I am having trouble understanding and getting the solution to this question:
Let $x : \mathbb Z \to \mathbb C$ denote a discrete-time sequence and for $N \in \mathbb N$ let $x |_N$ be the restrictin of $x$ to $\{0, \ldots, N-1\}$ i.e., the mapping $ x|_N : \{0, \ldots, N-1\} \to \mathbb C, \quad n \mapsto x[n]. $ Further, $X(e^{j\theta}) = \operatorname{DTFT} \{ x [n]\}$ denotes the DTFT of $x[n]$ and $X[k] = \operatorname{DFT}_N \{ x|_N[n] \}$ is the $N$-point DFT of $x|_N[n]$.
(a) Assume that the support* of the signal $x[n]$ is a subset of $\{ 0, \ldots, N-1 \}$ for some fixed $N$. Express the signal $x[n]$ by the signal $x|_N [n]$ (and some kind of zero-padding). Further, express the DFT $X[k]$ by the DTFT $X(e^{j \theta})$.
*The support of a function is the set of indices $n$ for which $x[n]$ is nonzero.
From what I understand, that is $x[n]=x|_N[n]$ for $n= 0,1, \ldots, N-1$. But there is $x[n] \in N$. So I have to define the value for $x[-1]$ or $x[N+1]$ and fill that up with zeros.
But I don't know if it is correct or how to write that correctly. Hope someone can help me with that.