# Problems with sampling

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?

• your formatted mathematical expressions don't make sense. Do you mean for the integral to be outside the piecewise expression (As it is in the mathematica form )? Dec 8, 2016 at 22:50
• @george2079 No, I don't, so thanks for pointing that out. However, using DiscretePlot[ Accumulate[ Table[Piecewise[{{NIntegrate[Exp[-1/(1 - s^2)], {s, 0, n*0, 1}, AccuracyGoal -> 20]*DiscreteDelta[t - n*0.1], Abs[s] < 1}, {0, Abs[s] >= 1}}], {n, -20, 20}]], {t, 0, 2}] also fails to yield what I want. Dec 8, 2016 at 22:57
• your first plot is this right? f[t_] := NIntegrate[Exp[-1/(1 - s^2)], {s, 0, Min[1, t]}]. That's not what your piecewise expression says.. Dec 8, 2016 at 23:06
• @george2079 I actually used Plot[NIntegrate[ Piecewise[{{Exp[-1/(1 - s^2)], Abs[s] < 1}, {0, Abs[s] >= 1}}], {s, 0, t}], {t, 0, 2}] Dec 8, 2016 at 23:36
• Do you mean discretizing like this: ListPlot[Table[{t, NIntegrate[ Piecewise[{{Exp[-1/(1 - s^2)], Abs[s] < 1}}], {s, 0, t}]}, {t, 0, 2, 1/10}]] ? Dec 9, 2016 at 13:28