Normalized frequency (signal processing): Difference between revisions

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Revision as of 13:50, 12 January 2023 editQuondum (talk \| contribs)Extended confirmed users36,927 edits →Alternative normalizations: ce statement← Previous edit		Revision as of 13:54, 12 January 2023 edit undoQuondum (talk \| contribs)Extended confirmed users36,927 edits →Alternative normalizations: defining ω′ in the text; we still need to do the same for ν′Next edit →
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	Some programs (such as ] toolboxes) that design filters with real-valued coefficients prefer the the ] as a characteristic frequency ({{math\|''f''<sub>s</sub>/2}}), which replaces the numeric range that represents frequencies of interest from {{math\|}} to {{math\|}}.		Some programs (such as ] toolboxes) that design filters with real-valued coefficients prefer the the ] as a characteristic frequency ({{math\|''f''<sub>s</sub>/2}}), which replaces the numeric range that represents frequencies of interest from {{math\|}} to {{math\|}}.

	], 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.		], 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.

	The following table shows examples of normalized frequencies for a 1 kHz signal, a sampling rate {{math\|''f''<sub>s</sub>}} = 44,100 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>}} = 44,100 samples/second (often denoted by ]), and 3 normalization options.

Revision as of 13:54, 12 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, space, or something else. 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 the Nyquist frequency as a characteristic frequency (f_s/2), which replaces the numeric range that represents frequencies of interest from to .

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 as ω′ = ω / f_s, 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 f_s = 44,100 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	(1000 cycles/second × 2 half-cycles/cycle) / 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 and 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 13:50, 12 January 2023 editQuondum (talk \| contribs)Extended confirmed users36,927 edits →Alternative normalizations: ce statement← Previous edit		Revision as of 13:54, 12 January 2023 edit undoQuondum (talk \| contribs)Extended confirmed users36,927 edits →Alternative normalizations: defining ω′ in the text; we still need to do the same for ν′Next edit →
Line 13:		Line 13:
	Some programs (such as ] toolboxes) that design filters with real-valued coefficients prefer the the ] as a characteristic frequency ({{math\|''f''<sub>s</sub>/2}}), which replaces the numeric range that represents frequencies of interest from {{math\|}} to {{math\|}}.		Some programs (such as ] toolboxes) that design filters with real-valued coefficients prefer the the ] as a characteristic frequency ({{math\|''f''<sub>s</sub>/2}}), which replaces the numeric range that represents frequencies of interest from {{math\|}} to {{math\|}}.

	], 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.		], 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.

	The following table shows examples of normalized frequencies for a 1 kHz signal, a sampling rate {{math\|''f''<sub>s</sub>}} = 44,100 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>}} = 44,100 samples/second (often denoted by ]), and 3 normalization options.

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