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Revision as of 13:14, 8 January 2023 editQuondum (talk | contribs)Extended confirmed users36,932 edits →Alternative normalizations: trying again← Previous edit Revision as of 15:56, 8 January 2023 edit undoQuondum (talk | contribs)Extended confirmed users36,932 edits Synchronizing in most of the cleaner version suggested at talkNext edit →
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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''' ({{math|'''''f''{{′}}'''}}) is a ] that is equal to the ratio of a ] and a characteristic frequency of a system.


An example of a normalized frequency is the sampling frequency in a system in which a signal is sampled at periodically, in which it equals {{math|1=''f''{{′}} = ''f'' / ''f''<sub>s</sub>}} (with the unit ''cycle per sample''), where {{math|''f''}} is a frequency and {{math|''f''<sub>s</sub>}} is the '']''. For regularly spaced sampling, the ] time variable, {{math|''t''}} (with unit ]), is replaced by a ] ''sampling ]'' variable, {{math|1=''n'' = ''t'' / ''T''<sub>s</sub>}} (with the unit sample), upon division by the sampling interval, {{math|1=''T''<sub>s</sub> = 1 / ''f''<sub>s</sub>}} (with the unit second per sample). 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, space, or something else. For example, when {{math|''f''}} is expressed in ] (''cycles per second''), {{math|''f''<sub>s</sub>}} is expressed in ''samples per second''.


The use of normalized frequency allows us to present concepts that are universal to all sample rates in a way that is independent of the sample rate. An example of 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|''T''<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|year=1992|publisher=©Houghton Mifflin Co|location=Boston,MA|isbn=8170232384|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 == == Alternative normalizations ==


Some programs (such as ] toolboxes) that design filters with real-valued coefficients use the ] ({{math|1=''f''<sub>Ny</sub> = ''f''<sub>s</sub> / 2}}) as the ] – that is, they use the (dimensionless) ratio of a frequency to the Nyquist frequency in place of the sampling rate. An alternative may to think of this is as expressing the normalized frequency {{math|1=''f''{{′}} = ''f'' / ''f''<sub>s</sub>}} in units of ''half-cycles per sample'' (or equivalently ''cycles per 2 samples''), which produces the same ''numeric'' result when the units (hertz per hertz and half-cycle per sample, respectively) are omitted. Some programs (such as ] toolboxes) that design filters with real-valued coefficients use the ] ({{math|''f''<sub>s</sub> / 2}}) as the ].


], denoted by {{math|''ω''}} and with the unit ], can be similarly normalized. When {{math|''ω''}} is normalized with reference to the sampling rate, the resulting unit is radian per sample. The normalized Nyquist angular frequency is ''π''&nbsp;radians/sample. ], denoted by {{math|''ω''}} and with the unit ], can be similarly normalized. When {{math|''ω''}} is normalized with reference to the sampling rate, the resulting unit is radian per sample. The normalized Nyquist angular frequency is ''π''&nbsp;radians/sample.


The following table shows examples of normalized frequencies for a 1&nbsp;kHz signal, a sampling rate {{math|''f''<sub>s</sub>}}&nbsp;=&nbsp;], and 3 different choices of normalized units. Also shown is the frequency region containing one cycle of the ], which is always a periodic function. The following table shows examples of normalized frequencies for a 1&nbsp;kHz signal, a sampling rate {{math|''f''<sub>s</sub>}}&nbsp;=&nbsp;], and 3 different choices of normalized units. Also shown is the frequency region containing one cycle of the ], which is always a periodic function.


{| class="wikitable" {| class="wikitable"
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| '''Value''' | '''Value'''
|- |-
| {{math|''f'' / ''f''<sub>s</sub>}} | {{math|''f''{{′}}}}
| cycle per sample | cycle per sample
| &nbsp;&nbsp; or &nbsp;&nbsp; | &nbsp;&nbsp; or &nbsp;&nbsp;
Line 31: Line 31:
| 0.02268 | 0.02268
|- |-
| {{math|''f'' / ''f''<sub>s</sub>}} | {{math|''f''{{′}}}}
| half-cycle per sample | half-cycle per sample
| &nbsp;&nbsp; or &nbsp;&nbsp; | &nbsp;&nbsp; or &nbsp;&nbsp;
Line 37: Line 37:
| 0.04535 | 0.04535
|- |-
| {{math|''f'' / ''f''<sub>Ny</sub>}} | {{math|''ω''{{′}}}}
| hertz per hertz
| &nbsp;&nbsp; or &nbsp;&nbsp;
| 1000 / 22050
| 0.04535
|-
| {{math|''ω''}}
| radian per sample | radian per sample
| &nbsp;&nbsp; or &nbsp;&nbsp; | &nbsp;&nbsp; or &nbsp;&nbsp;

Revision as of 15:56, 8 January 2023

Frequency divided by a characteristic frequency

In digital signal processing (DSP), a normalized frequency (f′) 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, typically has the unit cycle per sample regardless of whether the original signal is a function of time, space, or something else. 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. Formulas expressed in terms of fs (or Ts ≡ 1 / fs) 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, f, with f / fs or f Ts.

Alternative normalizations

Some programs (such as MATLAB toolboxes) that design filters with real-valued coefficients use the Nyquist frequency (fs / 2) as the normalization constant.

Angular frequency, denoted by ω and with the unit radian per second, can be similarly normalized. When ω is normalized with reference to the sampling rate, the resulting unit is radian per sample. 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 = 44,100 samples/second, and 3 different choices of normalized units. Also shown is the frequency region containing one cycle of the discrete-time Fourier transform, which is always a periodic function.

Quantity Unit Numeric range Computation Value
f′ cycle per sample    or    1000 / 44100 0.02268
f′ half-cycle per sample    or    1000 / 22050 0.04535
ω′ radian per sample    or    2π 1000 / 44100 0.1425

See also

Notes and 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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