Low-pass filter

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The exponentially weighed moving average (EWMA), a specific variant of a moving average is the name for an easy-to-realize (first-order) lowpass on discrete time-domain data.

Intuitively, you create an output that slowly follows new values, implicitly meaning it responds more sluggishly to quick changes (high-frequency content) while still following the overall tendency of the signal (lower-frequency content).

The influence of older data points reduces quickly (works out to be in an exponentally decaying way, hence the name), though it never quite reaches zero (in most computer implementations it eventually does because rounding errors), which is part of why it is a (first-order) infinite impulse response filter(verify).

There is a parameter α to vary its sensitivity.

In cases of not-so-regular sampling α is related only to speed of adaptation to newly sampled values
It's still relevant to frequency content, but not in a particularly quantifiable way
In applications that sample at a regular interval, you can relate α directly to frequency content.
For example sound. Also e.g. some sensor systems, where it's optional, but this filter can sometimes be be more meaningful this way.

When you want to calculate a filtered output series from an input series, you can loop through a list doing something like:

filtered_output[i] = α*raw_input[i] + (1-α)*filtered_output[i-1]

...or the equivalent:

filtered_output[i] = filtered_output[i-1] + α*(raw_input[i]-filtered_output[i-1])

The latter form may feel more intuitive/informative: the change in the filtered output is proportional to the amount of change, and to to the filter strength α.

Both may help consider how using the recent filtered output gives the system a sort of inertia:

  • A smaller α (larger 1-α in the former) (also makes for larger RC) means the output will adjust more sluggishly, and should show less noise (since the cutoff frequency is lower(verify)).
  • A larger α (smaller 1-α) (smaller RC) means that the output will adjust faster (have less inertia), but be more sensitive to noise (since the cutoff frequency is higher(verify))

Since the calculation is local and in one direction, you can choose to keep only the latest value. This make sense e.g. when your primarily goal is presenting a filtered version of a noisy or crude sensor

current_output  = α*current_input + (1-α)*previous_output
previous_output = current_output

You often want implement the above in floating point, even if you return ints, to avoid problems caused by rounding errors.

Most of the problem: when alpha*difference (itself a floating multiplication) is less than 1, it becomes 0 in a (truncating) cast to an integer.
For example, when alpha is 0.01, then a signal differences smaller than 100 will make for an adjustment of 0 (via integer truncation), so the filter would never adjust particularly close to the actual value.
You can do it in pure integer, it just takes more implementation care

Graphical example
A screenshot from a modified arduinoscope . Note the newest samples are on the left.
The top signal is the raw input (a few seconds's worth of an ADC sampling from a floating pin, with a finger touching it every now and then, e.g. picking up 50Hz).
The ones below are EWMA'd versions of it, at increasing strengths.

Some things to note about this plot:

  • the suppression of fast peaks and deviations
  • the filter has a shape is like a charging capacitor (most visible in the third onw down): proportional to the difference, so initially fast, then slower and slower
  • that it is certainly possible to filter too hard (fourth and probably third one down)
though that judgment depends on the nature of the content and your purpose
  • the filtered version of the the full-range oscillation comes out halfway not so much because of filtering, but also largely because most raw samples around there are effectively saturated at both ends of the ADC's numeric range. (This is essentially false signal - clipping, i.e. distortion)

On α, τ, and the cutoff frequency

α is the smoothing factor, theoretically between 0.0 and 1.0.

In practice typically <0.2 and often <0.1 or smaller, because above that you're barely doing any filtering (or sampling verry slowly).

In applications with relatively regular sampling, α is often based on:

  • Δt (dt), the time interval between samples (reciprocal of sampling rate)
  • a choice of time constant τ (tau), a.k.a. RC
(RC latter seems a reference to a resistor-and-capacitor circuit, which does lowpass in much the same way. Specifically, RC gives the time in which the capacitor charges to ~63% of the potential difference[1])(verify)


α = dt / (RC + dt)

In practice you'll often choose an RC that is at least a few multiples of dt, which means that α is on the order of 0.1 or less. (if using an RC close to dt you'll get alphas higher than ~0.5, which barely does anything)

In applications with strictly regular sampling interval (e.g. for sound), RC's relation to frequency content is well defined.

For example, the knee frequency (where it starts falling off, approximately the cutoff frequency) is:

1 / (2*pi*RC)

For example

when RC=0.002sec, the knee/cutoff is at ~80Hz.
at 200Hz, 2000Hz, and 20000Hz sampling, that makes for alphas of 0.7, 0.2, and 0.024, respectively.

(At the same sampling speed: the lower alpha is, the slower the adaptation to new values, and the lower the effective cutoff frequency)(verify)

For a first-order lowpass:

  • at lower frequencies, the response is almost completely flat,
  • at this knee frequency the response is -3dB (has started declining in a soft bend/knee)
  • at higher frequencies it it drops at 6db/octave (=20dB/decade)

(Higher-order variations fall off faster and have a harder knee)

Note there will also be a phase shift, which lags behind the input. It depends on the frequency; it starts earlier than the amplitude falloff, and will be -45 degrees at the knee frequency(verify).

Arduino example

Note: This is a single-piece-of-memory version, for when you're interested only in the (latest) output value, e.g. smoothing noisy sensor output.

// keeps values for each analog pin, assuming a basic arduino that muxes 6 pins. 
//   you can save a handful of bytes of RAM by putting your sensors on lower-numbered pins and lowering...
float   lowpass_prev_out[LOWPASS_ANALOG_PIN_AMT], 
int        lowpass_input[LOWPASS_ANALOG_PIN_AMT];
int adcsample_and_lowpass(int pin, int sample_rate, int samples, float alpha, char use_previous) {
  // pin:            arduino analog pin number to sample on   (should be < LOWPASS_ANALOG_PIN_AMT)
  // sample_rate:    approximate rate to sample at (less than ~9000 for default ADC settings)
  // samples:        how many samples to take in this call  (>1 if you want smoother results)
  // alpha:          lowpass alpha
  // use_previous:   If true,  we continue adjusting from the most recent output value.
  //                 If false, we do one extra analogRead here to prime the value.
  //   On noisy signals this non-priming value can be misleading, 
  //     and with few samples per call it may not quite adjust to a realistic value.
  //   If you want to continue with the value we saw last -- which is most valid when the
  //     value is not expected to change significantly between calls, you can use true.
  //   You may still want one initial sampling, possibly in setup(), to start from something real.
  float one_minus_alpha = 1.0-alpha;
  int micro_delay=max(100, (1000000/sample_rate) - 160); // 160 being our estimate of how long a loop takes 
               //(~110us for analogRead at the default ~9ksample/sec,  +50 grasped from thin air (TODO: test)  
  if (!use_previous) { 
    //prime with a real value (instead of letting it adjust from the value in the arrays)
    lowpass_input[pin] = analogRead(pin);
  //Do the amount of samples, and lowpass along the way  
  int i;
  for (i=samples;i>0;i--) {
    lowpass_input[pin]    = analogRead(pin);
    lowpass_cur_out[pin]  = alpha*lowpass_input[pin] + one_minus_alpha*lowpass_prev_out[pin];
    lowpass_prev_out[pin] = lowpass_cur_out[pin];
  return lowpass_cur_out[pin];
void setup() {
   //get an initial gauge on the pin. Assume it may do some initial weirdness so take some time
   //  Takes  approx 300ms (300 samples at approx 1000 samples/sec)
   adcsample_and_lowpass(0, 1000, 300, 0.015, false); // see below
void loop() {
   // updates can often be shorter (consider maximum adaptation speed to large changes, though)
   int resulting_value = adcsample_and_lowpass(0,     // sample A0 pin
                                               1000,  // aim for 1000Hz sampling
                                               50,    // follow 50 new samples
                                               0.015, // alpha
                                               true); // adapt from stored value for this pin (settled in setup())


Aliasing and oversampling

See also

Other filters you can use include:


RC filter