Normalized frequency (signal processing): Difference between revisions

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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.

	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''.		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 or space. For example, when {{math\|''f''}} is expressed in ] (''cycles per second''), {{math\|''f''<sub>s</sub>}} 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 ], 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>		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>
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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 ~~resulting unit is ''radian per sample''. 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 kHz signal, a sampling rate {{math\|''f''<sub>s</sub>}} = 44100 ''samples/second'' (often denoted by ]), and 3 normalization options.		The following table shows examples of normalized frequencies for a 1 kHz signal, a sampling rate {{math\|''f''<sub>s</sub>}} = 44100 ''samples/second'' (often denoted by ]), and 3 normalization options.

Revision as of 14:28, 14 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 (f_s) that is used to create the digital signal from a continuous one. The normalized quantity, f′ = f / f_s, typically has the unit cycle per sample regardless of whether the original signal is a function of time or space. For example, when f is expressed in Hz (cycles per second), f_s 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 f_s (or T_s ≡ 1 / f_s) 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 / f_s or f T_s.

Alternative normalizations

Some programs (such as MATLAB toolboxes) that design filters with real-valued coefficients prefer the Nyquist frequency (f_s/2) as the characteristic frequency, which changes the numeric range that represents frequencies of interest from to .

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 / f_s, 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 f_s = 44100 samples/second (often denoted by 44.1 kHz), and 3 normalization options.


Quantity	Computation	Value
f′ = f / f_s	1000 cycles/second / 44100 samples/second	0.02268 cycle/sample
ν′ = f / (f_s/2) = 2f / f_s	2000 cycles/second / 44100 samples/second	0.04535 half-cycle/sample
ω′ = ω / f_s	(1000 cycles/second × 2π radians/cycle) / 44100 samples/second	0.14250 radian/sample

Notes

The unit of normalized frequency is then cycles per two samples or half-cycles per sample. This is accounted for in the table by associating the cardinal number 2 with the label half-cycles per cycle.

Citations

Carlson, Gordon E. (1992). Signal and Linear System Analysis. Boston, MA: ©Houghton Mifflin Co. pp. 469, 490. ISBN 8170232384.

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Revision as of 12:15, 14 January 2023 editBob K (talk \| contribs)Extended confirmed users6,614 edits →Alternative normalizations: modify 3rd row (as per talk page)← Previous edit		Revision as of 14:28, 14 January 2023 edit undoBob K (talk \| contribs)Extended confirmed users6,614 edits a tad less wordyTag: Visual edit Next edit →
Line 5:		Line 5:
	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.

	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''.		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 or space. For example, when {{math\|''f''}} is expressed in ] (''cycles per second''), {{math\|''f''<sub>s</sub>}} 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 ], 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>		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>
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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 ~~resulting unit is ''radian per sample''. 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 kHz signal, a sampling rate {{math\|''f''<sub>s</sub>}} = 44100 ''samples/second'' (often denoted by ]), and 3 normalization options.		The following table shows examples of normalized frequencies for a 1 kHz signal, a sampling rate {{math\|''f''<sub>s</sub>}} = 44100 ''samples/second'' (often denoted by ]), and 3 normalization options.

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