Moving Average and Low-Pass Filters

Noisy sensor readings are usually smoothed before a controller uses them. A moving average averages the last few readings. A first-order low-pass filter blends each new reading into a running estimate. Both reduce noise, and both add delay.

Moving average window Low-pass smoothing α Sensor noise

Two filters

$$\text{moving average: } y_k = \frac{1}{N}\sum_{i=0}^{N-1} x_{k-i} \qquad \text{low-pass: } y_k = y_{k-1} + \alpha\,(x_k - y_{k-1})$$

The moving average delays the signal by about $(N-1)/2$ samples. The low-pass filter has time constant $\tau \approx T_s(1-\alpha)/\alpha$.

What to notice

Bigger windows and smaller $\alpha$ give smoother lines, but the step shows up later. That delay is the price of smoothing, and in a control loop it can cause oscillation, as in the discrete-time control concept. The low-pass filter needs only one stored number, which is why it is everywhere in embedded code. The moving average forgets old data completely after $N$ samples, while the low-pass filter forgets gradually.