MATLAB Code Implementation of Allan Variance

Allan variance is a crucial tool for analyzing noise characteristics in inertial sensors (such as gyroscopes and accelerometers). It enables identification and quantification of different noise components in sensor data, including angle random walk, rate random walk, bias instability, and more.

### Implementation Approach Data Preparation: Start by acquiring raw sensor output data, typically a sequence of angular velocity or acceleration samples collected over time. In MATLAB, this is usually stored as vectors or arrays. Segmentation Calculation: Divide the data into multiple subsets with varying lengths (different "cluster times"), where the subset length increases progressively. This can be implemented using nested loops or vectorized operations. Mean and Variance Calculation: For each cluster time, compute the difference between means of adjacent data subsets and calculate their variance. Key functions: `mean()`, `diff()`, `var()` operations. Plotting Results: Finally, plot the relationship between Allan variance and cluster time on a log-log scale using `loglog()` function for noise characteristic analysis.

### Core Algorithm Logic Cluster Time Selection: Typically employs exponential growth for cluster time selection (e.g., τ = τ0 × 2^k) to cover both short-term noise and long-term drift effects. Implementation tip: use geometric progression. Mean Difference Calculation: For each data segment, calculate mean differences between adjacent subsets, then compute the mean of squared differences to obtain Allan variance. Formula: σ²(τ) = ½⟨(θ_{k+1} - θ_k)²⟩ where θ_k represents subset means. Noise Identification: Different noise components (white noise, flicker noise, etc.) can be identified through slope variations in the Allan variance curve. Specific slopes correspond to different noise types.

### Extended Applications Allan variance is not limited to inertial sensors but can also be applied to other data sequences with random noise characteristics, such as oscillator frequency stability evaluation and precision timing analysis.

Implementing Allan variance calculation in MATLAB allows seamless integration into sensor calibration or data processing workflows, helping engineers optimize sensor performance evaluation and error compensation strategies through programmable analysis routines.