Inverse of exponential smoothing

Suppose that a time series $s_t$ it is known to be obtained via exponential smoothing of an underlying signal $x_t$, that is

$$s_{0}= x_0$$

and

$$s_{t} = (1-\alpha)\,x_t+\alpha\,s_{t-1}.$$

I am wondering wether there exist procedures/filters to re-construct or approximate the sequence of the $x_t$ having observed the sequence of the $s_t$ and having an estimate of the smoothing parameter $\alpha$.

If you know $\alpha$ and have $s_1,\dots,s_n$, if

$$s_{t} = (1-\alpha)\,x_t+\alpha\,s_{t-1}$$

then by simple arithmetic

$$x_t = \frac{s_{t} - \alpha\,s_{t-1}}{(1-\alpha)}$$

So you can reconstruct it exactly, not just approximately.

Below you can see the code example.

set.seed(123)

n <- 250
x <- runif(n)

alpha <- 0.6
s <- numeric(n)
s[1] <- x[1]
for (t in 2:n)
s[t] <- (1-alpha)*x[t] + alpha*s[t-1]

z <- numeric(n)
z[1] <- x[1] # since s[1] <- x[1]
for (t in n:2)
z[t] <- (s[t] - s[t-1]*alpha)/(1-alpha)

all.equal(z, x)
## [1] TRUE