Description

This equation describes how the radian frequency \(\htmlClass{sdt-0000000116}{\omega}\) is calculated.

Symbols Used:

Derivation

1. Consider the definition of frequency:
   

This symbol describes frequency. It represents the number of times something happens per second. It is measured in Hertz (\( \htmlClass{udt-0000000003}{Hz} \)) . This symbol describes frequency. It represents the number of times something happens per second. It is measured in Hertz (\( \htmlClass{udt-0000000003}{Hz} \)) . In physics, it might refer to the number of oscillations per second of a light wave. In signal processing, it might refer to how many times a signal oscillates per second.

and the definition of radian frequency:

The symbol \(\omega\) represents radian frequency, the speed of rotation. It is measured in radians per second. It is often used in equations used in signal processing. In the figure below, the blue arrow indicates the direction of rotation, the speed of rotation is described by \(\omega\) which is defined as the change in angle over the change in time \(\frac{\Delta \htmlClass{sdt-0000000024}{\theta}}{\Delta t}\)

1. In this case the specific definition of frequency becomes the number of full rotations of the unit circle per second.
2. Now consider the definition of a radian:
   

Radians are defined as the angle suspended by a segment of a circle along a circumference (\( \htmlClass{sdt-0000000128}{c} \)) that is one radius (\( \htmlClass{sdt-0000000063}{r} \)) long. (Will change soon, temporary definition)

1. Note that \(2 \htmlClass{sdt-0000000126}{\pi}\) radians is one full rotation. Mathematically this is then expressed as
   \[\text{one full rotation} = 2 \htmlClass{sdt-0000000126}{\pi} \: rad\]
2. From here it follows that:
   \[\htmlClass{sdt-0000000116}{\omega} = 2 \htmlClass{sdt-0000000126}{\pi} \htmlClass{sdt-0000000040}{f}\]