# Time scaling of discrete-time sequences and the DTFT

In the second edition of Signals and Systems by Alan Oppenheim, he discusses the DTFT of a "time-expanded" sequence that is effectively a slowed down version of the original sequence and can be thought of as the original sequence $x[n]$ with $k-1$ zeros placed between each sample. However, the DTFT for the sequence $x[an]$ where $a$ is an integer is not discussed. Why is that? Is it because information is lost when we talk about $x[an]$ for example (since it only contains samples evaluated at multiples of $a$)? One other thing that I noticed is that unlike the CTFT, Oppenheim only discusses differentiation in the frequency domain, unlike the CTFT where both integration and differentition in the frequency domain were presented. Why is that?

• You are asking two non-related questions in one same post. I'd suggest you leave just one of them and ask the other one separately, as they are two independent questions. I'll write an answer for the first one. Feb 23, 2018 at 13:26

The DTFT of some sequence $x(an)$ is discussed in the book you mention, but rather indirectly.
It's easy to see that $x(an)$ is a decimated sampled version of $x(n)$. This means that we take $x(n)$, we sample it with sampling period $a$, and then we decimate it (i.e. we take every $a$-th sample).
You can derive the DTFT of $x(an)$ knowing the DTFT of $x(n)$ with this in mind. If you have a copy of Oppenheim's Signals and Systems, check the section Sampling of discrete-time signals. Everything is explained there to answer your first question.