Sampling Theorem: T-Sampled Signal

Question

I am stuck in a small question which I am trying to solve.

The signal $f$ defined by $f(t) := \text{sinc}(t)$ is to be sampled. How should I draw the illustrations of the $T$-sampled signal $f_{T} = (\mathbb{Z} \ni n\longmapsto f(nT)) $ for $T \in \left\{1,\frac{1}{2},2 \right\} $?

Thanks.

Answer 1

I will solve it for T=1. You can solve the others on your own.

First, note that my definition of sinc is: $$sinc(t)=\frac{\sin(\pi t)}{\pi t}$$

Definitions of sinc tend do differ, so check what your definition is.

Now, for T=1, we have

$$ f_T: n \mapsto f(nT)=sinc(n) $$

So, $$ \begin{align} [\ldots, -3, -2, -1, 0, 1, 2, 3, \dots] &\mapsto [\ldots, sinc(-3), sinc(-2), sinc(-1), sinc(0), sinc(1), sinc(2), sinc(3), \ldots]\\ &=[\ldots, 0, 0, 0, 1, 0, 0, 0, \ldots] \end{align} $$

So, you will draw the following diagram:

Answer 2

This is the same answer as I posted here:

Sampling Theorem

Sampling in the Time Domain created replications on the Fourier Domain.
The distance between the replications center is according to the sampling rate.

If you sample at $ T = \frac{1}{2 \Omega} $ than the replications are $ 2 \Omega $ apart.
The point here is to understand the replications are added to each other.
Hence if they are not far away enough they are summed (What we call aliasing).

Now just draw the Fourier Transform of you signal and according to the sampling rate add to it the replications.