Indigital signal processing (DSP), anormalized frequency is a ratio of a variablefrequency (${\displaystyle f}$) and a constant frequency associated with a system (such as asampling rate,${\displaystyle f_s}$). 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.

A typical choice of characteristic frequency is thesampling rate (${\displaystyle f_s}$) that is used to create the digital signal from a continuous one. The normalized quantity,${\displaystyle f^{\prime }=\frac{f}{f_s},}$ has the unitcycle per sample regardless of whether the original signal is a function of time or distance. For example, when${\displaystyle f}$ is expressed inHz (cycles per second),${\displaystyle f_s}$ is expressed insamples per second.^[1]

Some programs (such asMATLAB toolboxes) that design filters with real-valued coefficients prefer theNyquist frequency${\displaystyle (f_s/2)}$ as the frequency reference, which changes the numeric range that represents frequencies of interest from${\displaystyle \left[0,\frac{1}{2}\right]}$cycle/sample to${\displaystyle [0,1]}$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${\displaystyle \frac{f_s}{N},}$ for some arbitrary integer${\displaystyle N}$ (see§ Sampling the DTFT). The samples (sometimes called frequencybins) are numbered consecutively, corresponding to a frequency normalization by${\displaystyle \frac{f_s}{N}.}$^{[2]: p.56 eq.(16) [3]} The normalized Nyquist frequency is${\displaystyle \frac{N}{2}}$ with the unit⁠1/N⁠^thcycle/sample.

Angular frequency, denoted by${\displaystyle \omega }$ and with the unitradians per second, can be similarly normalized. When${\displaystyle \omega }$ is normalized with reference to the sampling rate as${\displaystyle {\omega }^{\prime }=\frac{\omega }{f_s},}$ the normalized Nyquist angular frequency isπ radians/sample.

The following table shows examples of normalized frequency for${\displaystyle f=1}$kHz,${\displaystyle f_s=44100}$samples/second (often denoted by44.1 kHz), and 4 normalization conventions:

Quantity	Numeric range	Calculation	Reverse
${\displaystyle f^{\prime }=\frac{f}{f_s}}$	[0,⁠1/2⁠] cycle/sample	1000 / 44100 = 0.02268	${\displaystyle f=f^{\prime }\cdot f_s}$
${\displaystyle f^{\prime }=\frac{f}{f_s/2}}$	[0, 1] half-cycle/sample	1000 / 22050 = 0.04535	${\displaystyle f=f^{\prime }\cdot \frac{f_s}{2}}$
${\displaystyle f^{\prime }=\frac{f}{f_s/N}}$	[0,⁠N/2⁠] bins	1000 ×N / 44100 = 0.02268N	${\displaystyle f=f^{\prime }\cdot \frac{f_s}{N}}$
${\displaystyle {\omega }^{\prime }=\frac{\omega }{f_s}}$	[0, π] radians/sample	1000 × 2π / 44100 = 0.14250	${\displaystyle \omega ={\omega }^{\prime }\cdot f_s}$

• Prototype filter

• Digital signal processing
• Frequency

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