What is the name of this digital low pass filter?

Question

I found this digital filter in code I am working on. It is a low pass filter. In the code it is called an "alpha filter", but it is not the same as the alpha filter mentioned here.

I post the relevant code below:

float alpha = 0.7;
float prev = 0.0;

float filter(float sample) {
    float filtered = (sample * alpha) + (prev * (1.0 - alpha));
    prev = sample;
    return filtered;
}

It looks like alpha should be in range $[0, 1]$. By increasing alpha, the filter tracks the measurement with less delay, but lets more noise through.

I am tasked with writing library functions, and I would like to give this thing its proper name, and hopefully link to a Wikipedia article in the doc tags.

Answer 1

I'm new, so I can't add this comment to Matt L.'s answer.

It is not an exponential filter, the equation is actually:

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

So it is a very short FIR filter, not an IIR filter. I'm not expert enough to know a specific name.

Ced

============================================= Followup:

I want to thank everyone for their upvotes. Yes, I've contributed to comp.dsp and I am a blogger at dsprelated.com.

Like all of you, I suspect that the intent of the function was to be an exponential filter and was coded erroneously.

If I were to name the filter as coded, I would call it a "Linear Interpolation Filter", or perhaps a "Subsamplesize Time Shift filter" when $\alpha$ is between zero and one. It would make more sense as such if the $\alpha$ and $(1-\alpha)$ coefficients were reversed.

Answer 2

This is a correction of my old answer (see below). It turns out that all of us misread the question except for Cedron Dawg in his answer.

Since prev is not the previous output sample but the previous input sample, this is no recursion and, consequently, not a 1-pole IIR filter, but a simple two-tap (finite impulse response) FIR filter:

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

Since this is an FIR filter it is stable for any value of $\alpha$. For smoothing, a reasonable range for $\alpha$ is indeed $\alpha\in (0,1)$. Note that this filter doesn't really do that much, and it's much worse at smoothing than its IIR counterpart (see below).

Unlike the EWMA filter described below, this filter is no standard filter. It's just a two-tap FIR filter, and with $\alpha$ in the range $(0,1)$ it's a two-tap FIR smoothing filter. For $\alpha=0.5$ it would be a two-tap moving average filter.

I think what mislead most of us was the use of $\alpha$ in a convex combination of two samples. This is commonly done in a 1-pole IIR filter, but usually not in the FIR case. It could even be a bug, and that the author's intention actually was to code a 1-pole IIR filter.

---

OLD ANSWER

[I keep it because it's an explanation of a standard filter that might be a good alternative to the filter in the question.]

It's called exponential smoothing or exponentially weighted moving average (EWMA) filter.

The corresponding difference equation is

$$\begin{align}y[n]&=\alpha x[n]+(1-\alpha)y[n-1]\\&=\alpha (x[n]-y[n-1])+y[n-1]\end{align}\tag{1}$$

For stability, $\alpha$ must be in the range $\alpha\in (0,2)$. For this to be a smoothing filter, one must choose $\alpha\in (0,1)$.

Answer 3

As other contributors, I did not read the question correctly, and kept the initial answer at the bottom. So it is, generally and in special cases:

• a (stable) unit-sum 2-tap FIR filter
• $\alpha = 0$: a lazy filter, that does nothing
• $\alpha = 1$: a pure delay filter, with unit delay
• $\alpha =1/2$: the two-tap averaging filter
• $\alpha \in ]0,1[$: a low-pass filter
• $\alpha = 2$: the 2-tap linear extrapolating filter.

Details are as follows:

$$ \hat{s}[n] = \alpha s[n] + (1-\alpha) s[n-1] $$

is a 2-tap FIR filter. When $\alpha =1$ or $0$, it reduces to a one-tap filter, producing either the same output as the input or with a 1-unit delay (flat spectrum). Since the sum of the coefficients is always $1$, it leaves constant components unchanged : the frequency response is constant, equal to one. When $\alpha = 2$, $[2 - 1]$ is the 2-tap linear extrapolating filter.

Several frequency responses are given below:

[OLD ANSWER, KEPT IN CASE THE DESIGN WERE RECURSIVE] As this structure is one of the simplest/shortest one can imagine with a recursive form, it has likely emerged in different context, and got different names. Some of the names identified are:

• first order exponential averaging low-pass filter,
• exponential averager,
• exponential smoothing,
• exponential moving average (EMA),
• exponentially weighted moving average (EWMA),
• Brown's Simple (linear) Exponential Smoothing (sometimes called SES),
• ARIMA(0,1,1) model.

Other details, including historical notes on its emergence, can be retrieved at:

• Short: What is the name of these simple filter algorithms?,
• Longer: Is there a technical term for this simple method of smoothing out a signal?, with historical notes and paper excerpts.

Its behavior, as answered by Matt L., is driven by $\alpha$. For smoothing, the closer $\alpha$ to one, the higher the relative weight of the latest samples with respect to "filtered data". So the result is less immediate (more delay), but the larger the smoothing effect, as you benefit from the averaving of longer stationary sequences, weighted by an exponential window. Conversely, shortest delays correspond to less noise reduction. There is no free lunch in filtering: shorter filters, poorer frequency selection.

This filter does not deal too well with non stationarity (like trends) and was subsequently extended to double or triple exponential filters, giving rise to Holt-Winters smoothers, see for instance Exponential Smoothing for Time Series Forecasting:

If you notice that your time series is not stationary, you’ll have to find something other than a simple EWMA to do your forecasting.

In the late 1950s, Charles Holt recognized the issue with the simple EWMA model with time series with trend. He modified the simple exponential smoothing model to account for a linear trend. This is known as Holt’s exponential smoothing.