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In ] (DSP), a '''normalized frequency''' ({{math|'''''f''{{′}}'''}}) is a ] that is equal to the ratio of a ] and a characteristic frequency of a system. In ] (DSP), a '''normalized frequency''' is a ] that is equal to the ratio of a ] and a characteristic frequency of a system.


A typical choice of characteristic frequency is the '']'' ({{math|''f''<sub>s</sub>}}) that is used to create the digital signal from a continuous one. The normalized quantity, {{math|1=''f''{{′}} = ''f'' / ''f''<sub>s</sub>}}, typically has the unit ''cycle per sample'' regardless of whether the original signal is a function of time or space. For example, when {{math|''f''}} is expressed in ] (''cycles per second''), {{math|''f''<sub>s</sub>}} is expressed in ''samples per second''. A typical choice of characteristic frequency is the '']'' ({{math|''f''<sub>s</sub>}}) that is used to create the digital signal from a continuous one. The normalized quantity, {{math|1=''f''{{′}} = ''f'' / ''f''<sub>s</sub>}}, has the unit ''cycle per sample'' regardless of whether the original signal is a function of time or space. For example, when {{math|''f''}} is expressed in ] (''cycles per second''), {{math|''f''<sub>s</sub>}} is expressed in ''samples per second''. This allows us to present concepts that are universal to all sample rates in a way that is independent of the sample rate. Such a concept is a digital filter design whose bandwidth is specified not in ], but as a percentage of the sample rate of the data passing through it. The resultant set of filter coefficients provides that bandwidth ratio for any sample-rate.<ref>{{cite book |last=Carlson |first=Gordon E. |title=Signal and Linear System Analysis|publisher=©Houghton Mifflin Co |year=1992 |isbn=8170232384 |location=Boston, MA |pages=469, 490}}</ref>

This allows us to present concepts that are universal to all sample rates in a way that is independent of the sample rate. Such a concept is a digital filter design whose bandwidth is specified not in ], but as a percentage of the sample rate of the data passing through it. Formulas expressed in terms of {{math|''f''<sub>s</sub>}} (or {{math|1=''T''<sub>s</sub> &equiv; 1 / ''f''<sub>s</sub>}}) are readily converted to normalized frequency by setting those parameters to 1. The inverse operation is usually accomplished by replacing instances of the frequency parameter, {{math|''f''}}, with {{math|''f'' / ''f''<sub>s</sub>}} or {{math|''f'' ''T''<sub>s</sub>}}.<ref>{{cite book |last=Carlson |first=Gordon E. |title=Signal and Linear System Analysis|publisher=©Houghton Mifflin Co |year=1992 |isbn=8170232384 |location=Boston, MA |pages=469, 490}}</ref>

== Alternative normalizations ==


Some programs (such as ] toolboxes) that design filters with real-valued coefficients prefer the ] ({{math|''f''<sub>s</sub>/2}}) as the characteristic frequency, which changes the numeric range that represents frequencies of interest from {{math|}} ''cycle/sample'' to {{math|}} ''half-cycle/sample''. Some programs (such as ] toolboxes) that design filters with real-valued coefficients prefer the ] ({{math|''f''<sub>s</sub>/2}}) as the characteristic frequency, which changes the numeric range that represents frequencies of interest from {{math|}} ''cycle/sample'' to {{math|}} ''half-cycle/sample''.
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!'''Value''' !'''Value'''
|- |-
|{{math|1=''f''{{′}} = ''f'' / ''f''<sub>s</sub>}} |{{math|1=''f'' / ''f''<sub>s</sub>}}
|&nbsp;&nbsp;{{math||size=150%}}&nbsp; |&nbsp;&nbsp;{{math||size=150%}}&nbsp;
|1000&nbsp;cycles/second / 44100&nbsp;samples/second |1000&nbsp;cycles/second / 44100&nbsp;samples/second
|0.02268 cycle/sample |0.02268 cycle/sample
|- |-
|{{math|1=''ν''′ = ''f'' / (''f''<sub>s</sub>/2) = 2''f'' / ''f''<sub>s</sub>}} |{{math|1=''f'' / (''f''<sub>s</sub>/2) = 2''f'' / ''f''<sub>s</sub>}}
|&nbsp;&nbsp;&nbsp; |&nbsp;&nbsp;&nbsp;
|2000 half-cycles/second / 44100&nbsp;samples/second |2000 half-cycles/second / 44100&nbsp;samples/second
|0.04535&nbsp;half-cycle/sample |0.04535&nbsp;half-cycle/sample
|- |-
|{{math|''ω''′}} = {{math|''ω'' / ''f''<sub>s</sub>}} |{{math|''ω'' / ''f''<sub>s</sub>}}
|&nbsp;&nbsp;&nbsp; |&nbsp;&nbsp;&nbsp;
|(1000&nbsp;cycles/second × 2π&nbsp;radians/cycle) / 44100&nbsp;samples/second |(1000&nbsp;cycles/second × 2π&nbsp;radians/cycle) / 44100&nbsp;samples/second

Revision as of 12:14, 19 January 2023

Frequency divided by a characteristic frequency

In digital signal processing (DSP), a normalized frequency is a quantity that is equal to the ratio of a frequency and a characteristic frequency of a system.

A typical choice of characteristic frequency is the sampling rate (fs) that is used to create the digital signal from a continuous one. The normalized quantity, f′ = f / fs, has the unit cycle per sample regardless of whether the original signal is a function of time or space. For example, when f is expressed in Hz (cycles per second), fs is expressed in samples per second. This allows us to present concepts that are universal to all sample rates in a way that is independent of the sample rate. Such a concept is a digital filter design whose bandwidth is specified not in hertz, but as a percentage of the sample rate of the data passing through it. The resultant set of filter coefficients provides that bandwidth ratio for any sample-rate.

Some programs (such as MATLAB toolboxes) that design filters with real-valued coefficients prefer the Nyquist frequency (fs/2) as the characteristic frequency, which changes the numeric range that represents frequencies of interest from cycle/sample to half-cycle/sample.

Angular frequency, denoted by ω and with the unit radians per second, can be similarly normalized. When ω is normalized with reference to the sampling rate as ω′ = ω / fs, the normalized Nyquist angular frequency is π radians/sample.

The following table shows examples of normalized frequencies for a 1 kHz signal, a sampling rate fs = 44100 samples/second (often denoted by 44.1 kHz), and 3 normalization options.

Quantity Numeric range Computation Value
f / fs     1000 cycles/second / 44100 samples/second 0.02268 cycle/sample
f / (fs/2) = 2f / fs     2000 half-cycles/second / 44100 samples/second 0.04535 half-cycle/sample
ω / fs     (1000 cycles/second × 2π radians/cycle) / 44100 samples/second 0.14250 radian/sample


See also

Citations

  1. Carlson, Gordon E. (1992). Signal and Linear System Analysis. Boston, MA: ©Houghton Mifflin Co. pp. 469, 490. ISBN 8170232384.
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