Low pass filter algorithm origin I have been doing some research, because I used this algorithm that's very spread in the web:

y += a * (x - y)


or

y = (a * y) + (x - (a * x)


Where x is input, and y is the filter output, and a is actually alpha, which there's some math behind it to define it's value.

This algortihm is seen on wikipedia, this article and many more.

But none actually shows why this algorithm is a low pass filter and from which formula it actually came from(wikipedia has no source for it). I'm doing a small academic paper, so I need to know exactly from where it came.
I tried to make my derivation match up exactly with embeddedrelated's. The filter you cited uses an approximation, which further obfuscates intuitively understanding it. 

embeddedrelated does cite the frequency transfer function that the IIR filter is based off of:
$$
H(s)=\frac{1}{1+s\tau}
$$

The transfer function for a simple RC low pass filter is:
$$
H(s)_{RC}=\frac{1}{1+sRC}
$$
$$
\tau=RC
$$

Schematic:



We can derive a difference equation from the above schematic.

$$
I=C\frac{dV_{out}}{dt}
$$
$$
I = \frac{V_{in} - V_{out}}{R}
$$
$$
C\frac{dV_{out}}{dt}=\frac{V_{in} - V_{out}}{R}
$$
$$
C\frac{V_{out}[n]-V_{out}[n-1]}{ts}=\frac{V_{in}[n] - V_{out}[n]}{R}
$$
$$
V_{in}[n]+\tau f_s V_{out}[n-1]=\tau f_s V_{out}[n] + V_{out}[n]
$$
I have tested this form:
$$
\frac{V_{in}[n]+\tau f_s V_{out}[n-1]}{\tau f_s + 1}=V_{out}[n]
$$
$$
\tau f_s &gt;&gt; 1
$$
I have not tested this form:
$$
\frac{V_{in}[n]}{\tau f_s} + V_{out}[n-1]\approx V_{out}[n]
$$