Continued

Do discrete signals necessarily come from continuous signals?

Although intuition may suggest so, this is not necessarily the case. In one of the example above - we considered the daily rate of gold. Here, time is intrinsically a continuous variable, and we made a discrete variable by taking measurements after certain intervals. However, the marks as a function of roll numbers intrinsically form a discrete system - there is no continuous axis of roll numbers.

Then how do we define the neighbourhood?

Okay, by now it may seem that we are hiding some details here - we defined a discrete variable as one in which no other value exists in a certain neighbourhood of one. Now for roll numbers, a neighbourhood does not make sense. How do we formally define a discrete signal?

A discrete variable is one which can ultimately be indexed by integers.