Filter Algorithms

📌 Project Overview

This repository contains dataset2 collected from an IMU sensor and an implementation file Average_Filter.m that includes the following filtering algorithms:

• Average Filter
• Moving Average Filter
• Low-pass Filter

The objective of this project is to demonstrate fundamental filtering techniques for noise reduction and signal smoothing in sensor data.

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🧮 Filtering Algorithms

1. Average Filter

The average filter computes the arithmetic mean of the entire dataset to reduce random noise:

$$ y = \frac{1}{N} \sum_{i=1}^{N} x_i $$

• x_i: Input data
• N: Total number of samples
• y: Average output

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2. Moving Average Filter

The moving average filter calculates the average of the most recent (M) samples within a sliding window:

$$ y[n] = \frac{1}{M} \sum_{k=0}^{M-1} x[n-k] $$

• x[n]: Input signal at time step n
• M: Window size
• y[n]: Filter output at time step n

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3. Low-pass Filter

The low-pass filter attenuates high-frequency components while allowing low-frequency signals to pass, which is effective for reducing sensor noise.

A simple first-order discrete-time low-pass filter is defined as:

$$ y[n] = \alpha , x[n] + (1 - \alpha) , y[n-1] $$

• x[n]: Input signal at time step n
• y[n]: Output signal at time step n
• α ∈ (0,1): Filter coefficient (smaller values yield stronger smoothing)

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📊 Experimental Results

Original IMU Data (Offset Removed)

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Comparison of Moving Average Filter

In the figure above, the red curve corresponds to a window size of 20 samples, while the blue curve corresponds to a window size of 40 samples.

• A larger window size provides stronger noise reduction but introduces a delay, making the filter less responsive to rapid signal changes.
• A smaller window size follows the variations in the data more closely but results in higher noise levels.

Thus, M = 20 is more sensitive to signal variations compared to M = 40, but produces noisier results. On the other hand, M = 40 achieves better noise suppression but is less responsive to rapid changes.

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Comparison of Low-pass Filters with Different α Values

The figure shows the IMU dataset (offset removed) filtered with a first-order low-pass filter using two different values of the tuning parameter α:

• Red curve: α = 0.3
• Blue dashed curve: α = 0.9

Observations:

• With α = 0.3, the filter gives more weight to the current measurement, closely following the variations in the dataset.
• With α = 0.9, the filter relies more on past values, resulting in stronger smoothing but introducing a noticeable delay (phase shift).

Compared to the moving average filter, the low-pass filter allows different weighting between recent and past data points, making it more effective in balancing noise suppression and responsiveness to signal changes.

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🎯 Conclusion

• The moving average filter provides effective noise suppression but introduces a trade-off between smoothing and responsiveness depending on the window size.
• The low-pass filter (LPF) addresses this limitation by assigning different weights to recent and past values, offering a better compromise between noise reduction and signal tracking.

This repository demonstrates how different filtering algorithms can be applied to IMU sensor data, providing insights into the trade-offs between smoothing and responsiveness in signal processing.