussian assumptions, and multi-source information fusion. At the same time, it highlights limitations including dependence on linearity and Gaussianity, sensitivity to model parameters, and increased computational complexity in high-dimensional spaces. This helps readers accurately assess the scenarios where the algorithm is best suited.

Source of the article:Dev Resource Hub

The Kalman Filter algorithm revolves around a core concept: fusing predictive estimates with real-world observations. Using a recursive framework, it computes optimal state estimates even when dealing with system uncertainty.

1. Advantages and Limitations of the Kalman Filter

Advantages

• Efficient Recursion: It operates with low computational overhead, requiring only the current-time estimate and measurement data—no need to store large volumes of historical information. This makes it ideal for real-time processing tasks.
• Optimal Estimation: When the system is linear and noise follows a Gaussian distribution, the Kalman Filter delivers statistically optimal results, specifically minimum variance estimates.
• Robust Noise Handling: It effectively mitigates random noise in both system dynamics (e.g., unmodeled disturbances) and sensor measurements (e.g., sensor drift).
• Native Multi-Sensor Fusion: It is inherently designed for integrating data from multiple sensors, leveraging the strengths of each (e.g., high precision from one, high update rate from another) to produce more reliable estimates.

Limitations

• Linear and Gaussian Assumptions: The standard Kalman Filter relies on two key assumptions: the system’s dynamic and observation models must be linear, and both process noise (system uncertainty) and observation noise (sensor uncertainty) must be Gaussian white noise. In practice, many real-world systems exhibit non-linear behavior, which violates this constraint.
• Model Sensitivity: Filter performance is heavily dependent on the accuracy of the system model (defined by matrices F and H) and noise statistics (covariance matrices Q and R). Poorly tuned parameters can lead to biased estimates or even filter divergence (where estimates grow increasingly inaccurate over time).
• Dimensionality Scaling Issues: For high-dimensional state spaces, the computational cost of matrix operations (e.g., multiplication, inversion) increases significantly, potentially limiting its use in resource-constrained systems.

2. Application Areas of the Kalman Filter

Thanks to its low computational footprint, the Kalman Filter is widely adopted in scenarios where real-time performance is critical.

Application Field	Specific Use Cases	Role of the Kalman Filter
Target Tracking	Radar-based tracking, video surveillance, vehicle/pedestrian tracking in autonomous driving	Predicts the target’s next position, fuses multi-sensor data (e.g., radar and camera feeds), smooths erratic trajectories, and refines estimates of position, velocity, and acceleration.
Motor Control & Filtering	Servo motor feedback control, robotic joint actuation, UAV rotor speed regulation (e.g., STM32-based motor speed sensing)	Filters noise from encoder feedback signals to improve motor speed and position estimates; integrates data from current sensors to enable precise state monitoring and closed-loop control.
Navigation & Positioning	UAV navigation, autonomous vehicle localization, mobile robot guidance, smartphone GPS/IMU fusion	Serves as the backbone for multi-sensor fusion. For example, it combines GPS data (which provides absolute position but has slow update rates and noise) with IMU (Inertial Measurement Unit) data (high update rates but suffers from error accumulation over time) to deliver continuous, high-precision position, velocity, and attitude estimates.
Signal Processing & Economic Forecasting	Removing noise from time-series signals (e.g., sensor readings), stock price forecasting, macroeconomic indicator analysis	Extracts underlying trends from noisy time-series data; uses historical patterns to generate short-term predictions for financial or economic variables.
Aerospace	Satellite orbit determination, missile guidance systems, aircraft attitude control	Computes precise estimates of aircraft/satellite position, velocity, and orientation—critical for maintaining stable navigation and control in dynamic aerospace environments.

3. Kalman Filter Example Demonstration

The Kalman Filter’s combination of low computational cost, clear performance benefits, and no need for historical data storage makes it well-suited for real-time signal processing. The figure below compares the raw motor speed data (blue curve) with the Kalman-filtered speed data (orange curve) from a typical motor operation test. The filtered curve clearly reduces noise while preserving the underlying speed trend.

4. Kalman Filter Principles: 5 Core Equations

The Kalman Filter operates in two repeating phases—prediction and update—governed by five key equations:

• State Prediction Equation

• Error Covariance Prediction Equation

• Kalman Gain Calculation

• State Update Equation

• Error Covariance Update Equation

Below is a complete C language implementation based on these equations:

---

**

float Kalman_Filter(Kalman* p, float dat)
{if (!p) return 0; // Guard clause: return 0 if the Kalman structure pointer is null

    // Prediction phase: Estimate current state based on previous state
    p->X = p->A * p->X_last;                  // Predict current state (prior estimate)
    p->P = p->A * p->P_last + p->Q;            // Predict error covariance of the prior estimate

    // Update phase: Correct prior estimate with new measurement data
    p->kg = p->P / (p->P + p->R);              // Calculate Kalman gain (weighting factor)
    p->X_now = p->X + p->kg * (dat - p->X);    // Update state to get posterior estimate
    p->P_now = (1 - p->kg) * p->P;             // Update error covariance of the posterior estimate

    // Prepare for next iteration: Pass current posterior estimates to next time step
    p->P_last = p->P_now;
    p->X_last = p->X_now;

    return p->X_now; // Return the final filtered state estimate
}

Explanation of Variables in the Kalman Structure

• A: State transition matrix (or scalar coefficient in 1D cases) that defines how the system state evolves over time.
• Q: Process noise covariance matrix (or scalar) that quantifies uncertainty in the system model (e.g., unmodeled friction in a motor).
• R: Observation noise covariance matrix (or scalar) that quantifies uncertainty in sensor measurements (e.g., noise in an encoder reading).
• X_last: Posterior state estimate from the previous time step (denoted as k-1|k-1), representing the optimal estimate after incorporating the last measurement.
• P_last: Error covariance matrix from the previous time step (k-1|k-1), quantifying uncertainty in X_last.
• X: Prior state prediction for the current time step (k|k-1), estimated using only the system model (no measurement data yet).
• P: Error covariance of the prior prediction (k|k-1), quantifying uncertainty in X.
• kg: Kalman gain (scalar in 1D cases) that balances the trust between the prior prediction (X) and the new measurement (dat).
• X_now: Posterior state estimate for the current time step (k|k), the final filtered output after incorporating the new measurement.
• P_now: Error covariance of the posterior estimate (k|k), quantifying uncertainty in X_now.

5. Step-by-Step Interpretation: Kalman Filter Logic vs. C Code

(1) Prediction Phase

The prediction phase uses the system model to estimate the current state and its uncertainty, without any measurement data.

• p->X = p->A * p->X_last;This line computes the prior state prediction. It extrapolates the previous optimal estimate (X_last) forward in time using the state transition model (A). For example, if A = 1 (a stationary system like a constant motor speed), this simplifies to X = X_last—the predicted speed equals the previous optimal speed. Note: This code omits the control input term (BU(k)) for simplicity (common in 1D systems with no external control).
• p->P = p->A * p->P_last + p->Q;This line computes the prior error covariance. The term A * P_last propagates the uncertainty from the previous step forward in time. The process noise Q is added to account for new uncertainty introduced by the system model (e.g., sudden changes in load on a motor). In 1D cases, the transpose of A (Aᵀ) is equal to A (since it’s a scalar), so the full matrix equation (A * P_last * Aᵀ + Q) simplifies to the code above.

(2) Update Phase

The update phase incorporates the new measurement to correct the prior prediction, producing a more accurate posterior estimate.

• p->kg = p->P / (p->P + p->R);This line calculates the Kalman gain, the filter’s “trust balance” between prediction and measurement:

  • If R is large (unreliable sensor), the denominator grows, making kg small. The filter trusts the prior prediction (X) more.
  • If P is large (unreliable model), the numerator grows, making kg close to 1. The filter trusts the new measurement (dat) more.In 1D systems, the observation matrix H (which maps states to measurements) is typically 1, so the full gain equation (P * Hᵀ * (HPHᵀ + R)⁻¹) simplifies to the scalar division here.

• p->X_now = p->X + p->kg * (dat - p->X);This line computes the posterior state estimate—the core of the Kalman Filter. The term (dat - p->X) is the measurement residual (or “innovation”), representing the difference between what the sensor measured (dat) and what the model predicted (X). The Kalman gain kg weights this residual: a large kg means the residual has a big impact on the final estimate (trust the measurement), while a small kg means the residual has little impact (trust the prediction).

• p->P_now = (1 - p->kg) * p->P;This line updates the error covariance to reflect the reduced uncertainty after incorporating the measurement. Since we now have more information (the new measurement), P_now will always be smaller than P—the term (1 - p->kg) ensures this reduction. In 1D cases, the identity matrix I (used in the full matrix equation: (I – kg*H)*P) is 1, so the code simplifies to the scalar form above.

(3) Preparing for the Next Iteration

• p->P_last = p->P_now;
• p->X_last = p->X_now;These lines “pass forward” the current posterior estimates (X_now and P_now) to become the “previous” estimates (X_last and P_last) for the next time step. This recursive handoff is what allows the filter to run continuously, updating estimates as new measurements arrive.

6. Key Takeaways and Tuning Tips

A 1D Kalman Filter uses the prediction-update cycle to recursively fuse model-based predictions with sensor measurements, delivering optimal state estimates in uncertain environments.

The performance of the filter hinges on tuning two critical parameters:

• Process Noise Covariance (Q) : Think of Q as your “trust in the system model.” If your model is imprecise (e.g., a motor with variable friction), increase Q—this makes the filter rely more on sensor measurements to correct errors. If your model is highly accurate, decrease Q—the filter will trust the model’s predictions more.
• Observation Noise Covariance (R) : Think of R as your “trust in the sensor.” If the sensor is noisy (e.g., an old encoder with erratic readings), increase R—the filter will downweight the measurement and trust the model more. If the sensor is precise (e.g., a high-quality laser encoder), decrease R—the filter will prioritize the measurement.

Tuning Q and R is rarely a one-time task. Typically, you’ll adjust these parameters through iterative testing: start with small values, monitor filter performance (e.g., how well the filtered curve tracks the true state), and refine until the filter balances noise reduction and responsiveness.