Normalized frequency (signal processing) explained

In digital signal processing (DSP), a normalized frequency is a ratio of a variable frequency (

f ) and a constant frequency associated with a system (such as a sampling rate, f _{s} ). Some software applications require normalized inputs and produce normalized outputs, which can be re-scaled to physical units when necessary. Mathematical derivations are usually done in normalized units, relevant to a wide range of applications.

Examples of normalization

A typical choice of characteristic frequency is the sampling rate (

f _{s} ) that is used to create the digital signal from a continuous one. The normalized quantity, f' = \tfrac{f}{f _{s},} has the unit cycle per sample regardless of whether the original signal is a function of time or distance. For example, when f is expressed in Hz (cycles per second), f _{s} is expressed in samples per second.(f _{s/2)} as the frequency reference, which changes the numeric range that represents frequencies of interest from \left[ 0 , \tfrac{ 1 }{ 2 }\right] cycle/sample to [ 0 , 1 ] half-cycle/sample. Therefore, the normalized frequency unit is important when converting normalized results into physical units.

A common practice is to sample the frequency spectrum of the sampled data at frequency intervals of

\tfrac{f _{s}{N},} for some arbitrary integer N (see). The samples (sometimes called frequency bins) are numbered consecutively, corresponding to a frequency normalization by \tfrac{f _{s}{N}.} The normalized Nyquist frequency is \tfrac{N}{ 2 } with the unit ^th cycle/sample.

Angular frequency, denoted by

\omega and with the unit radians per second, can be similarly normalized. When \omega is normalized with reference to the sampling rate as \omega' = \tfrac{\omega}{f _{s},} the normalized Nyquist angular frequency is .

The following table shows examples of normalized frequency for

f = 1 kHz, f _{s} = 44100 samples/second (often denoted by 44.1 kHz), and 4 normalization conventions:

!Quantity!Numeric range!Calculation!Reverse

f' = \tfrac{f}{f _{s}}	0,  cycle/sample	1000 / 44100 = 0.02268	f = f' ⋅ f _{s}
f' = \tfrac{f}{f _{s} / 2 }	[0, 1] half-cycle/sample	1000 / 22050 = 0.04535	f = f' ⋅ \tfrac{f _{s}{2}}
f' = \tfrac{f}{f _{s} / N}	0,  bins	1000 × / 44100 = 0.02268	f = f ' ⋅ \tfrac{f _{s}{N}}
\omega' = \tfrac{\omega}{f _{s}}	[0, ''π''] radians/sample	1000 × 2π / 44100 = 0.14250	\omega = \omega' ⋅ f _{s}

See also

• Prototype filter