Problems with sampling

Question

Let $t\ge 0$ and let $f_{s}(t)$ be the sampled version of $f(t)$.

If $$f(t)=\begin{cases}\int_{0}^{t}\exp\left(-\frac{1}{1-s^{2}}\right)\,ds,&|s|<1, \\ 0,&\text{otherwise}.\end{cases}$$

I used

Plot[NIntegrate[
Piecewise[{{Exp[-1/(1 - s^2)], Abs[s] < 1}, {0, Abs[s] >= 1}}], {s, 
0, t}], {t, 0, 2}

which looks like this:

which is as I would expect.

Then we sample for arbitrary time interval $T=1/10$ to get

$$\begin{aligned}f_{s}(t)&=\sum_{n=-\infty}^{\infty}f(n/10)\delta(t-n/10) \\ &=\begin{cases}\int_{0}^{n/10}\exp\left(-\frac{1}{1-s^{2}}\right)\,ds\cdot\delta(t-n/10),&|s|<1, \\ 0,&\text{otherwise}.\end{cases} \end{aligned}$$

However, for the sampled function, when I use

DiscretePlot[
  Accumulate[ 
    Table[ 
      NIntegrate[ 
        Piecewise[
          {{Exp[-1/(1 - s^2)]*DiscreteDelta[t - n*0.1], Abs[s] < 1}, 
           {0, Abs[s] >= 1}}], 
        {s, 0, n*0.1}, 
        AccuracyGoal -> 20], 
      {n, -20, 20}]],
  {t, 0, 2}]

this yields:

which clearly isn't how it should look.

Moreover, when I attempt to perform a discrete time Fourier transform on the sampled data using

Plot[
  FourierSequenceTransform[
    NIntegrate[
      Piecewise[{{Exp[-1/(1 - s^2)], Abs[s] < 1}, {0, Abs[s] >= 1}}], 
      {s, 0, 0.1 n}, 
      AccuracyGoal -> 20], 
    {n, -20, 20}, 
    ω], 
  {ω, -20, 20}]

I fail to get the n in the FourierSequenceTransform argument, since it is in the integral limit, I think. But there must be a way to bypass this obstacle?