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Normalized Radian Frequency

Prerequisites

Normalized Frequency | \(\hat f = \frac{f}{f_{s}}\)
Definition of Sampling Frequency | \(f_{s} = \frac{1}{T_{s}}\)

Description

The normalized radian frequency is based on the more general concept of a normalized frequency (\(\htmlClass{sdt-0000000042}{\hat f}\)). It is described by the ratio between the radian frequency and the sampling frequency (\(\htmlClass{sdt-0000000055}{f_{s}}\)). Or in this case, the product of the radian frequency and the sampling period (\(\htmlClass{sdt-0000000048}{T_{s}}\)), which is the inverse of the sampling frequency.

\[\htmlClass{sdt-0000000016}{\hat \omega} = \htmlClass{sdt-0000000116}{\omega} \htmlClass{sdt-0000000048}{T_{s}}\]

Symbols Used:

This is the symbol for normalized radian frequency. It is measured in radians (\( \htmlClass{udt-0000000005}{rad} \))
\( T_{s} \)

This symbol represents sampling period, the amount of time between samples taken in an analog-to-digital converter.
\( \omega \)

This symbol represents radian frequency, the speed of rotation. It is measured in radians per second.

Derivation

  1. Consider the definition of normalized frequency:
    \[\htmlClass{sdt-0000000042}{\hat f} = \frac{\htmlClass{sdt-0000000040}{f}}{\htmlClass{sdt-0000000055}{f_{s}}}\]
  2. Now considering the fact that we this page is about the radian frequency, we can substitute \( \htmlClass{sdt-0000000116}{\omega} \) in for \( \htmlClass{sdt-0000000040}{f} \) to get:
    \[\htmlClass{sdt-0000000016}{\hat \omega} = \frac{\htmlClass{sdt-0000000116}{\omega}}{\htmlClass{sdt-0000000055}{f_{s}}}\]
  3. We now consider the definition of sampling frequency:
    \[\htmlClass{sdt-0000000055}{f_{s}} = \frac{1}{\htmlClass{sdt-0000000048}{T_{s}}}\]
  4. And we rewrite to get:
    \[\htmlClass{sdt-0000000016}{\hat \omega} = \htmlClass{sdt-0000000116}{\omega} \htmlClass{sdt-0000000048}{T_{s}}\]

Example

example coming soon...

References

  1. Wikipedia page