Please help me understand smoothing of data. This is a follow up to my previous question posted here. Especially the top answer by Junuxx where he says a way of smoothing a function $f(x)$ is:
$$ f'[t] = 0.1 f[t-1] + 0.8 f[t] + 0.1 f[t+1] $$
here we can see that for every point in $f[x]$, we are taking a weighted average of that point and its two adjacent points, to get a smoothed version of $f[t]$ called $f'[t]$.
A paper on speech enhancement explains that an equation of the form
$$ y[i] = a[i]y[i-1] + (1 - [i]) x[i] $$
helps us get the value of y as a recursive smoothing of x. Here $a[i]$ acts as a smoothing parameter and it is itself calculated as
$$ a[i] = \alpha + (1 - \alpha)p[i] $$
where $p[i]$ is calculated elsewhere and alpha is a constant. $y[i]$, $a[i]$, and $x[i]$ are all arrays with $i$ elements.
How can I relate this equation of $y[i]$ with the equation of $f'[t]$? Both of them are for smoothing data, however equation for $f'[t]$ contains weighted average of consecutive points in the array for $f[x]$ itself while the equation for $y[i]$ does not contain consecutive data points for $x[i]$. How can we comprehend this equation as a smoothing of data in $x$?
If this question is not relevant when the equations are taken out of context then I will be happy to provide more details.