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 (\(\htmlId{tooltip-normalizedFrequency}{\hat f}\)). It is described by the ratio between the radian frequency and the sampling frequency (\(\htmlId{tooltip-samplingFrequency}{f_{s}}\)). Or in this case, the product of the radian frequency and the sampling period (\(\htmlId{tooltip-samplingPeriod}{T_{s}}\)), which is the inverse of the sampling frequency.

Equation

\[\htmlId{tooltip-normalizedRadianFrequency}{\hat \omega} = \htmlId{tooltip-radianFrequency}{\omega} \htmlId{tooltip-samplingPeriod}{T_{s}}\]

Symbols Used

\(\hat \omega\)	This is the symbol for normalized radian frequency. It is measured in radians (\( \htmlId{tooltip-u-radians}{rad} \))
\(T_{s}\)	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

Consider the definition of normalized frequency:
\[\htmlId{tooltip-normalizedFrequency}{\hat f} = \frac{\htmlId{tooltip-frequency}{f}}{\htmlId{tooltip-samplingFrequency}{f_{s}}}\]
Now considering the fact that we this page is about the radian frequency, we can substitute \( \htmlId{tooltip-radianFrequency}{\omega} \) in for \( \htmlId{tooltip-frequency}{f} \) to get:
\[\htmlId{tooltip-normalizedRadianFrequency}{\hat \omega} = \frac{\htmlId{tooltip-radianFrequency}{\omega}}{\htmlId{tooltip-samplingFrequency}{f_{s}}}\]
We now consider the definition of sampling frequency:
\[\htmlId{tooltip-samplingFrequency}{f_{s}} = \frac{1}{\htmlId{tooltip-samplingPeriod}{T_{s}}}\]
And we rewrite to get:
\[\htmlId{tooltip-normalizedRadianFrequency}{\hat \omega} = \htmlId{tooltip-radianFrequency}{\omega} \htmlId{tooltip-samplingPeriod}{T_{s}}\]

Example

example coming soon...

References

Wikipedia page