{"id":154,"date":"2013-01-06T16:38:25","date_gmt":"2013-01-06T21:38:25","guid":{"rendered":"http:\/\/www.mcgurrin.com\/robots\/?p=154"},"modified":"2013-01-06T16:43:45","modified_gmt":"2013-01-06T21:43:45","slug":"filtering-noise-from-sensor-readings-a-simple-low-pass-filter","status":"publish","type":"post","link":"https:\/\/www.mcgurrin.info\/robots\/154\/","title":{"rendered":"Filtering Noise from Sensor Readings: A Simple Low Pass Filter"},"content":{"rendered":"

In a perfect world, a sensor\u2019s output would exactly match the input it is sensing.\u00a0 In the real world, there are a number of errors introduced, including bias, time lags in response, and noise.\u00a0 This article discusses one of the simplest filters for reducing noise: the Exponentially Weighted Moving Average (EWMA) filter.\u00a0 Despite its long name, it\u2019s about the simplest and most efficient filter you can implement.\u00a0 The name comes from the concept that the filter\u2019s output, y, is the weighted average of the current input and each previous inputs, with the weighting decreasing exponentially:<\/p>\n

Yt<\/sub> = \u03b1 (Xt <\/sub>+ (1 – \u03b1) Xt-1 <\/sub>+ (1-\u03b1)2 <\/sup>Xt-2 <\/sub>+ (1-\u03b1)3 <\/sup>Xt-3 + \u2026.<\/sub><\/p>\n

where Yt <\/sub>is the current output from the filter, and Xt <\/sub>is the current input, and the other X\u2019s are each preceding value.\u00a0 The weighting is shown in the figure:<\/p>\n

\"\"<\/p>\n

(graph courtesy of Kevin Ryde<\/a>, licensed under the Creative Commons<\/a> Attribution-Share Alike 3.0 Unported<\/a> license.)<\/p>\n

This averaging tends to smooth out the response and reduce noise.\u00a0 Of course, this formula would be cumbersome and time consuming to implement in an embedded system.\u00a0 Even if only a small number of past values were used.\u00a0 Some math can show that the formula can be rewritten as a recursive formula using the past output<\/i> from the filter:<\/p>\n

Yt<\/sub> = \u03b1 Yt-1 \u00a0<\/sub>+ (1 – \u03b1) Xt <\/sub>for all values of t > 0, or, re-arranging terms:<\/p>\n

Yt<\/sub> = \u00a0Xt \u00a0<\/sub>+\u00a0\u03b1 ( Yt-1 <\/sub>– Xt<\/sub>)<\/p>\n

This means that all you need to store is the immediate past output of the filter, which already has the weighted average of all previous values calculated in. This is simpler and takes less storage than, for example, averaging the current and past 4 values of the sensor.\u00a0 It will also respond more quickly to actual changes that you are trying to measure.<\/p>\n

Limits<\/h2>\n

You can choose a value for \u03b1 ranging between 0 and 1.\u00a0 At 0, the output is just the raw input: no filtering occurs.\u00a0 The output will respond instantly both to noise and to true changes in the system.<\/p>\n

At 1, the output becomes equal to the past value: it never changes, eliminating all noise and all real signal changes!\u00a0 Clearly one wants to select a value in between these extremes.<\/p>\n

Selecting \u03b1<\/h2>\n

The correct value will depend on your situation.\u00a0 The higher the a, the more noise if filtered out, but also the more lag that is introduced in reacting to real system changes.\u00a0 This is shown in the following figures:<\/p>\n

\"\"<\/p>\n

This figure shows the response of two filters to a unit step function.\u00a0 The red curve is for\u00a0\u03b1 = 0.3, and the green curve is for\u03b1 = 0.6.<\/p>\n

\"\"<\/p>\n

This graph shows the output of the filters to a noisy sensor, with Gaussian noise with a standard deviation of 0.1.\u00a0 One can see from the graph that a value of 0.3 reduces the noise somewhat, but leaves a lot in.\u00a0 A value of 0.6 greatly reduces the noise, at the expense of greater lag.<\/p>\n

Initializing the Filter<\/h2>\n

The formulas work for all values after the initial run of the filter.\u00a0 Like any recursive formula there needs to be a starting value.\u00a0 My rather rough and informal scanning of the literature identified three common methods:<\/p>\n