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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 {{mvar|f}} is expressed in ] (''cycles per second''), {{math|''f''<sub>s</sub>}} is expressed in ''samples per second''.<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 name=Carlson/> | ||
| 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. | ||
| ] | ] | ||
| 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 {{mvar|N}} (see {{slink|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 {{mvar|N}} (see {{slink|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''}}.<ref name=Harris/>{{rp|p.56 eq.(16)}} The normalized Nyquist frequency is {{math|''N''/2}} with the unit {{sfrac|1|N}}<sup>th</sup> ''cycle/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''. | ], 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''. | ||
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| |1000 × {{mvar|N}} / 44100 = 0.02268 {{mvar|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''}} | ||
| |- | |- | ||
| |{{math|1=''ω''{{′}} = ''ω'' / ''f''<sub>s</sub>}} | |{{math|1=''ω''{{′}} = ''ω'' / ''f''<sub>s</sub>}} | ||
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| *] | *] | ||
| == |
==References== | ||
| {{reflist |
{{reflist|1|refs= | ||
| <ref name=Carlson> | |||
| {{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> | |||
| <ref name=Harris> | |||
| {{cite journal | |||
| |doi=10.1109/PROC.1978.10837 | |||
| |last=Harris | |||
| |first=Fredric J. | |||
| |title=On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform | |||
| |journal=Proceedings of the IEEE | |||
| |volume=66 | |||
| |issue=1 | |||
| |page=51–83 | |||
| |date=Jan 1978 | |||
| |url=http://web.mit.edu/xiphmont/Public/windows.pdf|citeseerx=10.1.1.649.9880 | |||
| |bibcode=1978IEEEP..66...51H | |||
| |s2cid=426548 | |||
| }}</ref> | |||
| }} | |||
| {{DEFAULTSORT:Normalized Frequency (Digital Signal Processing)}} | {{DEFAULTSORT:Normalized Frequency (Digital Signal Processing)}} | ||
Revision as of 15:11, 7 February 2023
Frequency divided by a characteristic frequencyIn 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
References
- Carlson, Gordon E. (1992). Signal and Linear System Analysis. Boston, MA: ©Houghton Mifflin Co. pp. 469, 490. ISBN 8170232384.
- Harris, Fredric J. (Jan 1978). "On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform" (PDF). Proceedings of the IEEE. 66 (1): 51–83. Bibcode:1978IEEEP..66...51H. CiteSeerX 10.1.1.649.9880. doi:10.1109/PROC.1978.10837. S2CID 426548.