Unit interval

In mathematics, the unit interval is the closed interval [0,1], that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted I (capital letter I). In addition to its role in real analysis, the unit interval is used to study homotopy theory in the field of topology.

In the literature, the term "unit interval" is sometimes applied to the other shapes that an interval from 0 to 1 could take: (0,1], [0,1), and (0,1). However, the notation I is most commonly reserved for the closed interval [0,1].

Properties

The unit interval is a complete metric space, homeomorphic to the extended real number line. As a topological space it is compact, contractible, path connected and locally path connected. The Hilbert cube is obtained by taking a topological product of countably many copies of the unit interval.

In mathematical analysis, the unit interval is a one-dimensional analytical manifold whose boundary consists of the two points 0 and 1. Its standard orientation goes from 0 to 1.

Unit interval (data transmission)

The unit interval is the minimum time interval between condition changes of a data transmission signal, also known as the pulse time or symbol duration time.

A unit interval (UI) is the time taken in a data stream by each subsequent pulse (or symbol).

Time is a physical quantity. The UI is used as a unit of measurement of time and represents a definite predetermined time interval. Saying 10 unit intervals (or 10 UI), actually means 10 times the definite predetermined time called "UI".

When UI is used as a measurement unit of a time interval, the resulting measure of such time interval is dimensionless. It expresses the time interval in terms of UI.

Very often, but not always, the UI coincides with the bit time, i.e. with the time interval taken to transmit one bit (binary information digit).

The two coincide in fact in NRZ transmission; they do not coincide in a 2B1Q transmission, where one pulse takes the time of two bits.

For example, in a serial line with a baud rate of 2.5 Gbit/s, a unit interval is 1/(2.5 Gbit/s) = 0.4 ns/baud.