Discrete-time signals

Discrete variables are those in which there exists a neighborhood around each value in which no other value is present.

Intuitively, it means a variable like the natural numbers on the real line - we can isolate each instance of the discrete variable from the other instances.

Why should we bother about discrete variables?

Discrete variables come up intrinsically in several applications. Take for example, the cost of gold in the market every day. The dependent variable (cost) is a function of discrete time (incremented once every day). Another example is the marks scored by the students in class. Here the dependent variable (marks) is a function of the discrete variable roll number. While it is perfectly fine to talk about marks of 02007005, it makes no sense to talk of marks of roll no 02007011.67 - this system is inherently discrete.

Another point that should be noted here is that some results about signals and systems are common to both: continuous as well as discrete signals, but can be grasped more intuitively in one case as compared to the other. So, we shall pursue the study of both these cases simultaneously in this course.

Need the discrete variable be uniform?

No. though we imagine natural number or integers when we think of discrete signals, the points need not be equally spaced. For example, if the markets remained closed on Sundays, we would not record a price for gold on that day - so the spacing between the variables on this axis changes.

In most common cases, however, the independent variable is uniform - and throughout this course, we shall assume a uniform spacing of the variable unless otherwise stated explicitly. This assumption makes the analysis more intuitive and also yields several good theorems for our use, which we shall see as we proceed.