Moving Average Filter

The Moving Average filter is a classic Finite Impulse Response (FIR) filter. This means its output depends only on a finite number of past and present input samples. Its key characteristics include:

• Stability: Guaranteed, as it has no feedback loops.
• Linear Phase Response: Preserves the shape of waveforms within the passband, critical for timing-sensitive sensor data.
• Simple Coefficients: All filter coefficients are equal (typically 1/N), leading to computationally efficient implementations using additions and a single division (or bit-shift for power-of-two window sizes).

The filter acts as a low-pass filter, attenuating high-frequency noise. Its behavior is defined by the window size (N).

• Larger N: Provides greater smoothing and more aggressive high-frequency attenuation, but increases group delay and smears rapid signal transitions.
• Smaller N: Preserves faster signal changes but offers less noise reduction.
• The -3 dB cutoff frequency is approximately 0.443 / (N * T), where T is the sampling period. This quantifies the trade-off between noise removal and signal fidelity, a critical design choice for TinyML systems.

Its simplicity makes it ideal for microcontroller (MCU) deployment. Key optimizations include:

• Sliding Window Update: Recalculating the full sum for each sample is O(N). An efficient implementation maintains a running sum, adding the new sample and subtracting the oldest, reducing computation to O(1) per step.
• Fixed-Point Arithmetic: Eliminates the need for floating-point units (FPUs).
• Power-of-Two Window Sizes: Allows the division by N to be implemented as a fast bit-shift operation. This minimal compute footprint preserves battery life and leaves MCU resources free for other tasks like feature extraction or model inference.

Several variants address specific limitations of the simple moving average:

• Exponential Moving Average (EMA): A recursive (IIR-like) filter that applies exponentially decreasing weights to older samples. It is more responsive to recent changes and requires less memory (no buffer).
• Weighted Moving Average (WMA): Applies linearly or non-linearly decreasing weights within the window for tailored frequency response.
• Cumulative Moving Average (CMA): Averages all data from the start, useful for running statistics but not for real-time smoothing. Typical Sensor Applications: Denoising ADC readings, smoothing IMU data for orientation estimation, and pre-processing for activity recognition or anomaly detection pipelines.

While versatile, the MA filter has distinct drawbacks:

• Lag/Group Delay: Introduces a delay of (N-1)/2 samples, which can destabilize real-time control loops if not accounted for.
• Poor Stopband Attenuation: Its frequency response has high sidelobes (Gibbs phenomenon), allowing some high-frequency noise to leak through. It is less effective than optimized FIR or IIR filters for stringent filtering requirements.
• Step Response: It spreads a sudden step change over N samples, which may be undesirable.
• Window Artifacts: The rectangular window causes ringing in the frequency domain. Engineers must select N based on the desired trade-off between latency, smoothing, and computational cost.

In TinyML and sensor data processing workflows, the MA filter is often the first stage in a feature extraction chain. Typical Flow:

1. Raw sensor data is sampled via an ADC.
2. An MA filter smooths the stream to reduce high-frequency noise.
3. The smoothed signal is passed to downstream processes: peak detection, FFT for spectral analysis, or direct input to a lightweight machine learning model (e.g., for gesture recognition). Its deterministic, low-memory operation makes it a reliable building block in edge AI architectures where predictability and resource efficiency are paramount.

A Finite Impulse Response (FIR) filter is a type of digital filter whose output is a weighted sum of a finite number of past and present input samples. The Moving Average Filter is the simplest form of an FIR filter, where all weights within the window are equal.

• Key Property: Always stable and provides a linear phase response, meaning it does not distort the phase relationships between different frequencies in the signal.
• TinyML Relevance: FIR filters are computationally straightforward and deterministic, making them highly suitable for real-time implementation on microcontrollers with limited memory.

A sliding window is a fundamental data structure and algorithmic technique where a fixed-size buffer moves sequentially over a data stream. This is the core mechanism that enables real-time filters like the Moving Average.

• Operation: As a new sample arrives, the oldest sample in the window is discarded, and the new one is added, allowing continuous processing of the most recent data segment.
• Implementation: Efficient implementation is critical for low-power devices, often using a circular buffer to avoid shifting all elements, reducing the operation to O(1) complexity.

A low-pass filter attenuates high-frequency components of a signal while allowing low-frequency components to pass through. The Moving Average Filter acts as a simple, non-ideal low-pass filter.

• Frequency Response: Its frequency response is a sinc function, which has a gradual roll-off and significant side lobes (passband ripple).
• Trade-off: While easy to implement, a standard moving average has poor stopband attenuation compared to more sophisticated FIR or IIR designs. For TinyML, this simplicity often outweighs the performance penalty for basic noise smoothing.

An Exponential Moving Average (EMA) is a type of infinite impulse response (IIR) filter that applies exponentially decreasing weights to older samples, giving more importance to recent data.

• Formula: EMA_t = α * x_t + (1 - α) * EMA_{t-1}, where α is the smoothing factor (0 < α ≤ 1).
• Advantages: Requires only the previous EMA value and the new sample, making it extremely memory-efficient (O(1) memory) compared to the simple moving average's O(N) window storage. This is a major benefit for severely memory-constrained microcontrollers.

Sensor fusion is the process of combining data from multiple, disparate sensors to produce estimates with less uncertainty than those provided by any single sensor. Filtering is a prerequisite step before fusion.

• Role of Filtering: A Moving Average or similar filter is often applied to the raw output of each individual sensor (e.g., accelerometer, gyroscope) to reduce high-frequency noise before the data is fused using algorithms like the Kalman Filter.
• TinyML Context: On edge devices, lightweight filtering enables cleaner input for fusion algorithms, improving the accuracy of orientation, position, and activity recognition without excessive computational cost.

A median filter is a non-linear digital filtering technique that replaces each data point with the median of neighboring points within a specified window. It is highly effective at removing impulse noise (salt-and-pepper noise) while preserving sharp edges.

• Comparison with Moving Average: While a moving average blurs edges and is sensitive to outliers, the median filter is robust to outliers and preserves step changes in the signal.
• Computational Cost: Finding the median requires sorting the window, which is more computationally intensive (O(N log N)) than a moving average's sum. This trade-off is a key consideration in TinyML system design.