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Revision as of 17:15, 3 February 2023 editBob K (talk | contribs)Extended confirmed users6,614 edits add a normalization optionTag: Visual edit: Switched← Previous edit Revision as of 21:30, 3 February 2023 edit undoBob K (talk | contribs)Extended confirmed users6,614 edits →Examples of normalization: add a column to the tableNext edit →
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], denoted by {{math|''ω''}} and with the unit '']'', can be similarly normalized. When {{math|''ω''}} is normalized with reference to the sampling rate as {{math|1=''ω''′ = ''ω'' / ''f''<sub>s</sub>}}, the normalized Nyquist angular frequency is ''π 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 as {{math|1=''ω''′ = ''ω'' / ''f''<sub>s</sub>}}, the normalized Nyquist angular frequency is ''π radians/sample''.


The following table shows examples of normalized frequencies for a 1&nbsp;kHz signal (or filter bandwidth), a sampling rate {{math|''f''<sub>s</sub>}}&nbsp;=&nbsp;44100&nbsp;''samples/second'' (often denoted by ]), and 4 normalization options. The following table shows examples of normalized frequency for {{math|''f''}}&nbsp;=&nbsp;1&nbsp;kHz, &nbsp;{{math|''f''<sub>s</sub>}}&nbsp;=&nbsp;44100&nbsp;''samples/second'' (often denoted by ]), and 4 normalization conventions:


{| class="wikitable" {| class="wikitable"
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!'''Numeric range''' !'''Numeric range'''
!'''Calculation''' !'''Calculation'''
!'''Reverse'''
|- |-
|{{math|1=''f'' / ''f''<sub>s</sub>}} |{{math|1=''f''{{′}} = ''f'' / ''f''<sub>s</sub>}}
|&nbsp;&nbsp;{{math||size=150%}}&nbsp;''cycle/sample'' |&nbsp;&nbsp;{{math||size=150%}}&nbsp;''cycle/sample''
|1000 / 44100 = 0.02268 |1000 / 44100 = 0.02268
|{{math|1=''f'' = ''f''{{′}} × ''f''<sub>s</sub>}}
|- |-
|{{math|1=''f'' / (''f''<sub>s</sub>/2)}} |{{math|1=''f''{{′}} = ''f'' / (''f''<sub>s</sub>/2)}}
|&nbsp;&nbsp;&nbsp;''half-cycle/sample'' |&nbsp;&nbsp;&nbsp;''half-cycle/sample''
|1000 / 22050 = 0.04535 |1000 / 22050 = 0.04535
|{{math|1=''f'' = ''f''{{′}} × ''f''<sub>s</sub> / ''2''}}
|- |-
|{{math|1=''f'' / (''f''<sub>s</sub>/''N'')}} |{{math|1=''f''{{′}} = ''f'' / (''f''<sub>s</sub>/''N'')}}
|&nbsp;&nbsp;{{math||size=150%}}&nbsp;''bins'' |&nbsp;&nbsp;{{math||size=150%}}&nbsp;''bins''
|1000 × N / 44100 = 0.02268 N |1000 × ''N'' / 44100 = 0.02268 ''N''
|{{math|1=''f'' = ''f''{{′}} × ''f''<sub>s</sub> / ''N''}}

|- |-
|{{math|''ω'' / ''f''<sub>s</sub>}} |{{math|1=''ω''{{′}} = ''ω'' / ''f''<sub>s</sub>}}
|&nbsp;&nbsp;&nbsp;''radians/sample'' |&nbsp;&nbsp;&nbsp;''radians/sample''
|1000 × 2π / 44100 = 0.14250 |1000 × 2π / 44100 = 0.14250
|{{math|1=''ω'' = ''ω''{{′}} × ''f''<sub>s</sub>}}
|} |}


Revision as of 21:30, 3 February 2023

Frequency divided by a characteristic frequency

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, fs). 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 (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 distance. For example, when f is expressed in Hz (cycles per second), fs is expressed in samples per second.

Some programs (such as MATLAB toolboxes) that design filters with real-valued coefficients prefer the Nyquist frequency (fs/2) as the frequency reference, which changes the numeric range that represents frequencies of interest from cycle/sample to half-cycle/sample. Therefore, the normalized frequency unit is obviously 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 fs/N, for some arbitrary integer N (see § Sampling the DTFT). The samples (sometimes called frequency bins) are numbered consecutively, corresponding to a frequency normalization by fs/N. The normalized Nyquist frequency is N/2 with the unit ⁠1/N⁠ 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 frequency for f = 1 kHz,  fs = 44100 samples/second (often denoted by 44.1 kHz), and 4 normalization conventions:

Quantity Numeric range Calculation Reverse
f′ = f / fs    cycle/sample 1000 / 44100 = 0.02268 f = f′ × fs
f′ = f / (fs/2)    half-cycle/sample 1000 / 22050 = 0.04535 f = f′ × fs / 2
f′ = f / (fs/N)    bins 1000 × N / 44100 = 0.02268 N f = f′ × fs / N
ω′ = ω / fs    radians/sample 1000 × 2π / 44100 = 0.14250 ω = ω′ × fs

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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