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| {{Short description| |
{{Short description|Frequency divided by a characteristic frequency}} | ||
| {{Use American English|date=March 2021}} | |||
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| <!-- Hide the merge tag to avoid a loop, because Digital frequency now redirects here. | <!-- Hide the merge tag to avoid a loop, because Digital frequency now redirects here. | ||
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| In ] (DSP), '''normalized frequency''' ('''''f |
In ] (DSP), a '''normalized frequency''' ({{math|'''''f''{{′}}'''}}) is a ] that is equal to the ratio of a ] and a characteristic frequency of a system. | ||
| For regularly spaced sampling, the ] time variable, ''t'' (with units of ]), is replaced by a ] ''sampling ]'' variable, ''n=t/T'' (with units of "samples"), upon division by the sampling interval, ''T=1/f<sub>s</sub>'' (in "seconds per sample"). This practice is analogous to the concept of '']'' in physics, meaning that the natural unit of time in a DSP system is "samples". | |||
| 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). | |||
| The maximum frequency that can be unambiguously represented by digital data is <math>\textstyle f_s/2</math> (known as ]) when the samples are real numbers, and <math>\textstyle f_s</math> when the samples are complex numbers.<ref>See ]</ref> The normalized values of these limits are respectively 0.5 and 1.0 ''cycles/sample''. This has the advantage of simplicity, but (similar to ]) there is a potential disadvantage in terms of loss of clarity and understanding, as these constants <math>\textstyle T</math> and <math>\textstyle f_s</math> are then omitted from mathematical expressions of physical laws. | |||
| 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> | |||
| == Alternative normalizations == | == Alternative normalizations == | ||
| Some programs (such as ]) that design filters with real-valued coefficients use the Nyquist frequency (< |
Some programs (such as ]) that design filters with real-valued coefficients use the Nyquist frequency ({{math|''f''<sub>s</sub>/2}}) as the ]. The resultant normalized frequency has units of ''half-cycles/sample'' or equivalently ''cycles per 2 samples''. | ||
| ], 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'' radians/sample. | |||
| The following table shows examples of normalized frequencies for a 1 kHz signal, a |
The following table shows examples of normalized frequencies for a 1 kHz signal, a sampling rate {{math|''f''<sub>s</sub>}} = ], 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" | ||
| |- | |- | ||
| | ''' |
| '''Unit''' | ||
| | '''Domain''' | | '''Domain''' | ||
| | '''Computation''' | | '''Computation''' | ||
| | '''Value''' | | '''Value''' | ||
| |- | |- | ||
| | |
| cycle/sample | ||
| | or | | or | ||
| | 1000 / 44100 | | 1000 / 44100 | ||
| | 0.02268 | | 0.02268 | ||
| |- | |- | ||
| | half- |
| half-cycle/sample | ||
| | or | | or | ||
| | 1000 / 22050 | | 1000 / 22050 | ||
| | 0.04535 | | 0.04535 | ||
| |- | |- | ||
| | |
| radian/sample | ||
| | or | | or | ||
| | 2 |
| 2''π'' 1000 / 44100 | ||
| | 0.1425 | | 0.1425 | ||
| |} | |} | ||
Revision as of 22:02, 14 December 2022
Frequency divided by a characteristic frequencyIn 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.
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 f′ = f / fs (with the unit cycle per sample), where f is a frequency and fs is the sampling rate. For regularly spaced sampling, the continuous time variable, t (with unit second), is replaced by a discrete sampling count variable, n = t / Ts (with the unit sample), upon division by the sampling interval, Ts = 1 / fs (with the unit second per sample).
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 hertz, but as a percentage of the sample rate of the data passing through it. Formulas expressed in terms of fs or Ts 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) that design filters with real-valued coefficients use the Nyquist frequency (fs/2) as the normalization constant. The resultant normalized frequency has units of half-cycles/sample or equivalently cycles per 2 samples.
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 = 44.1 kHz, 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.
| Unit | Domain | Computation | Value |
| cycle/sample | or | 1000 / 44100 | 0.02268 |
| half-cycle/sample | or | 1000 / 22050 | 0.04535 |
| radian/sample | or | 2π 1000 / 44100 | 0.1425 |
See also
Notes and citations
- Carlson, Gordon E. (1992). Signal and Linear System Analysis. Boston,MA: ©Houghton Mifflin Co. pp. 469, 490. ISBN 8170232384.