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=== Examples of normalization === === Examples of normalization ===
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 distance. For example, when {{math|''f''}} is expressed in ] (''cycles per second''), {{math|''f''<sub>s</sub>}} is expressed in ''samples per second''.<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> 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 distance. For example, when {{mvar|f}} is expressed in ] (''cycles per second''), {{math|''f''<sub>s</sub>}} is expressed in ''samples per second''.<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>


Some programs (such as ] toolboxes) that design filters with real-valued coefficients prefer the ] ({{math|''f''<sub>s</sub>/2}}) as the frequency reference, which changes the numeric range that represents frequencies of interest from {{math|}} ''cycle/sample'' to {{math|}} ''half-cycle/sample''. Therefore, the normalized frequency unit is obviously important when converting normalized results into physical units. Some programs (such as ] toolboxes) that design filters with real-valued coefficients prefer the ] ({{math|''f''<sub>s</sub>/2}}) as the frequency reference, which changes the numeric range that represents frequencies of interest from {{math|}} ''cycle/sample'' to {{math|}} ''half-cycle/sample''. Therefore, the normalized frequency unit is obviously important when converting normalized results into physical units.
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A common practice is to sample the frequency spectrum of the sampled data at frequency intervals of {{math|''f''<sub>s</sub>/''N''}}, for some arbitrary integer ''N'' (see {{Section link|Discrete-time_Fourier_transform|Sampling_the_DTFT|nopage=y}}). The samples (sometimes called frequency ''bins'') are numbered consecutively, corresponding to a frequency normalization by {{math|''f''<sub>s</sub>/''N''}}. The normalized Nyquist frequency is {{math|''N''/2}} with the unit {{sfrac|1|N}}<sup>th</sup> ''cycle/sample''. A common practice is to sample the frequency spectrum of the sampled data at frequency intervals of {{math|''f''<sub>s</sub>/''N''}}, for some arbitrary integer ''N'' (see {{Section link|Discrete-time_Fourier_transform|Sampling_the_DTFT|nopage=y}}). The samples (sometimes called frequency ''bins'') are numbered consecutively, corresponding to a frequency normalization by {{math|''f''<sub>s</sub>/''N''}}. The normalized Nyquist frequency is {{math|''N''/2}} with the unit {{sfrac|1|N}}<sup>th</sup> ''cycle/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''. ], denoted by {{mvar|ω}} and with the unit '']'', can be similarly normalized. When {{mvar|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 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: The following table shows examples of normalized frequency for {{mvar|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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|{{math|1=''f''{{′}} = ''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 × {{mvar|N}} / 44100 = 0.02268 {{mvar|N}}
|{{math|1=''f'' = ''f''{{′}} × ''f''<sub>s</sub> / ''N''}} |{{math|1=''f'' = ''f''{{′}} × ''f''<sub>s</sub> / ''N''}}


Revision as of 22:09, 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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