# Is there a common name for the first order IIR averaging filter?

Is there a term to describe this IIR averaging/smoothing filter?

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

• it should be noted that $$0 < \alpha \le 1$$ for the leaky integrator to appear most like an integrator, then $$0 < \alpha \ll 1$$ this is the integrator to undo the differentiator of DC blocking filter. i like to represent it in terms of the single pole value which is $$p = 1-\alpha$$ so $$y[n] \ = \ p \, y[n-1] + (1-p) x[n]$$ and the transfer function is $$H(z) \triangleq \frac{Y(z)}{X(z)} = \frac{(1-p)z}{z-p}$$ – robert bristow-johnson Sep 21 '17 at 23:54
• also maybe related: dsp.stackexchange.com/questions/4699/… – user13267 Sep 22 '17 at 1:37

This is often called a leaky integrator, a special case of a first-order IIR lowpass filter. They are discussed in more detail in several previous questions:

It is called an exponentially weighted moving average (EWMA) filter. Here is a previous answer where I provided a Matlab script for computing $\alpha$ for a desired cutoff frequency: Exponential moving average cut-off frequency

• i think Andy is correct. – robert bristow-johnson Sep 21 '17 at 23:41
• I'm not gone on using Moving Average for describing an Autoregressive filter, though I've certainly heard the term before. – Peter K. Sep 22 '17 at 0:42

Wikipedia calls it "basic exponential smoothing".

I'd call it a first order smoother or first order lowpass filter.

Some combination of those words will probably work.

• sounds to me to be a Wikipedia neologism. but i dunno. i guess it implies a DC gain of 0 dB, which is important. – robert bristow-johnson Sep 21 '17 at 23:42
• @robertbristow-johnson not so sure about the "basic" but I'd seen it called exponential smoothing or an exponential filter before Wikipedia existed :) – hobbs Sep 22 '17 at 4:55